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Measuring Electron Correlation: The Impact of Symmetry and Orbital Transformations

Cite this: J. Chem. Theory Comput. 2023, 19, 10, 2703–2720
Publication Date (Web):April 6, 2023
https://doi.org/10.1021/acs.jctc.3c00122

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In this perspective, the various measures of electron correlation used in wave function theory, density functional theory and quantum information theory are briefly reviewed. We then focus on a more traditional metric based on dominant weights in the full configuration solution and discuss its behavior with respect to the choice of the N-electron and the one-electron basis. The impact of symmetry is discussed, and we emphasize that the distinction among determinants, configuration state functions and configurations as reference functions is useful because the latter incorporate spin-coupling into the reference and should thus reduce the complexity of the wave function expansion. The corresponding notions of single determinant, single spin-coupling and single configuration wave functions are discussed and the effect of orbital rotations on the multireference character is reviewed by analyzing a simple model system. In molecular systems, the extent of correlation effects should be limited by finite system size and in most cases the appropriate choices of one-electron and N-electron bases should be able to incorporate these into a low-complexity reference function, often a single configurational one.

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1. Electron Correlation

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Electron correlation has often been called the “chemical glue” of nature due to its ubiquitous influence in molecules and solids. (1,2) The problem was studied in the smallest two-electron systems, such as He and H2 already in the 1920s, and these early efforts are discussed in detail elsewhere. (3−6) The correlation concept usually presupposes an independent particle model, typically Hartree–Fock (HF) mean-field theory, that serves as a reference compared to which the exact solution is correlated. The associated idea of correlation energy goes back to Wigner’s work in the 1930s on the uniform electronic gas and metals, (7,8) and it is nowadays commonly defined, following Löwdin, as the difference between the exact and the HF energy. (3) There is a vast literature on the subject and here we will confine ourselves to some of the most pertinent reviews. Early work on the correlation problem is reviewed by Löwdin in the 1950s (3) and more recently, (9) as well as by Sinanoğlu (10,11) within the context of his many-electron theory. The early perspective of McWeeny is a short but useful discussion of the problem in terms of density matrices. (12) Kutzelnigg, Del Re and Berthier penned an overview on the statistical measures of correlation, (13) and, as P. v. Herigonte, they also reviewed the situation in the seventies. (14) This was later followed up several times by Kutzelnigg. (15,16) Bartlett and Stanton not only discuss correlation effects in molecules, but also offer a tutorial on post-HF wave function methods. (17) Similarly educational is the chapter written by Knowles, Schütz and Werner (18) and the feature articles of Tew, Klopper, Helgaker (19) and Martin. (2) More recent work on the homogeneous electron gas is reviewed by Senatore and March (20) as well as Loos and Gill, (21) the former authors also discuss orbital-based and quantum Monte Carlo methods. For a review taking an entanglement-based approach to the correlation problem, we refer the reader to Chan. (22) There are also several book-length monographs dedicated to the subject by Fulde, (23) March (24) and Wilson, (25) as well as a number of collections. (26,27) In the remainder of this section, we will consider some selected topics about correlation effects before stating the scope of this perspective.
An important aspect of electron correlation is its strength and the various measures proposed to quantify it. For the homogeneous electron gas, where some analytical energy expressions can be obtained at least under some circumstances, the discussion is often concerned with the relative magnitudes of kinetic and potential energies in the limiting cases of high and low electron densities. While Wigner was able to obtain an expression for the correlation energy at the low-density limit, (7,8) in general, perturbation theory diverges already at second order, (28) a problem that can only be remedied partly at the high-density limit following Gell-Mann and Brueckner. (29) In the high-density limit, the kinetic energy dominates over the repulsive potential energy, the electrons are delocalized and behave like a gas, thus, an independent particle model should provide a good description. In the low-density limit, it is the Coulomb interactions responsible for correlation effects that dominate, forcing the electrons to localize on site (form a lattice). Interestingly, the low- and high-density limits have their analogues in molecular systems: (20) in the hydrogen molecule, the high-density case would correspond to short bond lengths and a delocalized molecular orbital (5,30−32) (MO) description, while the low-density limit would correspond to large internuclear separation where the electrons are localized around the nuclei, better described by a local valence bond (5,33−35) (VB) approach. While this is a useful conceptual link, it must be remembered that the homogeneous electron gas is a better description of metallic solids than it is of more inhomogeneous systems like molecules. In the next section, the contrast between solids and molecules with regard to strong correlation will be discussed before we turn our attention fully to molecular systems and examine the various measures used in wave function theory, density functional theory and quantum information theory.
From a wave function point of view, correlation strength is often associated with the nature of the reference function which is usually the HF state, as mentioned above. An interesting point that draws attention to the significance of the chosen reference is the observation that from a statistical point of view HF already contains some correlation introduced by antisymmetrization and the exclusion principle. (13) Following Kutzelnigg, Del Re and Berthier, (13) two variables may be called uncorrelated if
⟨ r 1 · r 2 ⟩ i j = ⟨ r 1 ⟩ i ⟨ r 2 ⟩ j
(1)
where the quantities corresponding to the ith and jth particles in an N-electron system can be simply related to the usual expectation values as
⟨ r 1 ⟩ = N ⟨ r 1 ⟩ i , ⟨ r 1 · r 2 ⟩ = N ( N − 1 ) 2 ⟨ r 1 · r 2 ⟩ i j
(2)
Applying this to position vectors, the implication is that the positions of two electrons are uncorrelated. This is a weaker notion than that of independence, which is often taken to mean lack of correlation. Statistically, independence (or, correlation in the sense of dependence) could be defined (36) by requiring that the two-body cumulant λ2
λ 2 ( 1 , 1 ′ ; 2 , 2 ′ ) = γ 2 ( 1 , 1 ′ ; 2 , 2 ′ ) − γ 1 ( 1 , 1 ′ ) γ 1 ( 2 , 2 ′ )
(3)
obtained from the one- and two-body reduced density matrices (1- and 2-RDM), γ1 and γ2, should vanish. The fulfilment of this condition would imply a relationship like eq 1 for any observable. This is the sense in which Wigner and Seitz (37,38) understood correlation, and this would correspond to a simple Hartree product of orbitals as a reference function, or, more generally, to a direct product of subsystem wave functions. Antisymmetrization then implies switching from this product reference function to an antisymmetric Slater determinant constructed from spin–orbitals. More generally, it is also possible to use generalized (antisymmetrized/wedge) product functions (39) to obtain the total wave function from antisymmetric group functions of subsystems. Using an antisymmetrized reference, it is customary to talk about Fermi and Coulomb correlation, the former being excluded from the conventional definition of the correlation energy. To make this explicit, the cumulant can be rewritten as
λ 2 ( 1 , 1 ′ ; 2 , 2 ′ ) = γ 2 ( 1 , 1 ′ ; 2 , 2 ′ ) − γ 1 ( 1 , 1 ′ ) γ 1 ( 2 , 2 ′ ) + γ 1 ( 1 , 2 ′ ) γ 1 ( 2 , 1 ′ )
(4)
Here, the last term accounts for the exchange contribution, and thus, λ2 only contains the parts of γ2 that cannot be factorized in terms of γ1 in any way. In this sense, λ2 is the most general descriptor of correlation effects. It should also be noted that by using partial trace relations, (36) it can be shown that Tr(λ2) = Tr(γ12 – γ1), and hence correlation effects are also related to the idempotency of the 1-RDM. This notion goes beyond that of Wigner and Seitz and is closer to Löwdin’s idea of correlation. The significance of Fermi correlation in particular was discussed by several authors in the past, (12,13,19) and more recently in a pedagogical manner by Malrieu, Angeli and co-workers. (40) Nevertheless, as pointed out by Kutzelnigg and Mukherjee, (36) eq 4 is not identical to Löwdin’s definition either since it contains no reference to the Hartree–Fock state, apart from that fact that the closed-shell HF determinant, for example, would be enough to make λ2 vanish and would thus be uncorrelated in this sense. The definition in eq 4 is an intrinsic one in the sense that γ1 and γ2 can be calculated for any (reference) state. More generally, any second-quantized operator string can also be evaluated with respect to a general reference (vacuum) state using generalized normal order and the associated general form of Wick’s theorem that itself relies on n-body cumulants. (41,42)
The foregoing discussion indicates the role of the reference state in the definition of correlation effects and highlights the fact that reference states should conveniently incorporate certain correlation effects to make subsequent treatment easier. As antisymmetrization dictates using Slater determinants rather than simple orbital products, spin symmetry may be enforced by demanding that the reference be constructed as a fixed linear combination of determinants to account for spin symmetry, (19) and there might be other criteria. (22,43,44) In such cases, one may distinguish between correlation recovered by the reference and residual effects. From such an operational point of view, the question is whether the strongest correlation effects can already be captured at the reference level to ensure a qualitatively correct starting point. In practice, one may choose to build reference states and represent the wave function using the following N-electron basis states:

  • Determinant (DET): An antisymmetrized product of orbitals. They are eigenfunctions of Ŝz but not necessarily of Ŝ2.

  • Configuration State Function (CSF): A function that is an eigenstate of both Ŝz and Ŝ2. CSFs can be obtained from linear combinations of determinants, but there are at most as many CSFs as DETs for a given number of unpaired electrons and total spin.

  • Configuration (CFG): At an abstract level, a configuration is just a string of spatial orbital occupation numbers (0,1,2) for each orbital in a given N-electron wave function. By extension, a configuration is also the set of determinants or CSFs that share the same spatial occupation numbers, but differ in the spin–orbital occupation numbers.

We will discuss the role of this choice and offer a classification of wave function expansions and potential reference states based on the properties of the exact solution, which leads us to define what we call the multireference character of the wave function. We will also examine the role of the orbital basis and ask the question how orbital rotations influence this apparent multireference character in a simple model system. On the other hand, we are not concerned here with the practical construction of good reference states, even though the topics discussed undoubtedly have consequences in that regard. At this stage, we simply ask what the difference is between a canonical and a local orbital representation in terms of the apparent multireference character and what the chances are of incorporating the strongest correlation effects into the reference function, be that via enforcing spin symmetry or generating a new many-electron basis by rotating orbitals.

2. Solids versus Molecules

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Strong correlation is one of those important but elusive concepts that is often used in discussions of the properties of molecular and solid-state systems. A brief survey of the literature that follows below will reveal that it is often equated with various other notions, such as the multireference character of the wave function, the failure of density functional theory (DFT) and quantum entanglement. It is not immediately clear how these various notions are related to one another or to strong correlation, or whether any one should be preferred over the other. On the other hand, it seems relatively clear that a large part of the discussion on strong correlation focuses on extended solid-state systems, where density functional and dynamical mean-field (45,46) approaches are commonly used, (47) although many-body methods (48) and quantum Monte Carlo techniques (49,50) are also available as well as a number of exactly solvable model problems. (51) The theoretical tools used to describe periodic solids of infinite extension and finite-size molecular systems can be quite different in ways that reflect the underlying physics. Complexity and locality may have different manifestations for these systems in terms of the number of parameters or the type of basis sets required to describe them. In terms of symmetry, the obvious difference is the presence of translational symmetry in solids. Furthermore, symmetry for an infinite number of degrees of freedom also makes spontaneous symmetry breaking possible in ground-state solutions. (52) For all these reasons, it seems reasonable to separate the discussion of electron correlation in solids from that in molecules.
As mentioned before, in the homogeneous electron gas model strong correlation effects are related to the low density limit. Wigner argued that at this limit the electrons are localized at lattice points that minimize the potential energy (7) and vibrate around the lattice points (8) with just the zero point energy demanded by the uncertainty principle. Leaving the vibrational part aside, the total energy of the system at very low densities has the form (8,24,53,54)
E = ⟨ T ^ ⟩ + ⟨ V ^ ⟩ , ⁣ ⟨ T ^ ⟩ = α r 2 , ⁣ ⟨ V ^ ⟩ = β r
(5)
where α and β are constants and r is the Fermi radius, which in this context is the radius of a sphere available for a single electron. If the density is low, r is large, and that means that the mean kinetic energy ⟨T̂⟩ is negligible compared to the mean potential energy ⟨V̂⟩, which is sometimes taken to be a criterion for strong correlation. Wigner was able to obtain a correlation energy formula by assuming a vanishingly small kinetic energy at the low-density limit (7) and interpolating between the low- and high-density limits, and considered the zero point kinetic energy, which would add a term proportional to r − 3 / 2 in eq 5, only in a later study. (8,21,55) While the result works reasonably at density values actually present in some solids, some criticisms merit attention especially when application to molecules are concerned. While admitting the good agreement with measurements in metallic systems and similar results derived from plasma theory by Bohm and Pines, (56,57) Slater criticized (58) the assumption of uniform positive background, which should be an even more serious problem when considering more inhomogeneous systems, such as the dissociating H2 molecule. (3) We note in passing that it is possible to account for inhomogeneities in the electron gas, (59) but perhaps the conceptual simplicity of the homogeneous electron gas model is more relevant for the current discussion. Concerns were also raised by Löwdin (3) about the model not obeying the virial theorem (60) in the sense expected in molecular systems. It has since been shown that the uniform electron gas obeys the more general form of the virial theorem (61,62)
2 ⟨ T ^ ⟩ + ⟨ V ^ ⟩ = − r d E d r
(6)
which means that in the low-density limit, the total energy E is not stationary (although the potential energy may be in the sense needed to find an optimal lattice). In molecular systems, on the other hand, stationary points of the total electronic energy are of special interest as in the case of H2 at equilibrium and infinite separations. In these cases, the virial theorem takes the form 2⟨T̂⟩ + ⟨V̂⟩ = 0, which does not allow the kinetic energy to be vanishingly small. (3) In this regard, see also the work of Ruedenberg (63) for an analysis of the formation of the chemical bond that does respect the virial theorem. For the electron gas, the main use of the virial theorem is actually the separation of kinetic and potential energy contributions in the correlation energy. (61) Thus, it may be concluded that the criterion of the relative magnitude of the mean kinetic and potential energy is appropriate within the homogeneous electron gas model, which is a better approximation of metallic solids than molecules. Nevertheless, extensions of the same basic idea are possible. Within the Hubbard model, (64) for example, correlation effects are commonly considered as being strong if the on-site electronic repulsion is larger than the bandwidth (resonance energy) or the hopping term, which favors antiferromagnetic coupling between the various sites. (23) Repulsion is a potential energy term, while the hopping integral has to do with the kinetic energy of electrons. We will discuss the appropriate extension to the molecular Hamiltonian in later sections.
As in this perspective we are more concerned with molecular systems, it is especially interesting that solids containing d or f electrons are considered strongly correlated, (23) as this would prompt one to look for strong correlation in molecular systems containing transition metals or lanthanides. Before leaving the discussion of solid state behind, it should also be recalled that sometimes the nature of the problem at hand calls for the combination of solid-state and molecular approaches, as in studying strongly correlated (bond-breaking) problems in surface chemistry via embedding approaches, which pose challenges of their own. (65) The foregoing discussion also underlines the problem of finding quantitative measures of strong correlation, or, perhaps more precisely, of the various other properties associated with it. After reviewing the various attempts made along these lines, we will expound our own perspective on strongly correlated systems in chemistry as viewed through the lens of the multireference character of the wave function expansion. We will use a simple metric that can be obtained from the exact solution and examine its properties using the hydrogen molecule as a model system. As mentioned before, this problem has been studied since the early days of quantum chemistry (3−6) and also has some practical relevance in that it can be taken as a model for the antiferromagnetic coupling in transition metal compounds such as a typical Cu(II) dimer. We note in passing that the hydrogen chain is also a common model problem in the study of extended systems. (66,67)

3. Wave Function Theory

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Within the framework of wave function-based post-Hartree–Fock approaches, it is customary to distinguish between single and multireference methods to which the concepts of dynamic and static or nondynamic correlation also correspond. (11,17,18,68) Dynamic correlation is also called short-range correlation as it has to do with the electrostatic repulsion of electrons and the difficulty of describing the resulting Coulomb cusp (see this study (69) for a different perspective). When it comes to other correlation effects, some authors use static and nondynamic correlation interchangeably and sometimes refer to it as long-range correlation. (18,70) Others (17) prefer to reserve static correlation for effects induced by enforcing spin symmetry in order to construct a correct zeroth order description, and call those associated with other sources, such as bond breaking, nondynamical. Tew, Klopper and Helgaker follow a similar distinction except that they include spin symmetry in Fermi correlation and identify static correlation as the correlation associated with near degeneracy and thus needed for a correct zeroth order description. (19) Yet others (71) distinguish within static/nondynamical effects between those that can be captured by spin-unrestricted HF and those that cannot. A summary of more formal attempts to define static and dynamic correlations is also available. (72) The important conclusion from this discussion seems to be a point already raised by Wigner and Seitz (38) that the main difference is between correlation effects that arise from symmetry and those arising from interelectronic repulsion, regardless of how we choose to classify these further. Thus, within symmetry effects one might further distinguish between antisymmetry and spin symmetry, within repulsion effects between those associated with spatial (near) degeneracy and the rest.
Ignoring the question of spin symmetry for the moment, nondynamic/static correlation may be identified with strong correlation in the present context, and thus, it is also associated with multireference methods, where “reference” is a generic term for some function that serves as an initial state on which a more sophisticated Ansatz can be built. This also suggests that there are two ways of dealing with static correlation, either improving the reference or improving the wave function Ansatz built on it. In principle, single reference methods will yield the exact solution if all possible excitation classes are included in the Ansatz, and it was furthermore demonstrated that they can also be adapted to strong correlation by removing some of the terms from the working equations. (73) More pragmatically, one might ask at what excitation levels an Ansatz, such as coupled cluster (CC), needs to be truncated to yield acceptable results for some properties, (74,75) but this is costly on the one hand and does not offer a simple picture of strong correlation either. One way to improve the reference is to find the exact solution within at least a complete active space (CAS). (76) Unfortunately, this requires the construction of many-electron basis states the number of which scales exponentially with the size of the active space. However, as observed by Chan and Sharma, (77) this apparent exponential complexity does not always reflect the true complexity of the system since the principle of locality often forces a special structure on the wave function expansion. This entails that the amount of information needed to construct the wave function should be proportional to the system size, which agrees well with the observation that static correlation in molecules can typically be tackled with only a handful of terms in the wave function expansion. (18) As an illustration, the hydrogen molecule around the equilibrium bond length is well-described by a single-determinant built from canonical orbitals and associated with the configuration σg2, while at larger separations another contribution from σu2 becomes equally important. (18,77) In the latter limit, the wave function takes the following bideterminant form
Ψ 0 = 1 2 ( | i i ¯ | − | a a ¯ | ) = 1 2 ( | μ ν ¯ | − | μ ¯ ν | )
(7)
where i and a are the occupied and unoccupied molecular orbitals in the minimal basis HF solution, and the shorthand notation |ii| is used to denote determinants, with the overbar signifying a β spin. While these canonical orbitals are delocalized, μ and ν denote atomic orbitals, each localized on one of the hydrogens and orthonormal at large distances. As is well-known, (4,6,78) the molecular orbital (MO) and valence bond (VB) approaches offer equivalent descriptions of this wave function, but the locality of the VB picture leads to an interpretation in terms of “left–right” correlation, i.e., the phenomenon that at large internuclear separation an electron preferably stays close to the left nucleus if the other one is close to the right nucleus, and vice versa. This form of static correlation can be accounted for in a spin-unrestricted single-determinant framework leading to different orbitals for different spins beyond a certain bond distance, but, for molecular systems, approximations preserving the symmetries of the exact wave function are usually preferable. It is all the more interesting that the constrained-pairing mean-field theory of Scuseria and co-workers is designed to capture strong correlation (43) by breaking and restoring various symmetries, (79) while Kutzelnigg took a different route toward obtaining a similarly economic reference function by separating bond-breaking correlation effects via a generalized valence bond (GVB)/antisymmetrized product of strongly orthonormal geminals (APSG) approach. (44)
It is also apparent from these considerations that most traditional methods follow the “bottom-up” approach in which a systematic Ansatz is built on a simple reference function and then truncated using some simple criterion (e.g., the excitation level). The typical arsenal of this approach includes many body perturbation theory, CI and CC approaches in both the single and multireference cases. (17,18,80) The most popular of them is CC theory, (81) which was introduced into chemistry by Čižek, (82) Paldus and Shavitt, (83) was reformulated in terms of a variational principle by Arponen (84) and whose mathematical properties are still an active area of research. (85,86) Even more actively studied are its local variants (87,88) and its explicitly correlated, (89) excited state (90) and multireference (91) extensions. Beyond these methods, there is also room for a “top-down” approach in which one aims directly at the exact solution (full CI, full CC) and applies some strategy to neglect contributions that are less important while trying to save as much of the accuracy as possible. The relative sparsity illustrated above with the H2 example can be exploited in selected CI methods of the CIPSI (configuration interaction with perturbative selection through iterations) type (92) for much larger systems. The CIPSI method is the first of many selected CI (SCI) approaches that attempt to solve the problem of selecting relevant contributions in a deterministic fashion, but it is also possible to apply stochastic sampling processes, including QMC approaches (93) such as the FCIQMC method. (94,95) FCIQMC can be performed without approximation, in which case the method is exact but suffers from a sign problem in general. (96) Instead, the initiator approximation is typically applied, which involves truncating contributions between certain determinants with low weight. (97,98) These and other methods have been reviewed in more detail recently by Eriksen, (99) including Eriksen and Gauss’ own approach based on many body expansions. (100,101) Another approximate FCI method particularly suitable for strongly correlated systems, the density-matrix renormalization group (77,102) (DMRG) approach will be discussed later among entanglement-based approaches. We also note that some of these methods have been analyzed in terms of scaling and the gains due to the reduction in the number of important contributions in the wave function expansion (103−106) (see also some remarks on orbital rotations below).
It has been observed that multireference character, interpreted here as the fact that the wave function expansion contains more than one determinant of significant weight for some system, does not necessarily imply strong correlation, and the additional criteria of the breakdown of single reference perturbation theory and the appearance of degeneracies were suggested. (107) More explicitly, strong correlation could be said to occur whenever the perturbation (the difference between the full Hamiltonian and the Fockian) is larger than the gaps between the Hartree–Fock orbital energies (spectrum of the Fockian). (22) This could be regarded as an extension of the similar definition within the Hubbard model. For nonmetallic states, strong correlation would also be implied if the correlation energy is larger than the excitation energy to the first excited state of the same symmetry as the ground state. (108) Nevertheless, the multireference character can be used to characterize the exact wave function, even if its usefulness for approximate approaches may be questioned. This leads us toward a definition of multireference character via the full configuration interaction expansion
Ψ = ∑ I C I Φ I
(8)
where ΦI are some basis states, typically determinants. If any one coefficient CI dominates (see more later) the expansion, then the associated ΦI could be taken as the reference in a single reference (determinant) treatment. Thus, multireference character is associated with wave function coefficients CI which for orthonormal basis states are equivalent to the overlap between the exact state Ψ and the basis state ΦI. Unfortunately, this criterion is impractical to decide in advance whether a multireference treatment of a system is necessary, and other diagnostics, (109) such as the one relying on the norm of the singles vector (T1) in a CC calculation (110) truncated at the level of single and double excitations, were also found insufficient (109) or even misleading. (111) Metrics relying on natural orbital populations are more promising as they give information about the number of electrons uncoupled by correlation driven processes. (75,112,113) The deviation of the one-body reduced density matrix from idempotency has also been used as a measure (114,115) and well as measures associated directly with the two-body cumulant. (44,115,116) More pragmatically, the static correlation energy has also been defined as the exact correlation energy obtained from a minimal basis calculation involving valence electrons, the idea being that in such a calculation the most important configurations would be represented, although this approach also needs a hierarchy of approximations to be useful. (117) A few other schemes have also been suggested to decompose the correlation energy into dynamical and nondynamical parts using simple formulas, (70,115) and more recently, using cumulants. (115)
For the purposes of this perspective, the FCI-based criterion in eq 8 will be sufficient as it enables us to base our discussion on the properties of the exact wave function. While it is common to use determinants as basis states, this is not the only possibility. Using CSFs or CFGs for example would yield basis states that can be written as sums of determinants or constructed in some other way. (103,118) We will investigate their effects on the multireference character in a later section. Already in eq 7, while both the MO and VB descriptions are multideterminant expressions, multiple configurations only enter into the (canonical) MO picture. We will also examine the role of orbital transformations in defining the multireference character of the wave function. Again, in eq 7, the concept of left–right correlation seems to be connected more to the local VB picture than to the MO one. Our primary goal here is to comment on multireference character, rather than other notions about strong correlation, but before doing that, we briefly examine other approaches to the problem.

4. Density Functional Theory

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In their recent paper, Perdew et al. remark that strong correlation is sometimes taken to mean “everything that density functional theory gets wrong”. (119) Most practical DFT approaches rely on the Kohn–Sham procedure which in most cases corresponds to a single-determinantal implementation. Unlike the Hartree–Fock solution, however, the Kohn–Sham determinant is not an approximation to the exact wave function; rather, it is a tool to obtain the ground-state density or spin densities. There seem to be two opinions in the literature on what this signifies. Perdew et al. argue that as a consequence, the Kohn–Sham determinant need not exhibit the symmetries of the Hamiltonian as those will be reflected in the density in a more indirect fashion. This also implies that despite the appearance of a mean-field approach, Kohn–Sham DFT is in principle able to account for correlation effects. (120) Others argue (121−123) that spin-adapted reference functions should be used at least in some open-shell cases which may be multideterminantal, as it is done in the spin-adapted Hartree–Fock case in wave function theory. In any case, a single determinant may not always suffice and the question remains to what extent DFT can treat strongly correlated systems. In trying to answer this question, Perdew et al. (120) distinguish between strong correlation arising in condensed matter models and static correlation originating in near or exact degeneracies, such as the closing HOMO–LUMO gap in the H2 bond dissociation process. On problems of the latter type, they note the typical failure of common semilocal approximations.
While multideterminental techniques have been introduced in DFT, (121−123) it is more common to rely on broken symmetry DFT (78,124−126) in which multiplet energies are obtained from a combination of single-determinant energies. This scheme may or may not hold exactly, depending on the specific case. (121,126) Perhaps the simplest example of spin-symmetry breaking is the fact that the spin-unrestricted approach yields only one of the determinants |μν̅| or |μ̅ν| in eq 7 which can be interpreted as a resonance structure in the valence bond sense. Although such determinants are not pure spin states, they are degenerate. From such broken symmetry solutions and monodeterminantal pure spin states, it is often possible to calculate the energies of multideterminantal spin states: in H2, the appropriate combination of the energy of the broken symmetry solution and the triplet energy yields the energy of the excited-state singlet. (78) The use of this simple two-level model in the description of magnetic coupling is analyzed in detail elsewhere. (127) The more general broken symmetry (BS) DFT procedure starts with a mixture of canonical orbitals obtained from a closed-shell calculation and optimizes the mixing angles during the iterative solution to obtain a mixture of multiple configurations. (123) A similar idea underlies the permuted orbital (PO) approach where the broken symmetry states are generated by permuting orbitals, typically the HOMO and the LUMO, (123) and can be used to construct multideterminant spin eigenfunctions. Based on the capabilities of the BS and PO approaches, Cremer proposes a DFT-oriented multireference typology: (123) Type 0 wave functions need a single configuration or determinant and the spin-unrestricted approach can handle them; Type I consists of a single configuration with multiple determinants for which the PO approach is assigned; Type II is the case of multiple configurations each with a single determinant to be treated by BS-DFT; Type III systems with multiple configurations each with multiple determinants are the most difficult and can be handled by explicitly multireference methods, although possibly the PO approach might work. While we find the distinction between determinant and configuration/CSF references commendable, in the cases studied by Cremer there was no need to distinguish between configurations and CSFs, a feature that we will come back to in our own classification. As for interpreting broken symmetry solutions, it should be recalled that while the exact functional for finite systems is not symmetry broken, approximations are. Strong correlation in the exact wave function describes “fluctuations” in which the spins on the different nuclei are interchanged (|μν̅| vs |μ̅ν|). (128) Symmetry breaking in approximate methods is in fact taken to reveal strong correlations (119) with a mechanism associated with it: at large distances, when H2 becomes a pair of H atoms, an infinitesimal perturbation can reduce the wave function and the spin densities to a symmetry broken form, in which an electron with a given spin is localized around one H atom. (119,128) All this is in line with Anderson’s famed essay (129) on emergent phenomena on different scales involving increasingly larger numbers of particles, though it also implies that symmetry breaking may be more relevant for solids than for molecules. (52,119)
It may be worth pointing out that the physical contents of broken symmetry wave functions can be conveniently “read” by means of the corresponding orbital transformation (COT). (130) This transformation orders the spin-up and spin-down orbitals into pairs of maximum similarity. This results in a wave function that can be read analogously to a generalized valence bond (GVB) wave function (131,132) consisting of doubly occupied orbitals, singlet coupled electron pairs in two different orbitals and unpaired (spin-up) electrons. Obviously BS-DFT can properly represent the singlet coupled pairs, but the analysis shows that the spatially nonorthogonal pseudosinglet pairs obtained from the COT serve a similar purpose, namely to variationally adjust the ionic- and neutral components of the wave function. (127) This way of reading BS wave functions has found widespread application in the field of metal-radical coupling (e.g., examples (133−135)). A more thorough review is available. (136)
The theorems due to Hohenberg and Kohn (137) form the basis of the variational formulation of DFT, although questions about their precise meaning (138) remain pertinent enough to motivate further research, (139,140) not to mention the fact that these theorems do not generalize easily to excited states. (141) The favorable linear scaling property of DFT approaches is often derived from Kohn’s principle of nearsightedness, (142,143) i.e., that under certain conditions the density at some point is mostly determined by the external potential in a local region around that point. Practical approaches to density functional theory often consist of proposing new functional forms (144) in lieu of the unknown exact density functional of the energy. In contrast, the energy is a known functional of the 2-RDM, (114,145,146) although enforcing constraints (147) to actually optimize this functional is not trivial. Fortunately, the energy can also be written as a functional of the 1-RDM, (148) and different variants of density matrix functional theory are usually parametrized in terms of the 1-RDM and the 2-body cumulant. (149−152) Like DFT, this approach also relies on approximate functionals, but it manages to overcome some of the failings of the former. The appropriate constrains for the 1-RDM itself (153) are relatively easy to enforce, although the reconstructed 2-RDM should then still satisfy its own N-representability conditions. (151) One might then construct functionals (154) inspired by wave function approaches, such as geminal theories, or use auxiliary quantities to parametrize the cumulant. (155) In density cumulant functional theory (150,156) (DCFT), the functional is given in terms of the best idempotent estimate to the 1-RDM and the 2-body cumulant, and the critical N-representability conditions affect only the relatively small part due to the latter. Connections with traditional DFT can also be established, e.g., in the DCFT case, an analogue of the DFT exchange-correlation functional can be obtained by neglecting the Coulomb terms in which the nonidempotent part of the 1-RDM occur. (150) The nearsightedness principle applies to the 1-RDM as well, (142) and density matrix functionals have the advantage over DFT that the kinetic, Coulomb and exchange energies are all known functionals of the 1-RDM, only the correlation energy functional remains unknown. (157) Excited state applications of RDM functionals have also been considered. (152,158) A collection of studies on the various aspects of RDM-based methods is available, (159) while the relationship between wave function based methods, DFT and RDM functionals is discussed in Kutzelnigg’s review. (157) In the current context, it is relevant to mention that some density matrix functionals have been applied with success to strongly correlated problems. (152,155,160−164)
So far we have only considered functional approaches in which the variable is the density or the RDMs, but it is also possible to set up a variational formulation (165−167) for functionals of Green’s functions, which themselves can be regarded as generalizations of RDMs with their own cumulant expansion. (168) All three of these quantities form the basis for embedding theories enabling the treatment of large systems. (169) Dynamical mean-field theory (45,46,170) is a much applied method in solid state physics that can be derived from such a variational approach and has also been investigated from the perspective of molecular quantum chemistry. (171,172) Another method that falls into this category is the popular GW approximation of Hedin. (173−175) The properties of these and other similar methods have also been analyzed from the perspective of strong correlation, (170,172,176,177) excited states (174,178) and symmetry breaking. (177,179) Although both have some success in multireference systems, GW approaches are typically applied to weakly correlated systems while dynamical mean-field approaches have succeeded where DFT or other Green’s function approaches failed. (170,172,176)
It emerges from this discussion that the DFT concept of strong correlation is related on the one hand to its monodeterminant nature and to symmetry breaking on the other. Measures have been proposed along these lines. Thus, Yang and co-workers relate static correlation to fractional spins, and present an extension to DFT covering fractional spins including a measure based on this extension. (180,181) Grimme and Hansen offer a measure as well as a visualization tool for static correlation effects based on fractional occupation numbers obtained from finite temperature DFT calculations. (182) Head-Gordon and co-workers use spin (and sometimes spatial) symmetry breaking as a criterion and suggest a measure based on spin contamination, (107) which may be related to natural orbital occupation numbers, a criterion also used by others. (123) Still other authors use checks based on the idempotency of the one-body density, which is essentially used here as a check on the monodeterminant nature of the wave function under consideration. (114,183)

5. Quantum Information Theory

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Quantum information theory offers a different perspective on the same phenomenon, albeit one that can be related to more traditional quantum chemical concepts. (108) The starting point is a two-level quantum system, a qubit, that serves as an analogue of classical bits. We take spin-up and spin-down states to form the qubit in the following. A spin-up state can be labeled |0⟩ and a spin-down state as |1⟩, and a qubit can be any linear combination of these two basis states, (184) |m⟩ = α|0⟩ + β|1⟩. If the spin of an electron prepared in such a state is measured in a field oriented along an axis parallel to the spin-up and spin-down basis states, then the probability of finding the electrons in state |0⟩ or |1⟩ is |α|2 and |β|2, respectively. Along the same lines, one may prepare two electrons. Assuming at first that the two electrons do not interact before the measurement, the possible two-electron states may be described in the basis of the product states |0⟩ ⊗ |0⟩, |0⟩ ⊗ |1⟩, |1⟩ ⊗ |0⟩ and |1⟩ ⊗ |1⟩. They remain distinguishable and they are also independent in the sense that measuring the first qubit yields results that only depend on how the first electron was prepared and is not influenced by the second electron. Imagine now bringing spin-1/2 particles close enough together so that their spins align in an antiparallel fashion corresponding to the ground state and then separating them again far enough so that one is located at site A the other at site B. The wave function that describes this state is
| Ψ 0 ⟩ = 1 2 ( | 0 ⟩ A ⊗ | 1 ⟩ B − | 1 ⟩ A ⊗ | 0 ⟩ B )
(9)
In classical computing, the value of the first bit in a two-bit string does not yield information about the value of the second bit, in the same way as the independently prepared quantum systems do not. The two-qubit quantum state in eq 9 is different because measuring the spin only at site A also reveals what the result of the measurement will be at site B. Thus, the two qubits at site A and B are entangled (185) in the sense that the result of a measurement at A determines that at B. It is also often said that they are correlated, although entanglement is only a special kind of quantum correlation. (186) This is different from the definition of strong correlation in terms of perturbation strength, but has the advantage that it is more easily quantifiable. (22) Note that eq 9 describes a system of distinguishable spin-1/2 particles, such as an electron and a positron, or two electrons at a large distance. Although electrons are identical particles, in cases like the dissociating H2 molecule, each electron can be identified as the one localized around a specific nucleus if the two nuclei are separated enough so that the electronic wavefunctions no longer overlap. (187) Comparing eq 9 to the VB wave function of H2 at large bond distances in eq 7 and assuming that μ is at site A and ν at site B reveals that
| μ ν ¯ | = 1 2 ( | 0 ⟩ A ⊗ | 1 ⟩ B − | 1 ⟩ B ⊗ | 0 ⟩ A ) , ⁣ | μ ¯ ν | = 1 2 ( | 1 ⟩ A ⊗ | 0 ⟩ B − | 0 ⟩ B ⊗ | 1 ⟩ A )
(10)
since the tensor product here is ordered w.r.t. the particle labels, e.g., μ(2)α(2)ν(1)β(1) → |1⟩B ⊗ |0⟩A. Thus, eq 7 is the antisymmetrized version of eq 9. In a sense, this indicates another kind of symmetry breaking. In spin-unrestricted approaches discussed in the DFT context, only one of the determinants |μν̅| or |μ̅ν| in eq 7 are present at large bond distances, i.e., it is known which spin is at site A (or B) but not which particle carries that spin. On the other hand, eq 9 breaks antisymmetry as a natural consequence of the assumption of distinguishability: in this case it is known which particle is at site A (or B), but not the spin of that particle, which is the question a subsequent measurement can help resolve. As this discussion suggests, the established notion of entanglement (184) assumes distinguishable subsystems, although extensions to indistinguishable particle systems exist. (188) This should perhaps be kept in mind when comparing with quantum chemical notions, where indistinguishability of electrons in molecules is assumed (cf. also the discussion on product functions and generalized product functions (39) earlier). Still, the resonance structures of VB/GVB theories, for example, assume a degree of distinguishability and entanglement may be a useful tool in dealing with them. (22) It is also worth mentioning that when measuring the spin of a single electron as discussed above, it is always possible to rotate the measurement field so that, say, the spin-up state is measured with probability 1. This is not possible for the singlet described in eq 9, since the expectation value of one of the spins along any axis is zero. Due to this rotational invariance, the invariance of entanglement measures with respect to the choice of basis is often emphasized, (185) but this should not be confused with orbital rotations to be discussed below.
It may be said that the VB wave function is created by identifying the entangled local states (orbitals) associated with chemical bonds. In fact, just as traditional wave function approaches start from the independent particle model and build more accurate Ansätze by incorporating the most important excitations, one might start a similar hierarchy around independent subsystems as a starting point which explores the Hilbert space by encoding more entanglement. (22) This leads to matrix product states (MPS) in general and the density matrix renormalization group (DMRG) approach in particular. (22) The seniority-based CC theory of Scuseria and co-workers (189) can also be interpreted on a quantum information theory basis. (190,191) This approach is built on orbital pairs rather than single orbitals with seniority being the number of unpaired electrons in a determinant that can be used to select the determinants for inclusion in a wave function expansion. The claim is that strong correlation can be treated by low-seniority contributions in a suitable orbital basis, (189) and it is possible to set up an entanglement-based measure to be used as a cost function. (191)
One measure of entanglement is based on the von Neumann entropy (the “quantum Shannon entropy”) of a state and the quantum relative entropy (184) which provides a measure of the distinguishability of two states. (186) The relative entropy of entanglement can be given as (22)
S = − ∑ i σ i 2 log 2 ⁡ σ i 2
(11)
in terms of the singular values σi corresponding to a Schmidt decomposition: (184) if A and B are distinguishable subsystems of the whole system, then a pure state |Ψ⟩ can be expanded in the basis of functions of the type |ΨA⟩ ⊗ |ΨB⟩
| Ψ ⟩ = ∑ A B C A B | Ψ A ⟩ ⊗ | Ψ B ⟩
(12)
where |ΨA⟩, |ΨB⟩ are basis states in A and B, and the singular values σi are those of the expansion coefficient matrix CAB. If only one singular value is nonzero, the entropy is zero and |Ψ⟩ is separable; otherwise, it is entangled, and it is maximally entangled if all singular values are the same. (22,185) Observe that entanglement has to do with the characterization of subsystems and depends on the partitioning of the system. For example, in eq 9, it is the two subsystems at sites A and B that are entangled because the total state cannot be written in the simple form |m⟩A ⊗ |n⟩B. For further considerations on superselection rules and classical and quantum correlation types, see. (186,191) If the local states are chosen to be single orbitals, then we may speak about (single) orbital entropy (190,192) and use the eigenvalues of the orbital reduced density matrix instead of σi2 in the definition of the entropy above, with the summation running over unoccupied, singly occupied spin-up and spin-down, and doubly occupied states. Orbital entropy has been used to measure multiconfigurational character (193) and as a basis for automatic active space selection. (194) It has also been applied to the H2 model system (193) and to a number of other bond formation processes. (193,195) See also another entanglement-based study on 2-electron systems by Huang and Kais. (196) The dependence of entanglement-based measures on the orbital basis has been noted in the literature (186,193) and they have also been used to set up a qualitative distinction between nondynamic, static, dynamic and dispersion correlation effects (190,197) in the sense of Bartlett and Stanton. (17) In earlier approaches, natural orbital occupation numbers have also been substituted for σi2 to calculate the von Neumann entropy as a measure of correlation strength. (198) Recently, entropy-based measures have even served as a basis of a geometric interpretation and an upper bound to the correlation energy. (199) Finally, we note that the two-orbital entropy can be defined in an analogous manner, and serves as the basis for orbital interaction (mutual information) measures (190,192,200) that can be used to find an orbital ordering that ensures good convergence in DMRG by placing strongly interacting ones near each other. (200)
So far we have discussed the application of quantum information theoretical concepts in quantum chemistry. However, these concepts also underlie the theory of quantum computing in which chemistry has also been identified as one of the potential areas of breakthrough, (201) as exemplified by resource estimates in the study of Reiher et al. on FeMoco. (202) The latter compound served as an example of a strongly correlated system that would especially benefit from the fact that quantum computers are quantum systems themselves that need not explicitly generate the FCI coefficients in eq 8. Thus, we will also briefly mention recent developments in this field. In particular, the need that the initial states should have a good overlap with the target state for certain quantum algorithms to succeed is well-known. (203) The fact that the preparation of spin eigenfunctions on quantum computers has been investigated recently (204,205) can be seen as a step in this direction and it also underpins our efforts in the upcoming sections to provide a more nuanced classification of reference functions than the usual single vs multideterminant one. As far as the potential gains in scaling are concerned, a recent study suggests that the exponential nature of quantum advantage in quantum chemistry (206) cannot be taken for granted in most cases, although the authors are careful to point out that quantum computers may still be very useful in quantum chemistry. Thus, the search is on for specific chemical systems where quantum computers could be most beneficially applied. (207−209)

6. Symmetry Considerations

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We have already distinguished between DETs, CSFs and CFGs as various choices of basis states. It also follows from the previous discussion that symmetry plays a significant role in the classification of correlation strength. To get a better grip on these notions, let us remind ourselves about how an N-electron wave function is built from one electron functions (orbitals) and how various symmetries may be taken into account in the construction. The simplest approach is to approximate the N-electron wave function as a Hartree product (HPD) of N orbitals and enforce various symmetries. In general, the operator Q̂ encodes a symmetry of the Hamiltonian Ĥ if [Ĥ, Q̂] = 0, and thus the exact wave function is a simultaneous eigenstate of Ĥ and Q̂. If an approximate wave function does not have a particular symmetry, an appropriate projection operator ÔQ can be constructed that would produce an eigenfunction of Q̂ with eigenvalue Q. We are not concerned here with all possible symmetries (e.g., the particle number operator, time reversal), but may limit ourselves to the following scheme:
HPD → A ^ DET → O ^ S CSF ( CFG ) → O ^ R SA‐CSF
(13)
where antisymmetrization ( A ^ ) produces a Slater determinant (DET), spin projection (ÔS) a configuration state function (CSF), or, more generally, a number of CSFs belonging to the same configuration (CFG, i.e., a sequence of spatial occupation numbers), while spatial symmetry can also be enforced via ÔR of some symmetry operation R, producing a spatial symmetry-adapted CSF (SA-CSF). We will briefly examine these in the following. It only remains to note here that symmetry adaptation in general leads to higher energies in a variational procedure since it introduces constraints in the variational manifold, a phenomenon sometimes referred to as Löwdin’s dilemma. (210)
The original purpose of introducing the Slater determinant (211) was to find an N-electron basis that automatically satisfies the exclusion principle, or, rather, its generalization requiring a Fermionic wave function to be antisymmetric. For this purpose, the HPD is constructed from the product of spin orbitals ϕi(xi), which itself is the product of a spatial orbital φi(ri) and a spin function σi(si), and the antisymmetrization A ^ is carried out by permuting the coordinates xi = (ri, si) and summing over the permutations multiplied by their parity
Φ = A ^ ∏ i N ϕ i ( x i ) = A ^ ∏ i N φ i ( r i ) ∏ i N σ i ( s i )
(14)
Since A ^ is a projector multiplied by a constant, i.e., A ^ 2 = N ! A ^ , Φ is an eigenfunction of A ^ . The correlation introduced by this formulation is usually illustrated (19,40) using the case of two parallel electrons, where the spatial part φ1(r1)φ2(r2) – φ2(r1)φ1(r2) is obviously zero at r1 = r2. It has been observed that this Fermi hole is in fact nonlocal, because the spatial part also vanishes if φ1(r1) = φ1(r2) = ± φ2(r1) = ± φ2(r2). (40) Moreover, antisymmetrization also introduces anticorrelation in the opposite-spin case, which has been used to explain the fact that same-spin double excitations contribute less to the energy in post-Hartree–Fock calculations than opposite-spin ones. (40)
Given that the Slater determinant is a much simpler entity than a general group theoretical construction of the wave function, it could be claimed with some plausibility that “Slater had slain the “Gruppenpest”,” (212) i.e., the pestilence of groups that threatened the conceptual clarity of quantum mechanics in his view. Were it only about the antisymmetry requirement, Slater’s approach would be preferable to general group theoretical presentations given that it is much more transparent, a fact also responsible to a large extent for the establishment of the Slater determinant as the “default choice” as reference state. However, the question of spin adaptation still remains and is in some sense a natural extension of antisymmetrization. A spin-adapted CSF may be written as
Θ = A ^ ∏ i N φ i ( r i ) X S ( s 1 , ... , s N )
(15)
where we have retained the simplest Hartree product in the spatial part, but assumed a generalized spin part XS which is an eigenfunction of Ŝ2 with total spin S. Since XS can be written as a linear combination of spin function products of the type appearing in eq 14, it also follows that Θ can be written as a linear combination of determinants, although CSFs can be constructed directly without determinants as intermediates. (103,118) The general construction of XS remains an elaborate group theoretical exercise. Matsen’s spin-free approach (213) developed in the 1960s relied on the observation that enforcing the SU(2) spin symmetry in the spin part of the wave function is equivalent to representing the N-particle spatial part in terms of the irreducible representations of the symmetric group SN of permutations. (118) On the other hand, the spin-free representation of the Hamiltonian is given in terms of the generators of the unitary group U(n) in the n-orbital basis. The latter observation is the basis of the unitary group approach (214) (UGA) and its graphical application (215) (GUGA), which was recently discussed in detail by Dobrautz et al. (216) and the advantages of which have also been exploited in a number of publications. (104−106) We will return to some aspects of spin-adaptation and its consequences for the correlation problem in the next section, and similar considerations will also occur in the section on the role of orbitals.
Finally, we shall briefly consider the problem of spatial symmetry since it does appear in discussions on strong correlation. (107) The prototypical example is the He2+ system (217,218) which dissociates into a He atom and a He+ ion. At large distances, two electrons are localized on one of the He nuclei and one electron around the other. In the minimal basis, this means that the Hartree–Fock orbital associated with the two electrons has a larger spatial extent than the other one, and they do not reflect the point group symmetry of He2+. Moreover, there are two symmetry broken solutions depending on whether the two electrons are localized around nucleus A or B, corresponding in essence to two resonance structures. It is possible to enforce spatial symmetry by constructing orbitals of the type φA ± φB, where φA and φB are orbitals localized around A and B. A determinant built from such orbitals could be expanded in terms of the resonance structures mentioned, but would also include high-energy ionic configurations, giving rise to an artificially large energy at long bond distances. Thus, this is the spatial equivalent of the dissociation problem of the H2 molecule in which breaking spin (and spatial) symmetry leads to a lower energy. Usually, these problems are regarded multiconfigurational, (218) because more than one configuration is needed to give the correct energy without breaking symmetry. Unfortunately, at the level of approximate wave functions (often used as a reference), enforcing spatial symmetry may easily lead to additional complications. Since infinitesimal changes may disrupt the symmetric arrangement of the nuclei, this may lead to discontinuities in the energy or the need to use suboptimal symmetry-adapted orbitals. (219) Thus, using SA-CSFs may introduce more conceptual uncertainty about symmetry-related correlation effects than simply using DETs or even CSFs (CFGs). We will not consider SA-CSFs here any further, but the review of Davidson and Borden (220) discusses the issue of spatial symmetry breaking in all its aspects, be it “real or artifactual”, as they put it, and a series of papers by Čížek and Paldus on the HF stability conditions (221−223) also devotes considerable space to this problem.
It has been noted before that the various categories for correlation are operational, i.e., what counts as static or nondynamical correlation is practically whatever is covered by choice at the level of the reference calculation. Symmetry on the other hand introduces exact degeneracies that are known a priori and reference functions can be constructed to account for them. Thus, in line with the general thrust of many discussions on the subject, it seems worthwhile to distinguish between correlation effects due to various kinds of symmetry and the rest. We may call these symmetry effects static correlation. (12,17) In the next section, we will discuss accounting for spin symmetry in the basis states. The remaining correlation effects may be weak or strong depending on the weight of the symmetry-adapted reference state in the final calculation. In this regard, near degeneracies are of interest since they do not arise from the symmetry effects discussed so far. The case of the Be atom is instructive: (19) due to the near degeneracy of the 2s and 2p orbitals, the 1s22s2 configuration does not completely dominate the wave function expansion, the 1s22p2 configurations must also be considered. Such nondynamical effects are not captured by symmetry restrictions but are usually still included in the reference calculation, leaving any residual correlation effects, i.e., dynamic correlation, to be recovered by the wave function Ansatz built on that reference. Handy and co-workers (70) even go so far as to suggest that even this “angular” correlation in Be, so-called because s and p orbitals are of different angular types, might be regarded as dynamical based on a model in which explicit dependence on the interelectronic distance appears. The Be example also indicates that the weight of the reference state is of crucial importance and what is regarded as nondynamical or strong depends on a threshold value set on this weight. The dependence of weight-based measures on the orbital space will also be addressed in a subsequent section using a simple bond breaking process as an example, another situation in which strong correlation effects are typically expected.

7. Representing the Wave Function

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It has already been mentioned that DETs are only eigenfunctions of Ŝz, while CSFs are also eigenfunctions of the total spin. The total number of DETs (80) associated with a CFG and with a given spin-projection MS and number of unpaired electrons N is
f N M = ( N 1 2 N − M S )
(16)
from which the CSFs with different possible total spin values MS ≤ S can also be built. The number of CSFs (80) with S = MS is
f N S = ( N 1 2 N − S ) − ( N 1 2 N − S − 1 )
(17)
which is obviously less than the number of determinants. In fact, summing over all possible S values yields the number of determinants. (80)
To help ground these notions, consider some simple examples collected in Table 1, and discussed in detail elsewhere. (80) Taking the example of two electrons in two orbitals, the electron pair may be placed on the same orbital as in the configurations 20 and 02, or on different orbitals as in the configuration 11. In the former two cases, a single CSF consisting of a single determinant is enough to describe a closed-shell singlet, hence the success of determinants in many situations in chemistry. In the 11 case, a singlet and a triplet CSF can be built from two determinants, though the other two triplet states need only a single determinant. A slightly more complicated example involves distributing three electrons among three orbitals. For the configuration 111, the S = 3/2 cases yield nothing conceptually new, they are described by CSFs consisting of a single or multiple determinants. On the other hand, the S = 1/2, MS = 1/2 subspace can only be spanned by two degenerate CSFs, themselves consisting of multiple determinants. Since the reference function is often obtained at the Hartree–Fock level, it is worth recalling how this discussion affects such calculations. In more commonly occurring cases, the spin-adapted HF wave function can be constructed from either a single determinant, as in the typical closed-shell case, or from a single CSF, as in a low-spin open-shell case. (224) In those open-shell cases, such as S = 1/2, MS = 1/2 for 111, where a configuration cannot be uniquely described by a single CSF, (225) Edwards and Zerner recommend in a footnote that a HF calculation using a chosen CSF should be followed by a configuration interaction (CI) calculation. (225) This might minimally include only the CSFs belonging to the CFG in question, e.g., the two degenerate ones in the 111 case, possibly with or without orbital optimization, though the question remains whether such a small CI calculation would accurately reflect the exact solution. Indeed, Edwards and Zerner recommend CI in the active space, i.e., presumably including multiple configurations that might produce the same total and projected spin (e.g., 201 in the three-electron case). Nevertheless, the expansion coefficients of the CSFs of a single CFG will be of interest here for analytic purposes.
Table 1. Simple Examples of N-Electron Basis States, including Configurations (CFG), Configuration State Functions (CSF) and Determinants (DET)a
S MS CFG CSF DET
0 0 20 |11̅| |11̅|
0 0 02 |22̅| |22̅|
0 0 11 (|12̅|−|1̅2|) |12̅|, |1̅2|
1 1 11 |12| |12|
1 0 11 (|12̅|+|1̅2|) |12̅|, |1̅2|
1 –1 11 |1̅2̅| |1̅2̅|
3/2 3/2 111 |123| |123|
3/2 1/2 111 (|1̅23|+|12̅3|+|123̅|) |1̅23|, |12̅3|, |123̅|
1/2 1/2 111 (2|123̅|−|12̅3|−|1̅23|), |1̅23|, |12̅3|, |123̅|
a

Spatial orbitals are labelled simply as 1, 2, or 3 in DETs and CSFs, while the corresponding occupation numbers (0, 1 or 2) are indicated in CFGs.

It is worth noting that already at the HF level spin-restricted open-shell CSFs are considerably harder to optimize than the unrestricted determinants. This has led to several ways of circumventing the direct use of CSFs. A simplified version (130) of Löwdin’s projection operator (226) may be used to remove the spin-components that lie nearest to the desired state from the spin-unrestricted determinant, but in any case, spin-projection leads to artifacts if applied after optimization and falls short of size-consistency if made part of it. (227) More recent projection approaches constrain the eigenvalues of the spin-density to obtain results that are identical to the more cumbersome conventional spin-restricted approach. (228) Even more pragmatically, quasi-restricted approaches of various kinds have been proposed to obtain orbitals resembling restricted ones from some simple recipe. (229,230) More complicated Ansätze are usually obtained from reference states by (ideally spin-adapted) excitation (or ionization) operators, (214) and in a similar vein, it is also possible to define spin-flip operators that change the spin state of the reference. (231) In simple cases, such spin-flip operators may even be spin-adapted themselves. (232) Sometimes symmetry-broken solutions may be used creatively, e.g., to obtain some property of spin-states as in PO/BS-DFT discussed above, or in combining unrestricted determinants for quasi-spin-adaptation, (233) sometimes considerations about the CSF expansion are employed innovatively, e.g., to obtain singlet–triplet gaps from bideterminantal MS = 0 states (see Table 1) in cases in which methods with a single determinant bias encounter multideterminant situations. (234)
At this stage, there is some ambiguity whether we associate multireference with the requirement that the exact solution is dominated by more than a single determinant, CSF or configuration. In a normalized FCI expansion of the type in eq 8, a set of basis states dominates the expansion if their cumulative weight, ∑P |CP|2 is larger than the rest of the weights combined, i.e., 1 – ∑P |CP|2. Assuming knowledge of the exact solution, the following categories are proposed (cf. ref (123)):

  • Single Determinant (SD) Expansion: The weight of a single determinant dominates, and hence a single-determinant HF calculation is enough for a reference.

  • Single Spin-Coupling (SS) Expansion: It consists of a linear combination of determinants whose relative weights are fixed but the weight of the resulting CSF dominates the expansion. An restricted open-shell calculation should be a good reference.

  • Single Configurational (SC) Expansion: Such an expansion has multiple CSFs belonging to the same configuration. An expansion would fall into this category if the cumulative weight of CSFs of a single configuration dominates. A restricted open-shell calculation followed by a minimal CI calculation to determine the relative weight of the CSFs involved might be a good reference.

  • Multiconfiguration (MC) Expansion: No configuration alone dominates, and hence this is the true multireference case since no single reference function of the kinds discussed before can be found.

Note that the single reference (SR) categories above may overlap, since e.g., the closed-shell ground state of a two-electron system is at the same time SD, SS and SC. An open-shell singlet is SS and SC, but not SD. The three-electron doublet that has two CSFs would be SC but not SS or SD, if the cumulative weight of those CSFs dominate the expansion. All these are examples of various SR expansions. The simplest example of an MR expansion would be a combination of the type 02 + 20, as in the case of the dissociating H2 molecule. In the next section, we shall discuss how the choice of orbital space impacts this classification. Once spin-coupling is taken care of, an important source of correlation effects is addressed and the question remains how strong the remaining correlation effects are. This reflects the distinction between static and nondynamic correlation as used by Bartlett and Stanton, (17) with static correlation being eliminated by spin-adaptation. In this regard, this generalization of the excitation schemes can be compared with entanglement (22) and seniority (189) approaches which also aim at finding a good reference based on different criteria.

8. The Role of the Orbital Space

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The idea that the weight of the reference function in an approximate CI calculation could be used as an indication of multireference character is an old one, but it was also observed long ago that due to the bias of canonical orbitals toward the reference (HF) state, it is of limited use. (110) Furthermore, it was also suggested that such CI calculations should be complete within at least an active space for this metric to be of use. (109) In that case, the leading CAS coefficient in the basis of natural orbitals using determinants or CSFs was found to be a sensible metric. (235,236) The role of the orbital space has been discussed in connection to most approaches discussed in this perspective, (109,123,186,189,193) and orbital localization techniques form a part of practical DMRG (237) calculations as well as reducing the complexity of CAS (238) calculations. Within the DMRG context, orbital optimization techniques have been recently suggested as a means of reducing the multireference character of the wave function. (239) A similar result has been observed in SCI methods, where orbital optimization leads to a more rapid convergence to the FCI solution. (240) The recent work of Li Manni and co-workers on GUGA based FCIQMC (104−106) and its application as a stochastic CASCI solver (241) shed even more light on the role of the orbital space. This methodology has the following features relevant for the current discussion: a) using GUGA to calculate spin-adapted basis states efficiently; b) using a unitary orbital transformation to compress the wave function expansion; c) the combined use of unitary and symmetric group approaches means that the ordering of orbitals also affects the complexity of the wave function. The technique has been used to reduce the number of CSFs expected from the exponentially scaling combinatorial formula to a small number of CSFs, sometimes only a single one. (106) Moreover, not only is the wave function more compact, but the Hamiltonian assumes a convenient block-diagonal form that allows for the selective targeting of excited states. (106) Interestingly, it was also found that the best orbital ordering for a spin-adapted CSF is not identical to the best site-ordering in the DMRG sense. (242) More recent work has also attempted to place the orbital ordering process on a more systematic basis exploiting the commutation properties of cumulative spin operators and the Hamiltonian. (243) Here, the role of the orbital space will be illustrated via a simple example that will remain within a more traditional CAS/FCI context.
In this perspective, we are concerned with the a posteriori characterization of states rather than finding an a priori indicator of multireference character, and hence we will assume that an FCI solution is available. We will illustrate the role of orbital spaces by considering the H2 bond dissociation problem using canonical and localized orbitals. (244) We will define the multireference character M as
M = 1 − ∑ P | C P | 2 1 + ∑ P | C P | 2
(18)
where the summation runs over the DETs or CSFs belonging to the dominant CFG. This measure will converge to 1 for infinitely many coefficients assuming they are all equal, i.e., the worst MR case. In SR cases, M should be close to 0, and in the H2 case M = 1/3 corresponds to two configurations of equal weight. In practice, one might choose a different threshold on the cumulative weight of the configuration. For example, requiring the dominant cumulative weights to be above 80% leads to M ≤ 1/9. If such a dominant configuration exists (and if it were known in advance), it would be the ideal reference function on which a correlated Ansatz could be built.
Figure 1 shows the value of M at different internuclear separations D. In the canonical basis, a single configuration 20 is dominant around the equilibrium distance, while at infinite separation the CFGs 20 and 02 contribute equally. Thus, in the latter case, the expansion is multiconfigurational as well as multideterminantal. Evaluating M in the basis of local orbitals yields to the opposite results. The local orbitals are each essentially localized on one atom, and at long bond distance, a single CSF of the 11 configurations can describe the dissociation. Thus, the expansion is SC and SS, but not SD. At shorter bond lengths, the localized expansion is much less appropriate as seen from the rising multireference character. Changing from the canonical to the local orbital basis has a similar effect on entanglement-based metrics: (193) in the canonical basis entanglement changes between its minimal and maximal values as D grows and the other way around in the local basis since in the latter all occupation patterns are equally likely at small D. This ties in also with our earlier remarks on the relationship between VB and MO approaches.

Figure 1

Figure 1. Multireference character M as a function of internuclear distance D in the H2 dissociation problem for the FCI/STO-1G ground state. The blue curve corresponds to canonical orbitals (can), the red one to localized orbitals (loc).

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It has been argued earlier that an important source of correlation effects may be eliminated by spin adaptation. Yet, because of Löwdin’s dilemma, the constrained reference energy may increase and this could in turn lead to an increase in the correlation energy. In the H2 example, the spin-restricted closed-shell reference function has been found qualitatively adequate around the equilibrium bond length, but at large distances it leads to a catastrophic increase in the magnitude of the reference and correlation energies. Using canonical orbitals, the remedy involves combining the 20 and 02 configurations. Thus, it seems that enforcing spin symmetry makes the problem more complicated by requiring a multiconfigurational treatment whereas the spin-unrestricted single determinant also converges to the qualitatively correct energy at large bond lengths. This is because it reduces to one of the symmetry broken determinants |μν̅| or |μ̅ν| in this limit. In general, these broken symmetry determinants produce an energy that lies between the open shell singlet in eq 7 and triplet states that can also be constructed from them, and as such, they are not a good approximation to either. At large bond distances, however, the singlet–triplet gap closes and the energy converges to the correct limit while the wave function is not the exact one. (18,23) While symmetry breaking may be desirable under some circumstances, for finite molecular systems symmetric solutions are in general preferred. In the open-shell singlet case, for example, spin-adaptation is indispensable for a qualitatively accurate description. It is here that orbital rotations may play some role: by localizing the orbitals at large distances, it is possible to convert the multiconfigurational representation of the dissociating H2 molecule in the canonical basis into a single configurational open-shell singlet in the local basis.
The fact that the multireference character depends on the orbital representation warns against assigning the multiconfigurational nature of the wave function expansion as a physical property of a state of the system. At the very least, one should also demand that there be no unitary transformation originating in the orbital space that removes the multireference character of an expansion completely. Of course, the analysis as presented here is not a practical recipe for computation, since it assumes knowledge of the FCI solution, although it may be used fruitfully in interpreting less costly wave functions of the CASCI or CASSCF type. Furthermore, it does underline the potential of orbital rotations in reducing the multireference character. This important subject will be analyzed in more detail in a forthcoming publication.

9. Outlook

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The distinctions we introduced among single-determinantal, single spin-coupling, single configurational and multiconfigurational expansions should have a relevance in any chemical system, as also pointed out by other authors. (17,19) These categories form a clear hierarchy of SR measures in the sense that they are increasingly less restrictive: SD implies SS and both SD and SS imply SC. Unlike in extended systems, where symmetry broken solutions are often preferred, symmetry-adapted wave functions are often needed to reliably predict certain chemical properties of molecules. Again, unlike in periodic solids, strongly interacting regimes are usually easier to localize, and hence the complexity of the wave function can often be reduced. If spin coupling effects can be incorporated into a single reference function, then the remaining correlation effects might be weak, and in cases when they are not, such as in certain regions of bond dissociation curves, they can be typically taken care of by small active space calculations. As observed by Malrieu et al., (245) the fact that one can always build a small complete active space representation by placing two electrons in a pair of localized bonding and antibonding orbitals that accounts for such correlation effects is perhaps the best unbiased confirmation of Lewis’ idea of the chemical bond consisting of an electron pair. (246) We have discussed the effect of orbital rotations on the multireference character in a simple bond dissociation process and pointed out that such orbital rotations have the potential of reducing it. If by strong correlation we also mean the large size of the active space necessary to remedy these nondynamical effects, then it seems likely that most molecular systems are not strongly correlated, with the possible exception of multisite antiferromagnetically coupled systems. (107,247−251) Thus, the complexity of most finite chemical systems should be well below exponential and should in most cases involve only a few configurations. Devising an appropriate metric that can help find these a priori is a challenge, but measures based on leading coefficients, density matrices, natural orbitals and orbital entanglement have been used for this purpose in the literature.

Author Information

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Acknowledgments

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The authors thank Earl T. Cambell, Giovanni Li Manni and Pavel Pokhilko for useful discussions on the paper.

References

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With this understanding, a suitable choice of quantum nos. for a diatomic mol. appears to be one corresponding to an atom in a strong elec. field, namely, quantum nos. nr, lr, σr and Sr (Sr = 1/2 always) for the rth electron and quantum nos. s σl and σs for the mol. as a whole (σlr and σs represent quantized components of lr, and s, resp., with reference to the line joining the nuclei). These quantum nos. may be thought of as those assocd. with the imagined "united atom" formed by bringing the nuclei of the mol. together. A notation is proposed whereby the state of each electron and of the mol. as a whole can be designated, e. g., (1 sε)2 (2 sρ)2 (2sε)2 (2pρ), 2P for a seven-electron mol. with σ = 1, s = 1/2, in a symbol such as 2 sρ the superscript denotes lr, the main letter, σlr, thus 2 sP means that the electron in question has nr = 2, lr = 1, σlr = 0. Electrons with σlr = 0, 1, 2,-are referred to as s, p, d-electrons. It is shown that in a mol. it is usually natural to define a group of equiv. electrons giving a resultant σl = 0, s = 0 as a closed shell; in this sense, two s electrons, or four p, or d, f-, electrons form a closed shell. The possible mol. states corresponding to various electron configurations are deduced by means of the Pauli principle. Electrons which undergo an increase in their n values (principal quantum nos.) when atoms unite to form a mol. (Hund) are here called promoted electrons. The electrons in a mol. may be classified according to their bonding power, positive, zero, or negative. Electrons whose presence tends to hold the mol. together, as judged by the fact that their removal from a stable mol. causes a decrease in the energy of dissociation D or an increase in the equil. internuclear sepn. r0 may be said to have positive bonding power, and are identified with, or defined as, bonding electrons. Bonding power in terms of changes of D and of changes of r0 are distinguished as "energy-bonding-power" and "distance-bonding-power." On the whole, promoted electrons should tend to show negative energy-bonding-power, unpromoted electrons positive energy-bonding-power, but much should depend on "orbit dimensions." Certain rules governing the relations of the electronic states of a mol. to those of its dissociation products are discussed; in addn. to theoretical rules established by Hund in regard to σl and s values, another rule is here proposed, namely, that the σlr values of all the at. electrons before union should be preserved in the mol. (σlr conservation rule). Selection rules for electronic transitions are also discussed; in addn. to rules given by Hund, the following are proposed: Δlr = ±1 for intense transitions: Δσlr = 0, ±1. Results. The key to the assignment of quantum nos. made here is found in the fact that the mols. BO, CO+ and CN show an inverted 2P state instead of the normal 2P which should occur if this state were analogous to the ordinary 2P states of the Na atom. The existence of such a low-lying inverted 2P indicates that in these mols. there exists a closed shell of p electrons from which one is easily excited. It is concluded that this is a (2 pp)4 shell. The identification of 2 other closed shells, of s electrons, very likely (3 sρ)2 and (3 sε)2, follows; the electrons in these and the (2 pρ)4 shell are roughly equal in energy of binding. According to this interpretation, the electron jumps involved in the band spectra of BO, CN, CO+ and N+ are more analogous to x-ray than to optical electron transitions. From this beginning, proceeding to CO, N2, O2, O2+, F2, C2, etc., a self-consistent assignment of quantum nos. is built up for most of the known states of the various mols. treated in this paper. The spectroscopic analogies of CN, N2, NO, etc., to Na, Mg, Al are justified and the partial failure of these analogies such as the chem. resemblance of CN to a halogen, are explained. Nearly all the hitherto observed ionization potentials of the mols. discussed can be accounted for by the removal of a single electron from one or another of the various closed shells supposed to be present. The N2+ band fluorescence produced by short wave-length ultra-violet light (Oldenberg) is accounted for as the expected result of photo-ionization of a 3 SP electron. The steadily decreasing heat of dissocn. in the series, N2-NO-O2-F2, is accounted for by the successive addn. of promoted 3pP electrons with strong neg. bonding power. Starting from N2, whose normal state corresponds to a 1S configuration of closed shells, we add one 3 pP electron to give the 2p normal state of NO, and O2+, two to give the 3S normal state of O2, four to give a closed shell, (3 pP)4, which accounts for the 1S normal state of F2. In N2 (probably also in O2 and the other homopolar mols.) band systems for which Δlr ≠ 1 are notably lacking, thus giving support to Hund's predicted selection rule for homopolar mols., in the analogous heteropolar mol. CO2, many systems occur with Δlr = 0 than those for which Δlr = ±1. On account of this strict selection rule in N2 certain levels should be metastable, in particular the final level of the α afterglow bands of active nitrogen. There is evidence for the existence of a strict selection rule Δs = 1 in homopolar mols.
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    The electronic structure of some diatomic molecules
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    Transactions of the Faraday Society (1929), 25 (), 668-86CODEN: TFSOA4; ISSN:0014-7672.
    A review of the ideas of Franck and Herzberg, Heisenberg, Heitler and London, Hund and Mulliken on the formation of mols. and their dissocn. energy. J. criticizes Hund's application of the Pauli exclusion principle for the definition of mol. states. A notation is given which will make it possible to distinguish between mol, and at. levels in the same mol. The transitions encountered in mol. formation are given.
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    Heitler, W.; London, F. Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Z. Phys. 1927, 44, 455– 472,  DOI: 10.1007/BF01397394
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    Interaction of neutral atoms and homopolar binding according to the quantum mechanics
    Heitler, W.; London, F.
    Zeitschrift fuer Physik (1927), 44 (), 455-72CODEN: ZEPYAA; ISSN:0044-3328.
    The action of forces between neutral atoms has a characteristic ambiguity in the quantum mechanics. The ambiguity seems capable of including the different modes of behavior actually found, i. e., for H either homopolar binding or elastic reflection, but for the rare gases only reflection. It also permits an evaluation of the elastic reflection effects in He, giving results of the right order of magnitude. For the selection and discussion of the various possible interactions the Pauli principle is here applied to a system of several atoms.
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    Pauling, L. The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules. J. Am. Chem. Soc. 1931, 53, 1367– 1400,  DOI: 10.1021/ja01355a027
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    The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules
    Pauling, Linus
    Journal of the American Chemical Society (1931), 53 (), 1367-1400CODEN: JACSAT; ISSN:0002-7863.
    The electron-pair bond is discussed and from quantum mechanics a set of rules is presented which describes the properties of the bond with special ref. to the strength of the bond and the nature of the single-electron proper functions. These rules give information about the relative strengths of bonds formed by different atoms, the angles between bonds, properties of tetrahedral atoms with single and double bonds, cis and trans forms, the no. and spatial configuration of bonds and other properties. Transitions from electron-pair to ionic bonds are also discussed. A theory of the magnetic moments of mol. and complex ions is also developed. For the transition elements the proper functions involved in bond formation show that compds. with CN have electron-pair bonds, those with F have ionic bonds, and those with H2O, ion-dipole bonds. Electron structure, bond angles and other properties of mol. and complex ions can also be detd.from the magnetic data.
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    Coulson, C. A.; Fischer, I.
    Philosophical Magazine (1798-1977) (1949), 40 (), 386-93CODEN: PHMAA4; ISSN:0031-8086.
    The fundamentals are investigated in more detail than formerly. Configurational interaction consts. are calcd. Energy curves for the H2 mol. are derived. Failure at large internuclear distances is discussed.
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    Kutzelnigg, W.; Mukherjee, D. Cumulant expansion of the reduced density matrices. J. Chem. Phys. 1999, 110, 2800– 2809,  DOI: 10.1063/1.478189
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    Cumulant expansion of the reduced density matrixes
    Kutzelnigg, Werner; Mukherjee, Debashis
    Journal of Chemical Physics (1999), 110 (6), 2800-2809CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    K-particle cumulants λk (for 2≤k≤n) corresponding to the k-particle reduced d. matrixes γk for an n-fermion system are defined via a generating function. The two-particle cumulant λ2 describes two-particle correlations (excluding exchange), λ3 genuine three-particle correlations etc. The properties of these cumulants are analyzed. Conditions for vanishing of certain λk are formulated. Necessary and sufficient for λ2=0 is the well-known idempotency condition γ2=γ for γ γ1. For λ3=0 to hold, a general necessary condition is Tr{2γ3-3γ2+γ}=0, for three special forms of the wave function (arbitrary two-electron state, antisymmetrized product of strongly orthogonal geminals on antisymmetrized geminal power wave function of extreme type) 2γ3-3γ2+γ=0 turns out to be necessary and sufficient. For a multiconfiguration SCF wave function the only nonvanishing matrix elements of the cumulants are those where all labels refer to active (partially occupied) spin orbitals. Spin-free cumulants Λk corresponding to the spin-free reduced d. matrixes Γk are also defined and analyzed. The main interest in the d. cumulants is in connection with the recently formulated normal ordering and the corresponding Wick theorem for arbitrary ref. functions, but they are also useful for an anal. of electron correlation.
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    Constitution of metallic sodium
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    The lattice const., binding energy and compressibility of metallic Na are calcd. from a theoretical treatment of the energy of the free electrons.
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    Wigner, E.; Seitz, F. On the Constitution of Metallic Sodium. II. Phys. Rev. 1934, 46, 509– 524,  DOI: 10.1103/PhysRev.46.509
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    The constitution of metallic sodium. II
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    cf. C. A. 27, 3135. Calcns. including interactions between electrons with parallel spins give a lattice energy of only 9 kg.-cal., against the exptl. value of 26.9 kg.-cal. When interactions between electrons with antiparallel spins are included the calcd. value becomes 23.2 kg.-cal.
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    McWeeny, R. The density matrix in many-electron quantum mechanics I. Generalized product functions. Factorization and physical interpretation of the density matrices. Proc. R. Soc. A 1959, 253, 242– 259,  DOI: 10.1098/rspa.1959.0191
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    The density matrix in many-electron quantum mechanics. I. Generalized product functions. Factorization and physical interpretation of the density matrixes
    McWeeny, R.
    Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences (1959), 253 (), 242-59CODEN: PRLAAZ; ISSN:1364-5021.
    Many-electron wave functions are usually constructed from antisymmetrized products of 1-electron orbitals (determinants) and energy calcns. are based on the matrix element expressions due to Slater. The orbitals in such a product are replaced by "group functions,'' each describing any no. of electrons, and the necessary generalization of Slater's results is carried out. It is 1st necessary to develop the matrix theory of N-particle systems and to show that, for systems described by generalized product functions, the matrixes of the whole system can be expressed in terms of those of the component electron groups. The matrix elements of the Hamiltonian between generalized product functions are then given by expressions which resemble those of Slater, the "Coulomb" and "exchange" integrals being replaced by integrals contg. the 1-electron matrixes of the various groups. By setting up an "effective" Hamiltonian for each electron group in the presence of the others, the discussion of a many-particle system in which groups or "shells" can be distinguished (e.g. at. K, L, M, ..., shells) can rigorously be reduced to a discussion of smaller subsystems. A single generalized product (cf. the single determinant of Hartree-Fock theory) provides a convenient 1st approxn. and the effect of admitting "excited" products (cf. configuration interaction) can be estd. by a perturbation method. The energy expression can then be discussed in terms of the electron d. and "pair" functions. The energy is a sum of group energies supplemented by interaction terms which represent electrostatic repulsions between charge clouds, the polarization of each group in the field of the others, and dispersion effects of the type defined by London. All these terms can be calcd. for group functions of any kind, in terms of the d. matrixes of the sep. groups. Approxns. to the theory of intermol. forces and to π-electron systems are discussed.
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    Giner, E.; Tenti, L.; Angeli, C.; Malrieu, J.-P. The “Fermi hole” and the correlation introduced by the symmetrization or the anti-symmetrization of the wave function. J. Chem. Phys. 2016, 145, 124114,  DOI: 10.1063/1.4963018
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    The "Fermi hole" and the correlation introduced by the symmetrization or the anti-symmetrization of the wave function
    Giner, Emmanuel; Tenti, Lorenzo; Angeli, Celestino; Malrieu, Jean-Paul
    Journal of Chemical Physics (2016), 145 (12), 124114/1-124114/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The impact of the antisymmetrization is often addressed as a local property of the many-electron wave function, namely that the wave function should vanish when two electrons with parallel spins are in the same position in space. In this paper, we emphasize that this presentation is unduly restrictive: we illustrate the strong non-local character of the antisymmetrization principle, together with the fact that it is a matter of spin symmetry rather than spin parallelism. To this aim, we focus our attention on the simplest representation of various states of two-electron systems, both in at. (helium atom) and mol. (H2 and the π system of the ethylene mol.) cases. We discuss the non-local property of the nodal structure of some two-electron wave functions, both using anal. derivations and graphical representations of cuttings of the nodal hypersurfaces. The attention is then focused on the impact of the antisymmetrization on the maxima of the two-body d., and we show that it introduces strong correlation effects (radial and/or angular) with a non-local character. These correlation effects are analyzed in terms of inflation and depletion zones, which are easily identifiable, thanks to the nodes of the orbitals composing the wave function. Also, we show that the correlation effects induced by the antisymmetrization occur also for anti-parallel spins since all Ms components of a given spin state have the same N-body densities. Finally, we illustrate that these correlation effects occur also for the singlet states, but they have strictly opposite impacts: the inflation zones in the triplet become depletion zones in the singlet and vice versa. (c) 2016 American Institute of Physics.
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    Kutzelnigg, W.; Mukherjee, D. Normal order and extended Wick theorem for a multiconfiguration reference wave function. J. Chem. Phys. 1997, 107, 432– 449,  DOI: 10.1063/1.474405
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    Normal order and extended Wick theorem for a multiconfiguration reference wave function
    Kutzelnigg, Werner; Mukherjee, Debashis
    Journal of Chemical Physics (1997), 107 (2), 432-449CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A generalization of normal ordering and of Wick's theorem with respect to an arbitrary ref. function Φ as some generalized "phys. vacuum" is formulated in a different (but essentially equiv.) way than that suggested previously by one of the present authors. Guiding principles are that normal order operators with respect to any ref. state must be expressible as linear combinations of those with respect to the genuine vacuum, that the vacuum expectation value of a normal order operator must vanish (with respect to the vacuum to which it is in normal order), and that the well-known formalism for a single Slater determinant as phys. vacuum must be contained as a special case. The derivation is largely based on the concepts of "Quantum Chem. in Fock space", which means that particle-no.-conserving operators (excitation operators) play a central role. Nevertheless, the contraction rules in the frame of the generalized Wick theorem are derived, that hold for non-particle-no.-conserving operators as well. The contraction rules are formulated and illustrated in terms of diagrams. The contractions involve the "residual n-particle d. matrixes" λ, which are the irreducible (non-factorizable) parts of the conventional n-particle d. matrixes γ, in the sense of a cumulant expansion for the d. A spin-free formulation is presented as well. The expression of the Hamiltonian in normal order with respect to a multiconfiguration ref. function leads to a natural definition of a generalized Fock operator. MC-SCF-theory is easily worked out in this context. The paper concludes with a discussion of the excited configurations and the first-order interacting space, that underlies a perturbative coupled cluster type correction to the MCSCF function for an arbitrary ref. function, and with general implications of the new formalism, that is related to "internally contracted multireference CI". The present generalization of normal ordering is not only valid for arbitrary ref. functions, but also if the ref. state is an ensemble state.
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    Evangelista, F. A. Automatic derivation of many-body theories based on general Fermi vacua. J. Chem. Phys. 2022, 157, 064111,  DOI: 10.1063/5.0097858
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    Automatic derivation of many-body theories based on general Fermi vacua
    Evangelista, Francesco A.
    Journal of Chemical Physics (2022), 157 (6), 064111CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    This paper describes WIC&D, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated refs. and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. WICK&D employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since WICK&D can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of WICK&D, we use it to evaluate the single-ref. coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multi-ref. perturbation theory. (c) 2022 American Institute of Physics.
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    Tsuchimochi, T.; Scuseria, G. E. Strong correlations via constrained-pairing mean-field theory. J. Chem. Phys. 2009, 131, 121102,  DOI: 10.1063/1.3237029
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    Strong correlations via constrained-pairing mean-field theory
    Tsuchimochi, Takashi; Scuseria, Gustavo E.
    Journal of Chemical Physics (2009), 131 (12), 121102/1-121102/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We present a mean-field approach for accurately describing strong correlations via electron no. fluctuations and pairings constrained to an active space. Electron no. conservation is broken and correct only on av., but both spin and spatial symmetries are preserved. Optimized natural orbitals and occupations are detd. by diagonalization of a mean-field Hamiltonian. This constrained-pairing mean-field theory (CPMFT) yields a two-particle d. matrix ansatz that exclusively describes strong correlations. We demonstrate CPMFT accuracy with applications to the metal-insulator transition of large hydrogen clusters and mol. dissocn. curves. (c) 2009 American Institute of Physics.
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    Kutzelnigg, Werner
    Chemical Physics (2012), 401 (), 119-124CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)
    A CC (coupled-cluster) ansatz based on a GVB (generalized valence bond) or an APSG (antisymmetrized product of strongly orthogonal geminals) ref. function arises naturally if one tries to treat strong correlations exactly (to infinite order), and weak correlations by TCC (traditional coupled cluster) theory. This ansatz is proposed as an alternative to MC-CC (multi-configuration coupled cluster) theory. One uses esp. that APSG and GVB are of CC type, but allow to combine separability with the variation principle. The energy and the stationarity conditions are formulated in terms of spin-free d. cumulants. The replacement operators corresponding to the APSG ansatz generate a Lie algebra which is a sub-algebra of that of all replacement operators.
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    Georges, Antoine; Kotliar, Gabriel; Krauth, Werner; Rozenberg, Marcelo J.
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    A review with many refs. is given on the dynamic mean-field theory of strongly correlated electron, systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the std. mean-field construction from classical statistical mechanics to quantum problems. We discuss the phys. ideas underlying this theory and its math. derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamic mean-field equations are reviewed and compared to each other. The method can be used for the detn. of phase diagrams (by comparing the stability of various types of long-range order), and the calcn. of thermodn. properties, one-particle Green functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to expts. on three-dimensional transition metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are discussed. Computer programs for the numerical implementation of this method are also provided with this article.
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    Quantum Monte Carlo simulations of solids
    Foulkes, W. M. C.; Mitas, L.; Needs, R. J.; Rajagopal, G.
    Reviews of Modern Physics (2001), 73 (1), 33-83CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
    This review with many refs. describes the variational and fixed-node diffusion quantum Monte Carlo methods and how they may be used to calc. the properties of many-electron systems. These stochastic wave-function-based approaches provide a very direct treatment of quantum many-body effects and serve as benchmarks against which other techniques may be compared. They complement the less demanding d.-functional approach by providing more accurate results and a deeper understanding of the physics of electronic correlation in real materials. The algorithms are intrinsically parallel, and currently available high-performance computers allow applications to systems contg. a thousand or more electrons. With these tools one can study complicated problems such as the properties of surfaces and defects, while including electron correlation effects with high precision. The authors provide a pedagogical overview of the techniques and describe a selection of applications to ground and excited states of solids and clusters.
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    Zhang, S.; Malone, F. D.; Morales, M. A. Auxiliary-field quantum Monte Carlo calculations of the structural properties of nickel oxide. J. Chem. Phys. 2018, 149, 164102,  DOI: 10.1063/1.5040900
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    Auxiliary-field quantum Monte Carlo calculations of the structural properties of nickel oxide
    Zhang, Shuai; Malone, Fionn D.; Morales, Miguel A.
    Journal of Chemical Physics (2018), 149 (16), 164102/1-164102/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Auxiliary-field quantum Monte Carlo (AFQMC) has repeatedly demonstrated itself as one of the most accurate quantum many-body methods, capable of simulating both real and model systems. We investigate the application of AFQMC to realistic strongly correlated materials in periodic Gaussian basis sets. Using nickel oxide (NiO) as an example, we investigate the importance of finite size effects and basis set errors on the structural properties of the correlated solid. We provide benchmark calcns. for NiO and compare our results to both exptl. measurements and existing theor. methods. (c) 2018 American Institute of Physics.
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    The role of symmetry in fundamental physics is reviewed with no refs.
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    Collective description of electron interactions. I. Magnetic interactions
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    Math.-theoretical. A new approach to the treatment of the interaction in a collection of electrons is developed, which is called the collective description. The collective description is based on the organized behavior produced by the interactions in an electron gas of high d.; this organized behavior results in oscillations of the system as a whole, the so-called plasma oscillations. The collective description, in contrast to the usual individual particle description, describes in a natural way the long-range correlations in electron positions brought about by their mutual interaction. Here, attention is confined to the magnetic interactions between the electrons. Both a classical and a quantum-mech. treatment are given, and the criteria for the validity of the collective description are discussed.
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    The virial and molecular structure
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    cf. C. A. 26, 5799. The virial theorem is applied to a mol. if external forces are applied to keep the nuclei fixed. It permits sep. detns. of the kinetic and potential energies for all configurations of the nuclei from the total-energy curves as derived from expt. or theory. Such potential- and kinetic-energy curves are derived for simple forms of the total-energy curves for diat. mols. These can be readily interpreted as indicating bond formation in attractive forces. The method can be extended to apply to more complicated mols. and to solids.
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    Kinetic and potential energies of an electron gas
    March, N. H.
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    cf. Gell-Mann and Brueckner, C.A. 51, 17393h. The kinetic and potential energy values of an electron gas may be obtained exactly in the high-d. limit by applying the virial theorem to the correlation energy.
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    Virial theorem for the homogeneous electron gas
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    A proof is presented of the virial theorem for the interacting electron gas in a uniform pos. background with the boundary conditions used in actual calcns. of the total energy.
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    Physical nature of the chemical bond
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    Mol. energy as well as all other observable quantities are completely detd. by 2 functions: the d. (1st-order d. kernel) and the pair d. (2nd-order d. kernel). These 2 are chosen as the starting point for an interpretive analysis of mols. A simultaneous regional and phys. partitioning of the mol. d., the mol. pair d., and the mol. energy is attempted such that meaningful concepts can be assocd. with the proposed fragments. An analysis of how electron-sharing affects ds. and energies is included. It is suggested that a mol. differs from the juxtaposed atoms in 3 major aspects characterized by the concepts of interference, penetration, and charge transfer. Interference contributions embody the precise connections existing between overlap and chem. binding. Penetration contributions describe how electron sharing modifies electronic correlations. From an analysis of the H2 mol.-H2+ ion it is concluded that electron sharing leads to chem. binding as the result of a subtle interplay between the uncertainty principle and nuclear attractions.
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    Hubbard, J. Electron correlations in narrow energy bands. Proc. R. Soc. A 1963, 276, 238– 257,  DOI: 10.1098/rspa.1963.0204
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    Janesko, B. G. Strong correlation in surface chemistry. Mol. Simul. 2017, 43, 394– 405,  DOI: 10.1080/08927022.2016.1261136
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    Strong correlation in surface chemistry
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    Molecular Simulation (2017), 43 (5-6), 394-405CODEN: MOSIEA; ISSN:0892-7022. (Taylor & Francis Ltd.)
    D. functional theory (DFT) simulations of surface chem. have emerged as a valuable complement to expt. However, std. DFT methods do not always accurately model the 'strong' electron correlation effects seen in stretched covalent bonds. Such systems' ground-state wavefunctions are not well-described by single MO configurations. I review some of the challenges of strong correlation, and some methods used to simulate it in surface chem. I also use the electron delocalisation range function EDR(), which quantifies the extent to which electrons at point delocalise over distance d, to highlight how a nearby metal cluster affects strong correlation in a dissocg. covalent bond.
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    Motta, M.; Ceperley, D. M.; Chan, G. K.-L.; Gomez, J. A.; Gull, E.; Guo, S.; Jiménez- Hoyos, C. A.; Lan, T. N.; Li, J.; Ma, F.; Millis, A. J.; Prokof’ev, N. V.; Ray, U.; Scuseria, G. E.; Sorella, S.; Stoudenmire, E. M.; Sun, Q.; Tupitsyn, I. S.; White, S. R.; Zgid, D.; Zhang, S. Towards the Solution of the Many-Electron Problem in Real Materials: Equation of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods. Phys. Rev. X 2017, 7, 031059,  DOI: 10.1103/PhysRevX.7.031059
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    Towards the solution of the many-electron problem in real materials: equation of state of the hydrogen chain with state-of-the-art many-body methods
    Motta, Mario; Ceperley, David M.; Chan, Garnet Kin-Lic; Gomez, John A.; Gull, Emanuel; Guo, Sheng; Jimenez-Hoyos, Carlos A.; Lan, Tran Nguyen; Li, Jia; Ma, Fengjie; Millis, Andrew J.; Prokof'ev, Nikolay V.; Ray, Ushnish; Scuseria, Gustavo E.; Sorella, Sandro; Stoudenmire, Edwin M.; Sun, Qiming; Tupitsyn, Igor S.; White, Steven R.; Zgid, Dominika; Zhang, Shiwei
    Physical Review X (2017), 7 (3), 031059/1-031059/28CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
    We present numerical results for the equation of state of an infinite chain of hydrogen atoms. A variety of modern many-body methods are employed, with exhaustive cross-checks and validation. Approaches for reaching the continuous space limit and the thermodn. limit are investigated, proposed, and tested. The detailed comparisons provide a benchmark for assessing the current state of the art in many-body computation, and for the development of new methods. The ground-state energy per atom in the linear chain is accurately detd. vs. bond length, with a confidence bound given on all uncertainties.
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    Motta, M.; Genovese, C.; Ma, F.; Cui, Z.-H.; Sawaya, R.; Chan, G. K.-L.; Chepiga, N.; Helms, P.; Jiménez-Hoyos, C.; Millis, A. J.; Ray, U.; Ronca, E.; Shi, H.; Sorella, S.; Stoudenmire, E. M.; White, S. R.; Zhang, S. Ground-State Properties of the Hydrogen Chain: Dimerization, Insulator-to-Metal Transition, and Magnetic Phases. Phys. Rev. X 2020, 10, 031058
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    Ground-State Properties of the Hydrogen Chain: Dimerization, Insulator-to-Metal Transition, and Magnetic Phases
    Motta, Mario; Genovese, Claudio; Ma, Fengjie; Cui, Zhi-Hao; Sawaya, Randy; Chan, Garnet Kin-Lic; Chepiga, Natalia; Helms, Phillip; Jimenez-Hoyos, Carlos; Millis, Andrew J.; Ray, Ushnish; Ronca, Enrico; Shi, Hao; Sorella, Sandro; Stoudenmire, Edwin M.; White, Steven R.; Zhang, Shiwei
    Physical Review X (2020), 10 (3), 031058CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
    Accurate and predictive computations of the quantum-mech. behavior of many interacting electrons in realistic at. environments are crit. for the theor. design of materials with desired properties, and they require solving the grand-challenge problem of the many-electron Schr.ovrddot.odinger equation. An infinite chain of equispaced hydrogen atoms is perhaps the simplest realistic model for a bulk material, embodying several central themes of modern condensed-matter physics and chem. while retaining a connection to the paradigmatic Hubbard model. Here, we report a combined application of cutting-edge computational methods to det. the properties of the hydrogen chain in its quantum-mech. ground state. Varying the sepn. between the nuclei leads to a rich phase diagram, including a Mott phase with quasi-long-range antiferromagnetic order, electron d. dimerization with power-law correlations, an insulator-to-metal transition, and an intricate set of intertwined magnetic orders.
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    Sinanoğlu, O.; Tuan, D. F.-t. Many-Electron Theory of Atoms and Molecules. III. Effect of Correlation on Orbitals. J. Chem. Phys. 1963, 38, 1740– 1748,  DOI: 10.1063/1.1776948
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    Many-electron theory of atoms and molecules. III. Effect of correlation on orbitals
    Sinanoglu, Oktay; Tuan, Debbie Fu-tai
    Journal of Chemical Physics (1963), 38 (), 1740-8CODEN: JCPSA6; ISSN:0021-9606.
    cf. CA 57, 13299c. The exact wave function of an N-electron atom or mol. contains, after the Hartree-Fock (H.F.) part, correlation terms involving successively 1, 2. . . . N electrons at a time. Particularly in closed shells, 1-electron terms fi result mainly from pair correlations. The fi were previously neglected in the many-electron theory. Reasons for the smallness of fi are summarized. Different types of correlation effects are classified, and methods for estg. each type of fi are given. fi in closed form, i.e., including infinitely many single excitations, is <2.8% of the H.F. orbital in He with an energy contribution 0.0001 a.u. (63 cal./mole). In the H2 mol. fi is negligible for (R/Re) < 2. At. larger R, as (1σ0)2 becomes degenerate with (1σu)2, the fi effect increases to ∼0.4 e.v. at dissocn. However, in such cases and in actual nonclosed shells, these nondynamical fi are removed if H.F. orbitals are obtained after the removal of degeneracies. Dynamic correlation effects give negligible fi, and so, generalized S.C.F. methods are not necessary. Qual. quantum chemistry can be based on just H.F. orbitals or approxns. to them, though energies include localized pair correlations.
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    Prendergast, D.; Nolan, M.; Filippi, C.; Fahy, S.; Greer, J. Impact of electron–electron cusp on configuration interaction energies. J. Chem. Phys. 2001, 115, 1626– 1634,  DOI: 10.1063/1.1383585
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    Impact of electron-electron cusp on configuration interaction energies
    Prendergast, David; Nolan, M.; Filippi, Claudia; Fahy, Stephen; Greer, J. C.
    Journal of Chemical Physics (2001), 115 (4), 1626-1634CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The effect of the electron-electron cusp on the convergence of CI wave functions is examd. By analogy with the pseudopotential approach for electron-ion interactions, an effective electron-electron interaction is developed which closely reproduces the scattering of the Coulomb interaction but is smooth and finite at zero electron-electron sepn. The exact many-electron wave function for this smooth effective interaction has no cusp at zero electron-electron sepn. We perform CI and quantum Monte Carlo calcns. for He and Be atoms, both with the Coulomb electron-electron interaction and with the smooth effective electron-electron interaction. We find that convergence of the CI expansion of the wave function for the smooth electron-electron interaction is not significantly improved compared with that for the divergent Coulomb interaction for energy differences on the order of 1 mHartree. This shows that, contrary to popular belief, description of the electron-electron cusp is not a limiting factor, to within chem. accuracy, for CI calcns.
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    Mok, D. K.; Neumann, R.; Handy, N. C. Dynamical and nondynamical correlation. J. Phys. Chem. 1996, 100, 6225– 6230,  DOI: 10.1021/jp9528020
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    Dynamical and Nondynamical Correlation
    Mok, Daniel K. W.; Neumann, Ralf; Handy, Nicholas C.
    Journal of Physical Chemistry (1996), 100 (15), 6225-30CODEN: JPCHAX; ISSN:0022-3654. (American Chemical Society)
    The variation is studied of correlation energies with bond distances of various first row diat. mols. SCF and complete active space SCF potential curves of these mols. are calcd. Exact potential energy curves are constructed from exptl. data using the Rydberg-Klein-Rees method. With appropriate definitions, the dynamical and nondynamical correlation energies are obtained and the variation of these with bond distance is calcd. Two definitions of nondynamical correlation are examd. Classifying the angular correlation as dynamical seems to be a better way to partition the correlation energy. The correlation functionals of d. functional theory, VWN, LYP, and P86, are also evaluated and compared with the ab initio dynamical correlation energies. LYP appears to give the closest agreement with the dynamical correlation energy.
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    Hollett, J. W.; Gill, P. M. The two faces of static correlation. J. Chem. Phys. 2011, 134, 114111,  DOI: 10.1063/1.3570574
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    The two faces of static correlation
    Hollett, Joshua W.; Gill, Peter M. W.
    Journal of Chemical Physics (2011), 134 (11), 114111/1-114111/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    RHF and UHF wavefunctions for beryllium-like ions with nuclear charge 3 ≤ Z ≤ 5 are found using a near-complete Slater basis set. The triplet (RHF → UHF) instability and correlation energy are investigated as a function of Z and we find that the instability vanishes for Z > 4.5. We reproduce this surprising behavior using a minimal-basis model and, by comparing with the stretched H2 mol., conclude that "static" (also known as nondynamical, near-degeneracy, first-order, or strong) correlation comes in two flavors: one that can be captured by UHF and another that cannot. In the former (Type A), there is an "abs. near-degeneracy"; in the latter (Type B), there is a "relative near-degeneracy." This dichotomy clarifies discussions of static correlation effects. (c) 2011 American Institute of Physics.
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    Benavides-Riveros, C. L.; Lathiotakis, N. N.; Marques, M. A. Towards a formal definition of static and dynamic electronic correlations. Phys. Chem. Chem. Phys. 2017, 19, 12655– 12664,  DOI: 10.1039/C7CP01137G
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    Towards a formal definition of static and dynamic electronic correlations
    Benavides-Riveros, Carlos L.; Lathiotakis, Nektarios N.; Marques, Miguel A. L.
    Physical Chemistry Chemical Physics (2017), 19 (20), 12655-12664CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    Some of the most spectacular failures of d.-functional and Hartree-Fock theories are related to an incorrect description of the so-called static electron correlation. Motivated by recent progress in the N-representability problem of the one-body d. matrix for pure states, we propose a method to quantify the static contribution to the electronic correlation. By studying several mol. systems we show that our proposal correlates well with our intuition of static and dynamic electron correlation. Our results bring out the paramount importance of the occupancy of the highest occupied natural spin-orbital in such quantification.
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    Bulik, I. W.; Henderson, T. M.; Scuseria, G. E. Can single-reference coupled cluster theory describe static correlation?. J. Chem. Theory Comput. 2015, 11, 3171– 3179,  DOI: 10.1021/acs.jctc.5b00422
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    Can Single-Reference Coupled Cluster Theory Describe Static Correlation?
    Bulik, Ireneusz W.; Henderson, Thomas M.; Scuseria, Gustavo E.
    Journal of Chemical Theory and Computation (2015), 11 (7), 3171-3179CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    While restricted single-ref. coupled cluster theory truncated to singles and doubles (CCSD) provides very accurate results for weakly correlated systems, it usually fails in the presence of static or strong correlation. This failure is generally attributed to the qual. breakdown of the ref., and can accordingly be cor. by using a multideterminant ref., including higher-body cluster operators in the ansatz, or allowing symmetry breaking in the ref. None of these solns. are ideal; multireference coupled cluster is not black box, including higher-body cluster operators is computationally demanding, and allowing symmetry breaking leads to the loss of good quantum nos. It has long been recognized that quasidegeneracies can instead be treated by modifying the coupled cluster ansatz. The recently introduced pair coupled cluster doubles (pCCD) approach is one such example which avoids catastrophic failures and accurately models strong correlations in a symmetry-adapted framework. Here, we generalize pCCD to a singlet-paired coupled cluster model (CCD0) intermediate between coupled cluster doubles and pCCD, yielding a method that possesses the invariances of the former and much of the stability of the latter. Moreover, CCD0 retains the full structure of coupled cluster theory, including a fermionic wave function, antisym. cluster amplitudes, and well-defined response equations and d. matrixes.
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    Karton, A.; Rabinovich, E.; Martin, J. M.; Ruscic, B. W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions. J. Chem. Phys. 2006, 125, 144108,  DOI: 10.1063/1.2348881
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    W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions
    Karton, Amir; Rabinovich, Elena; Martin, Jan M. L.; Ruscic, Branko
    Journal of Chemical Physics (2006), 125 (14), 144108/1-144108/17CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    In an attempt to improve on our earlier W3 theory [A. D. Boese et al., J. Chem. Phys. 120, 4129 (2004)] we consider such refinements as more accurate ests. for the contribution of connected quadruple excitations (T4), inclusion of connected quintuple excitations (T5), diagonal Born-Oppenheimer corrections (DBOC), and improved basis set extrapolation procedures. Revised exptl. data for validation purposes were obtained from the latest version of the Active Thermochem. Tables thermochem. network. The recent CCSDT(Q) method offers a cost-effective way of estg. T4, but is insufficient by itself if the mol. exhibits some nondynamical correlation. The latter considerably slows down basis set convergence for T4, and anomalous basis set convergence in highly polar systems makes two-point extrapolation procedures unusable. However, we found that the CCSDTQ-CCSDT(Q) difference converges quite rapidly with the basis set, and that the formula 1.10[CCSDT(Q)/cc-pVTZ + CCSDTQ/cc-pVDZ - CCSDT(Q)/cc-pVDZ] offers a very reliable as well as fairly cost-effective est. of the basis set limit T4 contribution. The T5 contribution converges very rapidly with the basis set, and even a simple double-zeta basis set appears to be adequate. The largest T5 contribution found in the present work is on the order of 0.5 kcal/mol (for ozone). DBOCs are significant at the 0.1 kcal/mol level in hydride systems. Post-CCSD(T) contributions to the core-valence correlation energy are only significant at that level in systems with severe nondynamical correlation effects. Based on the accumulated experience, a new computational thermochem. protocol for first- and second-row main-group systems, to be known as W4 theory, is proposed. Its computational cost is not insurmountably higher than that of the earlier W3 theory, while performance is markedly superior. Our W4 atomization energies for a no. of key species are in excellent agreement (better than 0.1 kcal/mol on av., 95% confidence intervals narrower than 1 kJ/mol) with the latest exptl. data obtained from Active Thermochem. Tables. Lower-cost variants are proposed: the sequence W1 → W2.2 → W3.2 → W4lite → W4 is proposed as a converging hierarchy of computational thermochem. methods. A simple a priori est. for the importance of post-CCSD(T) correlation contributions (and hence a pessimistic est. for the error in a W2-type calcn.) is proposed.
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    Hait, D.; Tubman, N. M.; Levine, D. S.; Whaley, K. B.; Head-Gordon, M. What levels of coupled cluster theory are appropriate for transition metal systems? A study using near-exact quantum chemical values for 3d transition metal binary compounds. J. Chem. Theory Comput. 2019, 15, 5370– 5385,  DOI: 10.1021/acs.jctc.9b00674
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    What Levels of Coupled Cluster Theory Are Appropriate for Transition Metal Systems? A Study Using Near-Exact Quantum Chemical Values for 3d Transition Metal Binary Compounds
    Hait, Diptarka; Tubman, Norman M.; Levine, Daniel S.; Whaley, K. Birgitta; Head-Gordon, Martin
    Journal of Chemical Theory and Computation (2019), 15 (10), 5370-5385CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Transition metal compds. are traditionally considered to be challenging for std. quantum chem. approxns. like coupled cluster (CC) theory, which are usually employed to validate lower level methods like d. functional theory (DFT). To explore this issue, we present a database of bond dissocn. energies (BDEs) for 74 spin states of 69 diat. species contg. a 3d transition metal atom and a main group element, in the moderately sized def2-SVP basis. The presented BDEs appear to have an (estd.) 3σ error less than 1 kJ/mol relative to the exact solns. to the nonrelativistic Born-Oppenheimer Hamiltonian. These benchmark values were used to assess the performance of a wide range of std. single ref. CC models, as the results should be beneficial for understanding the limitations of these models for transition metal systems. We find that interactions between metals and monovalent ligands like hydride and fluoride are well described by CCSDT. Similarly, CCSDTQ appears to be adequate for bonds between metals and nominally divalent ligands like oxide and sulfide. However, interactions with polyvalent ligands like nitride and carbide are more challenging, with even CCSDTQ(P)Λ yielding errors on the scale of a few kJ/mol. We also find that many perturbative and iterative approxns. to higher order terms either yield disappointing results or actually worsen the performance relative to the baseline low level CC method, indicating that complexity does not always guarantee accuracy.
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    Roos, B. O.; Taylor, P. R.; Sigbahn, P. E. A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys. 1980, 48, 157– 173,  DOI: 10.1016/0301-0104(80)80045-0
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    A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach
    Roos, Bjoern O.; Taylor, Peter R.; Siegbahn, E. M.
    Chemical Physics (1980), 48 (2), 157-73CODEN: CMPHC2; ISSN:0301-0104.
    A d. matrix formulation of the super-CI MCSCF method is presented. The MC expansion is assumed to be complete in an active subset of the orbital space, and the corresponding CI secular problem is solved by a direct scheme using the unitary group approach. With a d. matrix formulation the orbital optimization step becomes independent of the size of the CI expansion. It is possible to formulate the super-CI in terms of d. matrices defined only in the small active subspace; the doubly occupied orbitals (the inactive subspace) do not enter. Further, in the unitary group formalism it is straightforward and simple to obtain the necessary d. matrices from the symbolic formula list. It then becomes possible to treat very long MC expansions, the largest so far comprising 726 configurations. The method is demonstrated in a calcn. of the potential curves for the 3 lowest states (1.sum.g+, 3.sum.u+ and 3πg) of the N2 mol., using a medium-sized gaussian basis set. 7 Active orbitals were used yielding the following results: Dc:8.76(9.90), 2.43(3.68) and 3.39 (4.90) eV; rc:1.108 (1.098), 1.309(1.287) and 1.230 (1.213) Å; ωe: 2333 (2359), 1385 (1461) and 1680 (1733) cm-1, for the 3 states (exptl. values within parentheses). The results of these calcns. indicate that it is important to consider not only the dissocn. limit but also the united atom limit in partitioning the occupied orbital space into an active and an inactive part.
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    A review. The d. matrix renormalization group is a method that is useful for describing mols. that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the d. matrix renormalization group works, a survey of its existing applications to mol. problems, and some thoughts on the future of the method.
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    Scuseria, G. E.; Jiménez-Hoyos, C. A.; Henderson, T. M.; Samanta, K.; Ellis, J. K. Projected quasiparticle theory for molecular electronic structure. J. Chem. Phys. 2011, 135, 124108,  DOI: 10.1063/1.3643338
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    Projected quasiparticle theory for molecular electronic structure
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    Journal of Chemical Physics (2011), 135 (12), 124108/1-124108/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We derive and implement symmetry-projected Hartree-Fock-Bogoliubov (HFB) equations and apply them to the mol. electronic structure problem. All symmetries (particle no., spin, spatial, and complex conjugation) are deliberately broken and restored in a self-consistent variation-after-projection approach. We show that the resulting method yields a comprehensive black-box treatment of static correlations with effective one-electron (mean-field) computational cost. The ensuing wave function is of multireference character and permeates the entire Hilbert space of the problem. The energy expression is different from regular HFB theory but remains a functional of an independent quasiparticle d. matrix. All reduced d. matrixes are expressible as an integration of transition d. matrixes over a gauge grid. We present several proof-of-principle examples demonstrating the compelling power of projected quasiparticle theory for quantum chem. (c) 2011 American Institute of Physics.
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    A method is suggested for the calcn. of the matrix elements of the logarithm of an operator which gives the exact wave function when operating on the wave function in the 1-electron approxn. The method is based on the use of the creation and annihilation operators, hole-particle formalism, Wick's theorem, and the technique of Feynman-like diagrams. The connection of this method with the configuration interaction method as well as with the perturbation theory in the quantum-field theoretical form is discussed. The method is applied to the simple models of N and C6H6 mols. The results are compared with those obtained with the configuration-interaction method considering all possible configurations within the chosen basis of 1-electron functions.
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    The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed by using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using 2 different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the av. values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes.
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    Faulstich, F. M.; Laestadius, A.; Legeza, O.; Schneider, R.; Kvaal, S. Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry. SIAM J. Numer. Anal. 2019, 57, 2579– 2607,  DOI: 10.1137/18M1171436
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    Schütz, M.; Werner, H.-J. Low-order scaling local electron correlation methods. IV. Linear scaling local coupled-cluster (LCCSD). J. Chem. Phys. 2001, 114, 661– 681,  DOI: 10.1063/1.1330207
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    Schutz, Martin; Werner, Hans-Joachim
    Journal of Chemical Physics (2001), 114 (2), 661-681CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A new implementation of local coupled-cluster theory with single and double excitations (LCCSD) is presented for which asymptotically all computational resources (CPU, memory, and disk) scale only linearly with the mol. size. This is achieved by: (i) restricting the correlation space for each electron pair to domains that are independent of mol. size; (ii) classifying the pairs according to a distance criterion and treating only strong pairs at the highest level; (iii) using efficient pre-screening algorithms in the integral transformation and other integral-direct procedures; and (iv) neglect of small couplings of electron pairs that are far apart from each other. The errors caused by the various approxns. are negligible. LCCSD calcns. on mols. including up to 300 correlated electrons and over 1000 basis functions in C1 symmetry are reported, all carried out on a workstation.
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    Riplinger, C.; Neese, F. An efficient and near linear scaling pair natural orbital based local coupled cluster method. J. Chem. Phys. 2013, 138, 034106,  DOI: 10.1063/1.4773581
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    An efficient and near linear scaling pair natural orbital based local coupled cluster method
    Riplinger, Christoph; Neese, Frank
    Journal of Chemical Physics (2013), 138 (3), 034106/1-034106/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    In previous publications, it was shown that an efficient local coupled cluster method with single- and double excitations can be based on the concept of pair natural orbitals (PNOs) . The resulting local pair natural orbital-coupled-cluster single double (LPNO-CCSD) method has since been proven to be highly reliable and efficient. For large mols., the no. of amplitudes to be detd. is reduced by a factor of 105-106 relative to a canonical CCSD calcn. on the same system with the same basis set. In the original method, the PNOs were expanded in the set of canonical virtual orbitals and single excitations were not truncated. This led to a no. of fifth order scaling steps that eventually rendered the method computationally expensive for large mols. (e.g., >100 atoms). In the present work, these limitations are overcome by a complete redesign of the LPNO-CCSD method. The new method is based on the combination of the concepts of PNOs and projected AOs (PAOs). Thus, each PNO is expanded in a set of PAOs that in turn belong to a given electron pair specific domain. In this way, it is possible to fully exploit locality while maintaining the extremely high compactness of the original LPNO-CCSD wavefunction. No terms are dropped from the CCSD equations and domains are chosen conservatively. The correlation energy loss due to the domains remains below <0.05%, which implies typically 15-20 but occasionally up to 30 atoms per domain on av. The new method has been given the acronym DLPNO-CCSD ("domain based LPNO-CCSD"). The method is nearly linear scaling with respect to system size. The original LPNO-CCSD method had three adjustable truncation thresholds that were chosen conservatively and do not need to be changed for actual applications. In the present treatment, no addnl. truncation parameters have been introduced. Any addnl. truncation is performed on the basis of the three original thresholds. There are no real-space cutoffs. Single excitations are truncated using singles-specific natural orbitals. Pairs are prescreened according to a multipole expansion of a pair correlation energy est. based on local orbital specific virtual orbitals (LOSVs). Like its LPNO-CCSD predecessor, the method is completely of black box character and does not require any user adjustments. It is shown here that DLPNO-CCSD is as accurate as LPNO-CCSD while leading to computational savings exceeding one order of magnitude for larger systems. The largest calcns. reported here featured >8800 basis functions and >450 atoms. In all larger test calcns. done so far, the LPNO-CCSD step took less time than the preceding Hartree-Fock calcn., provided no approxns. have been introduced in the latter. Thus, based on the present development reliable CCSD calcns. on large mols. with unprecedented efficiency and accuracy are realized. (c) 2013 American Institute of Physics.
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    Shiozaki, T.; Valeev, E. F.; Hirata, S. Explicitly correlated coupled-cluster methods. In Annual Reports in Computational Chemistry; Elsevier: Amsterdam, 2009; Vol. 5, Chapter 6, pp 131– 148.
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    Izsák, R. Single-reference coupled cluster methods for computing excitation energies in large molecules: The efficiency and accuracy of approximations. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2020, 10, e1445,  DOI: 10.1002/wcms.1445
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    Single-reference coupled cluster methods for computing excitation energies in large molecules: The efficiency and accuracy of approximations
    Izsak, Robert
    Wiley Interdisciplinary Reviews: Computational Molecular Science (2020), 10 (3), e1445CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)
    A review. While methodol. developments in the last decade made it possible to compute coupled cluster (CC) energies including excitations up to a perturbative triples correction for mols. contg. several hundred atoms, a similar breakthrough has not yet been reported for excited state computations. Accurate CC methods for excited states are still expensive, although some promising candidates for an efficient and accurate excited state CC method have emerged recently. This review examines the various approxn. schemes with particular emphasis on their performance for excitation energies and summarizes the best state-of-the-art results which may pave the way for a robust excited state method applicable to mols. of hundreds of atoms. Among these, special attention will be given to exploiting the techniques of similarity transformation, perturbative approxns. as well as integral decompn., local and embedding techniques within the equation of motion CC framework. This article is categorized under:Electronic Structure Theory > Ab Initio Electronic Structure Methods Structure and Mechanism > Mol. Structures.
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    A review. The following topics are discussed: Exponential era of electron correlation theory; Genuine MR CC theory in Hilbert space and in Fock space; Alternative MR CC methods. Numerical illustrations are presented.
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    Huron, B.; Malrieu, J. P.; Rancurel, P. Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions. J. Chem. Phys. 1973, 58, 5745– 5759,  DOI: 10.1063/1.1679199
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    Huron, B.; Malrieu, J. P.; Rancurel, P.
    Journal of Chemical Physics (1973), 58 (12), 5745-59CODEN: JCPSA6; ISSN:0021-9606.
    A method is proposed for calcg. the effect of configuration interaction by a Rayleigh Schroedinger perturbation expansion when starting from a multiconfigurational wavefunction. A careless choice of H0 may lead to absurd transition energies between 2 states, at the 1st orders of the perturbation, even when the perturbation converges for both states. A barycentric definition of H0 is proposed, which ensures the cancellation of common diagrams in the calcd. transition energies. A practical iterative procedure is defined which allows a progressive improvement of the unperturbed wavefunction ψ0; the CI (configuration interaction) matrix restricted to a subspace S of strongly interacting determinants is diagonalized. The desired eigenvector ψ0 of this matrix is perturbed by the determinants which do not belong to S. The most important determinants in ψ1 are added to S, etc. The energy thus obtained after the 2nd-order correction is compared with the ordinary perturbation series where ψ0 is a single determinant. For the ground state, this procedure includes, besides the whole 2nd-order correction, the most important terms of the 3rd and 4th orders. The question of orthogonality of excited states is discussed. This technique was tested on the ground and several excited states of H2, Ne, and MgO, showing both a rapid convergence of the calcd. transition energy and the importance of correlation effects on transition energy.
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    A review. The following topics are discussed: Variational Monte Carlo, Fixed-node diffusion Monte Carlo, Self-healing diffusion Monte Carlo, Auxiliary field quantum Monte Carlo, Reptation quantum Monte Carlo, Full CI quantum Monte Carlo, Time-dependent quantum Monte Carlo; Trial electronic wave functions (antisym., backflow transformed, Jastrow), Trial wave function optimization, Effective core potentials; Computational considerations (Linear scaling quantum Monte Carlo, Parallelization and hardware acceleration, Advances in algorithms and software); Applications.
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    Booth, G. H.; Thom, A. J.; Alavi, A. Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space. J. Chem. Phys. 2009, 131, 054106,  DOI: 10.1063/1.3193710
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    Booth, George H.; Thom, Alex J. W.; Alavi, Ali
    Journal of Chemical Physics (2009), 131 (5), 054106/1-054106/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We have developed a new quantum Monte Carlo method for the simulation of correlated many-electron systems in full configuration-interaction (Slater determinant) spaces. The new method is a population dynamics of a set of walkers, and is designed to simulate the underlying imaginary-time Schrodinger equation of the interacting Hamiltonian. The walkers (which carry a pos. or neg. sign) inhabit Slater determinant space, and evolve according to a simple set of rules which include spawning, death and annihilation processes. We show that this method is capable of converging onto the full configuration-interaction (FCI) energy and wave function of the problem, without any a priori information regarding the nodal structure of the wave function being provided. Walker annihilation is shown to play a key role. The pattern of walker growth exhibits a characteristic plateau once a crit. (system-dependent) no. of walkers has been reached. At this point, the correlation energy can be measured using two independent methods-a projection formula and a energy shift; agreement between these provides a strong measure of confidence in the accuracy of the computed correlation energies. We have verified the method by performing calcns. on systems for which FCI calcns. already exist. In addn., we report on a no. of new systems, including CO, O2, CH4, and NaH-with FCI spaces ranging from 109 to 1014, whose FCI energies we compute using modest computational resources. (c) 2009 American Institute of Physics.
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    Petruzielo, F. R.; Holmes, A. A.; Changlani, H. J.; Nightingale, M. P.; Umrigar, C. J. Semistochastic Projector Monte Carlo Method. Phys. Rev. Lett. 2012, 109, 230201,  DOI: 10.1103/PhysRevLett.109.230201
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    Physical Review Letters (2012), 109 (23), 230201/1-230201/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    We introduce a semistochastic implementation of the power method to compute, for very large matrixes, the dominant eigenvalue and expectation values involving the corresponding eigenvector. The method is semistochastic in that the matrix multiplication is partially implemented numerically exactly and partially stochastically with respect to expectation values only. Compared to a fully stochastic method, the semistochastic approach significantly reduces the computational time required to obtain the eigenvalue to a specified statistical uncertainty. This is demonstrated by the application of the semistochastic quantum Monte Carlo method to systems with a sign problem: the fermion Hubbard model and the carbon dimer.
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    Spencer, J. S.; Blunt, N. S.; Foulkes, W. M. The sign problem and population dynamics in the full configuration interaction quantum Monte Carlo method. J. Chem. Phys. 2012, 136, 054110,  DOI: 10.1063/1.3681396
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    The sign problem and population dynamics in the full configuration interaction quantum Monte Carlo method
    Spencer, J. S.; Blunt, N. S.; Foulkes, W. M. C.
    Journal of Chemical Physics (2012), 136 (5), 054110/1-054110/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The recently proposed full CI quantum Monte Carlo method allows access to essentially exact ground-state energies of systems of interacting fermions substantially larger than previously tractable without knowledge of the nodal structure of the ground-state wave function. We investigate the nature of the sign problem in this method and how its severity depends on the system studied. We explain how cancellation of the pos. and neg. particles sampling the wave function ensures convergence to a stochastic representation of the many-fermion ground state and accounts for the characteristic population dynamics obsd. in simulations. (c) 2012 American Institute of Physics.
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  97. 97
    Cleland, D.; Booth, G. H.; Alavi, A. Communications: Survival of the fittest: Accelerating convergence in full configuration-interaction quantum Monte Carlo. J. Chem. Phys. 2010, 132, 041103,  DOI: 10.1063/1.3302277
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    Communications: Survival of the fittest: Accelerating convergence in full configuration-interaction quantum Monte Carlo
    Cleland, Deidre; Booth, George H.; Alavi, Ali
    Journal of Chemical Physics (2010), 132 (4), 041103/1-041103/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We provide a very simple adaptation of our recently published quantum Monte Carlo algorithm in full configuration-interaction (Slater determinant) spaces which dramatically reduces the no. of walkers required to achieve convergence. A survival criterion is imposed for newly spawned walkers. We define a set of initiator determinants such that progeny of walkers spawned from such determinants onto unoccupied determinants are able to survive, while the progeny of walkers not in this set can survive only if they are spawned onto determinants which are already occupied. The set of initiators is originally defined to be all determinants constructable from a subset of orbitals, in analogy with complete-active spaces. This set is dynamically updated so that if a non-initiator determinant reaches an occupation larger than a preset limit, it becomes an initiator. The new algorithm allows sign-coherent sampling of the FCI space to be achieved with relatively few walkers. Using the N2 mol. as an illustration, we show that rather small initiator spaces and nos. of walkers can converge with submilli-Hartree accuracy to the known full configuration-interaction (FCI) energy (in the cc-pVDZ basis), in both the equil. geometry and the multiconfigurational stretched case. We use the same method to compute the energy with cc-pVTZ and cc-pVQZ basis sets, the latter having an FCI space of over 1015 with very modest computational resources. (c) 2010 American Institute of Physics.
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    Cleland, D. M.; Booth, G. H.; Alavi, A. A study of electron affinities using the initiator approach to full configuration interaction quantum Monte Carlo. J. Chem. Phys. 2011, 134, 024112,  DOI: 10.1063/1.3525712
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    A study of electron affinities using the initiator approach to full configuration interaction quantum Monte Carlo
    Cleland, D. M.; Booth, George H.; Alavi, Ali
    Journal of Chemical Physics (2011), 134 (2), 024112/1-024112/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    For the atoms with Z ≤ 11, energies obtained using the "initiator" extension to full CI quantum Monte Carlo (i-FCIQMC) come to within statistical errors of the FCIQMC results. As these FCIQMC values have been shown to converge onto FCI results, the i-FCIQMC method allows similar accuracy to be achieved while significantly reducing the scaling with the size of the Slater determinant space. The i-FCIQMC electron affinities of the Z ≤ 11 atoms in the aug-cc-pVXZ basis sets are presented here. In every case, values are obtained to well within chem. accuracy the mean abs. deviation (MAD) from the relativistically cor. exptl. values is 0.41 mEh, and significantly improve on coupled cluster with singles, doubles and perturbative triples CCSD(T) results. Since the only remaining source of error is basis set incompleteness, we have investigated using CCSD(T)-F12 contributions to correct the i-FCIQMC results. By doing so, much faster convergence with respect to basis set size may be achieved for both the electron affinities and the FCIQMC ionization potentials presented in a previous paper. With this F12 correction, the MAD can be further reduced to 0.13 mEh for the electron affinities and 0.31 mEh for the ionization potentials. (c) 2011 American Institute of Physics.
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    Eriksen, J. J. The shape of full configuration interaction to come. J. Phys. Chem. Lett. 2021, 12, 418– 432,  DOI: 10.1021/acs.jpclett.0c03225
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    The Shape of Full Configuration Interaction to Come
    Eriksen, Janus J.
    Journal of Physical Chemistry Letters (2021), 12 (1), 418-432CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
    A review. We present a Perspective on what the future holds for full CI (FCI) theory, with an emphasis on conceptual rather than tech. details. Upon revisiting the early history of FCI, a no. of its key contemporary approxns. are compared on as equal a footing as possible, using a recent blind challenge on the benzene mol. as a testbed [Eriksen et al., J. Phys. Chem. Lett, 2020, 11, 8920]. In the process, we review the scope of applications for which FCI continues to prove indispensable, and the required traits in terms of robustness, efficacy, and reliability its modern approxns. must satisfy are discussed. We close by conveying a no. of general observations on the merits offered by the state-of-the-art alongside some of the challenges still faced to this day. While the field has altogether seen immense progress over the years-the past decade, in particular-it remains clear that our community as a whole has a substantial way to go in enhancing the overall applicability of near-exact electronic structure theory for systems of general compn. and increasing size.
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    Eriksen, J. J.; Gauss, J. Many-body expanded full configuration interaction. I. Weakly correlated regime. J. Chem. Theory Comput. 2018, 14, 5180– 5191,  DOI: 10.1021/acs.jctc.8b00680
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    Many-Body Expanded Full Configuration Interaction. I. Weakly Correlated Regime
    Eriksen, Janus J.; Gauss, Juergen
    Journal of Chemical Theory and Computation (2018), 14 (10), 5180-5191CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Over the course of the past few decades, the field of computational chem. has managed to manifest itself as a key complement to more traditional lab-oriented chem. This is particularly true in the wake of the recent renaissance of full CI (FCI)-level methodologies, albeit only if these can prove themselves sufficiently robust and versatile to be routinely applied to a variety of chem. problems of interest. In the present series of works, performance and feature enhancements of one such avenue toward FCI-level results for medium to large one-electron basis sets, the recently introduced many-body expanded full CI (MBE-FCI) formalism [J. Phys. Chem. Lett. 2017, 8, 4633], will be presented. Specifically, in this opening part of the series, the capabilities of the MBE-FCI method in producing near-exact ground state energies for weakly correlated mols. of any spin multiplicity will be demonstrated.
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    Eriksen, J. J.; Gauss, J. Many-body expanded full configuration interaction. II. Strongly correlated regime. J. Chem. Theory Comput. 2019, 15, 4873– 4884,  DOI: 10.1021/acs.jctc.9b00456
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    Many-Body Expanded Full Configuration Interaction. II. Strongly Correlated Regime
    Eriksen, Janus J.; Gauss, Juergen
    Journal of Chemical Theory and Computation (2019), 15 (9), 4873-4884CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    In this second part of our series on the recently proposed many-body expanded full CI (MBE-FCI) method, we introduce the concept of multideterminantal expansion refs. Through theor. arguments and numerical validations, the use of this class of starting points is shown to result in a focused compression of the MBE decompn. of the FCI energy, thus allowing chem. problems dominated by strong correlation to be addressed by the method. The general applicability and performance enhancements of MBE-FCI are verified for std. stress tests such as the bond dissocns. in H2O, N2, C2, and a linear H10 chain. Furthermore, the benefits of employing a multideterminantal expansion ref. in accelerating calcns. of high accuracy are discussed, with an emphasis on calcns. in extended basis sets. As an illustration of this latter quality of the MBE-FCI method, results for H2O and C2 in basis sets ranging from double- to pentuple-ζ quality are presented, demonstrating near-ideal parallel scaling on up to almost 25000 processing units.
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    White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 1992, 69, 2863– 2866,  DOI: 10.1103/PhysRevLett.69.2863
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    Density matrix formulation for quantum renormalization groups
    White
    Physical review letters (1992), 69 (19), 2863-2866 ISSN:.
    There is no expanded citation for this reference.
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    Chilkuri, V. G.; Neese, F. Comparison of many-particle representations for selected-CI I: A tree based approach. J. Comput. Chem. 2021, 42, 982– 1005,  DOI: 10.1002/jcc.26518
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    Comparison of many-particle representations for selected-CI I: A tree based approach
    Chilkuri, Vijay Gopal; Neese, Frank
    Journal of Computational Chemistry (2021), 42 (14), 982-1005CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    The full CI (FCI) method is only applicable to small mols. with few electrons in moderate size basis sets. One of the main alternatives to obtain approx. FCI energies for bigger mols. and larger basis sets is selected CI. However, due to: (a) the lack of a well-defined structure in a selected CI Hamiltonian, (b) the potentially large no. of electrons together with (c) potentially large orbital spaces, a computationally and memory efficient algorithm is difficult to construct. In the present series of papers, we describe our attempts to address these issues by exploring tree-based approaches. At the same time, we devote special attention to the issue of obtaining eigenfunctions of the total spin squared operator since this is of particular importance in tackling magnetic properties of complex open shell systems. Dedicated algorithms are designed to tackle the CI problem in terms of determinant, configuration (CFG) and configuration state function many-particle bases by effective use of the tree representation. In this paper we describe the underlying logic of our algorithm design and discuss the advantages and disadvantages of the different many particle bases. We demonstrate by the use of small examples how the use of the tree simplifies many key algorithms required for the design of an efficient selected CI program. Our selected CI algorithm, called the iterative configuration expansion, is presented in the penultimate part. Finally, we discuss the limitations and scaling characteristics of the present approach.
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    Li Manni, G.; Dobrautz, W.; Alavi, A. Compression of spin-adapted multiconfigurational wave functions in exchange-coupled polynuclear spin systems. J. Chem. Theory Comput. 2020, 16, 2202– 2215,  DOI: 10.1021/acs.jctc.9b01013
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    104
    Compression of Spin-Adapted Multiconfigurational Wave Functions in Exchange-Coupled Polynuclear Spin Systems
    Li Manni, Giovanni; Dobrautz, Werner; Alavi, Ali
    Journal of Chemical Theory and Computation (2020), 16 (4), 2202-2215CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We present a protocol based on unitary transformations of MOs to reduce the no. of nonvanishing coeffs. of spin-adapted CI expansions. Methods that exploit the sparsity of the Hamiltonian matrix and compactness of its eigensolns., such as the full CI quantum Monte Carlo (FCIQMC) algorithm in its spin-adapted implementation, are well suited to this protocol. The wave function compression resulting from this approach is particularly attractive for antiferromagnetically coupled polynuclear spin systems, such as transition-metal cubanes in biocatalysis, and Mott and charge-transfer insulators in solid-state physics. Active space CI calcns. on N2 and CN- at various bond lengths, the stretched square N4 compds., the chromium dimer, and a [Fe2S2]2- model system are presented as a proof-of-concept. For the Cr2 case, large and intermediate bond distances are discussed, showing that the approach is effective in cases where static and dynamic correlations are equally important. The [Fe2S2]2- case shows the general applicability of the method.
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    Li Manni, G. Modeling magnetic interactions in high-valent trinuclear [Mn3(IV)O4]4+ complexes through highly compressed multi-configurational wave functions. Phys. Chem. Chem. Phys. 2021, 23, 19766– 19780,  DOI: 10.1039/D1CP03259C
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    105
    Modeling magnetic interactions in high-valent trinuclear [Mn3(IV)O4]4+ complexes through highly compressed multi-configurational wave functions
    Li Manni, Giovanni
    Physical Chemistry Chemical Physics (2021), 23 (35), 19766-19780CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    In this work we apply a quantum chem. framework, recently designed in our labs., to rationalize the low-energy electronic spectrum and the magnetic properties of an homo-valent trinuclear [Mn3(IV)O4]4+ model of the oxygen-evolving center in photosystem II. The method is based on chem. motivated MO unitary transformations, and the optimization of spin-adapted many-body wave functions, both for ground- and excited-states, in the transformed MO basis. In this basis, the CI Hamiltonian matrix of exchange-coupled multi-center clusters is extremely sparse and characterized by a unique block diagonal structure. This property leads to highly compressed wave functions (oligo- or single-ref.) and crucially enables state-specific optimizations. This work is the first showing that compression and selective targeting of ground- and excited-states wave functions is possible for systems with three magnetic centers that are not exactly half-filled, and that potentially exhibit frustrated spin interactions. The reduced multi-ref. character of the wave function greatly simplifies the interpretation of the ground- and excited-state electronic structures, and provides a route for the direct rationalization of magnetic interactions in these compds., often considered a challenge in polynuclear transition-metal chem. In this study, strong electron correlation effects have explicitly been described by conventional and stochastic multiconfigurational methodologies, while dynamic correlation effects have been accounted for by multiconfigurational second order perturbation theory, CASPT2. Ab initio results for the [Mn3(IV)O4]4+ system have been mapped to a three-site Heisenberg model with two magnetic coupling consts. The magnetic coupling consts. and the temp. dependence of the effective magnetic moment predicted by the ab initio calcns. are in good agreement with the available exptl. data, and confirm the antiferromagnetic interaction among the three magnetic centers, while providing a simple and rigorous description of the noncollinearity of the local spins, that characterize most of the low-energy states for this system.
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  106. 106
    Li Manni, G.; Dobrautz, W.; Bogdanov, N. A.; Guther, K.; Alavi, A. Resolution of low-energy states in spin-exchange transition-metal clusters: Case study of singlet states in [Fe(III)4S4] cubanes. J. Phys. Chem. A 2021, 125, 4727– 4740,  DOI: 10.1021/acs.jpca.1c00397
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    Resolution of Low-Energy States in Spin-Exchange Transition-Metal Clusters: Case Study of Singlet States in [Fe(III)4S4] Cubanes
    Li Manni, Giovanni; Dobrautz, Werner; Bogdanov, Nikolay A.; Guther, Kai; Alavi, Ali
    Journal of Physical Chemistry A (2021), 125 (22), 4727-4740CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    Polynuclear transition-metal (PNTM) clusters owe their catalytic activity to numerous energetically low-lying spin states and stable oxidn. states. The characterization of their electronic structure represents one of the greatest challenges of modern chem. We propose a theor. framework that enables the resoln. of targeted electronic states with ease and apply it to two [Fe(III)4S4] cubanes. Through direct access to their many-body wave functions, we identify important correlation mechanisms and their interplay with the geometrical distortions obsd. in these clusters, which are core properties in understanding their catalytic activity. The simulated magnetic coupling consts. predicted by our strategy allow us to make qual. connections between spin interactions and geometrical distortions, demonstrating its predictive power. Moreover, despite its simplicity, the strategy provides magnetic coupling consts. in good agreement with the available exptl. ones. The complexes are intrinsically frustrated anti-ferromagnets, and the obtained spin structures together with the geometrical distortions represent two possible ways to release spin frustration (spin-driven Jahn-Teller distortion). Our paradigm provides a simple, yet rigorous, route to uncover the electronic structure of PNTM clusters and may be applied to a wide variety of such clusters.
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    Shee, J.; Loipersberger, M.; Hait, D.; Lee, J.; Head-Gordon, M. Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition. J. Chem. Phys. 2021, 154, 194109,  DOI: 10.1063/5.0047386
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    Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition
    Shee, James; Loipersberger, Matthias; Hait, Diptarka; Lee, Joonho; Head-Gordon, Martin
    Journal of Chemical Physics (2021), 154 (19), 194109CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    In this work, we provide a nuanced view of electron correlation in the context of transition metal complexes, reconciling computational characterization via spin and spatial symmetry breaking in single-ref. methods with qual. concepts from ligand-field and MO theories. These insights provide the tools to reliably diagnose the multi-ref. character, and our anal. reveals that while strong (i.e., static) correlation can be found in linear mols. (e.g., diatomics) and weakly bound and antiferromagnetically coupled (monometal-noninnocent ligand or multi-metal) complexes, it is rarely found in the ground-states of mono-transition-metal complexes. This leads to a picture of static correlation that is no more complex for transition metals than it is, e.g., for org. biradicaloids. In contrast, the ability of organometallic species to form more complex interactions, involving both ligand-to-metal σ-donation and metal-to-ligand π-backdonation, places a larger burden on a theory's treatment of dynamic correlation. We hypothesize that chem. bonds in which inter-electron pair correlation is non-negligible cannot be adequately described by theories using MP2 correlation energies and indeed find large errors vs expt. for carbonyl-dissocn. energies from double-hybrid d. functionals. A theory's description of dynamic correlation (and to a less important extent, delocalization error), which affects relative spin-state energetics and thus spin symmetry breaking, is found to govern the efficacy of its use to diagnose static correlation. (c) 2021 American Institute of Physics.
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    Benavides-Riveros, C. L.; Lathiotakis, N. N.; Schilling, C.; Marques, M. A. Relating correlation measures: The importance of the energy gap. Phys. Rev. A 2017, 95, 032507,  DOI: 10.1103/PhysRevA.95.032507
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    Relating correlation measures: the importance of the energy gap
    Benavides-Riveros, Carlos L.; Lathiotakis, Nektarios N.; Schillin, Christian; Marques, Miguel A. L.
    Physical Review A (2017), 95 (3), 032507/1-032507/6CODEN: PRAHC3; ISSN:2469-9934. (American Physical Society)
    The concept of correlation is central to all approaches that attempt the description of many-body effects in electronic systems. Multipartite correlation is a quantum information theor. property that is attributed to quantum states independent of the underlying physics. In quantum chem., however, the correlation energy (the energy not seized by the Hartree-Fock ansatz) plays a more prominent role.We show that these two different viewpoints on electron correlation are closely related. The key ingredient turns out to be the energy gap within the symmetry-adapted subspace. We then use a few-site Hubbard model and the stretched H2 to illustrate this connection and to show how the corresponding measures of correlation compare.
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    Jiang, W.; DeYonker, N. J.; Wilson, A. K. Multireference character for 3d transition-metal-containing molecules. J. Chem. Theory Comput. 2012, 8, 460– 468,  DOI: 10.1021/ct2006852
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    Multireference Character for 3d Transition-Metal-Containing Molecules
    Jiang, Wanyi; DeYonker, Nathan J.; Wilson, Angela K.
    Journal of Chemical Theory and Computation (2012), 8 (2), 460-468CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Coupled cluster and CI diagnostics have been examd. in order to assess the reliability of single ref. quantum methods for a series of 3d transition metal species including hydrides, nitrides, chalcogenides, halides, small clusters, coordination complexes, and metal dimers. Several means of diagnostics have been considered including T1 and D1 diagnostics (the Frobenius norm and matrix 2-norm of coupled cluster amplitudes for single excitations, resp.), C02 (the wt. of leading configuration of a complete active space wave function), and %TAE (percent total atomization energy). T1 and D1 diagnostics are strongly correlated for certain metal-ligand bonding types. The use of T1 and D1 together with %TAE can provide more reliable assessment of the severity of nondynamical correlation than a single indicator can provide. New criteria, namely T1 > 0.05, D1 > 0.15, and |%TAE| > 10, are suggested to identify inorg. species with substantial nondynamical correlation. For these systems, energies and spectroscopic properties computed using single ref. electronic correlation methods may suffer from large errors and unpredictable behavior. Conversely, a computation where a mol. is below one or more of these thresholds does not always imply domination by a single ref. Some historically pathol. mols. such as Mn2 and Cr2 show T1 < 0.05 and D1 < 0.15. Current implementations of coupled cluster diagnostics may still be insufficient for categorization of mols. that have pronounced nondynamical correlation.
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    Lee, T. J.; Rice, J. E.; Scuseria, G. E.; Schaefer, H. F. Theoretical investigations of molecules composed only of fluorine, oxygen and nitrogen: determination of the equilibrium structures of FOOF, (NO)2 and FNNF and the transition state structure for FNNF cis-trans isomerization. Theor. Chim. Acta 1989, 75, 81– 98,  DOI: 10.1007/BF00527711
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    Theoretical investigations of molecules composed only of fluorine, oxygen and nitrogen: determination of the equilibrium structures of difluoroperoxide (FOOF), nitric oxide dimer, and difluorodiazene (FNNF) and the transition state structure for difluorodiazene cis-trans
    Lee, Timothy J.; Rice, Julia E.; Scuseria, Gustavo E.; Schaefer, Henry F., III
    Theoretica Chimica Acta (1989), 75 (2), 81-98CODEN: TCHAAM; ISSN:0040-5744.
    The deficiencies of common ab initio methods were studied for the reliable prediction of the equil. structures of compds. composed of only the fluorine, oxygen and nitrogen atoms. Specifically, the importance of using large one-particle basis sets with multiple sets of polarization functions was studied. Addnl., the need for a set of f basis functions was investigated. Several different single-ref. electron correlation methods were tested to det. whether it is possible for a single-ref.-based method to be used routinely on such chem. systems. These electron-correlation methods include second-order Moeller-Plesset perturbation theory (MP2), singles and doubles CI (CISD), the coupled-pair-functional (CPF) approach, and singles and doubles coupled-cluster (CCSD) theory. The mol. systems studied included difluoroperoxide (FOOF), the cis form of the NO dimer, cis and trans difluorodiazene (FNNF), and the transition state to interconversion of the cis and trans isomers of FNNF. This is the first time that the cis-trans isomerization transition state has been reported. At the highest level of theory employed, the equil. structures of cis and trans FNNF agreed very well with the exptl. structures. However, the barrier to interconversion was predicted to be 65 kcal/mol. which is substantially higher than the exptl. activation energy of 32 kcal/mol. Potential sources of error are discussed. A new diagnostic method for detg. a priori the reliability of single-ref.-based electron correlation methods is suggested, and discussed.
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    Liakos, D. G.; Neese, F. Interplay of Correlation and Relativistic Effects in Correlated Calculations on Transition-Metal Complexes: The (Cu2O2)2+ Core Revisited. J. Chem. Theory Comput. 2011, 7, 1511– 1523,  DOI: 10.1021/ct1006949
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    Interplay of Correlation and Relativistic Effects in Correlated Calculations on Transition-Metal Complexes: The (Cu2O2)2+ Core Revisited
    Liakos, Dimitrios G.; Neese, Frank
    Journal of Chemical Theory and Computation (2011), 7 (5), 1511-1523CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Owing to the availability of large-scale computing facilities and the development of efficient new algorithms, wave function-based ab initio calcns. are becoming more common in bioinorg. chem. In principle they offer a systematic route toward high accuracy. However, these calcns. are by no means trivial. In this contribution we address some pertinent points through a systematic theor. study for the equil. between the peroxo- and bis-(μ-oxo) isomers of the [{Cu(C2H8N2)}2O2]2+ complex. While this system is often regarded as a prototypical multireference case, we treat it with the single ref. local-pair natural orbital coupled cluster method and reiterate that the multireference character in this system is very limited. A set of intermediate structures, for the interconversion between the two isomers, is calcd. through a relaxed surface scan thus allowing the calcn. of an energetic profile that cleanly connects the bis-(μ-oxo) and side-on peroxo min. on the ground-state potential energy surface. Only at the highest level of theory involving complete basis set extrapolation, triple excitation contributions as well as relativistic and solvent effects, the bis-(μ-oxo) isomer is found to be slightly more stable than the peroxo structure. This is in agreement with the exptl. findings. The effects of basis set, triples excitation, relativity, and solvent contribution have all been analyzed in detail. Finally, the ab initio results are compared with d. functional calcns. using various functionals. It is demonstrated that the largest part of the discrepancies of the results reported in the literature are due to an inconsistent handling of relativistic effects, which are large in both ab initio and d. functional theory calcns.
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    Head-Gordon, M. Characterizing unpaired electrons from the one-particle density matrix. Chem. Phys. Lett. 2003, 372, 508– 511,  DOI: 10.1016/S0009-2614(03)00422-6
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    Characterizing unpaired electrons from the one-particle density matrix
    Head-Gordon, Martin
    Chemical Physics Letters (2003), 372 (3,4), 508-511CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
    A new definition of the unpaired electrons in a mol. is proposed, which derives from the one-particle reduced d. matrix. It yields lower ests. of the no. of radical electrons than the widely discussed distribution of effectively unpaired electrons', with a max. possible difference of a factor of two. Unlike the existing definition, the new definition cannot yield nos. of unpaired electrons higher than the total no. of electrons, and also recovers the intuitively expected result for the dissocn. of O2.
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    Ramos-Cordoba, E.; Matito, E. Local Descriptors of Dynamic and Nondynamic Correlation. J. Chem. Theory Comput. 2017, 13, 2705– 2711,  DOI: 10.1021/acs.jctc.7b00293
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    Local Descriptors of Dynamic and Nondynamic Correlation
    Ramos-Cordoba, Eloy; Matito, Eduard
    Journal of Chemical Theory and Computation (2017), 13 (6), 2705-2711CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Quant. accurate electronic structure calcns. rely on the proper description of electron correlation. A judicious choice of the approx. quantum chem. method depends upon the importance of dynamic and nondynamic correlation, which is usually assesed by scalar measures. Existing measures of electron correlation do not consider sep. the regions of the Cartesian space where dynamic or nondynamic correlation are most important. We introduce real-space descriptors of dynamic and nondynamic electron correlation that admit orbital decompn. Integration of the local descriptors yields global nos. that can be used to quantify dynamic and nondynamic correlation. Illustrative examples over different chem. systems with varying electron correlation regimes are used to demonstrate the capabilities of the local descriptors. Since the expressions only require orbitals and occupation nos., they can be readily applied in the context of local correlation methods, hybrid methods, d. matrix functional theory, and fractional-occupancy d. functional theory.
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    Löwdin, P.-O. Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction. Phys. Rev. 1955, 97, 1474– 1489,  DOI: 10.1103/PhysRev.97.1474
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    Ramos-Cordoba, E.; Salvador, P.; Matito, E. Separation of dynamic and nondynamic correlation. Phys. Chem. Chem. Phys. 2016, 18, 24015– 24023,  DOI: 10.1039/C6CP03072F
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    Separation of dynamic and nondynamic correlation
    Ramos-Cordoba, Eloy; Salvador, Pedro; Matito, Eduard
    Physical Chemistry Chemical Physics (2016), 18 (34), 24015-24023CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    The account of electron correlation and its efficient sepn. into dynamic and nondynamic parts plays a key role in the development of computational methods. In this paper we suggest a phys.-sound matrix formulation to split electron correlation into dynamic and nondynamic parts using the two-particle cumulant matrix and a measure of the deviation from idempotency of the first-order d. matrix. These matrixes are applied to a two-electron model, giving rise to a simplified electron correlation index that (i) depends only on natural orbitals and their occupancies, (ii) can be straightforwardly decompd. into orbital contributions and (iii) splits into dynamic and nondynamic correlation parts that (iv) admit a local version. These expressions are shown to account for dynamic and nondynamic correlation in a variety of systems contg. different electron correlation regimes, thus providing the first sepn. of dynamic and nondynamic correlation using solely natural orbital occupancies.
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    Juhász, T.; Mazziotti, D. A. The cumulant two-particle reduced density matrix as a measure of electron correlation and entanglement. J. Chem. Phys. 2006, 125, 174105,  DOI: 10.1063/1.2378768
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    The cumulant two-particle reduced density matrix as a measure of electron correlation and entanglement
    Juhasz, Tamas; Mazziotti, David A.
    Journal of Chemical Physics (2006), 125 (17), 174105/1-174105/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Several measures of electron correlation are compared based on two criteria: (i) the presence of a unique mapping between the reduced variables in the measure and the many-electron wave function and (ii) the linear scaling of the measure and its variables with system size. We propose the squared Frobenius norm of the cumulant part of the two-particle reduced d. matrix (2-RDM) as a measure of electron correlation that satisfies these criteria. An advantage of this cumulant-based norm is its ability to measure the correlation from spin entanglement, which is not contained in the correlation energy. Alternative measures based on the 2-RDM, such as the von Neumann entropy, do not scale linearly with system size. Properties of the measures are demonstrated with Be, F2, HF, N2, and a hydrogen chain.
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    Crittenden, D. L. A Hierarchy of Static Correlation Models. J. Phys. Chem. A 2013, 117, 3852– 3860,  DOI: 10.1021/jp400669p
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    A Hierarchy of Static Correlation Models
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    Journal of Physical Chemistry A (2013), 117 (18), 3852-3860CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    It is commonly accepted in the scientific literature that the static correlation energy, Estat, of a system can be defined as the exact correlation energy of its valence electrons in a minimal basis. Unfortunately, the computational cost of calcg. the exact correlation energy within a fully optimized minimal basis grows exponentially with system size, making such calcns. intractable for all but the smallest systems. However, analogous to single-ref. methods, it is possible to systematically approx. both the treatment of electron correlation and flexibility of the minimal basis to reduce computational cost. This yields a hierarchy of methods for calcg. Estat, ranging from coupled cluster methods in a minimal at. basis up to full valence complete active space methods with a minimal MO basis constructed from a near-complete AO basis. By examg. a variety of dissocg. diatomics, along with equil. and transition structures for polyat. systems, we show that std. coupled cluster models with minimal at. basis sets (e.g., STO-3G) offer a convenient and cost-effective hierarchy of black box ests. for Estat in small- to medium-sized systems near their equil. geometries. To properly describe homolytic bond dissocn., it is better to use a more flexible basis set expansion so that each AO can effectively adapt to its mol. environment.
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    Perdew, J. P.; Ruzsinszky, A.; Sun, J.; Nepal, N. K.; Kaplan, A. D. Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories. Proc. Nat. Acad. Sci. 2021, 118, e2017850118,  DOI: 10.1073/pnas.2017850118
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    Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories
    Perdew, John P.; Ruzsinszky, Adrienn; Sun, Jianwei; Nepal, Niraj K.; Kaplan, Aaron D.
    Proceedings of the National Academy of Sciences of the United States of America (2021), 118 (4), e2017850118CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
    Strong correlations within a symmetry-unbroken ground-state wavefunction can show up in approx. d. functional theory as symmetry-broken spin densities or total densities, which are sometimes observable. They can arise from soft modes of fluctuations (sometimes collective excitations) such as spin-d. or charge-d. waves at nonzero wavevector. In this sense, an approx. d. functional for exchange and correlation that breaks symmetry can be more revealing (albeit less accurate) than an exact functional that does not. The examples discussed here include the stretched H2 mol., antiferromagnetic solids, and the static charge-d. wave/Wigner crystal phase of a low-d. jellium. Time-dependent d. functional theory is used to show quant. that the static charge-d. wave is a soft plasmon. More precisely, the frequency of a related d. fluctuation drops to zero, as found from the frequency moments of the spectral function, calcd. from a recent constraint-based wavevector- and frequency-dependent jellium exchange-correlation kernel.
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    Perdew, J. P.; Ruzsinszky, A.; Constantin, L. A.; Sun, J.; Csonka, G. I. Some fundamental issues in ground-state density functional theory: A guide for the perplexed. J. Chem. Theory Comput. 2009, 5, 902– 908,  DOI: 10.1021/ct800531s
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    Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for the Perplexed
    Perdew, John P.; Ruzsinszky, Adrienn; Constantin, Lucian A.; Sun, Jianwei; Csonka, Gabor I.
    Journal of Chemical Theory and Computation (2009), 5 (4), 902-908CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Some fundamental issues in ground-state d. functional theory are discussed without equations: (1) The std. Hohenberg-Kohn and Kohn-Sham theorems were proven for a Hamiltonian that is not quite exact for real atoms, mols., and solids. (2) The d. functional for the exchange-correlation energy, which must be approximated, arises from the tendency of electrons to avoid one another as they move through the electron d. (3) In the absence of a magnetic field, either spin densities or total electron d. can be used, although the former choice is better for approxns. (4) "Spin contamination" of the determinant of Kohn-Sham orbitals for an open-shell system is not wrong but right. (5) Only to the extent that symmetries of the interacting wave function are reflected in the spin densities should those symmetries be respected by the Kohn-Sham noninteracting or determinantal wave function. Functionals below the highest level of approxns. should however sometimes break even those symmetries, for good phys. reasons. (6) Simple and commonly used semilocal (lower-level) approxns. for the exchange-correlation energy as a functional of the d. can be accurate for closed systems near equil. and yet fail for open systems of fluctuating electron no. (7) The exact Kohn-Sham noninteracting state need not be a single determinant, but common approxns. can fail when it is not. (8) Over an open system of fluctuating electron no., connected to another such system by stretched bonds, semilocal approxns. make the exchange-correlation energy and hole-d. sum rule too neg. (9) The gap in the exact Kohn-Sham band structure of a crystal underestimates the real fundamental gap but may approx. the first exciton energy in the large-gap limit. (10) D. functional theory is not really a mean-field theory, although it looks like one. The exact functional includes strong correlation, and semilocal approxns. often overestimate the strength of static correlation through their semilocal exchange contributions. (11) Only under rare conditions can excited states arise directly from a ground-state theory.
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    Filatov, M.; Shaik, S. Spin-restricted density functional approach to the open-shell problem. Chem. Phys. Lett. 1998, 288, 689– 697,  DOI: 10.1016/S0009-2614(98)00364-9
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    Filatov, Michael; Shaik, Sason
    Chemical Physics Letters (1998), 288 (5,6), 689-697CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
    Open-shell one-electron equations are derived by application of Roothaan's coupling operator technique to the variational procedure of finding the Kohn-Sham orbitals and minimizing the energy of an open-shell system, represented within the d. functional vector coupling scheme. The final equations are presented in a form suitable for std. quantum-chem. codes using finite basis set Kohn-Sham calcns. Examples of multiplets for which the theory is applicable are discussed.
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    Gagliardi, L.; Truhlar, D. G.; Li Manni, G.; Carlson, R. K.; Hoyer, C. E.; Bao, J. L. Multiconfiguration pair-density functional theory: A new way to treat strongly correlated systems. Acc. Chem. Res. 2017, 50, 66– 73,  DOI: 10.1021/acs.accounts.6b00471
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    Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems
    Gagliardi, Laura; Truhlar, Donald G.; Li Manni, Giovanni; Carlson, Rebecca K.; Hoyer, Chad E.; Bao, Junwei Lucas
    Accounts of Chemical Research (2017), 50 (1), 66-73CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)
    A review. The electronic energy of a system provides the Born-Oppenheimer potential energy for internuclear motion and thus dets. mol. structure and spectra, bond energies, conformational energies, reaction barrier heights, and vibrational frequencies. The development of more efficient and more accurate ways to calc. the electronic energy of systems with inherently multiconfigurational electronic structure is essential for many applications, including transition metal and actinide chem., systems with partially broken bonds, many transition states, and most electronically excited states. Inherently multiconfigurational systems are called strongly correlated systems or multireference systems, where the latter name refers to the need for using more than one ("multiple") configuration state function to provide a good zero-order ref. wave function. The present account describes (MC-PDFT), which was developed as a way to combine the advantages of wave function theory (WFT) and d. functional theory (DFT) to provide a better treatment of strongly correlated systems. First we review background material: the widely used Kohn-Sham DFT (which uses only a single Slater determinant as ref. wave function), multiconfiguration WFT methods that treat inherently multiconfigurational systems based on an active space, and previous attempts to combine multiconfiguration WFT with DFT. Then we review the formulation of MC-PDFT. MC-PDFT is a generalization of Kohn-Sham DFT in that the electron kinetic energy and classical electrostatic energy are calcd. from a ref. wave function, with the rest of the energy obtained from a d. functional. However, there are two main differences: (i) The ref. wave function is multiconfigurational rather than being a single Slater determinant. (ii) The d. functional is a function of the total d. and the on-top pair d. rather than being a function of the spin-up and spin-down densities. In work carried out so far, the multiconfigurational wave function is a multiconfiguration self-consistent-field wave function. The new formulation has the advantage that the ref. wave function has the correct spatial and spin symmetry and can describe bond dissocn. (of both single and multiple bonds) and electronic excitations in a formally and phys. correct way. We then review the formulation of d. functionals in terms of the on-top pair d. Finally we review successful applications of the theory to bond energies and bond dissocn. potential energy curves of main-group and transition metal bonds, to barrier heights (including pericyclic reactions), to proton affinities, to the hydrogen bond energy of water dimer, to ground- and excited-state charge transfer, to valence and Rydberg excitations of mols., and to singlet-triplet splittings of radicals. We find that MC-PDFT can give accurate results not only with complete-active-space multiconfiguration wave functions, but also with generalized-active-space multiconfiguration wave functions, which are practical for larger nos. of active electrons and active orbitals than are complete-active-space wave functions. The sepd.-pair approxn., which is a special case of generalized active space self-consistent-field theory, is esp. promising. MC-PDFT, because it requires much less computer time and storage than previous WFT methods, has the potential to open larger and more complex strongly correlated systems to accurate simulation.
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    Density functional theory: coverage of dynamic and non-dynamic electron correlation effects
    Cremer, Dieter
    Molecular Physics (2001), 99 (23), 1899-1940CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
    The electron correlation effects covered by d. functional theory (DFT) can be assessed qual. by comparing DFT densities ρ(r) with suitable ref. densities obtained with wavefunction theory (WFT) methods that cover typical electron correlation effects. The anal. of difference densities ρ(DFT)-ρ(WFT) reveals that LDA and GGA exchange (X) functionals mimic non-dynamic correlation effects in an unspecified way. It is shown that these long range correlation effects are caused by the self-interaction error (SIE) of std. X functionals. Self-interaction cor. (SIC) DFT exchange gives, similar to exact exchange, for the bonding region a delocalized exchange hole, and does not cover any correlation effects. Hence, the exchange SIE is responsible for the fact that DFT densities often resemble MP4 or MP2 densities. The correlation functional changes X-only DFT densities in a manner obsd. when higher order coupling effects between lower order N-electron correlation effects are included. Hybrid functionals lead to changes in the d. similar to those caused by SIC-DFT, which simply reflects the fact that hybrid functionals have been developed to cover part of the SIE and its long range correlation effects in a balanced manner. In the case of spin-unrestricted DFT (UDFT), non-dynamic electron correlation effects enter the calcn. both via the X functional and via the wavefunction, which may cause a double-counting of correlation effects. The use of UDFT in the form of permuted orbital and broken-symmetry DFT (PO-UDFT, BS-UDFT) can lead to reasonable descriptions of multireference systems provided certain conditions are fulfilled. More reliable, however, is a combination of DFT and WFT methods, which makes the routine description of multireference systems possible. The development of such methods implies a sepn. of dynamic and non-dynamic correlation effects. Strategies for accomplishing this goal are discussed in general and tested in practice for CAS (complete active space)-DFT.
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    Theoretica Chimica Acta (1977), 43 (3), 261-71CODEN: TCHAAM; ISSN:0040-5744.
    A consistent application of the statistical-exchange approxn. (J. C. Slater, 1974; B., et at., 1973) of the Hartree-Fock-Slater method requires use of the sum method for calcn. of the energy Es1 of singlet excited states of closed-shell mols. Values of Es1 were calcd. in satisfactory agreement with the available exptl. data for a no. of mols. Multiplet splittings other than singlet-triplet were also calcd. with the Hartree-Fock-Slater method.
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    Journal of Chemical Physics (1981), 74 (10), 5737-43CODEN: JCPSA6; ISSN:0021-9606.
    A single configuration model contg. nonorthogonal magnetic orbitals is developed to represent the important features of the antiferromagnetic state of a transition metal dimer. A state of mixed spin symmetry and lowered space symmetry is constructed which has both conceptual and practical computational value. Either UHF theory or spin polarized d. functional theory, e.g., Xα theory, can be used to generate the mixed spin state wave function. The most important consequence of the theory is that the Heisenberg exchange coupling const. J can be calcd. simply from the energies of the mixed spin state and the highest pure spin multiplet.
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    Coordination compds. are usually sym. mols. with degenerate orbitals. Hence, the individual multiplet states arising from open-shell configurations can, in general, not be expressed by a single determinant. We have, therefore, exploited symmetry to the largest possible extent in order to simplify the relation between the multiplet splitting and single-determinant energies, and, thus, developed a new method based on vector coupling to keep the computational effort to a min. A system of computer programs, working on both mainframe and personal computers, was developed, carrying out (for any desired point group) the required group-theor. manipulations. The description of the method is illustrated by considering three practical examples.
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    The broken symmetry (BS) concept is an extremely useful tool for the prediction of exchange coupling consts. in mols. with interacting paramagnetic centers. An anal. of the BS wave functions is presented and the relationship between the overlap of magnetic orbitals and the exchange coupling is stressed. The corresponding orbital transformation is introduced as a useful tool in order to det. the non-orthogonal valence bond-like magnetic orbital pairs in many electron systems.
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    The spin-d.-functional (SDF) formalism (e.g., G., et al., 1974-5) was extended to apply to generalized Hamiltonians and to lowest excited states with different types of symmetry. A relation between the exchange-correlation functional and the pair-correlation function was derived, and was used to interpret approx. versions of the theory, esp. the local-spin-d. (LSD) approxn., which can be used in calcn. of the exchange-correlation energy (Exc) in rather inhomogeneous systems. Calcns. done on the homogeneous spin-polarized electron liq., where the charge-d. fluctuations were described by using a plasmon model, provide interpolation formulas for detg. Exc and the exchange-correlation potentials in the LSD approxn. Other properties calcd. for the electron liq. include: bulk modulus at const. magnetization, compressibility at const. magnetic field, and magnetic susceptibility. Applications of the SDF formalism in calcns. of the properties of atoms, mols., and metals are discussed.
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    Science (Washington, DC, United States) (1972), 177 (4047), 393-6CODEN: SCIEAS; ISSN:0036-8075.
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    Ladner, R. C.; Goddard III, W. A. Improved Quantum Theory of Many-Electron Systems. V. The Spin-Coupling Optimized GI Method. J. Chem. Phys. 1969, 51, 1073– 1087,  DOI: 10.1063/1.1672106
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    Improved quantum theory of many-electron systems. V. The spin-coupling optimized GI method
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    Journal of Chemical Physics (1969), 51 (3), 1073-87CODEN: JCPSA6; ISSN:0021-9606.
    The previously developed GI (gauge invariant) methods have an arbitrary aspect since they are based on a particular representation of the symmetric group. Here, the authors remove this arbitrariness by optimizing the representation, that is, optimizing the spin-coupling scheme simultaneously with the optimization of the orbitals. The resulting wavefunctions, called the spin-coupling optimized GI or SOGI wavefunctions, have all of the general properties of GI wavefunctions including the independent particle interpretation and are found as the solns. to a set of coupled differential equations which differ from the GI equations only in that the equations are constructed from a different representation of the symmetric group. The authors have applied this method to the ground state and some excited states of Li, to the ground states of Be+ and B++ and to the ground state of LiH. In each of these cases, they found that the SOGI wavefunction was only slightly different from the GI wavefunction and led to very similar energies and other spatial properties. For the spin d. at the nucleus, however, SOGI led to much better results. To illustrate the effects of spatial symmetry on the SOGI orbitals, the authors examd. the lowest 1B1g, 3A2g, and 3Eu states of sq. H4 and the 2Σu+ state of linear sym. H3. They find that in 3 of these cases optimization of the spin representation is crucial to providing an adequate description of the state. To investigate how the SOGI method would describe chem. reactions, the SOGI wavefunctions were computed for several other nuclear configurations of the H3 system along the reaction path. These calcns. showed that the spin coupling changed significantly during the reaction H2 + H .dblharw. H + H2 and that the variation of the SOGI orbitals provides a clear description of the changes in bonding which occur during this reaction.
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    Ghosh, P.; Bill, E.; Weyhermüller, T.; Neese, F.; Wieghardt, K. Noninnocence of the Ligand Glyoxal-bis (2-mercaptoanil). The Electronic Structures of [Fe(gma)]2,[Fe(gma)(py)].py,[Fe(gma)(CN)]1–/0,[Fe(gma)I], and [Fe(gma)(PR3)n] (n= 1, 2). Experimental and Theoretical Evidence for “Excited State” Coordination. J. Am. Chem. Soc. 2003, 125, 1293– 1308,  DOI: 10.1021/ja021123h
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    Noninnocence of the Ligand Glyoxal-bis(2-mercaptoanil). The Electronic Structures of [Fe(gma)]2, [Fe(gma)(py)]·py, [Fe(gma)(CN)]1-/0, [Fe(gma)I], and [Fe(gma)(PR3)n] (n = 1, 2). Experimental and Theoretical Evidence for "Excited State" Coordination
    Ghosh, Prasanta; Bill, Eckhard; Mueller, Thomas Weyherm; Neese, Frank; Wieghardt, Karl
    Journal of the American Chemical Society (2003), 125 (5), 1293-1308CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    The electronic structure of the known Fe complexes [Fe(gma)]2 (St = 0) (1) (D. Sellmann, 1992) and [Fe(gma)(py)]·py (St = 1) (2) (P. Karsten, 1997) (H2(gma) = glyoxal-bis(2-mercaptoanil)) was shown by x-ray crystallog., Mossbauer spectroscopy, and d. functional theory calcns. to be best described as ferric (SFe = 3/2) complexes contg. a coordinated open-shell π radical trianion (gma•)3- and not as previously reported as ferrous species with a coordinated closed-shell dianion (gma)2-. Compd. 1 (or 2) can be oxidized by I2 yielding [FeIII(gma)I] (St = 1/2) (3). With cyanide anions, complex 1 forms [Bu4N][FeIII(gma•)(CN)] (St = 1) (4), which can be 1-electron oxidized with I yielding the neutral species [FeIII(gma)(CN)] (St = 1/2) (5). With phosphines complex 1 also forms adducts7 of which [FeIII(gma•)(P(n-propyl)3)] (St = 1) (6) was isolated and characterized by x-ray crystallog. [FeII(gma)(P(n-propyl)3)2] (St = 0) (7) represents the only genuine ferrous species of the series. D. functional theory (DFT) calcns. at the BP86 and B3LYP levels were applied to calc. the structural as well as the EPR and Mossbauer spectroscopic parameters of the title compds. and of [Zn(gma)]0/- and [Ni(gma)]0/-. Overall, the calcns. give excellent agreement with the available spectroscopic information, thus lending support to the following electronic structure descriptions: The gma ligand features an unusually low lying LUMO, which readily accepts an electron to give (gma•)3-. The 1-electron redn. of [Zn(gma)] and [Ni(gma)] is strictly ligand centered and differences in the phys. properties of [Zn(gma•)]- and [Ni(gma•)]- are readily accounted for in terms of a model that features enhanced back-bonding from the metal to the gma LUMO in the case of [Ni(gma•)]-. In the case of [Fe(gma)(PH3)], [Fe(gma)(py)], and [Fe(gma)(CN)]- an electron transfer from the Fe to the gma LUMO takes place to give strong antiferromagnetic coupling between an intermediate spin Fe(III) (SFe = 3/2) and (gma•)3- (Sgma = 1/2), yielding a total spin St = 1. Broken symmetry DFT calcns. take properly account of this exptl. calibrated electronic structure description. By contrast, [Fe(gma)(PH3)2] and [Fe(PhBMA)] feature closed-shell ligands with a low-spin Fe(II) (SFe = St = 0) and an intermediate spin central Fe(II) (SFe = St = 1), resp. The most interesting case is provided by the 1-electron oxidized species [Fe(gma)(py)]+, [Fe(gma)I], and [Fe(gma)(CN)]. Here the combination of theory and expt. suggests the coupling of an intermediate spin Fe(III) (SFe = 3/2) to the dianionic ligand (gma)2- formally in its 1st excited triplet state (Sgma = 1) to give a resulting St = 1/2. All phys. properties are in accord with this interpretation. Probably this unique excited state coordination is energetically driven by the strong antiferromagnetic exchange interaction between the metal and the ligand, which cannot occur for the closed-shell form of the ligand.
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    Herebian, D.; Wieghardt, K. E.; Neese, F. Analysis and interpretation of metal-radical coupling in a series of square planar nickel complexes: correlated ab initio and density functional investigation of [Ni(LISQ)2](LISQ= 3,5-di-tert-butyl-o-diiminobenzosemiquinonate(1-)). J. Am. Chem. Soc. 2003, 125, 10997– 11005,  DOI: 10.1021/ja030124m
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    Analysis and Interpretation of Metal-Radical Coupling in a Series of Square Planar Nickel Complexes: Correlated Ab Initio and Density Functional Investigation of [Ni(LISQ)2] (LISQ=3,5-di-tert-butyl-o-diiminobenzosemiquinonate (1-))
    Herebian, Diran; Wieghardt, Karl E.; Neese, Frank
    Journal of the American Chemical Society (2003), 125 (36), 10997-11005CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    The author report a detailed theor. study of the interaction between a central low-spin d8 nickel ion and two N,N-coordinating diiminobenzosemiquinonate(1-) ligands in a square planar arrangement. Such complexes have recently attracted much attention due to their unusual bonding patterns, structures, optical, and magnetic properties. Geometry optimizations using various levels of d. functional theory (DFT) result in excellent agreement with the exptl. detd. structure and in particular reproduce the quinoidal distortions in the arom. rings well. A detailed anal. of the orbital structure reveals that the complex features essentially two strongly interacting ligand radicals which interact with each other via an efficient superexchange mechanism that is mediated by a back-bonding interaction to the central metal. An anal. of the broken symmetry DFT wave function is presented and a new index for the diradical character is proposed which shows that [Ni(LISQ)2] has a diradical character of ∼77%. These results are in full agreement with elaborate multireference post-Hartree-Fock ab initio calcns. for [Ni(LISQ)2] using the difference dedicated CI (DDCI) method as well as second-order multireference Moller-Plesset (MR-MP2) theory, which give diradical characters of 65-80%. On the basis of these calcns. our best est. for the singlet-triplet gap in this system is 3096 cm-1. This very large value results from an efficient mixing of the ionic configurations into the mainly singlet diradical ground state which is feasible because the semiquinonate SOMOs are delocalized and, therefore, have moderate on-site Coulomb repulsion parameters. As pointed out in the discussion, this represents an interesting difference to the case of magnetically interacting transition metal ions which typically show much smaller magnetic exchange couplings.
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    Chłopek, K.; Muresan, N.; Neese, F.; Wieghardt, K. Electronic Structures of Five-Coordinate Complexes of Iron Containing Zero, One, or Two π-Radical Ligands: A Broken-Symmetry Density Functional Theoretical Study. Chem. Eur. J. 2007, 13, 8390– 8403,  DOI: 10.1002/chem.200700897
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    Electronic structures of five-coordinate complexes of iron containing zero, one, or two π-radical ligands: a broken-symmetry density functional theoretical study
    Chlopek, Krzysztof; Muresan, Nicoleta; Neese, Frank; Wieghardt, Karl
    Chemistry - A European Journal (2007), 13 (30), 8390-8403CODEN: CEUJED; ISSN:0947-6539. (Wiley-VCH Verlag GmbH & Co. KGaA)
    The electronic structures of a series of five-coordinate complexes of iron contg. zero, one, or two bidentate, org. π-radical ligands and a monodentate ligand (pyridine, iodide) have been studied by broken-symmetry (BS) d. functional theor. (DFT) methods. By analyzing the set of corresponding orbitals (CO) a convenient division of the spin-up and spin-down orbitals into (1) essentially doubly-occupied MOs , (2) exactly singly-occupied MOs, (3) spin-coupled pairs, and (4) virtual orbitals can be achieved and a clear picture of the spin coupling between the ligands (non-innocence vs. innocence) and the central metal ion (dN configuration) can be generated. We have identified three classes of complexes which all contain a ferric ion (d5) with an intrinsic intermediate spin (SFe = 3/2) that yield (1) an St = 3/2 ground spin state if the two bidentate ligands are closed-shell species (innocent ligands); (2) if one π-radical ligand is present, an St = 1 ground state is obtained through intramol. antiferromagnetic coupling; (3) if two such radicals are present, an St = 1/2 ground state is obtained. We show unambiguously for the first time that the pentane-2,4-dione-bis(S-alkylisothiosemicarbazonato) ligand can bind as π-radical dianion (L•TSC)2- in [FeIII(L•TSC)I](St = 1); the description as [FeIV-(LTSC3-)I] is incorrect. Similarly, the diamagnetic monoanion in 14 must be described as [FeIII(CN)2(L•TSC)]-(St = 0) with a low-spin ferric ion (d5, SFe = 1/2) coupled antiferromagnetically to a π-radical ligand; [FeII(CN)2(LTSC-)]- is an incorrect description.
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    Neese, F. Prediction of molecular properties and molecular spectroscopy with density functional theory: From fundamental theory to exchange-coupling. Coord. Chem. Rev. 2009, 253, 526– 563,  DOI: 10.1016/j.ccr.2008.05.014
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    Prediction of molecular properties and molecular spectroscopy with density functional theory: From fundamental theory to exchange-coupling
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    This review provides a detailed account of d. functional theory (DFT) and its application to the calcn. of mol. properties of inorg. compds. After introducing some fundamental quantum mech. concepts, the foundations of DFT and their realization in the framework of the Kohn-Sham construction are described. Following a brief exposition of the computational machinery required to carry out large-scale DFT calcns., the application of analytic deriv. theory to DFT is developed in some detail. The cases covered include geometric, elec., magnetic, and time-dependent perturbations. The developed theor. app. is then applied to the calcns. of mol. structures, vibrational energies as well as a wide variety of properties including absorption, CD, magnetic CD, resonance Raman, x-ray absorption, Moessbauer, and ESR spectroscopies. Finally, the important subjects of spin state energetics and exchange couplings in oligomeric transition metal clusters is discussed.
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    Density functional theory in terms of a Legendre transformation for beginners
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    The derivation of the Hohenberg-Kohn (HK) theorem by means of a simplified version of Lieb's Legendre transformation is presented, with the stress on phys. problematic aspects, and caring about the definition of the domains of the resp. functionals only to the extent, that this is absolutely necessary. The take-home lesson consists in a crit. anal. of some statements often found in the DFT literature. As a simple illustration for a Legendre transformation we discuss that in parameter space for the family of n-electron isoelectronic at. ions, with the nuclear charge as parameter.
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    Penz, M.; Tellgren, E. I.; Csirik, M. A.; Ruggenthaler, M.; Laestadius, A. structure of the density-potential mapping. Part I: Standard density-functional theory. arXiv:2211.16627 2022, DOI: 10.48550/arXiv.2211.16627 .
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    Casida, M. E.; Huix-Rotllant, M. Progress in time-dependent density-functional theory. Annu. Rev. Phys. Chem. 2012, 63, 287– 323,  DOI: 10.1146/annurev-physchem-032511-143803
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    Annual Review of Physical Chemistry (2012), 63 (), 287-323CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)
    The classic d.-functional theory (DFT) formalism introduced by Hohenberg, Kohn, and Sham in the mid-1960s is based on the idea that the complicated N-electron wave function can be replaced with the math. simpler 1-electron charge d. in electronic structure calcns. of the ground stationary state. As such, ordinary DFT cannot treat time-dependent (TD) problems nor describe excited electronic states. In 1984, Runge and Gross proved a theorem making TD-DFT formally exact. Information about electronic excited states may be obtained from this theory through the linear response (LR) theory formalism. Beginning in the mid-1990s, LR-TD-DFT became increasingly popular for calcg. absorption and other spectra of medium- and large-sized mols. Its ease of use and relatively good accuracy has now brought LR-TD-DFT to the forefront for this type of application. As the no. and the diversity of applications of TD-DFT have grown, so too has our understanding of the strengths and weaknesses of the approx. functionals commonly used for TD-DFT. The objective of this article is to continue where a previous review of TD-DFT in Vol. 55 of the Annual Review of Phys. Chem. left off and highlight some of the problems and solns. from the point of view of applied phys. chem. Because doubly-excited states have a particularly important role to play in bond dissocn. and formation in both thermal and photochem., particular emphasis is placed on the problem of going beyond or around the TD-DFT adiabatic approxn., which limits TD-DFT calcns. to nominally singly-excited states.
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    Physical Review Letters (1996), 76 (17), 3168-71CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    A widely applicable "nearsightedness" principle is first discussed as the phys. basis for the existence of computational methods scaling linearly with the no. of atoms. This principle applies to the one particle d. matrix n(r,r') but not to individual eigenfunctions. A variational principle for n(r,r') is derived in which, by the use of a penalty functional P[n(r,r')], the (difficult) idempotency of n(r,r') need not be assured in advance but is automatically achieved. The method applies to both insulators and metals.
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    In an earlier paper, W. Kohn had qual. introduced the concept of "nearsightedness" of electrons in many-atom systems. It can be viewed as underlying such important ideas as Pauling's "chem. bond," "transferability," and Yang's computational principle of "divide and conquer.". It describes the fact that, for fixed chem. potential, local electronic properties, such as the d. n(r), depend significantly on the effective external potential only at nearby points. Changes of that potential, no matter how large, beyond a distance R have limited effects on local electronic properties, which rapidly tend to zero as a function of R. In the present paper, the concept is first sharpened for representative models of uncharged fermions moving in external potentials, and then the effects of electron-electron interactions and of perturbing external charges are discussed.
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    We present a constructive soln. to the N-representability problem: a full characterization of the conditions for constraining the two-electron reduced d. matrix to represent an N-electron d. matrix. Previously known conditions, while rigorous, were incomplete. Here, we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompns. of model Hamiltonians. Existing conditions D, Q, G, T1, and T2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.
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    Density-cumulant functional theory
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    Journal of Chemical Physics (2006), 125 (17), 171101/1-171101/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Starting point is the energy expectation value as a functional of the one-particle d. matrix γ and the two-particle d. cumulant λ2. We decomp. γ into a best idempotent approxn. κ and a correction τ, that is entirely expressible in terms of λ2. So we get the energy E as a functional of κ and λ2, which can be varied independently. Approx. n-representability conditions, derived by perturbation theory are imposed on the variation of λ2. A nonlinear system of equations satisfied by λ2 is derived, the linearized version of which turns out to be equiv. to the CEPA, variant zero. The start for κ is Hartree-Fock, but κ is then updated to become the best idempotent approxn. of γ. Relations to d. matrix functional theory and Kohn-Sham type d. functional theory are discussed.
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    Piris, M.; Ugalde, J. M. Perspective on natural orbital functional theory. Int. J. Quantum Chem. 2014, 114, 1169– 1175,  DOI: 10.1002/qua.24663
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    Perspective on natural orbital functional theory
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    International Journal of Quantum Chemistry (2014), 114 (18), 1169-1175CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
    A review. The natural orbital functional (NOF) theory is briefly reviewed. The meaning of the top-down and bottom-up approaches for the construction of a NOF is analyzed. A particular reconstruction of the two-particle reduced d. matrix (2-RDM) based on the cumulant expansion is discussed. The cumulant is expressed by two auxiliary matrixes, which are constrained to certain bounds due to the N-representabilty conditions of the 2-RDM. Appropriate forms of these matrixes lead to different implementations known in the literature as PNOFi (i = 1-5). The strengths and weaknesses of PNOF5 are assessed. Its main strength is its ability to deal with the intrapair electron correlation at a reasonable computational cost. Its main limitation is the absence of the interpair electron correlation. The inclusion of the missing correlation via a multiconfigurational perturbation theory is shortly described. The growing interest in methods based on NOF theory points to a promising future in this field. © 2014 Wiley Periodicals, Inc.
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    Approximate one-matrix functionals for the electron-electron repulsion energy from geminal theories
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    Journal of Chemical Physics (2003), 119 (13), 6443-6447CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A simple extension of the antisymmetrized product of strongly orthogonal geminals theory produces a "JK-only" one-matrix functional for the electron-electron repulsion energy of a closed-shell system that is exact for two-electron singlet ground states, size-extensive, and incorporates some intergeminal correlation and thus dispersion effects. The functional is defined only for one-matrixes with occupation nos. that can be arranged into sets with elements that sum up to two. Its possible generalizations are discussed.
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    A natural orbital functional based on an explicit approach of the two-electron cumulant
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    A review. The cumulant expansion gives rise to an useful decompn. of the two-matrix in which the pair correlated matrix (cumulant) is disconnected from the antisym. product of the one-matrixes. The cumulant can be approximated in terms of two matrixes, Δ and Π, which are explicit functions of the occupation nos. of the natural orbitals. It produces a natural orbital functional (NOF) that reduces to the exact expression for the total energy in two-electron systems. The N-representability positivity necessary conditions of the two-matrix impose several bounds on the matrixes Δ and Π. Appropriate forms of these matrixes lead to different implementations of the NOF known in the literature as PNOFi (i = 1-5). The basic features of these functionals are reviewed here. The strengths and weaknesses of the different PNOFs are assessed. © 2012 Wiley Periodicals, Inc.
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    Density cumulant functional theory: First implementation and benchmark results for the DCFT-06 model
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    D. cumulant functional theory is implemented for the first time. Benchmark results are provided for atoms and diat. mols., demonstrating the performance of DCFT-06 for both nonbonded and bonded interactions. The results show that DCFT-06 appears to perform similarly to coupled cluster theory with single and double excitations (CCSD) in describing dispersion. For covalently bound systems, the phys. properties predicted by DCFT-06 appear to be at least of CCSD quality around equil. geometries. The computational scaling of both DCFT-06 and CCSD is O(N6), but the former has reduced nonlinearities among the variables and a Hermitian energy functional, making it an attractive alternative. (c) 2010 American Institute of Physics.
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    Copan, A. V.; Sokolov, A. Y. Linear-response density cumulant theory for excited electronic states. J. Chem. Theory Comput. 2018, 14, 4097– 4108,  DOI: 10.1021/acs.jctc.8b00326
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    Linear-Response Density Cumulant Theory for Excited Electronic States
    Copan, Andreas V.; Sokolov, Alexander Yu.
    Journal of Chemical Theory and Computation (2018), 14 (8), 4097-4108CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We present a linear-response formulation of d. cumulant theory (DCT) that provides a balanced and accurate description of many electronic states simultaneously. In the original DCT formulation, only information about a single electronic state (usually, the ground state) is obtained. We discuss the derivation of linear-response DCT, present its implementation for the ODC-12 method (LR-ODC-12), and benchmark its performance for excitation energies in small mols. (N2, CO, HCN, HNC, C2H2, and H2CO), as well as challenging excited states in ethylene, butadiene, and hexatriene. For small mols., LR-ODC-12 shows smaller mean abs. errors in excitation energies than equation-of-motion coupled cluster theory with single and double excitations (EOM-CCSD), relative to the ref. data from EOM-CCSDT. In a study of butadiene and hexatriene, LR-ODC-12 correctly describes the relative energies of the singly excited 11Bu and the doubly excited 21Ag states, in excellent agreement with highly accurate semistochastic heat-bath CI results, while EOM-CCSD overestimates the energy of the 21Ag state by almost 1 eV. Our results demonstrate that linear-response DCT is a promising theor. approach for excited states of mols.
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    Gidofalvi, G.; Mazziotti, D. A. Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the N-electron Hamiltonian. J. Chem. Phys. 2008, 129, 134108,  DOI: 10.1063/1.2983652
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    Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the N-electron Hamiltonian
    Gidofalvi, Gergely; Mazziotti, David A.
    Journal of Chemical Physics (2008), 129 (13), 134108/1-134108/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Mol. systems in chem. often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the no. N of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-d.-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calcn. where the 2-RDM is constrained by approx. N-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as ra6 where ra is the no. of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calcn. of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, H2O, and N2 as well as n-acene chains for n=2-8. When n°4, the acenes cannot be treated by conventional CASSCF methods; for example, when n=8, CASSCF requires optimization over approx. 1.47×1017 configuration state functions. The natural occupation nos. of the n-acenes show the emergence of bi- and polyradical character with increasing chain length. (c) 2008 American Institute of Physics.
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    Mullinax, J. W.; Sokolov, A. Y.; Schaefer, H. F., III Can density cumulant functional theory describe static correlation effects?. J. Chem. Theory Comput. 2015, 11, 2487– 2495,  DOI: 10.1021/acs.jctc.5b00346
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    Can Density Cumulant Functional Theory Describe Static Correlation Effects?
    Mullinax, J. Wayne; Sokolov, Alexander Yu.; Schaefer, Henry F.
    Journal of Chemical Theory and Computation (2015), 11 (6), 2487-2495CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We evaluate the performance of d. cumulant functional theory (DCT) for capturing static correlation effects. In particular, we examine systems with significant multideterminant character of the electronic wave function, such as the beryllium dimer, diat. carbon, m-benzyne, 2,6-pyridyne, twisted ethylene, as well as the barrier for double-bond migration in cyclobutadiene. We compute mol. properties of these systems using the ODC-12 and DC-12 variants of DCT and compare these results to multireference CI and multireference coupled-cluster theories, as well as single-ref. coupled-cluster theory with single, double (CCSD), and perturbative triple excitations [CCSD(T)]. For all systems the DCT methods show intermediate performance between that of CCSD and CCSD(T), with significant improvement over the former method. In particular, for the beryllium dimer, m-benzyne, and 2,6-pyridyne, the ODC-12 method along with CCSD(T) correctly predict the global min. structures, while CCSD predictions fail qual., underestimating the multireference effects. Our results suggest that the DC-12 and ODC-12 methods are capable of describing emerging static correlation effects but should be used cautiously when highly accurate results are required. Conveniently, the appearance of multireference effects in DCT can be diagnosed by analyzing the DCT natural orbital occupations, which are readily available at the end of the energy computation.
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    Lathiotakis, N. N.; Helbig, N.; Rubio, A.; Gidopoulos, N. I. Local reduced-density-matrix-functional theory: Incorporating static correlation effects in Kohn-Sham equations. Phys. Rev. A 2014, 90, 032511,  DOI: 10.1103/PhysRevA.90.032511
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    Local reduced-density-matrix-functional theory: incorporating static correlation effects in Kohn-Sham equations
    Lathiotakis, Nektarios N.; Helbig, Nicole; Rubio, Angel; Gidopoulos, Nikitas I.
    Physical Review A: Atomic, Molecular, and Optical Physics (2014), 90 (3-A), 032511/1-032511/8CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
    We propose a scheme to bring reduced-d.-matrix-functional theory into the realm of d. functional theory (DFT) that preserves the accurate d. functional description at equil., while incorporating accurately static and left-right correlation effects in mols. and keeping the good computational performance of DFT-based schemes. The key ingredient is to relax the requirement that the local potential is the functional deriv. of the energy with respect to the d. Instead, we propose to restrict the search for the approx. natural orbitals within a domain where these orbitals are eigenfunctions of a single-particle Hamiltonian with a local effective potential. In this way, fractional natural occupation nos. are accommodated into Kohn-Sham equations allowing for the description of mol. dissocn. without breaking spin symmetry. Addnl., our scheme provides a natural way to connect an energy eigenvalue spectrum to the approx. natural orbitals and this spectrum is found to represent accurately the ionization potentials of atoms and small mols.
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    Global method for electron correlation
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    A review. The current work presents a new single-ref. method for capturing at the same time the static and dynamic electron correlation. The starting point is a determinant wave function formed with natural orbitals obtained from a new interacting-pair model. The latter leads to a natural orbital functional (NOF) capable of recovering the complete intrapair, but only the static interpair correlation. Using the soln. of the NOF, two new energy functionals are defined for both dynamic (Edyn) and static (Esta) correlation. Edyn is derived from a modified second-order Moller-Plesset perturbation theory (MP2), while Esta is obtained from the static component of the new NOF. Double counting is avoided by introducing the amt. of static and dynamic correlation in each orbital as a function of its occupation. As a result, the total energy is represented by the sum EHF + Edyn + Esta, where EHF is the Hartree-Fock energy obtained with natural orbitals. The new procedure called NOF-MP2 scales formally as O(M5) (where M is the no. of basis functions), and is applied successfully to the homolytic dissocn. of a selected set of diat. mols., paradigmatic cases of near-degeneracy effects. The size consistency has been numerically demonstrated for singlets. The values obtained are in good agreement with the exptl. data.
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    Gibney, D.; Boyn, J.-N.; Mazziotti, D. A. Density Functional Theory Transformed into a One-Electron Reduced-Density-Matrix Functional Theory for the Capture of Static Correlation. J. Phys. Chem. Lett. 2022, 13, 1382– 1388,  DOI: 10.1021/acs.jpclett.2c00083
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    Density Functional Theory Transformed into a One-Electron Reduced-Density-Matrix Functional Theory for the Capture of Static Correlation
    Gibney, Daniel; Boyn, Jan-Niklas; Mazziotti, David A.
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    D. Functional Theory (DFT), the most widely adopted method in modern computational chem., fails to describe accurately the electronic structure of strongly correlated systems. Here we show that DFT can be formally and practically transformed into a one-electron reduced-d.-matrix (1-RDM) functional theory, which can address the limitations of DFT while retaining favorable computational scaling compared to wave function based approaches. In addn. to relaxing the idempotency restriction on the 1-RDM in the kinetic energy term, we add a quadratic 1-RDM-based term to DFT's d.-based exchange-correlation functional. Our approach, which we implement by quadratic semidefinite programming at DFT's computational scaling of O(r3), yields substantial improvements over traditional DFT in the description of static correlation in chem. structures and processes such as singlet biradicals and bond dissocns.
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    Sun, Q.; Chan, G. K.-L. Quantum embedding theories. Acc. Chem. Res. 2016, 49, 2705– 2712,  DOI: 10.1021/acs.accounts.6b00356
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    Quantum Embedding Theories
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    Accounts of Chemical Research (2016), 49 (12), 2705-2712CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)
    A review. In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations - a high level calcn. on the active region of interest, and a low level calcn. on its environment - formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: d. functional embedding, Green's function embedding, and d. matrix embedding, resp., use the single-particle d., single-particle Green's function, and single-particle d. matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used, functional equations for d. functional embedding, diagrammatics for Green's function embedding, and entanglement arguments for d. matrix embedding, make the three formulations appear vastly different. In this Account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theor. perspective, we also give a brief overview of recent applications and possible future developments. The first section starts with d. functional embedding, where we introduce the key embedding potential via the Euler equation. We then discuss recent work concerning the treatment of the nonadditive kinetic potential, before describing mean-field d. functional embedding and wave function in d. functional embedding. We finish the section with extensions to time-dependence and excited states. The second section is devoted to Green's function embedding. Here, we use the Dyson equation to obtain equations that parallel as closely as possible the d. functional embedding equations, with the hybridization playing the role of the embedding potential. Embedding a high-level self-energy within a low-level self-energy is treated analogously to wave function in d. functional embedding. The numerical computation of the high-level self-energy allows us to briefly introduce the bath representation in the quantum impurity problem. We then consider translationally invariant systems to bring in the important dynamical mean-field theory. Recent developments to incorporate screening and long-range interactions are discussed.The third section concerns d. matrix embedding. Here, we first highlight some math. complications assocd. with a simple Euler equation derivation, arising from the open nature of fragments. This motivates the d. matrix embedding theory, where we use the Schmidt decompn. to represent the entanglement through bath orbitals. The resulting impurity plus bath formulation resembles that of dynamical mean-field theory. We discuss the numerical self-consistency assocd. with using a high-level correlated wave function with a mean-field low-level treatment, and connect the resulting numerical inversion to that used in d. functional embedding. We finish with perspectives on the future of all three methods.
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    Kotliar, G.; Savrasov, S. Y.; Haule, K.; Oudovenko, V. S.; Parcollet, O.; Marianetti, C. Electronic structure calculations with dynamical mean-field theory. Rev. Mod. Phys. 2006, 78, 865– 951,  DOI: 10.1103/RevModPhys.78.865
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    Kotliar, G.; Savrasov, S. Y.; Haule, K.; Oudovenko, V. S.; Parcollet, O.; Marianetti, C. A.
    Reviews of Modern Physics (2006), 78 (3), 865-951CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
    A review of the basic ideas and techniques of the spectral d.-functional theory is presented. This method is currently used for electronic structure calcns. of strongly correlated materials where the one-electron description breaks down. The method is illustrated with several examples where interactions play a dominant role: systems near metal-insulator transitions, systems near vol. collapse transitions, and systems with local moments.
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    Lin, N.; Marianetti, C.; Millis, A. J.; Reichman, D. R. Dynamical mean-field theory for quantum chemistry. Phys. Rev. Lett. 2011, 106, 096402,  DOI: 10.1103/PhysRevLett.106.096402
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    Dynamical Mean-Field Theory for Quantum Chemistry
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    Physical Review Letters (2011), 106 (9), 096402/1-096402/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    The dynamical mean-field concept of approximating an unsolvable many-body problem in terms of the soln. of an auxiliary quantum impurity problem, introduced to study bulk materials with a continuous energy spectrum, is here extended to mols., i.e., finite systems with a discrete energy spectrum. The application to small clusters of hydrogen atoms yields ground state energies which are competitive with leading quantum chem. approaches at intermediate and large interat. distances as well as good approxns. to the excitation spectrum.
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    Zgid, D.; Chan, G. K.-L. Dynamical mean-field theory from a quantum chemical perspective. J. Chem. Phys. 2011, 134, 094115,  DOI: 10.1063/1.3556707
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    Dynamical mean-field theory from a quantum chemical perspective
    Zgid, Dominika; Chan, Garnet Kin-Lic
    Journal of Chemical Physics (2011), 134 (9), 094115/1-094115/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We investigate the dynamical mean-field theory (DMFT) from a quantum chem. perspective. Dynamical mean-field theory offers a formalism to extend quantum chem. methods for finite systems to infinite periodic problems within a local correlation approxn. In addn., quantum chem. techniques can be used to construct new ab initio Hamiltonians and impurity solvers for DMFT. Here, we explore some ways in which these things may be achieved. First, we present an informal overview of dynamical mean-field theory to connect to quantum chem. language. Next, we describe an implementation of dynamical mean-field theory where we start from an ab initio Hartree-Fock Hamiltonian that avoids double counting issues present in many applications of DMFT. We then explore the use of the CI hierarchy in DMFT as an approx. solver for the impurity problem. We also investigate some numerical issues of convergence within DMFT. Our studies are carried out in the context of the cubic hydrogen model, a simple but challenging test for correlation methods. Finally, we finish with some conclusions for future directions. (c) 2011 American Institute of Physics.
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    Hedin, L. New method for calculating the one-particle Green’s function with application to the electron-gas problem. Phys. Rev. 1965, 139, A796– A823,  DOI: 10.1103/PhysRev.139.A796
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    New method for calculating the one-particle Green's function with application to the electron-gas problem
    Hedin, Lars
    Physical Review (1965), 139 (3A), 796-823CODEN: PHRVAO; ISSN:0031-899X.
    A set of successively more accurate self-consistent equations for the 1-electron Green's function were derived. They correspond to an expansion in a screened potential rather than the bare Coulomb potential. The 1st equation is adequate for many purposes. Each equation follows from the demand that a corresponding expression for the total energy be stationary with respect to variations in the Green's function. The main information to be obtained, besides the total energy, is 1-particle-like excitation spectra, i.e., spectra characterized by the quantum nos. of a single particle. This includes the low-excitation spectra in metals as well as configurations in atoms, mols., and solids with one electron outside or one electron missing from a closed-shell structure. In the latter cases, an approx. description is obtained by a modified Hartree-Fock equation involving a "Coulomb hole" and a static screened potential in the exchange term. As an example, spectra of some atoms are discussed. To investigate the convergence of successive approxn. for the Green's function, extensive calcns. were made for the electron gas at a range of metallic ds. The results are expressed in terms of quasiparticle energies Ε(k) and quasiparticle interactions f(k,k'). The very 1st approxn. gives a good value for the magnitude of Ε(k.). To est. the deriv. of Ε(k), both the 1st- and the 2nd-order terms are needed. The derivative, and thus the sp. heat, differs from the free-particle value by only a few percent. The correction to the sp. heat keeps the same sign down to the lowest alkalimetal ds., and is smaller than those obtained recently by Silverstein (CA 59, 144d) and by Rice (CA 62, 7218f). The results for the paramagnetic susceptibility are unreliable in the alkalimetal-d.-region owing to poor convergence of the expansion for f. Besides the proof of a modified Luttinger-Ward-Klein variational principle and a related self-consistency idea, there is not much new in principle but emphasis is on the development of a numerically manageable approxn. scheme.
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    Aryasetiawan, F.; Gunnarsson, O. The GW method. Rep. Prog. Phys. 1998, 61, 237– 312,  DOI: 10.1088/0034-4885/61/3/002
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    The GW method
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    Reports on Progress in Physics (1998), 61 (3), 237-312CODEN: RPPHAG; ISSN:0034-4885. (Institute of Physics Publishing)
    A review with many refs. Calcns. of ground-state and excited-state properties of materials have been one of the major goals of condensed matter physics. Ground-state properties of solids have been extensively investigated for several decades within the std. d. functional theory. Excited-state properties, on the other hand, were relatively unexplored in ab initio calcns. until a decade ago. The most suitable approach up to now for studying excited-state properties of extended systems is the Green function method. To calc. the Green function one requires the self-energy operator which is non-local and energy dependent. In this article we describe the GW approxn. which has turned out to be a fruitful approxn. to the self-energy. The Green function theory, numerical methods for carrying out the self-energy calcns., simplified schemes, and applications to various systems are described. Self-consistency issues and new developments beyond the GW approxn. are also discussed as well as the success and shortcomings of the GW approxn.
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    Reining, L. The GW approximation: content, successes and limitations. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2018, 8, e1344,  DOI: 10.1002/wcms.1344
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    Phillips, J. J.; Zgid, D. Communication: The description of strong correlation within self-consistent Green’s function second-order perturbation theory. J. Chem. Phys. 2014, 140, 241101,  DOI: 10.1063/1.4884951
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    Communication: The description of strong correlation within self-consistent Green's function second-order perturbation theory
    Phillips, Jordan J.; Zgid, Dominika
    Journal of Chemical Physics (2014), 140 (24), 241101/1-241101/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We report an implementation of self-consistent Green's function many-body theory within a second-order approxn. (GF2) for application with mol. systems. This is done by iterative soln. of the Dyson equation expressed in matrix form in an AO basis, where the Green's function and self-energy are built on the imaginary frequency and imaginary time domain, resp., and fast Fourier transform is used to efficiently transform these quantities as needed. We apply this method to several archetypical examples of strong correlation, such as a H32 finite lattice that displays a highly multireference electronic ground state even at equil. lattice spacing. In all cases, GF2 gives a phys. meaningful description of the metal to insulator transition in these systems, without resorting to spin-symmetry breaking. Our results show that self-consistent Green's function many-body theory offers a viable route to describing strong correlations while remaining within a computationally tractable single-particle formalism. (c) 2014 American Institute of Physics.
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  177. 177
    Pokhilko, P.; Zgid, D. Interpretation of multiple solutions in fully iterative GF2 and GW schemes using local analysis of two-particle density matrices. J. Chem. Phys. 2021, 155, 024101,  DOI: 10.1063/5.0055191
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    Interpretation of multiple solutions in fully iterative GF2 and GW schemes using local analysis of two-particle density matrices
    Pokhilko, Pavel; Zgid, Dominika
    Journal of Chemical Physics (2021), 155 (2), 024101CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Due to the presence of non-linear equations, iterative Green's function methods can result in multiple different solns. even for simple mol. systems. In contrast to the wave-function methods, a detailed and careful anal. of such mol. solns. was not performed before. In this work, we use two-particle d. matrixes to investigate local spin and charge correlators that quantify the charge resonance and covalent characters of these solns. When applied within the unrestricted orbital set, spin correlators elucidate the broken symmetry of the solns., contg. necessary information for building effective magnetic Hamiltonians. Based on GW and GF2 calcns. of simple mols. and transition metal complexes, we construct Heisenberg Hamiltonians, four-spin-four-center corrections, and biquadratic spin-spin interactions. These Hamiltonian parameterizations are compared to previous wave-function calcns. (c) 2021 American Institute of Physics.
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  178. 178
    Blase, X.; Duchemin, I.; Jacquemin, D. The Bethe–Salpeter equation in chemistry: relations with TD-DFT, applications and challenges. Chem. Soc. Rev. 2018, 47, 1022– 1043,  DOI: 10.1039/C7CS00049A
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    178
    The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges
    Blase, Xavier; Duchemin, Ivan; Jacquemin, Denis
    Chemical Society Reviews (2018), 47 (3), 1022-1043CODEN: CSRVBR; ISSN:0306-0012. (Royal Society of Chemistry)
    We review the many-body Green's function Bethe-Salpeter equation (BSE) formalism that is rapidly gaining importance for the study of the optical properties of mol. org. systems. We emphasize in particular its similarities and differences with time-dependent d. functional theory (TD-DFT), both methods sharing the same formal O(N4) computing time scaling with system size. By comparison with higher level wavefunction based methods and exptl. results, the advantages of BSE over TD-DFT are presented, including an accurate description of charge-transfer states and an improved accuracy for the challenging cyanine dyes. We further discuss the models that have been developed for including environmental effects. Finally, we summarize the challenges to be faced so that BSE reaches the same popularity as TD-DFT.
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  179. 179
    Pokhilko, P.; Iskakov, S.; Yeh, C.-N.; Zgid, D. Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green’s function methods: Theory and application to GW and GF2. J. Chem. Phys. 2021, 155, 024119,  DOI: 10.1063/5.0054661
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    179
    Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green's function methods: Theory and application to GW and GF2
    Pokhilko, Pavel; Iskakov, Sergei; Yeh, Chia-Nan; Zgid, Dominika
    Journal of Chemical Physics (2021), 155 (2), 024119CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    One-particle Green's function methods can model mol. and solid spectra at zero or non-zero temps. One-particle Green's functions directly provide electronic energies and one-particle properties, such as dipole moment. However, the evaluation of two-particle properties, such as 〈S2〉 and 〈N2〉, can be challenging because they require a soln. of the computationally expensive Bethe-Salpeter equation to find two-particle Green's functions. We demonstrate that the soln. of the Bethe-Salpeter equation can be completely avoided. Applying the thermodn. Hellmann-Feynman theorem to self-consistent one-particle Green's function methods, we derive expressions for two-particle d. matrixes in a general case and provide explicit expressions for GF2 and GW methods. Such d. matrixes can be decompd. into an antisymmetrized product of correlated one-electron d. matrixes and the two-particle electronic cumulant of the d. matrix. Cumulant expressions reveal a deviation from ensemble representability for GW, explaining its known deficiencies. We analyze the temp. dependence of 〈S2〉 and 〈N2〉 for a set of small closed-shell systems. Interestingly, both GF2 and GW show a non-zero spin contamination and a non-zero fluctuation of the no. of particles for closed-shell systems at the zero-temp. limit. (c) 2021 American Institute of Physics.
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    Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Fractional spins and static correlation error in density functional theory. J. Chem. Phys. 2008, 129, 121104,  DOI: 10.1063/1.2987202
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    Fractional spins and static correlation error in density functional theory
    Cohen, Aron J.; Mori-Sanchez, Paula; Yang, Weitao
    Journal of Chemical Physics (2008), 129 (12), 121104/1-121104/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Electronic states with fractional spins arise in systems with large static correlation (strongly correlated systems). Such fractional-spin states are shown to be ensembles of degenerate ground states with normal spins. It is proven here that the energy of the exact functional for fractional-spin states is a const., equal to the energy of the comprising degenerate pure-spin states. Dramatic deviations from this exact constancy condition exist with all approx. functionals, leading to large static correlation errors for strongly correlated systems, such as chem. bond dissocn. and band structure of Mott insulators. This is demonstrated with numerical calcns. for several mol. systems. Approximating the constancy behavior for fractional spins should be a major aim in functional constructions and should open the frontier for d. functional theory to describe strongly correlated systems. The key results are also shown to apply in reduced d.-matrix functional theory. (c) 2008 American Institute of Physics.
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    Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Insights into current limitations of density functional theory. Science 2008, 321, 792– 794,  DOI: 10.1126/science.1158722
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    Insights into Current Limitations of Density Functional Theory
    Cohen, Aron J.; Mori-Sanchez, Paula; Yang, Weitao
    Science (Washington, DC, United States) (2008), 321 (5890), 792-794CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
    A review. D. functional theory of electronic structure is widely and successfully applied in simulations throughout engineering and sciences. However, for many predicted properties, there are spectacular failures that can be traced to the delocalization error and static correlation error of commonly used approxns. These errors can be characterized and understood through the perspective of fractional charges and fractional spins introduced recently. Reducing these errors will open new frontiers for applications of d. functional theory.
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    Grimme, S.; Hansen, A. A practicable real-space measure and visualization of static electron-correlation effects. Angew. Chem., Int. Ed. 2015, 54, 12308– 12313,  DOI: 10.1002/anie.201501887
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    A Practicable Real-Space Measure and Visualization of Static Electron-Correlation Effects
    Grimme, Stefan; Hansen, Andreas
    Angewandte Chemie, International Edition (2015), 54 (42), 12308-12313CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
    The inclusion of dynamical and static electron correlation (SEC) is mandatory for accurate quantum chem. (QC). SEC is particularly difficult to calc. and hence a qual. understanding is important to judge the applicability of approx. QC methods. Existing scalar SEC diagnostics, however, lack the important information where the SEC effects occur in a mol. We introduce an anal. tool based on a fractional occupation no. weighted electron d. (ρFOD) that is plotted in 3D for a pre-defined contour surface value. The scalar field is obtained by finite-temp. DFT calcns. with pre-defined electronic temp. (e.g. TPSS at 5000 K). FOD plots only show the contribution of the "hot" (strongly correlated) electrons. We discuss illustrative plots for a broad range of chem. systems from small mols. to large conjugated mols. with polyradicaloid character. Spatial integration yields a single no. which can be used to globally quantify SEC.
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    Muechler, L.; Badrtdinov, D. I.; Hampel, A.; Cano, J.; Rösner, M.; Dreyer, C. E. Quantum embedding methods for correlated excited states of point defects: Case studies and challenges. Phys. Rev. B 2022, 105, 235104,  DOI: 10.1103/PhysRevB.105.235104
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    Quantum embedding methods for correlated excited states of point defects: Case studies and challenges
    Muechler, Lukas; Badrtdinov, Danis I.; Hampel, Alexander; Cano, Jennifer; Rosner, Malte; Dreyer, Cyrus E.
    Physical Review B (2022), 105 (23), 235104CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    A quant. description of the excited electronic states of point defects and impurities is crucial for understanding materials properties, and possible applications of defects in quantum technologies. This is a considerable challenge for computational methods, since Kohn-Sham d. functional theory (DFT) is inherently a ground-state theory, while higher-level methods are often too computationally expensive for defect systems. Recently, embedding approaches have been applied that treat defect states with many-body methods, while using DFT to describe the bulk host material. We implement such an embedding method, based on Wannierization of defect orbitals and the constrained RPA approach, and perform systematic characterization of the method for three distinct systems with current technol. relevance: a carbon dimer replacing a B and N pair in bulk hexagonal BN (CBCN), the neg. charged nitrogen-vacancy center in diamond (NV-), and an Fe impurity on the Al site in wurtzite AlN (FeAl). In the context of these test-case defects, we demonstrate that crucial considerations of the methodol. include convergence of the bulk screening of the active-space Coulomb interaction, the choice of exchange-correlation functional for the initial DFT calcn., and the treatment of the "double-counting" correction. For CBCN we show that the embedding approach gives many-body states in agreement with anal. results on the Hubbard dimer model, which allows us to elucidate the effects of the DFT functional and double-counting correction. For the NV- center, our method demonstrates good quant. agreement with expts. for the zero-phonon line of the triplet-triplet transition. Finally, we illustrate challenges assocd. with this method for detg. the energies and orderings of the complex spin multiplets in FeAl.
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    Nielsen, M. A.; Chuang, I. L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, 2010.
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    Almeida, M.; Omar, Y.; Rocha Vieira, V. Introduction to entanglement and applications to the simulation of many-body quantum systems. In Strongly Correlated Systems, Coherence And Entanglement; World Scientific: Singapore, 2007; Chapter 19, pp 525– 547.
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    Ding, L.; Knecht, S.; Zimborás, Z.; Schilling, C. Quantum Correlations in Molecules: from Quantum Resourcing to Chemical Bonding. Quantum Sci. Technol. 2023, 8, 015015,  DOI: 10.1088/2058-9565/aca4ee
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    Omar, Y. Particle statistics in quantum information processing. Int. J. Quantum Inf. 2005, 3, 201– 205,  DOI: 10.1142/S021974990500075X
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    Benatti, F.; Floreanini, R.; Franchini, F.; Marzolino, U. Entanglement in indistinguishable particle systems. Phys. Rep. 2020, 878, 1– 27,  DOI: 10.1016/j.physrep.2020.07.003
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    Henderson, T. M.; Bulik, I. W.; Stein, T.; Scuseria, G. E. Seniority-based coupled cluster theory. J. Chem. Phys. 2014, 141, 244104,  DOI: 10.1063/1.4904384
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    Seniority-based coupled cluster theory
    Henderson, Thomas M.; Bulik, Ireneusz W.; Stein, Tamar; Scuseria, Gustavo E.
    Journal of Chemical Physics (2014), 141 (24), 244104/1-244104/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Doubly occupied CI (DOCI) with optimized orbitals often accurately describes strong correlations while working in a Hilbert space much smaller than that needed for full CI. However, the scaling of such calcns. remains combinatorial with system size. Pair coupled cluster doubles (pCCD) is very successful in reproducing DOCI energetically, but can do so with low polynomial scaling (N3, disregarding the two-electron integral transformation from at. to MOs). We show here several examples illustrating the success of pCCD in reproducing both the DOCI energy and wave function and show how this success frequently comes about. What DOCI and pCCD lack are an effective treatment of dynamic correlations, which we here add by including higher-seniority cluster amplitudes which are excluded from pCCD. This frozen pair coupled cluster approach is comparable in cost to traditional closed-shell coupled cluster methods with results that are competitive for weakly correlated systems and often superior for the description of strongly correlated systems. (c) 2014 American Institute of Physics.
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    Boguslawski, K.; Tecmer, P. Orbital entanglement in quantum chemistry. Int. J. Quantum Chem. 2015, 115, 1289– 1295,  DOI: 10.1002/qua.24832
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    Orbital entanglement in quantum chemistry
    Boguslawski, Katharina; Tecmer, Pawel
    International Journal of Quantum Chemistry (2015), 115 (19), 1289-1295CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
    The basic concepts of orbital entanglement and its application to chem. are briefly reviewed. The calcn. of orbital entanglement measures from correlated wavefunctions is discussed in terms of reduced n-particle d. matrixes. Possible simplifications in their evaluation are highlighted in case of seniority-zero wavefunctions. Specifically, orbital entanglement allows us to dissect electron correlation effects in its strong and weak contributions, to det. bond orders, to assess the quality and stability of active space calcns., to monitor chem. reactions, and to identify points along the reaction coordinate where electronic wavefunctions change drastically. Thus, orbital entanglement represents a useful and intuitive tool to interpret complex electronic wavefunctions and to facilitate a qual. understanding of electronic structure and how it changes in chem. processes. © 2014 Wiley Periodicals, Inc.
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    Ding, L.; Zimboras, Z.; Schilling, C. Quantifying Electron Entanglement Faithfully. arXiv:2207.03377 2022, DOI: 10.48550/arXiv.2207.03377 .
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    Legeza, Ö.; Sólyom, J. Optimizing the density-matrix renormalization group method using quantum information entropy. Phys. Rev. B 2003, 68, 195116,  DOI: 10.1103/PhysRevB.68.195116
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    Optimizing the density-matrix renormalization group method using quantum information entropy
    Legeza, O.; Solyom, J.
    Physical Review B: Condensed Matter and Materials Physics (2003), 68 (19), 195116/1-195116/19CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    In order to optimize the ordering of the lattice sites in the momentum space and quantum chem. versions of the d.-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various mols. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chem. DMRG to a great extent. The effect of lattice site ordering on the no. of block states to be kept during the RG procedure is also investigated.
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    Stein, C. J.; Reiher, M. Measuring multi-configurational character by orbital entanglement. Mol. Phys. 2017, 115, 2110– 2119,  DOI: 10.1080/00268976.2017.1288934
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    Measuring multi-configurational character by orbital entanglement
    Stein, Christopher J.; Reiher, Markus
    Molecular Physics (2017), 115 (17-18), 2110-2119CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
    One of the most crit. tasks at the very beginning of a quantum chem. investigation is the choice of either a multi- or single-configurational method. Naturally, many proposals exist to define a suitable diagnostic of the multi-configurational character for various types of wave functions in order to assist this crucial decision. Here, we present a new orbital-entanglement-based multi-configurational diagnostic termed Zs(1). The correspondence of orbital entanglement and static (or non-dynamic) electron correlation permits the definition of such a diagnostic. We chose our diagnostic to meet important requirements such as well-defined limits for pure single-configurational and multi-configurational wave functions. The Zs(1) diagnostic can be evaluated from a partially converged, but qual. correct, and therefore inexpensive d. matrix renormalization group wave function as in our recently presented automated active orbital selection protocol. Its robustness and the fact that it can be evaluated at low cost make this diagnostic a practical tool for routine applications.
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    Stein, C. J.; Reiher, M. Automated selection of active orbital spaces. J. Chem. Theory Comput. 2016, 12, 1760– 1771,  DOI: 10.1021/acs.jctc.6b00156
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    Automated Selection of Active Orbital Spaces
    Stein, Christopher J.; Reiher, Markus
    Journal of Chemical Theory and Computation (2016), 12 (4), 1760-1771CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    One of the key challenges of quantum-chem. multi-configuration methods is the necessity to manually select orbitals for the active space. This selection requires both expertise and experience and can therefore impose severe limitations on the applicability of this most general class of ab initio methods. A poor choice of the active orbital space may yield even qual. wrong results. This is obviously a severe problem, esp. for wave function methods that are designed to be systematically improvable. Here, we show how the iterative nature of the d. matrix renormalization group combined with its capability to include up to about 100 orbitals in the active space can be exploited for a systematic assessment and selection of active orbitals. These benefits allow us to implement an automated approach for active orbital space selection, which can turn multi-configuration models into black box approaches.
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  195. 195
    Boguslawski, K.; Tecmer, P.; Barcza, G.; Legeza, O.; Reiher, M. Orbital entanglement in bond-formation processes. J. Chem. Theory Comput. 2013, 9, 2959– 2973,  DOI: 10.1021/ct400247p
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    Orbital Entanglement in Bond-Formation Processes
    Boguslawski, Katharina; Tecmer, Pawel; Barcza, Gergely; Legeza, Ors; Reiher, Markus
    Journal of Chemical Theory and Computation (2013), 9 (7), 2959-2973CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    The accurate calcn. of the (differential) correlation energy is central to the quantum chem. description of bond-formation and bond-dissocn. processes. In order to est. the quality of single- and multireference approaches for this purpose, various diagnostic tools have been developed. In this work, we elaborate on our previous observation that one- and two-orbital-based entanglement measures provide quant. means for the assessment and classification of electron correlation effects among MOs. The dissocn. behavior of some prototypical diat. mols. features all types of correlation effects relevant for chem. bonding. We demonstrate that our entanglement anal. is convenient to dissect these electron correlation effects and to provide a conceptual understanding of bond-forming and bond-breaking processes from the point of view of quantum information theory.
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  196. 196
    Huang, Z.; Kais, S. Entanglement as measure of electron–electron correlation in quantum chemistry calculations. Chem. Phys. Lett. 2005, 413, 1– 5,  DOI: 10.1016/j.cplett.2005.07.045
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    Entanglement as measure of electron-electron correlation in quantum chemistry calculations
    Huang, Zhen; Kais, Sabre
    Chemical Physics Letters (2005), 413 (1-3), 1-5CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)
    In quantum chem. calcns., the correlation energy is defined as the difference between the Hartree-Fock limit energy and the exact soln. of the nonrelativistic Schroedinger equation. With this definition, the electron correlation effects are not directly observable. In this report, we show that the entanglement can be used as an alternative measure of the electron correlation in quantum chem. calcns. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics. As an example we calc. the entanglement for He atom and H2 mol. with different basis sets.
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    Boguslawski, K.; Tecmer, P.; Legeza, O.; Reiher, M. Entanglement measures for single-and multireference correlation effects. J. Phys. Chem. Lett. 2012, 3, 3129– 3135,  DOI: 10.1021/jz301319v
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    Entanglement Measures for Single- and Multireference Correlation Effects
    Boguslawski, Katharina; Tecmer, Pawel; Legeza, Ors; Reiher, Markus
    Journal of Physical Chemistry Letters (2012), 3 (21), 3129-3135CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
    Electron correlation effects are essential for an accurate ab initio description of mols. A quant. a priori knowledge of the single- or multireference nature of electronic structures as well as of the dominant contributions to the correlation energy can facilitate the decision regarding the optimum quantum chem. method of choice. We propose concepts from quantum information theory as orbital entanglement measures that allow us to evaluate the single- and multireference character of any mol. structure in a given orbital basis set. By studying these measures we can detect possible artifacts of small active spaces.
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    The correlation present in the nondegenerate ground state of an interacting Fermi system is discussed in terms of reduced d. matrixes and their cumulant expansion. By generalizing a result obtained for the interacting uniform electron gas (correlation induced exchange-hole narrowing), possible measures of the correlation strength in terms of natural occupation nos. (the eigenvalues of the true one-particle d. matrix) are introduced. These quantities, the ν-order nonidempotency and the information entropy of the natural occupation nos., result from the correlated many-body wave function and characterize the ground-state correlation in addn. to the usual correlation energy. The uniform electron gas serves as a first illustrative example.
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    Ghosh, K. J.; Kais, S.; Herschbach, D. R. Geometrical picture of the electron–electron correlation at the large-D limit. Phys. Chem. Chem. Phys. 2022, 24, 9298– 9307,  DOI: 10.1039/D2CP00438K
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    Geometrical picture of the electron-electron correlation at the large-D limit
    Ghosh, Kumar J. B.; Kais, Sabre; Herschbach, Dudley R.
    Physical Chemistry Chemical Physics (2022), 24 (16), 9298-9307CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    In electronic structure calcns., the correlation energy is defined as the difference between the mean field and the exact soln. of the non relativistic Schroedinger equation. Such an error in the different calcns. is not directly observable as there is no simple quantum mech. operator, apart from correlation functions, that correspond to such quantity. Here, we use the dimensional scaling approach, in which the electrons are localized at the large-dimensional scaled space, to describe a geometric picture of the electronic correlation. Both, the mean field, and the exact solns. at the large-D limit have distinct geometries. Thus, the difference might be used to describe the correlation effect. Moreover, correlations can be also described and quantified by the entanglement between the electrons, which is a strong correlation without a classical analog. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics and bounded by the area law for local gapped Hamiltonians of interacting many-body systems. This study opens up the possibility of presenting a geometrical picture of the electron-electron correlations and might give a bound on the correlation energy. The results at the large-D limit and at D = 3 indicate the feasibility of using the geometrical picture to get a bound on the electron-electron correlations.
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    Rissler, J.; Noack, R. M.; White, S. R. Measuring orbital interaction using quantum information theory. Chem. Phys. 2006, 323, 519– 531,  DOI: 10.1016/j.chemphys.2005.10.018
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    Quantum information theory gives rise to a straightforward definition of the interaction of electrons Ip,q in two orbitals p,q for a given many-body wave function. A convenient way to calc. the von Neumann entropies needed is presented in this work, and the orbital interaction Ip,q is successfully tested for different types of chem. bonds. As an example of an application of Ip,q beyond the interpretation of wave functions, Ip,q is then used to investigate the ordering problem in the d.-matrix renormalization group.
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    Cao, Y.; Romero, J.; Olson, J. P.; Degroote, M.; Johnson, P. D.; Kieferová, M.; Kivlichan, I. D.; Menke, T.; Peropadre, B.; Sawaya, N. P. D.; Sim, S.; Veis, L.; Aspuru-Guzik, A. Quantum Chemistry in the Age of Quantum Computing. Chem. Rev. 2019, 119, 10856– 10915,  DOI: 10.1021/acs.chemrev.8b00803
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    Chemical Reviews (Washington, DC, United States) (2019), 119 (19), 10856-10915CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
    A review. Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chem. communities over the past century. Although many approxn. methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging complexity landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chem. such as the electronic structure of mols. In the past two decades significant advances have been made in developing algorithms and phys. hardware for quantum computing, heralding a revolution in simulation of quantum systems. This article is an overview of the algorithms and results that are relevant for quantum chem. The intended audience is both quantum chemists who seek to learn more about quantum computing, and quantum computing researchers who would like to explore applications in quantum chem.
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    Reiher, M.; Wiebe, N.; Svore, K. M.; Wecker, D.; Troyer, M. Elucidating reaction mechanisms on quantum computers. Proc. Nat. Acad. Sci. 2017, 114, 7555– 7560,  DOI: 10.1073/pnas.1619152114
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    Elucidating reaction mechanisms on quantum computers
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    Proceedings of the National Academy of Sciences of the United States of America (2017), 114 (29), 7555-7560CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
    With rapid recent advances in quantum technol., we are close to the threshold of quantum devices whose computational powers can exceed those of classical supercomputers. Here, we show that a quantum computer can be used to elucidate reaction mechanisms in complex chem. systems, using the open problem of biol. nitrogen fixation in nitrogenase as an example. We discuss how quantum computers can augment classical computer simulations used to probe these reaction mechanisms, to significantly increase their accuracy and enable hitherto intractable simulations. Our resource ests. show that, even when taking into account the substantial overhead of quantum error correction, and the need to compile into discrete gate sets, the necessary computations can be performed in reasonable time on small quantum computers. Our results demonstrate that quantum computers will be able to tackle important problems in chem. without requiring exorbitant resources.
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    Tubman, N. M.; Mejuto-Zaera, C.; Epstein, J. M.; Hait, D.; Levine, D. S.; Huggins, W.; Jiang, Z.; McClean, J. R.; Babbush, R.; Head-Gordon, M.; Whaley, K. B. Postponing the orthogonality catastrophe: efficient state preparation for electronic structure simulations on quantum devices. arXiv:1809.05523 2018, DOI: 10.48550/arXiv.1809.05523 .
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    Carbone, A.; Galli, D. E.; Motta, M.; Jones, B. Quantum circuits for the preparation of spin eigenfunctions on quantum computers. Symmetry 2022, 14, 624,  DOI: 10.3390/sym14030624
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    Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers
    Carbone, Alessandro; Galli, Davide Emilio; Motta, Mario; Jones, Barbara
    Symmetry (2022), 14 (3), 624CODEN: SYMMAM; ISSN:2073-8994. (MDPI AG)
    The application of quantum algorithms to the study of many-particle quantum systems requires the ability to prep. wave functions that are relevant in the behavior of the system under study. Hamiltonian symmetries are important instruments used to classify relevant many-particle wave functions and to improve the efficiency of numerical simulations. In this work, quantum circuits for the exact and approx. prepn. of total spin eigenfunctions on quantum computers are presented. Two different strategies are discussed and compared: exact recursive construction of total spin eigenfunctions based on the addn. theorem of angular momentum, and heuristic approxn. of total spin eigenfunctions based on the variational optimization of a suitable cost function. The construction of these quantum circuits is illustrated in detail, and the prepn. of total spin eigenfunctions is demonstrated on IBM quantum devices, focusing on three- and five-spin systems on graphs with triangle connectivity.
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    Lacroix, D.; Ruiz Guzman, E. A.; Siwach, P. Symmetry breaking/symmetry preserving circuits and symmetry restoration on quantum computers. Eur. Phys. J. A 2023, 59, 3,  DOI: 10.1140/epja/s10050-022-00911-7
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    Symmetry breaking/symmetry preserving circuits and symmetry restoration on quantum computers - A quantum many-body perspective
    Lacroix, Denis; Ruiz Guzman, Edgar Andres; Siwach, Pooja
    European Physical Journal A: Hadrons and Nuclei (2023), 59 (1), 3CODEN: EPJAFV; ISSN:1434-601X. (Springer International Publishing AG)
    Abstr.: We discuss here some aspects related to the symmetries of a quantum many-body problem when trying to treat it on a quantum computer. Several features related to symmetry conservation, symmetry breaking, and possible symmetry restoration are reviewed. After briefly discussing some of the std. symmetries relevant for many-particle systems, we discuss the advantage of encoding some symmetries directly in quantum ansatze, esp. to reduce the quantum register size. It is, however, well-known that the use of symmetry-breaking states can also be a unique way to incorporate specific internal correlations when a spontaneous symmetry breaking occurs. These aspects are discussed in the quantum computing context. Ultimately, an accurate description of quantum systems can be achieved only when the initially broken symmetries are properly restored. We review several methods explored previously to perform symmetry restoration on a quantum computer, for instance, the ones based on symmetry filtering by quantum phase estn. and by an iterative independent set of Hadamard tests. We propose novel methods that pave the new directions to perform symmetry restoration, like those based on the purifn. of the state employing the linear combination of unitaries (LCU) approach.
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    Dobrautz, W.; Smart, S. D.; Alavi, A. Efficient formulation of full configuration interaction quantum Monte Carlo in a spin eigenbasis via the graphical unitary group approach. J. Chem. Phys. 2019, 151, 094104,  DOI: 10.1063/1.5108908
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    Efficient formulation of full configuration interaction quantum Monte Carlo in a spin eigenbasis via the graphical unitary group approach
    Dobrautz, Werner; Smart, Simon D.; Alavi, Ali
    Journal of Chemical Physics (2019), 151 (9), 094104/1-094104/33CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We provide a spin-adapted formulation of the Full CI Quantum Monte Carlo (FCIQMC) algorithm, based on the Graphical Unitary Group Approach (GUGA), which enables the exploitation of SU(2) symmetry within this stochastic framework. Random excitation generation and matrix element calcn. on the Shavitt graph of GUGA can be efficiently implemented via a biasing procedure on the branching diagram. The use of a spin-pure basis explicitly resolves the different spin-sectors and ensures that the stochastically sampled wavefunction is an eigenfunction of the total spin operator ̂S2. The method allows for the calcn. of states with low or intermediate spin in systems dominated by Hund's first rule, which are otherwise generally inaccessible. Furthermore, in systems with small spin gaps, the new methodol. enables much more rapid convergence with respect to walker no. and simulation time. Some illustrative applications of the GUGA-FCIQMC method are provided: computation of the 2F - 4F spin gap of the cobalt atom in large basis sets, achieving chem. accuracy to expt., and the 1Σg+, 3Σg+, 5Σg+, and 7Σg+ spin-gaps of the stretched N2 mol., an archetypal strongly correlated system. (c) 2019 American Institute of Physics.
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    McLean, A. D.; Lengsfield, B. H., III; Pacansky, J.; Ellinger, Y.
    Journal of Chemical Physics (1985), 83 (7), 3567-76CODEN: JCPSA6; ISSN:0021-9606.
    A systematic approach is given to symmetry breaking in mol. calcns., based on MCSCF and multiref. CI (MRCI) wave functions. A series of MCSCF expansions is generated by successively incorporating resonance effects and size effects into the wave functions. The character of the potential surface obtained at each level is analyzed. As an example, the potential energy curves of the ground state (σ) and the 1st excited state (π) of the HCO2 radical are characterized. The σ and π equil. structures are sym., with an adiabatic σ-π excitation energy of 9.2 kcal/mol. Unlike earlier theor. studies, present MCSCF model produces a qual. correct potential surface. Reliable vibrational frequencies are calcd. from the MRCI potential surface.
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    Ayala, P. Y.; Schlegel, H. B. A nonorthogonal CI treatment of symmetry breaking in sigma formyloxyl radical. J. Chem. Phys. 1998, 108, 7560– 7567,  DOI: 10.1063/1.476190
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    Ayala, Philippe Y.; Schlegel, H. Bernhard
    Journal of Chemical Physics (1998), 108 (18), 7560-7567CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Spatial symmetry breaking can occur in Hartree-Fock (HF) wavefunctions when there are ≥2 close-lying configurations that can mix strongly, such as in HCO2•, NO2 and allyl radical. Like spin contamination, spatial symmetry breaking can cause sizeable errors when perturbation theory is used to est. the correlation energy. With conventional methodol., very large MCSCF and MRCI calcns. are necessary to overcome the spatial symmetry-breaking problem. This paper explores an alternative approach in which a 2×2 nonorthogonal CI is used to recombine the 2 symmetry broken HF determinants. The necessary matrix elements closely resemble those used in spin-projection calcns. Second-order perturbation theory is used to include electron correlation energy in this approach. With perturbative corrections for correlation energy, this approach predicts that the 2B2 structure is a min., in agreement with the best available calcns.
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    Manne, Rolf
    Molecular Physics (1972), 24 (5), 935-44CODEN: MOPHAM; ISSN:0026-8976.
    The underlying assumptions of Roothaan's symmetry-restricted SCF method for open-shell systems are considered. A restricted Brillouin theorem is formulated and applied in a discussion of total energy discontinuities arising in restricted SCF calcns. of systems which exhibit Jahn-Teller instability, such as the tetrahedral 2T2 state of CH4+.
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    Davidson, Ernest R.; Borden, Weston Thatcher
    Journal of Physical Chemistry (1983), 87 (24), 4783-90CODEN: JPCHAX; ISSN:0022-3654.
    A review is presented with over 40 refs. of recent work by the authors and their collaborators on mols. with broken symmetry.
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    Čížek, J.; Paldus, J. Stability Conditions for the Solutions of the Hartree–Fock Equations for Atomic and Molecular Systems. Application to the Pi-Electron Model of Cyclic Polyenes. J. Chem. Phys. 1967, 47, 3976– 3985,  DOI: 10.1063/1.1701562
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    Stability conditions for the solutions of the Hartree-Fock equations for atomic and molecular systems. Application to the π-electron model of cyclic polyenes
    Cizek, Jiri; Paldus, Josef
    Journal of Chemical Physics (1967), 47 (10), 3976-85CODEN: JCPSA6; ISSN:0021-9606.
    The stability conditions which ensure that the Hartree-Fock determinant minimizes the energy expectation value are rederived by using the language familiar in quantum chemistry. These stability conditions are then specified for the case of closed-shell electronic systems which allow addnl. simplification of the conditions as well as a certain classification of the instabilities. Examples of the instabilities of different types are presented and the case of the so-called singlet instabilities (most interesting from the phys. point of view) is studied in detail for the π-electron model of cyclic polyenes.
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    Čížek, J.; Paldus, J. Stability Conditions for the Solutions of the Hartree–Fock Equations for Atomic and Molecular Systems. III. Rules for the Singlet Stability of Hartree–Fock Solutions of π-Electronic Systems. J. Chem. Phys. 1970, 53, 821– 829,  DOI: 10.1063/1.1674065
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    Cizek, Jiri; Paldus, Josef
    Journal of Chemical Physics (1970), 53 (2), 821-9CODEN: JCPSA6; ISSN:0021-9606.
    The singlet stability conditions for closed-shell electronic systems, which ensure that the Hartree-Fock (H.F.) determinant with doubly occupied orbitals minimizes the energy expectation value, are applied to the symmetry adapted H.F. solns. of linear polyacenes, by using the Pariser-Parr-Pople-type semiempirical Hamiltonian. The symmetry adapted H.F. solns. for linear polyacenes contg. an even no. of benzene rings are always singlet stable, while the H.F. solns. for linear polyacenes having an odd no. of benzene rings may exhibit singlet instability if the coupling const. is large enough. For cases where singlet instability was found, new H.F. solns. having lower energy than the symmetry-adapted H.F. solns. were calcd. These new H.F. solns. violate the space symmetry conservation laws as usual. Furthermore, the qual. rules for the existence of singlet stability of the symmetry adapted H.F. soln. of π-electronic systems with conjugated double bonds are derived. These rules are formulated through the simple symmetry properties of possible Kekule structures of the system. These rules are used to explain the results of stability calcns. for linear polyacenes as well as further illustrated on other examples.
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    Restricted open-shell Hartree-Fock (ROHF) theory is formulated as a projected self-consistent unrestricted HF (UHF) model by math. constraining spin d. eigenvalues. This constrained UHF (CUHF) wave function is identical to that obtained from Roothaan's effective Fock operator. The α and β CUHF Fock operators are parameter-free and have eigenvalues (orbital energies) that are phys. meaningful as in UHF, except for eliminating spin contamination. This new way of solving ROHF leads to orbitals that turn out to be identical to semicanonical orbitals. The present approach removes ambiguities in ROHF orbital energies. (c) 2010 American Institute of Physics.
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    Journal of Physical Chemistry (1988), 92 (11), 3033-6CODEN: JPCHAX; ISSN:0022-3654.
    To circumvent the problem of spin contamination in UHF based coupled cluster calcns., a new method of calcn. is given for certain classes of open-shell systems. The approach ensures that the proper spin component of the resulting correlated wave function is projected out in the energy evaluation by the use of a ref. function constructed from suitably chosen restricted open-shell Hartree-Fock or closed-shell Hartree-Fock orbitals. This single-ref. open-shell spin-restricted CC method is applied to the calcn. of ionization potentials in the N2 mol., and it is shown that highly accurate results can be obtained in a 5s4p1d basis. The mean error for all the principal ionization potentials of N2 compared to expt. is 0.45%.
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    Neese, F. Importance of direct spin-spin coupling and spin-flip excitations for the zero-field splittings of transition metal complexes: A case study. J. Am. Chem. Soc. 2006, 128, 10213– 10222,  DOI: 10.1021/ja061798a
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    Importance of Direct Spin-Spin Coupling and Spin-Flip Excitations for the Zero-Field Splittings of Transition Metal Complexes: A Case Study
    Neese, Frank
    Journal of the American Chemical Society (2006), 128 (31), 10213-10222CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    This work reports the evaluation of several theor. approaches to the zero-field splitting (ZFS) in transition metal complexes. The exptl. well-known complex [Mn(acac)3] is taken as an example. The direct spin-spin contributions to the ZFS have been calcd. on the basis of d. functional theory (DFT) or complete active space SCF (CASSCF) wave functions and have been found to be much more important than previously assumed. The contributions of the direct term may exceed ∼1 cm-1 in magnitude and therefore cannot be neglected in any treatment that aims at a realistic quant. modeling of the ZFS. In the DFT framework, two different variants to treat the spin-orbit coupling (SOC) term have been evaluated. The first approach is based on previous work by Pederson, Khanna, and Kortus, and the second is based on a "quasi-restricted" DFT treatment which is rooted in our previous work on ZFS. Both approaches provide very similar results and underestimate the SOC contribution to the ZFS by a factor of 2 or more. The SOC is represented by an accurate multicenter spin-orbit mean-field (SOMF) approxn. which is compared to the popular effective DFT potential-derived SOC operator. In addn. to the DFT results, direct "infinite order" ab initio calcns. of the SOC contribution to the ZFS based on CASSCF wave functions, the spectroscopy-oriented CI (SORCI), and the difference-dedicated CI (DDCI) approach are reported. In general, the multireference ab initio results provide a more realistic description of the ZFS in [Mn(acac)3]. The conclusions likely carry over to many other systems. This is attributed to the explicit treatment of the multiplet effects which are of dominant importance, since the calcns. demonstrate that, even in the high-spin d4 system Mn(III), the spin-flip excitations make the largest contribution to the SOC. It is demonstrated that the ab initio methods can be used even for somewhat larger mols. (the present calcns. were done with more than 500 basis functions) in a reasonable time frame. Much more economical but still fairly reasonable results have been achieved with the INDO/S treatment based on CASSCF and SOC-CI wave functions.
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    Spin-flip methods in quantum chemistry
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    Physical Chemistry Chemical Physics (2020), 22 (8), 4326-4342CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    A review. This Perspective discusses salient features of the spin-flip approach to strong correlation and describes different methods that sprung from this idea. The spin-flip treatment exploits the different physics of low-spin and high-spin states and is based on the observation that correlation is small for same-spin electrons. By using a well-behaved high-spin state as a ref., one can access problematic low-spin states by deploying the same formal tools as in the excited-state treatments (i.e., linear response, propagator, or equation-of-motion theories). The Perspective reviews applications of this strategy within wave function and d. functional theory frameworks as well as the extensions for mol. properties and spectroscopy. The utility of spin-flip methods is illustrated by examples. Limitations and proposed future directions are also discussed.
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    Roemelt, M.; Neese, F. Excited states of large open-shell molecules: an efficient, general, and spin-adapted approach based on a restricted open-shell ground state wave function. J. Phys. Chem. A 2013, 117, 3069– 3083,  DOI: 10.1021/jp3126126
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    Excited States of Large Open-Shell Molecules: An Efficient, General, and Spin-Adapted Approach Based on a Restricted Open-Shell Ground State Wave function
    Roemelt, Michael; Neese, Frank
    Journal of Physical Chemistry A (2013), 117 (14), 3069-3083CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    A spin-adapted CI with singles method that is based on a restricted open-shell ref. function (ROCIS) with general total spin S is presented. All excited configuration state functions (CSFs) are generated with the aid of a spin-free second quantization formalism that only leads to CSFs within the first order interacting space. By virtue of the CSF construction, the formalism involves higher than singly excited determinants but not higher than singly excited configurations. Matrix elements between CSFs are evaluated on the basis of commutator relationships using a symbolic algebra program. The final equations were, however, hand-coded in order to maximize performance. The method can be applied to fairly large systems with more than 100 atoms in reasonable wall-clock times and also parallelizes well. Test calcns. demonstrate that the approach is far superior to UHF-based CI with single excitations but necessarily falls somewhat short of quant. accuracy due to the lack of dynamic correlation contributions. In order to implicitly account for dynamic correlation in a crude way, the program optionally allows for the use of Kohn-Sham orbitals in combination with a modest downscaling of two-electron integrals (DFT/ROCIS). All two-electron integrals of Kohn-Sham orbitals that appear in the Hamiltonian matrix are reduced by a total of three scaling parameters that are suitable for a wide range of mols. Test calcns. on open-shell org. radicals as well as transition metal complexes demonstrate the wide applicability of the method and its ability to calc. the electronic spectra of large mol. systems.
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    Izsák, R. Second quantisation for unrestricted references: formalism and quasi-spin-adaptation of excitation and spin-flip operators. Mol. Phys. 2022, e2126802,  DOI: 10.1080/00268976.2022.2126802
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    Veis, L.; Antalik, A.; Legeza, O.; Alavi, A.; Pittner, J. The intricate case of tetramethyleneethane: A full configuration interaction quantum Monte Carlo benchmark and multireference coupled cluster studies. J. Chem. Theory Comput. 2018, 14, 2439– 2445,  DOI: 10.1021/acs.jctc.8b00022
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    The Intricate Case of Tetramethyleneethane: A Full Configuration Interaction Quantum Monte Carlo Benchmark and Multireference Coupled Cluster Studies
    Veis, Libor; Antalik, Andrej; Legeza, Ors; Alavi, Ali; Pittner, Jiri
    Journal of Chemical Theory and Computation (2018), 14 (5), 2439-2445CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We have performed a full CI (FCI) quality benchmark calcn. for the tetramethyleneethane mol. in the cc-pVTZ basis set employing a subset of complete active space second order perturbation theory, CASPT2(6,6), natural orbitals for the FCI quantum Monte Carlo calcn. The results are in an excellent agreement with the previous large scale diffusion Monte Carlo calcns. by Pozun et al. and available exptl. results. Our computations verified that there is a max. on the potential energy surface (PES) of the ground singlet state (1A) 45° torsional angle, and the corresponding vertical singlet-triplet energy gap is 0.01 eV. We have employed this benchmark for the assessment of the accuracy of Mukherjee's coupled clusters with up to triple excitations (MkCCSDT) and CCSD tailored by the d. matrix renormalization group method (DMRG). Multireference MkCCSDT with CAS(2,2) model space, though giving good values for the singlet-triplet energy gap, is not able to properly describe the shape of the multireference singlet PES. Similarly, DMRG(24,25) is not able to correctly capture the shape of the singlet surface, due to the missing dynamic correlation. On the other hand, the DMRG-tailored CCSD method describes the shape of the ground singlet state with excellent accuracy but for the correct ordering requires computation of the zero-spin-projection component of the triplet state (3B1).
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    Sears, J. S.; Sherrill, C. D. Assessing the Performance of Density Functional Theory for the Electronic Structure of Metal-Salens: The 3d0-Metals. J. Phys. Chem. A 2008, 112, 3466– 3477,  DOI: 10.1021/jp711595w
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    Assessing the Performance of Density Functional Theory for the Electronic Structure of Metal-Salens: The 3d0-Metals
    Sears, John S.; Sherrill, C. David
    Journal of Physical Chemistry A (2008), 112 (15), 3466-3477CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    A series of metal-salen complexes of the 3d0 metals Sc(III), Ti(IV), V(V), Cr(VI), and Mn(VII) have been explored using high-level electronic structure methods including coupled-cluster theory with singles, doubles, and perturbative triples as well as complete active-space third-order perturbation theory. The performance of three common d. functional theory approaches has been assessed for both the geometries and the relative energies of the low-lying electronic states. The nondynamical correlation effects are demonstrated to be extremely large in all of the systems examd. Although d. functional theory provides reasonable results for some of the systems, the overall agreement is quite poor. This said, the d. functional theory approaches are shown to outperform the single-ref. perturbation theory and coupled-cluster theory approaches for cases of strong nondynamical correlation.
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    Sears, J. S.; Sherrill, C. D. Assessing the performance of Density Functional Theory for the electronic structure of metal-salens: the d2-metals. J. Phys. Chem. A 2008, 112, 6741– 6752,  DOI: 10.1021/jp802249n
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    Assessing the Performance of Density Functional Theory for the Electronic Structure of Metal-Salens: The d2-Metals
    Sears, John S.; Sherrill, C. David
    Journal of Physical Chemistry A (2008), 112 (29), 6741-6752CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    The performance of three common combinations of d. functional theory has been evaluated for the geometries and relative energies of a commonly-employed model complex of the salen ligand [salen = bis(salicylaldehydo)ethylenediamine] with the d2-metals Ti(II), V(III), Cr(IV), Zr(II), Nb(III), and Mo(IV). High-level ab initio methods including complete active-space third-order perturbation theory have been employed both as benchmarks for the d. functional theory results and to examine the multireference character of the low-lying electronic states in these systems. The strong multireference character of the systems has been clearly demonstrated. All of the functionals examd. provide geometries that are typically within 0.2 Å least root mean square deviation from the benchmark geometries. The performance of the d. functionals for the relative energies of the low-lying electronic states is significantly worse, providing qual. different descriptions in some instances. Of the systems explored, no significant difference is obsd. in the multireference character or in the reliability of the d. functional results when comparing 3d vs 4d transition-metal systems.
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    Olivares-Amaya, R.; Hu, W.; Nakatani, N.; Sharma, S.; Yang, J.; Chan, G. K.-L. The ab-initio density matrix renormalization group in practice. J. Chem. Phys. 2015, 142, 034102,  DOI: 10.1063/1.4905329
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    The ab-initio density matrix renormalization group in practice
    Olivares-Amaya, Roberto; Hu, Weifeng; Nakatani, Naoki; Sharma, Sandeep; Yang, Jun; Chan, Garnet Kin-Lic
    Journal of Chemical Physics (2015), 142 (3), 034102/1-034102/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The ab-initio d. matrix renormalization group (DMRG) is a tool that can be applied to a wide variety of interesting problems in quantum chem. Here, we examine the d. matrix renormalization group from the vantage point of the quantum chem. user. What kinds of problems is the DMRG well-suited to. What are the largest systems that can be treated at practical cost. What sort of accuracies can be obtained, and how do we reason about the computational difficulty in different mols.. By examg. a diverse benchmark set of mols.: π-electron systems, benchmark main-group and transition metal dimers, and the Mn-oxo-salen and Fe-porphine organometallic compds., we provide some answers to these questions, and show how the d. matrix renormalization group is used in practice. (c) 2015 American Institute of Physics.
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    Pandharkar, R.; Hermes, M. R.; Cramer, C. J.; Gagliardi, L. Localized Active Space-State Interaction: a Multireference Method for Chemical Insight. J. Chem. Theory Comput. 2022, 18, 6557– 6566,  DOI: 10.1021/acs.jctc.2c00536
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    Localized Active Space-State Interaction: a Multireference Method for Chemical Insight
    Pandharkar, Riddhish; Hermes, Matthew R.; Cramer, Christopher J.; Gagliardi, Laura
    Journal of Chemical Theory and Computation (2022), 18 (11), 6557-6566CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Multireference electronic structure methods, like the complete active space (CAS) SCF model, have long been used to characterize chem. interesting processes. Important work has been done in recent years to develop modifications having a lower computational cost than CAS, but typically these methods offer no more chem. insight than that from the CAS soln. being approximated. In this paper, we present the localized active space-state interaction (LASSI) method that can be used not only to lower the intrinsic cost of the multireference calcn. but also to improve interpretability. The localized active space (LAS) approach utilizes the local nature of the electron-electron correlation to express a composite wave function as an antisymmetrized product of unentangled wave functions in local active subspaces. LASSI then uses these LAS states as a basis from which to express complete mol. wave functions. This not only makes the mol. wave function more compact but also permits flexibility in choosing those states to be included in the basis. Such selective inclusion of states translates to the selective inclusion of specific types of interactions, thereby allowing a quant. anal. of these interactions. We demonstrate the use of LASSI to study charge migration and spin-flip excitations in multireference org. mols. We also compute the J coupling parameter for a bimetallic compd. using various LAS bases to construct the Hamiltonian to provide insights into the coupling mechanism.
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    Máté, M.; Petrov, K.; Szalay, S.; Legeza, Ö. Compressing multireference character of wave functions via fermionic mode optimization. J. Math. Chem. 2023, 61, 362– 375,  DOI: 10.1007/s10910-022-01379-y
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    Compressing multireference character of wave functions via fermionic mode optimization
    Mate, Mihaly; Petrov, Klara; Szalay, Szilard; Legeza, Ors
    Journal of Mathematical Chemistry (2023), 61 (2), 362-375CODEN: JMCHEG; ISSN:0259-9791. (Springer)
    Abstr.: In this work, we present a brief overview of the fermionic mode optimization within the framework of tensor network state methods (Krumnow et al. in Phys Rev Lett 117:210402, 2016, https://doi.org/10.1103/PhysRevLett.117.210402), and demonstrate that it has the potential to compress the multireference character of the wave functions after finding optimal MOs (modes), based on entanglement minimization. Numerical simulations have been performed for the nitrogen dimer in the cc-pVDZ basis for the equil. and for stretched geometries.
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    Smith, J. E. T.; Mussard, B.; Holmes, A. A.; Sharma, S. Cheap and Near Exact CASSCF with Large Active Spaces. J. Chem. Theory Comput. 2017, 13, 5468– 5478,  DOI: 10.1021/acs.jctc.7b00900
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    Cheap and Near Exact CASSCF with Large Active Spaces
    Smith, James E. T.; Mussard, Bastien; Holmes, Adam A.; Sharma, Sandeep
    Journal of Chemical Theory and Computation (2017), 13 (11), 5468-5478CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We use the recently-developed Heat-bath CI (HCI) algorithm as an efficient active-space solver to perform multi-configuration SCF calcns. (HCISCF) with large active spaces. We give a detailed derivation of the theory and show that difficulties assocd. with non-variationality of the HCI procedure can be overcome by making use of the Lagrangian formulation to calc. the HCI relaxed two body reduced d. matrix. HCISCF is then used to study the electronic structure of butadiene, pentacene, and Fe-porphyrin. One of the most striking results of our work is that the converged active space orbitals obtained from HCISCF are relatively insensitive to the accuracy of the HCI calcn. This allows us to obtain nearly converged CASSCF energies with an estd. error of less than 1 mHa using the orbitals obtained from the HCISCF procedure in which the integral transformation is the dominant cost. For example, an HCISCF calcn. on Fe-Porphyrin model complex with an active space of (44e, 44o) took only 412 s per iteration on a single node contg. 28 cores, out of which 185 s were spent in the HCI calcn. and the remaining 227 s were mainly used for integral transformation. Finally, we also show that active-space orbitals can be optimized using HCISCF to substantially speed up the convergence of the HCI energy to the Full CI limit because HCI is not invariant to unitary transformations within the active space.
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    Dobrautz, W.; Weser, O.; Bogdanov, N. A.; Alavi, A.; Li Manni, G. Spin-Pure Stochastic-CASSCF via GUGA-FCIQMC Applied to Iron–Sulfur Clusters. J. Chem. Theory Comput. 2021, 17, 5684– 5703,  DOI: 10.1021/acs.jctc.1c00589
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    Spin-Pure Stochastic-CASSCF via GUGA-FCIQMC Applied to Iron-Sulfur Clusters
    Dobrautz, Werner; Weser, Oskar; Bogdanov, Nikolay A.; Alavi, Ali; Li Manni, Giovanni
    Journal of Chemical Theory and Computation (2021), 17 (9), 5684-5703CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    In this work, we demonstrate how to efficiently compute the one- and two-body reduced d. matrixes within the spin-adapted full CI quantum Monte Carlo (FCIQMC) method, which is based on the graphical unitary group approach (GUGA). This allows us to use GUGA-FCIQMC as a spin-pure CI eigensolver within the complete active space SCF (CASSCF) procedure and hence to stochastically treat active spaces far larger than conventional CI solvers while variationally relaxing orbitals for specific spin-pure states. We apply the method to investigate the spin ladder in iron-sulfur dimer and tetramer model systems. We demonstrate the importance of the orbital relaxation by comparing the Heisenberg model magnetic coupling parameters from the CASSCF procedure to those from a CI-only (CASCI) procedure based on restricted open-shell Hartree-Fock orbitals. We show that the orbital relaxation differentially stabilizes the lower-spin states, thus enlarging the coupling parameters with respect to the values predicted by ignoring orbital relaxation effects. Moreover, we find that, while CASCI results are well fit by a simple bilinear Heisenberg Hamiltonian, the CASSCF eigenvalues exhibit deviations that necessitate the inclusion of biquadratic terms in the model Hamiltonian.
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    Dobrautz, W.; Katukuri, V. M.; Bogdanov, N. A.; Kats, D.; Li Manni, G.; Alavi, A. Combined unitary and symmetric group approach applied to low-dimensional Heisenberg spin systems. Phys. Rev. B 2022, 105, 195123,  DOI: 10.1103/PhysRevB.105.195123
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    Combined unitary and symmetric group approach applied to low-dimensional Heisenberg spin systems
    Dobrautz, Werner; Katukuri, Vamshi M.; Bogdanov, Nikolay A.; Kats, Daniel; Li Manni, Giovanni; Alavi, Ali
    Physical Review B (2022), 105 (19), 195123CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    A novel combined unitary and sym. group approach is used to study the spin-12 Heisenberg model and related Fermionic systems in a total spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground-state wave function representation-indicated by a larger leading ground-state coeff.-is obtained when combining the sym. group Sn, in the form of permutations of the underlying lattice site ordering, with the cumulative spin coupling based on the unitary group U(n). In one-dimensional systems the obsd. compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full CI quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest wt. already captures with high accuracy the spin-spin correlation function of the exact ground-state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chem. models, exemplified by one-dimensional hydrogen chains. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equiv. to the optimal ordering obtained for methods based on matrix-product states, such as the d.-matrix renormalization group approach.
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    Li Manni, G.; Kats, D.; Liebermann, N. Resolution of Electronic States in Heisenberg Cluster Models within the Unitary Group Approach. ChemRxiv 2022, DOI: 10.26434/chemrxiv-2022-rfmhk-v2 .
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    A review. This paper first comments on the surprisingly poor status that Quantum Chem. has offered to the fantastic intuition of Lewis concerning the distribution of the electrons in the mol. Then, it advocates in favor of a hierarchical description of the mol. wave-function, distinguishing the physics taking place in the valence space (in the bond and between the bonds), and the dynamical correlation effects. It is argued that the clearest pictures of the valence electronic population combine two localized views, namely the bond (and lone pair) MOs and the Valence Bond decompn. of the wave-function, preferably in the orthogonal version directly accessible from the complete active space self consistent field method. Such a reading of the wave function enables one to understand the work of the nondynamical correlation as an enhancement of the wt. of the low-energy VB components, i.e. as a better compromise between the electronic delocalization and the energetic preferences of the atoms. It is suggested that regarding the bond building, the leading dynamical correlation effect may be the dynamical polarization phenomenon. It is shown that most correlation effects do not destroy the bond electron pairs and remain compatible with Lewis' vision. A certain no. of free epistemol. considerations have been introduced in the development of the argument.
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    cf. C. A. 71 3865 and Bray and Branch, C. A. 7, 3865. Compds. should be classed as polar and nonpolar rather than inorg. and org. These classes are roughly the same. A nonpolar mol. is one in which the electrons belonging to the individual atom are held by such restraints that they do not move far from their normal positions, while in the polar mols. the electrons, being more mobile, so move as to sep. the mol. into positive and negative parts. In an extremely polar mol. such as NaCl it is probable that in the great majority of the mols. the Cl atom has acquired a unit negative charge and therefore the Na atom a unit positive charge, and the process of ionization probably consists only in a further sepn. of these charged parts. If a weakly polar mol. comes into the neighborhood of a more polar one it becomes itself more polar. In this process the weaker bipole stretches and its moment increases. A "cubical atom" is proposed as a basis of a new theory of atomic structure. Thus Li is a cube with a single electron on one corner, Be has 2 electrons, B 3, C 4, N 5, O 6, and F 7. This view is in harmony with the theory developed by Parson, C. A. 10, 406. An atom is considered as having an unalterable kernel which possesses an excess of positive charges corresponding in number to the ordinal number of the group in the periodic table to which the element belongs (cf. Thomson, C. A. 8, 824). There is a shell of electrons around the kernel which, in the case of a neutral atom, contains negative electrons equal in number to the excess of positive charges of the kernel, but the number of electrons in the shell may vary during chem. changes between zero and 8. The atom tends to hold an even number of electrons in the shell (especially 8 at the corners of the cube) but the electrons may ordinarily pass from one position to another in this shell. Two atomic shells are mutually interpenetrable. The paper is a discussion of these ideas applied to the structure of atoms and compds.
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    A review. Transition metal chem. is a challenging playground for quantum chem. methods owing to the simultaneous presence of static and dynamic electron correlation effects in many systems. Wavefunction based multireference (MR) methods constitute a phys. sound and systematically improvable Ansatz to deal with this complexity but suffer from some conceptual difficulties and high computational costs. The latter problem partially arises from the unfavorable scaling of the Full CI (Full-CI) problem which in the majority of MR methods is solved for a subset of the MO space, the so-called active space. In the last years multiple methods such as modern variants of selected CI, Full-CI Quantum Monte Carlo (FCIQMC) and the d. matrix renormalization group (DMRG) have been developed that solve the Full-CI problem approx. for a fraction of the computational cost required by conventional techniques thereby significantly extending the range of applicability of modern MR methods. This perspective review outlines recent advancements in the field of MR electronic structure methods together with the resulting chances and challenges for theor. studies in the field of transition metal chem. In light of its emerging importance a special focus is put on the selection of adequate active spaces and the concomitant development of numerous selection aides in recent years.
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    Tarrago, M.; Römelt, C.; Nehrkorn, J.; Schnegg, A.; Neese, F.; Bill, E.; Ye, S. Experimental and Theoretical Evidence for an Unusual Almost Triply Degenerate Electronic Ground State of Ferrous Tetraphenylporphyrin. Inorg. Chem. 2021, 60, 4966– 4985,  DOI: 10.1021/acs.inorgchem.1c00031
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    Experimental and Theoretical Evidence for an Unusual Almost Triply Degenerate Electronic Ground State of Ferrous Tetraphenylporphyrin
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    Iron porphyrins exhibit unrivalled catalytic activity for electrochem. CO2-to-CO conversion. Despite intensive exptl. and computational studies in the last four decades, the exact nature of the prototypical square-planar [FeII(TPP)] complex (1; TPP2- = tetraphenylporphyrinate dianion) remained highly debated. Specifically, its intermediate spin (S = 1) ground state was contradictorily assigned to either a nondegenerate 3A2g state with (dxy)2(dz2)2(dxz,yz)2 configuration or a degenerate 3Eθg state with (dxy)2(dxz,yz)3(dz2)1/(dz2)2(dxy)1(dxz,yz)3 configuration. To address this question, we present herein a comprehensive, spectroscopy-based theor. and exptl. electronic-structure investigation on complex 1. Highly correlated wave function-based computations predicted that 3A2g and 3Egθ are well-isolated from other triplet states by ca. 4000 cm-1, whereas their splitting ΔA-E is on par with the effective spin-orbit coupling (SOC) const. of iron(II) (≈ 400 cm-1). In order to model the entire manifold of the nine magnetic sublevels arising from SOC between the 3A2g and 3Eθg states explicitly, we invoked an effective Hamiltonian (EH) operating on the corresponding nine-dimensional Hilbert space. This approach enabled us to successfully simulate all spectroscopic data of 1 obtained by variable temp. and variable field magnetization, applied-field 57Fe Mossbauer, and THz-EPR measurements. Remarkably, the EH contains only three adjustable parameters, namely, the energy gap without SOC, ΔA-E, an angle θ that describes the mixing of (dxy)2(dxz,yz)3(dz2)1 and (dz2)2(dxy)1(dxz,yz)3 configurations, and the 〈rd-3〉 expectation value of the iron d-orbitals that is necessary to est. the 57Fe magnetic hyperfine coupling tensor. The simulations revealed ΔA-E = +950 cm-1, rendering 3Eθg slightly above 3A2g in energy. Hence, 1 has a triplet ground state with substantial parentage of both 3A2g ( < 88%) and 3Eθg ( > 12%). Thus, the electronic ground state cannot simply be interpreted as either 3A2g or 3Eθg, but is genuinely multiconfigurational, arising from accidental near-triple degeneracy. Consequently, although this low-lying triplet is isolated from other states by ca. 900 cm-1, the magnetic properties of 1 cannot be adequately understood by the conventional S = 1 spin Hamiltonian (SH), which is valid only for orbitally nondegenerate states. Instead, the EH treatment easily explains the obsd. huge effective magnetic moment (4.2μB at 300 K), strong temp.-independent paramagnetism and large and pos. axial zero-field splitting within the triplet, giving rise to a nondegenerate magnetic sublevel being lowest in energy. Application of an external magnetic field demonstrates that the three magnetic sublevels carry substantial orbital angular momentum in the xy plane. This results in a large magnetization and a large and pos. internal field at the 57Fe nucleus aligned in the xy plane. (In the alternative SH description, the magnetic anisotropy manifests itself in an unusually large g anisotropy for an S = 1 system, with g.perp. ≈ 3 and g|| vbr ≈ 1.7). Further in-depth analyses suggested that such strong easy-plane anisotropy is a general spectroscopic signature of near-triple orbital degeneracy with more than half filled pseudodegenerate orbital sets. Implications of the unusual electronic structure of 1 for its CO2 redn. reactivity are discussed.
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    Figure 1

    Figure 1. Multireference character M as a function of internuclear distance D in the H2 dissociation problem for the FCI/STO-1G ground state. The blue curve corresponds to canonical orbitals (can), the red one to localized orbitals (loc).

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      The simple theory of the energy bands in metals is broadened, by application of the Hartree-Fock equations, to take into account the fact that the electrons in metals do not move independently of each other, and that the field in which the electrons move is not preformed, but owes its origin partly to the movement of those selfsame electrons.
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      The uniform electron gas or UEG (also known as jellium) is one of the most fundamental models in condensed-matter physics and the cornerstone of the most popular approxn.-the local-d. approxn.-within d.-functional theory. In this article, we provide a detailed review on the energetics of the UEG at high, intermediate, and low densities, and in one, two, and three dimensions. We also report the best quantum Monte Carlo and symmetry-broken Hartree-Fock calcns. available in the literature for the UEG and discuss the phase diagrams of jellium. WIREs Comput Mol Sci 2016, 6:410-429. doi: 10.1002/wcms.1257 For further resources related to this article, please visit the .
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      Wilson, S. Electron Correlation in Molecules; Dover: Mineola, 2007.
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    26. 26
      Advances in Chemical Physics; Vol. 14; Lefebvre, R., Moser, C., Eds.; John Wiley & Sons, Ltd: London, 1969 Correlation Effects in Atoms and Molecules.
      There is no corresponding record for this reference.
    27. 27
      Electron Correlation in the Solid State; March, N. H., Ed.; Imperial Collage Press: London, 1999.
      There is no corresponding record for this reference.
    28. 28
      Raimes, S. Many-electron Theory; North-Holland: Amsterdam, 1972.
      There is no corresponding record for this reference.
    29. 29
      Gell-Mann, M.; Brueckner, K. A. Correlation energy of an electron gas at high density. Phys. Rev. 1957, 106, 364– 368,  DOI: 10.1103/PhysRev.106.364
      29
      Correlation energy of an electron gas at high density
      Gell-Mann, Murray; Brueckner, Keith A.
      Physical Review (1957), 106 (), 364-8CODEN: PHRVAO; ISSN:0031-899X.
      cf. following 2 abstrs. Two consts. in the formula for the correlation energy are detd. by computation.
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    30. 30
      Hund, F. Zur Deutung der Molekelspektren. Z. Phys. 1927, 40, 742– 764,  DOI: 10.1007/BF01400234
      30
      Interpretation of the spectra of molecules
      Hund, I. F.
      Zeitschrift fuer Physik (1927), 40 (), 742-64CODEN: ZEPYAA; ISSN:0044-3328.
      The aim of this work is to indicate ways whereby a qual. understanding may be attained of those features of band spectra which depend on electron motions. The wave mechanics of schr.ovrddot.odinger (cf. Phys. Rev. 2S, 1049-70(1926)) are applied first to systems with one degree of freedom, and then to systems of several degrees of freedom, to det. their stationary states. The results of this analysis are in turn used to find the spectral terms of: (1) mols. with 2 unequal nuclei and 1 electron; (2) mols. with 2 equal nuclei and 1 electron; (3) mols. with 2 equal nuclei and 2 electrons. An illustration is afforded by the NaCl mol. The low spectral terms of NaCl, on sepn. of the nuclei, go over into the low terms of Na + Cl and Na+ + Cl-, the former corresponding to large. sepns. of the nuclei, the latter to small sepns. In harmony with the conception of NaCl as a polar mol.-i. e., one which on sepn. of the nuclei goes over into 2 oppositely charged ions-the lowest term of its spectrum must be ascribed to Na+ + Cl-.
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    31. 31
      Mulliken, R. S. The assignment of quantum numbers for electrons in molecules. I. Phys. Rev. 1928, 32, 186– 222,  DOI: 10.1103/PhysRev.32.186
      31
      The assignment of quantum numbers for electrons in molecules. I
      Mulliken, Robert S.
      Physical Review (1928), 32 (), 186-222CODEN: PHRVAO; ISSN:0031-899X.
      The problem of making a complete assignment of quantum nos. for the electrons in a (non-rotating) diatomic mol. is considered. A tentative assignment of such quantum nos. is made in this paper for most of the known electronic states of diatomic mols. composed of atoms of the first short period of the periodic system. The assignments are based mainly on band-spectrum and to a lesser extent on ionization-potential and positive-ray data. The methods used involve the application and extension of Hund's theoretical work on the electronic states of mols. Although the actual state of the electrons in a mol., as contrasted with an atom, cannot ordinarily be expected to be described accurately by quantum nos. corresponding to simple mech. quantities, such quantum nos. can nevertheless be assigned formally, with the understanding that their mech. interpretation in the real mol. (obtained by adiabatic correlation) may differ markedly from that corresponding to a literal interpretation. With this understanding, a suitable choice of quantum nos. for a diatomic mol. appears to be one corresponding to an atom in a strong elec. field, namely, quantum nos. nr, lr, σr and Sr (Sr = 1/2 always) for the rth electron and quantum nos. s σl and σs for the mol. as a whole (σlr and σs represent quantized components of lr, and s, resp., with reference to the line joining the nuclei). These quantum nos. may be thought of as those assocd. with the imagined "united atom" formed by bringing the nuclei of the mol. together. A notation is proposed whereby the state of each electron and of the mol. as a whole can be designated, e. g., (1 sε)2 (2 sρ)2 (2sε)2 (2pρ), 2P for a seven-electron mol. with σ = 1, s = 1/2, in a symbol such as 2 sρ the superscript denotes lr, the main letter, σlr, thus 2 sP means that the electron in question has nr = 2, lr = 1, σlr = 0. Electrons with σlr = 0, 1, 2,-are referred to as s, p, d-electrons. It is shown that in a mol. it is usually natural to define a group of equiv. electrons giving a resultant σl = 0, s = 0 as a closed shell; in this sense, two s electrons, or four p, or d, f-, electrons form a closed shell. The possible mol. states corresponding to various electron configurations are deduced by means of the Pauli principle. Electrons which undergo an increase in their n values (principal quantum nos.) when atoms unite to form a mol. (Hund) are here called promoted electrons. The electrons in a mol. may be classified according to their bonding power, positive, zero, or negative. Electrons whose presence tends to hold the mol. together, as judged by the fact that their removal from a stable mol. causes a decrease in the energy of dissociation D or an increase in the equil. internuclear sepn. r0 may be said to have positive bonding power, and are identified with, or defined as, bonding electrons. Bonding power in terms of changes of D and of changes of r0 are distinguished as "energy-bonding-power" and "distance-bonding-power." On the whole, promoted electrons should tend to show negative energy-bonding-power, unpromoted electrons positive energy-bonding-power, but much should depend on "orbit dimensions." Certain rules governing the relations of the electronic states of a mol. to those of its dissociation products are discussed; in addn. to theoretical rules established by Hund in regard to σl and s values, another rule is here proposed, namely, that the σlr values of all the at. electrons before union should be preserved in the mol. (σlr conservation rule). Selection rules for electronic transitions are also discussed; in addn. to rules given by Hund, the following are proposed: Δlr = ±1 for intense transitions: Δσlr = 0, ±1. Results. The key to the assignment of quantum nos. made here is found in the fact that the mols. BO, CO+ and CN show an inverted 2P state instead of the normal 2P which should occur if this state were analogous to the ordinary 2P states of the Na atom. The existence of such a low-lying inverted 2P indicates that in these mols. there exists a closed shell of p electrons from which one is easily excited. It is concluded that this is a (2 pp)4 shell. The identification of 2 other closed shells, of s electrons, very likely (3 sρ)2 and (3 sε)2, follows; the electrons in these and the (2 pρ)4 shell are roughly equal in energy of binding. According to this interpretation, the electron jumps involved in the band spectra of BO, CN, CO+ and N+ are more analogous to x-ray than to optical electron transitions. From this beginning, proceeding to CO, N2, O2, O2+, F2, C2, etc., a self-consistent assignment of quantum nos. is built up for most of the known states of the various mols. treated in this paper. The spectroscopic analogies of CN, N2, NO, etc., to Na, Mg, Al are justified and the partial failure of these analogies such as the chem. resemblance of CN to a halogen, are explained. Nearly all the hitherto observed ionization potentials of the mols. discussed can be accounted for by the removal of a single electron from one or another of the various closed shells supposed to be present. The N2+ band fluorescence produced by short wave-length ultra-violet light (Oldenberg) is accounted for as the expected result of photo-ionization of a 3 SP electron. The steadily decreasing heat of dissocn. in the series, N2-NO-O2-F2, is accounted for by the successive addn. of promoted 3pP electrons with strong neg. bonding power. Starting from N2, whose normal state corresponds to a 1S configuration of closed shells, we add one 3 pP electron to give the 2p normal state of NO, and O2+, two to give the 3S normal state of O2, four to give a closed shell, (3 pP)4, which accounts for the 1S normal state of F2. In N2 (probably also in O2 and the other homopolar mols.) band systems for which Δlr ≠ 1 are notably lacking, thus giving support to Hund's predicted selection rule for homopolar mols., in the analogous heteropolar mol. CO2, many systems occur with Δlr = 0 than those for which Δlr = ±1. On account of this strict selection rule in N2 certain levels should be metastable, in particular the final level of the α afterglow bands of active nitrogen. There is evidence for the existence of a strict selection rule Δs = 1 in homopolar mols.
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    32. 32
      Lennard-Jones, J. E. The electronic structure of some diatomic molecules. Trans. Faraday Soc. 1929, 25, 668– 686,  DOI: 10.1039/tf9292500668
      32
      The electronic structure of some diatomic molecules
      Lennard-Jones, J. E.
      Transactions of the Faraday Society (1929), 25 (), 668-86CODEN: TFSOA4; ISSN:0014-7672.
      A review of the ideas of Franck and Herzberg, Heisenberg, Heitler and London, Hund and Mulliken on the formation of mols. and their dissocn. energy. J. criticizes Hund's application of the Pauli exclusion principle for the definition of mol. states. A notation is given which will make it possible to distinguish between mol, and at. levels in the same mol. The transitions encountered in mol. formation are given.
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    33. 33
      Heitler, W.; London, F. Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Z. Phys. 1927, 44, 455– 472,  DOI: 10.1007/BF01397394
      33
      Interaction of neutral atoms and homopolar binding according to the quantum mechanics
      Heitler, W.; London, F.
      Zeitschrift fuer Physik (1927), 44 (), 455-72CODEN: ZEPYAA; ISSN:0044-3328.
      The action of forces between neutral atoms has a characteristic ambiguity in the quantum mechanics. The ambiguity seems capable of including the different modes of behavior actually found, i. e., for H either homopolar binding or elastic reflection, but for the rare gases only reflection. It also permits an evaluation of the elastic reflection effects in He, giving results of the right order of magnitude. For the selection and discussion of the various possible interactions the Pauli principle is here applied to a system of several atoms.
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    34. 34
      Pauling, L. The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules. J. Am. Chem. Soc. 1931, 53, 1367– 1400,  DOI: 10.1021/ja01355a027
      34
      The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules
      Pauling, Linus
      Journal of the American Chemical Society (1931), 53 (), 1367-1400CODEN: JACSAT; ISSN:0002-7863.
      The electron-pair bond is discussed and from quantum mechanics a set of rules is presented which describes the properties of the bond with special ref. to the strength of the bond and the nature of the single-electron proper functions. These rules give information about the relative strengths of bonds formed by different atoms, the angles between bonds, properties of tetrahedral atoms with single and double bonds, cis and trans forms, the no. and spatial configuration of bonds and other properties. Transitions from electron-pair to ionic bonds are also discussed. A theory of the magnetic moments of mol. and complex ions is also developed. For the transition elements the proper functions involved in bond formation show that compds. with CN have electron-pair bonds, those with F have ionic bonds, and those with H2O, ion-dipole bonds. Electron structure, bond angles and other properties of mol. and complex ions can also be detd.from the magnetic data.
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    35. 35
      Coulson, C. A.; Fischer, I. Notes on the molecular orbital treatment of the hydrogen molecule. Philos. Mag. 1949, 40, 386– 393,  DOI: 10.1080/14786444908521726
      35
      Notes on the molecular-orbital treatment of the hydrogen molecule
      Coulson, C. A.; Fischer, I.
      Philosophical Magazine (1798-1977) (1949), 40 (), 386-93CODEN: PHMAA4; ISSN:0031-8086.
      The fundamentals are investigated in more detail than formerly. Configurational interaction consts. are calcd. Energy curves for the H2 mol. are derived. Failure at large internuclear distances is discussed.
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    36. 36
      Kutzelnigg, W.; Mukherjee, D. Cumulant expansion of the reduced density matrices. J. Chem. Phys. 1999, 110, 2800– 2809,  DOI: 10.1063/1.478189
      36
      Cumulant expansion of the reduced density matrixes
      Kutzelnigg, Werner; Mukherjee, Debashis
      Journal of Chemical Physics (1999), 110 (6), 2800-2809CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      K-particle cumulants λk (for 2≤k≤n) corresponding to the k-particle reduced d. matrixes γk for an n-fermion system are defined via a generating function. The two-particle cumulant λ2 describes two-particle correlations (excluding exchange), λ3 genuine three-particle correlations etc. The properties of these cumulants are analyzed. Conditions for vanishing of certain λk are formulated. Necessary and sufficient for λ2=0 is the well-known idempotency condition γ2=γ for γ γ1. For λ3=0 to hold, a general necessary condition is Tr{2γ3-3γ2+γ}=0, for three special forms of the wave function (arbitrary two-electron state, antisymmetrized product of strongly orthogonal geminals on antisymmetrized geminal power wave function of extreme type) 2γ3-3γ2+γ=0 turns out to be necessary and sufficient. For a multiconfiguration SCF wave function the only nonvanishing matrix elements of the cumulants are those where all labels refer to active (partially occupied) spin orbitals. Spin-free cumulants Λk corresponding to the spin-free reduced d. matrixes Γk are also defined and analyzed. The main interest in the d. cumulants is in connection with the recently formulated normal ordering and the corresponding Wick theorem for arbitrary ref. functions, but they are also useful for an anal. of electron correlation.
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    37. 37
      Wigner, E.; Seitz, F. On the Constitution of Metallic Sodium. Phys. Rev. 1933, 43, 804– 810,  DOI: 10.1103/PhysRev.43.804
      37
      Constitution of metallic sodium
      Wigner, E.; Seitz, F.
      Physical Review (1933), 43 (), 804-10CODEN: PHRVAO; ISSN:0031-899X.
      The lattice const., binding energy and compressibility of metallic Na are calcd. from a theoretical treatment of the energy of the free electrons.
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    38. 38
      Wigner, E.; Seitz, F. On the Constitution of Metallic Sodium. II. Phys. Rev. 1934, 46, 509– 524,  DOI: 10.1103/PhysRev.46.509
      38
      The constitution of metallic sodium. II
      Wigner, E.; Seitz, F.
      Physical Review (1934), 46 (), 509-24CODEN: PHRVAO; ISSN:0031-899X.
      cf. C. A. 27, 3135. Calcns. including interactions between electrons with parallel spins give a lattice energy of only 9 kg.-cal., against the exptl. value of 26.9 kg.-cal. When interactions between electrons with antiparallel spins are included the calcd. value becomes 23.2 kg.-cal.
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    39. 39
      McWeeny, R. The density matrix in many-electron quantum mechanics I. Generalized product functions. Factorization and physical interpretation of the density matrices. Proc. R. Soc. A 1959, 253, 242– 259,  DOI: 10.1098/rspa.1959.0191
      39
      The density matrix in many-electron quantum mechanics. I. Generalized product functions. Factorization and physical interpretation of the density matrixes
      McWeeny, R.
      Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences (1959), 253 (), 242-59CODEN: PRLAAZ; ISSN:1364-5021.
      Many-electron wave functions are usually constructed from antisymmetrized products of 1-electron orbitals (determinants) and energy calcns. are based on the matrix element expressions due to Slater. The orbitals in such a product are replaced by "group functions,'' each describing any no. of electrons, and the necessary generalization of Slater's results is carried out. It is 1st necessary to develop the matrix theory of N-particle systems and to show that, for systems described by generalized product functions, the matrixes of the whole system can be expressed in terms of those of the component electron groups. The matrix elements of the Hamiltonian between generalized product functions are then given by expressions which resemble those of Slater, the "Coulomb" and "exchange" integrals being replaced by integrals contg. the 1-electron matrixes of the various groups. By setting up an "effective" Hamiltonian for each electron group in the presence of the others, the discussion of a many-particle system in which groups or "shells" can be distinguished (e.g. at. K, L, M, ..., shells) can rigorously be reduced to a discussion of smaller subsystems. A single generalized product (cf. the single determinant of Hartree-Fock theory) provides a convenient 1st approxn. and the effect of admitting "excited" products (cf. configuration interaction) can be estd. by a perturbation method. The energy expression can then be discussed in terms of the electron d. and "pair" functions. The energy is a sum of group energies supplemented by interaction terms which represent electrostatic repulsions between charge clouds, the polarization of each group in the field of the others, and dispersion effects of the type defined by London. All these terms can be calcd. for group functions of any kind, in terms of the d. matrixes of the sep. groups. Approxns. to the theory of intermol. forces and to π-electron systems are discussed.
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    40. 40
      Giner, E.; Tenti, L.; Angeli, C.; Malrieu, J.-P. The “Fermi hole” and the correlation introduced by the symmetrization or the anti-symmetrization of the wave function. J. Chem. Phys. 2016, 145, 124114,  DOI: 10.1063/1.4963018
      40
      The "Fermi hole" and the correlation introduced by the symmetrization or the anti-symmetrization of the wave function
      Giner, Emmanuel; Tenti, Lorenzo; Angeli, Celestino; Malrieu, Jean-Paul
      Journal of Chemical Physics (2016), 145 (12), 124114/1-124114/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The impact of the antisymmetrization is often addressed as a local property of the many-electron wave function, namely that the wave function should vanish when two electrons with parallel spins are in the same position in space. In this paper, we emphasize that this presentation is unduly restrictive: we illustrate the strong non-local character of the antisymmetrization principle, together with the fact that it is a matter of spin symmetry rather than spin parallelism. To this aim, we focus our attention on the simplest representation of various states of two-electron systems, both in at. (helium atom) and mol. (H2 and the π system of the ethylene mol.) cases. We discuss the non-local property of the nodal structure of some two-electron wave functions, both using anal. derivations and graphical representations of cuttings of the nodal hypersurfaces. The attention is then focused on the impact of the antisymmetrization on the maxima of the two-body d., and we show that it introduces strong correlation effects (radial and/or angular) with a non-local character. These correlation effects are analyzed in terms of inflation and depletion zones, which are easily identifiable, thanks to the nodes of the orbitals composing the wave function. Also, we show that the correlation effects induced by the antisymmetrization occur also for anti-parallel spins since all Ms components of a given spin state have the same N-body densities. Finally, we illustrate that these correlation effects occur also for the singlet states, but they have strictly opposite impacts: the inflation zones in the triplet become depletion zones in the singlet and vice versa. (c) 2016 American Institute of Physics.
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    41. 41
      Kutzelnigg, W.; Mukherjee, D. Normal order and extended Wick theorem for a multiconfiguration reference wave function. J. Chem. Phys. 1997, 107, 432– 449,  DOI: 10.1063/1.474405
      41
      Normal order and extended Wick theorem for a multiconfiguration reference wave function
      Kutzelnigg, Werner; Mukherjee, Debashis
      Journal of Chemical Physics (1997), 107 (2), 432-449CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A generalization of normal ordering and of Wick's theorem with respect to an arbitrary ref. function Φ as some generalized "phys. vacuum" is formulated in a different (but essentially equiv.) way than that suggested previously by one of the present authors. Guiding principles are that normal order operators with respect to any ref. state must be expressible as linear combinations of those with respect to the genuine vacuum, that the vacuum expectation value of a normal order operator must vanish (with respect to the vacuum to which it is in normal order), and that the well-known formalism for a single Slater determinant as phys. vacuum must be contained as a special case. The derivation is largely based on the concepts of "Quantum Chem. in Fock space", which means that particle-no.-conserving operators (excitation operators) play a central role. Nevertheless, the contraction rules in the frame of the generalized Wick theorem are derived, that hold for non-particle-no.-conserving operators as well. The contraction rules are formulated and illustrated in terms of diagrams. The contractions involve the "residual n-particle d. matrixes" λ, which are the irreducible (non-factorizable) parts of the conventional n-particle d. matrixes γ, in the sense of a cumulant expansion for the d. A spin-free formulation is presented as well. The expression of the Hamiltonian in normal order with respect to a multiconfiguration ref. function leads to a natural definition of a generalized Fock operator. MC-SCF-theory is easily worked out in this context. The paper concludes with a discussion of the excited configurations and the first-order interacting space, that underlies a perturbative coupled cluster type correction to the MCSCF function for an arbitrary ref. function, and with general implications of the new formalism, that is related to "internally contracted multireference CI". The present generalization of normal ordering is not only valid for arbitrary ref. functions, but also if the ref. state is an ensemble state.
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    42. 42
      Evangelista, F. A. Automatic derivation of many-body theories based on general Fermi vacua. J. Chem. Phys. 2022, 157, 064111,  DOI: 10.1063/5.0097858
      42
      Automatic derivation of many-body theories based on general Fermi vacua
      Evangelista, Francesco A.
      Journal of Chemical Physics (2022), 157 (6), 064111CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      This paper describes WIC&D, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated refs. and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. WICK&D employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since WICK&D can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of WICK&D, we use it to evaluate the single-ref. coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multi-ref. perturbation theory. (c) 2022 American Institute of Physics.
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    43. 43
      Tsuchimochi, T.; Scuseria, G. E. Strong correlations via constrained-pairing mean-field theory. J. Chem. Phys. 2009, 131, 121102,  DOI: 10.1063/1.3237029
      43
      Strong correlations via constrained-pairing mean-field theory
      Tsuchimochi, Takashi; Scuseria, Gustavo E.
      Journal of Chemical Physics (2009), 131 (12), 121102/1-121102/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present a mean-field approach for accurately describing strong correlations via electron no. fluctuations and pairings constrained to an active space. Electron no. conservation is broken and correct only on av., but both spin and spatial symmetries are preserved. Optimized natural orbitals and occupations are detd. by diagonalization of a mean-field Hamiltonian. This constrained-pairing mean-field theory (CPMFT) yields a two-particle d. matrix ansatz that exclusively describes strong correlations. We demonstrate CPMFT accuracy with applications to the metal-insulator transition of large hydrogen clusters and mol. dissocn. curves. (c) 2009 American Institute of Physics.
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    44. 44
      Kutzelnigg, W. Separation of strong (bond-breaking) from weak (dynamical) correlation. Chem. Phys. 2012, 401, 119– 124,  DOI: 10.1016/j.chemphys.2011.10.020
      44
      Separation of strong (bond-breaking) from weak (dynamical) correlation
      Kutzelnigg, Werner
      Chemical Physics (2012), 401 (), 119-124CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)
      A CC (coupled-cluster) ansatz based on a GVB (generalized valence bond) or an APSG (antisymmetrized product of strongly orthogonal geminals) ref. function arises naturally if one tries to treat strong correlations exactly (to infinite order), and weak correlations by TCC (traditional coupled cluster) theory. This ansatz is proposed as an alternative to MC-CC (multi-configuration coupled cluster) theory. One uses esp. that APSG and GVB are of CC type, but allow to combine separability with the variation principle. The energy and the stationarity conditions are formulated in terms of spin-free d. cumulants. The replacement operators corresponding to the APSG ansatz generate a Lie algebra which is a sub-algebra of that of all replacement operators.
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    45. 45
      Georges, A.; Kotliar, G. Hubbard model in infinite dimensions. Phys. Rev. B 1992, 45, 6479– 6483,  DOI: 10.1103/PhysRevB.45.6479
      45
      Hubbard model in infinite dimensions
      Georges; Kotliar
      Physical review. B, Condensed matter (1992), 45 (12), 6479-6483 ISSN:0163-1829.
      There is no expanded citation for this reference.
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    46. 46
      Georges, A.; Kotliar, G.; Krauth, W.; Rozenberg, M. J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 1996, 68, 13– 125,  DOI: 10.1103/RevModPhys.68.13
      46
      Dynamic mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions
      Georges, Antoine; Kotliar, Gabriel; Krauth, Werner; Rozenberg, Marcelo J.
      Reviews of Modern Physics (1996), 68 (1), 13-125CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
      A review with many refs. is given on the dynamic mean-field theory of strongly correlated electron, systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the std. mean-field construction from classical statistical mechanics to quantum problems. We discuss the phys. ideas underlying this theory and its math. derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamic mean-field equations are reviewed and compared to each other. The method can be used for the detn. of phase diagrams (by comparing the stability of various types of long-range order), and the calcn. of thermodn. properties, one-particle Green functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to expts. on three-dimensional transition metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are discussed. Computer programs for the numerical implementation of this method are also provided with this article.
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    47. 47
      Anisimov, V.; Izyumov, Y. Electronic Structure of Strongly Correlated Materials; Springer: Berlin, 2010.
      There is no corresponding record for this reference.
    48. 48
      Kuramoto, Y. Quantum Many-Body Physics; Springer: Tokyo, 2020.
      There is no corresponding record for this reference.
    49. 49
      Foulkes, W. M. C.; Mitas, L.; Needs, R. J.; Rajagopal, G. Quantum Monte Carlo simulations of solids. Rev. Mod. Phys. 2001, 73, 33– 83,  DOI: 10.1103/RevModPhys.73.33
      49
      Quantum Monte Carlo simulations of solids
      Foulkes, W. M. C.; Mitas, L.; Needs, R. J.; Rajagopal, G.
      Reviews of Modern Physics (2001), 73 (1), 33-83CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
      This review with many refs. describes the variational and fixed-node diffusion quantum Monte Carlo methods and how they may be used to calc. the properties of many-electron systems. These stochastic wave-function-based approaches provide a very direct treatment of quantum many-body effects and serve as benchmarks against which other techniques may be compared. They complement the less demanding d.-functional approach by providing more accurate results and a deeper understanding of the physics of electronic correlation in real materials. The algorithms are intrinsically parallel, and currently available high-performance computers allow applications to systems contg. a thousand or more electrons. With these tools one can study complicated problems such as the properties of surfaces and defects, while including electron correlation effects with high precision. The authors provide a pedagogical overview of the techniques and describe a selection of applications to ground and excited states of solids and clusters.
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    50. 50
      Zhang, S.; Malone, F. D.; Morales, M. A. Auxiliary-field quantum Monte Carlo calculations of the structural properties of nickel oxide. J. Chem. Phys. 2018, 149, 164102,  DOI: 10.1063/1.5040900
      50
      Auxiliary-field quantum Monte Carlo calculations of the structural properties of nickel oxide
      Zhang, Shuai; Malone, Fionn D.; Morales, Miguel A.
      Journal of Chemical Physics (2018), 149 (16), 164102/1-164102/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Auxiliary-field quantum Monte Carlo (AFQMC) has repeatedly demonstrated itself as one of the most accurate quantum many-body methods, capable of simulating both real and model systems. We investigate the application of AFQMC to realistic strongly correlated materials in periodic Gaussian basis sets. Using nickel oxide (NiO) as an example, we investigate the importance of finite size effects and basis set errors on the structural properties of the correlated solid. We provide benchmark calcns. for NiO and compare our results to both exptl. measurements and existing theor. methods. (c) 2018 American Institute of Physics.
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    51. 51
      Exactly solvable models of strongly correlated electrons; Korepin, V. E., Essler, F. H., Eds.; World Scientific: Singapore, 1994.
      There is no corresponding record for this reference.
    52. 52
      Gross, D. J. The role of symmetry in fundamental physics. Proc. Nat. Acad. Sci. 1996, 93, 14256– 14259,  DOI: 10.1073/pnas.93.25.14256
      52
      The role of symmetry in fundamental physics
      Gross, David J.
      Proceedings of the National Academy of Sciences of the United States of America (1996), 93 (25), 14256-14259CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      The role of symmetry in fundamental physics is reviewed with no refs.
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    53. 53
      Raimes, S. The wave mechanics of electrons in metals; North-Holland: Amsterdam, 1963.
      There is no corresponding record for this reference.
    54. 54
      Fulde, P. Solids with weak and strong electron correlations. In Electron Correlation in the Solid State; Imperial Collage Press: London, 1999; Chapter 2, pp 47– 102.
      There is no corresponding record for this reference.
    55. 55
      Coldwell-Horsfall, R. A.; Maradudin, A. A. Zero-Point Energy of an Electron Lattice. J. Math. Phys. 1960, 1, 395– 404,  DOI: 10.1063/1.1703670
      There is no corresponding record for this reference.
    56. 56
      Bohm, D.; Pines, D. A Collective Description of Electron Interactions. I. Magnetic Interactions. Phys. Rev. 1951, 82, 625– 634,  DOI: 10.1103/PhysRev.82.625
      56
      Collective description of electron interactions. I. Magnetic interactions
      Bohm, David; Pines, David
      Physical Review (1951), 82 (), 625-34CODEN: PHRVAO; ISSN:0031-899X.
      Math.-theoretical. A new approach to the treatment of the interaction in a collection of electrons is developed, which is called the collective description. The collective description is based on the organized behavior produced by the interactions in an electron gas of high d.; this organized behavior results in oscillations of the system as a whole, the so-called plasma oscillations. The collective description, in contrast to the usual individual particle description, describes in a natural way the long-range correlations in electron positions brought about by their mutual interaction. Here, attention is confined to the magnetic interactions between the electrons. Both a classical and a quantum-mech. treatment are given, and the criteria for the validity of the collective description are discussed.
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    57. 57
      Pines, D. Electron Interaction in Metals. In Solid State Physics; Academic Press: New York, 1955; Vol. 1, pp 367– 450.
      There is no corresponding record for this reference.
    58. 58
      Slater, J. C. The electronic structure of solids. In Encyclopedia of Physics/Handbuch der Physik; Springer: Berlin, 1956; Vol. 19, pp 1– 136.
      There is no corresponding record for this reference.
    59. 59
      Brueckner, K. A. The Correlation Energy of a Non-Uniform Electron Gas. In Advances in Chemical Physics; John Wiley & Sons, Ltd: London, 1969; Vol. 14, Chapter 7, pp 215– 236.
      There is no corresponding record for this reference.
    60. 60
      Slater, J. C. The Virial and Molecular Structure. J. Chem. Phys. 1933, 1, 687– 691,  DOI: 10.1063/1.1749227
      60
      The virial and molecular structure
      Slater, J. C.
      Journal of Chemical Physics (1933), 1 (), 687-91CODEN: JCPSA6; ISSN:0021-9606.
      cf. C. A. 26, 5799. The virial theorem is applied to a mol. if external forces are applied to keep the nuclei fixed. It permits sep. detns. of the kinetic and potential energies for all configurations of the nuclei from the total-energy curves as derived from expt. or theory. Such potential- and kinetic-energy curves are derived for simple forms of the total-energy curves for diat. mols. These can be readily interpreted as indicating bond formation in attractive forces. The method can be extended to apply to more complicated mols. and to solids.
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    61. 61
      March, N. H. Kinetic and Potential Energies of an Electron Gas. Phys. Rev. 1958, 110, 604– 605,  DOI: 10.1103/PhysRev.110.604
      61
      Kinetic and potential energies of an electron gas
      March, N. H.
      Physical Review (1958), 110 (), 604-5CODEN: PHRVAO; ISSN:0031-899X.
      cf. Gell-Mann and Brueckner, C.A. 51, 17393h. The kinetic and potential energy values of an electron gas may be obtained exactly in the high-d. limit by applying the virial theorem to the correlation energy.
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    62. 62
      Argyres, P. N. Virial Theorem for the Homogeneous Electron Gas. Phys. Rev. 1967, 154, 410– 413,  DOI: 10.1103/PhysRev.154.410
      62
      Virial theorem for the homogeneous electron gas
      Argyres, Petros N.
      Physical Review (1967), 154 (2), 410-13CODEN: PHRVAO; ISSN:0031-899X.
      A proof is presented of the virial theorem for the interacting electron gas in a uniform pos. background with the boundary conditions used in actual calcns. of the total energy.
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    63. 63
      Ruedenberg, K. The Physical Nature of the Chemical Bond. Rev. Mod. Phys. 1962, 34, 326– 376,  DOI: 10.1103/RevModPhys.34.326
      63
      Physical nature of the chemical bond
      Ruedenberg, Klaus
      Reviews of Modern Physics (1962), 34 (), 326-76CODEN: RMPHAT; ISSN:0034-6861.
      Mol. energy as well as all other observable quantities are completely detd. by 2 functions: the d. (1st-order d. kernel) and the pair d. (2nd-order d. kernel). These 2 are chosen as the starting point for an interpretive analysis of mols. A simultaneous regional and phys. partitioning of the mol. d., the mol. pair d., and the mol. energy is attempted such that meaningful concepts can be assocd. with the proposed fragments. An analysis of how electron-sharing affects ds. and energies is included. It is suggested that a mol. differs from the juxtaposed atoms in 3 major aspects characterized by the concepts of interference, penetration, and charge transfer. Interference contributions embody the precise connections existing between overlap and chem. binding. Penetration contributions describe how electron sharing modifies electronic correlations. From an analysis of the H2 mol.-H2+ ion it is concluded that electron sharing leads to chem. binding as the result of a subtle interplay between the uncertainty principle and nuclear attractions.
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    64. 64
      Hubbard, J. Electron correlations in narrow energy bands. Proc. R. Soc. A 1963, 276, 238– 257,  DOI: 10.1098/rspa.1963.0204
      There is no corresponding record for this reference.
    65. 65
      Janesko, B. G. Strong correlation in surface chemistry. Mol. Simul. 2017, 43, 394– 405,  DOI: 10.1080/08927022.2016.1261136
      65
      Strong correlation in surface chemistry
      Janesko, Benjamin G.
      Molecular Simulation (2017), 43 (5-6), 394-405CODEN: MOSIEA; ISSN:0892-7022. (Taylor & Francis Ltd.)
      D. functional theory (DFT) simulations of surface chem. have emerged as a valuable complement to expt. However, std. DFT methods do not always accurately model the 'strong' electron correlation effects seen in stretched covalent bonds. Such systems' ground-state wavefunctions are not well-described by single MO configurations. I review some of the challenges of strong correlation, and some methods used to simulate it in surface chem. I also use the electron delocalisation range function EDR(), which quantifies the extent to which electrons at point delocalise over distance d, to highlight how a nearby metal cluster affects strong correlation in a dissocg. covalent bond.
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    66. 66
      Motta, M.; Ceperley, D. M.; Chan, G. K.-L.; Gomez, J. A.; Gull, E.; Guo, S.; Jiménez- Hoyos, C. A.; Lan, T. N.; Li, J.; Ma, F.; Millis, A. J.; Prokof’ev, N. V.; Ray, U.; Scuseria, G. E.; Sorella, S.; Stoudenmire, E. M.; Sun, Q.; Tupitsyn, I. S.; White, S. R.; Zgid, D.; Zhang, S. Towards the Solution of the Many-Electron Problem in Real Materials: Equation of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods. Phys. Rev. X 2017, 7, 031059,  DOI: 10.1103/PhysRevX.7.031059
      66
      Towards the solution of the many-electron problem in real materials: equation of state of the hydrogen chain with state-of-the-art many-body methods
      Motta, Mario; Ceperley, David M.; Chan, Garnet Kin-Lic; Gomez, John A.; Gull, Emanuel; Guo, Sheng; Jimenez-Hoyos, Carlos A.; Lan, Tran Nguyen; Li, Jia; Ma, Fengjie; Millis, Andrew J.; Prokof'ev, Nikolay V.; Ray, Ushnish; Scuseria, Gustavo E.; Sorella, Sandro; Stoudenmire, Edwin M.; Sun, Qiming; Tupitsyn, Igor S.; White, Steven R.; Zgid, Dominika; Zhang, Shiwei
      Physical Review X (2017), 7 (3), 031059/1-031059/28CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
      We present numerical results for the equation of state of an infinite chain of hydrogen atoms. A variety of modern many-body methods are employed, with exhaustive cross-checks and validation. Approaches for reaching the continuous space limit and the thermodn. limit are investigated, proposed, and tested. The detailed comparisons provide a benchmark for assessing the current state of the art in many-body computation, and for the development of new methods. The ground-state energy per atom in the linear chain is accurately detd. vs. bond length, with a confidence bound given on all uncertainties.
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    67. 67
      Motta, M.; Genovese, C.; Ma, F.; Cui, Z.-H.; Sawaya, R.; Chan, G. K.-L.; Chepiga, N.; Helms, P.; Jiménez-Hoyos, C.; Millis, A. J.; Ray, U.; Ronca, E.; Shi, H.; Sorella, S.; Stoudenmire, E. M.; White, S. R.; Zhang, S. Ground-State Properties of the Hydrogen Chain: Dimerization, Insulator-to-Metal Transition, and Magnetic Phases. Phys. Rev. X 2020, 10, 031058
      67
      Ground-State Properties of the Hydrogen Chain: Dimerization, Insulator-to-Metal Transition, and Magnetic Phases
      Motta, Mario; Genovese, Claudio; Ma, Fengjie; Cui, Zhi-Hao; Sawaya, Randy; Chan, Garnet Kin-Lic; Chepiga, Natalia; Helms, Phillip; Jimenez-Hoyos, Carlos; Millis, Andrew J.; Ray, Ushnish; Ronca, Enrico; Shi, Hao; Sorella, Sandro; Stoudenmire, Edwin M.; White, Steven R.; Zhang, Shiwei
      Physical Review X (2020), 10 (3), 031058CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
      Accurate and predictive computations of the quantum-mech. behavior of many interacting electrons in realistic at. environments are crit. for the theor. design of materials with desired properties, and they require solving the grand-challenge problem of the many-electron Schr.ovrddot.odinger equation. An infinite chain of equispaced hydrogen atoms is perhaps the simplest realistic model for a bulk material, embodying several central themes of modern condensed-matter physics and chem. while retaining a connection to the paradigmatic Hubbard model. Here, we report a combined application of cutting-edge computational methods to det. the properties of the hydrogen chain in its quantum-mech. ground state. Varying the sepn. between the nuclei leads to a rich phase diagram, including a Mott phase with quasi-long-range antiferromagnetic order, electron d. dimerization with power-law correlations, an insulator-to-metal transition, and an intricate set of intertwined magnetic orders.
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    68. 68
      Sinanoğlu, O.; Tuan, D. F.-t. Many-Electron Theory of Atoms and Molecules. III. Effect of Correlation on Orbitals. J. Chem. Phys. 1963, 38, 1740– 1748,  DOI: 10.1063/1.1776948
      68
      Many-electron theory of atoms and molecules. III. Effect of correlation on orbitals
      Sinanoglu, Oktay; Tuan, Debbie Fu-tai
      Journal of Chemical Physics (1963), 38 (), 1740-8CODEN: JCPSA6; ISSN:0021-9606.
      cf. CA 57, 13299c. The exact wave function of an N-electron atom or mol. contains, after the Hartree-Fock (H.F.) part, correlation terms involving successively 1, 2. . . . N electrons at a time. Particularly in closed shells, 1-electron terms fi result mainly from pair correlations. The fi were previously neglected in the many-electron theory. Reasons for the smallness of fi are summarized. Different types of correlation effects are classified, and methods for estg. each type of fi are given. fi in closed form, i.e., including infinitely many single excitations, is <2.8% of the H.F. orbital in He with an energy contribution 0.0001 a.u. (63 cal./mole). In the H2 mol. fi is negligible for (R/Re) < 2. At. larger R, as (1σ0)2 becomes degenerate with (1σu)2, the fi effect increases to ∼0.4 e.v. at dissocn. However, in such cases and in actual nonclosed shells, these nondynamical fi are removed if H.F. orbitals are obtained after the removal of degeneracies. Dynamic correlation effects give negligible fi, and so, generalized S.C.F. methods are not necessary. Qual. quantum chemistry can be based on just H.F. orbitals or approxns. to them, though energies include localized pair correlations.
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    69. 69
      Prendergast, D.; Nolan, M.; Filippi, C.; Fahy, S.; Greer, J. Impact of electron–electron cusp on configuration interaction energies. J. Chem. Phys. 2001, 115, 1626– 1634,  DOI: 10.1063/1.1383585
      69
      Impact of electron-electron cusp on configuration interaction energies
      Prendergast, David; Nolan, M.; Filippi, Claudia; Fahy, Stephen; Greer, J. C.
      Journal of Chemical Physics (2001), 115 (4), 1626-1634CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The effect of the electron-electron cusp on the convergence of CI wave functions is examd. By analogy with the pseudopotential approach for electron-ion interactions, an effective electron-electron interaction is developed which closely reproduces the scattering of the Coulomb interaction but is smooth and finite at zero electron-electron sepn. The exact many-electron wave function for this smooth effective interaction has no cusp at zero electron-electron sepn. We perform CI and quantum Monte Carlo calcns. for He and Be atoms, both with the Coulomb electron-electron interaction and with the smooth effective electron-electron interaction. We find that convergence of the CI expansion of the wave function for the smooth electron-electron interaction is not significantly improved compared with that for the divergent Coulomb interaction for energy differences on the order of 1 mHartree. This shows that, contrary to popular belief, description of the electron-electron cusp is not a limiting factor, to within chem. accuracy, for CI calcns.
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    70. 70
      Mok, D. K.; Neumann, R.; Handy, N. C. Dynamical and nondynamical correlation. J. Phys. Chem. 1996, 100, 6225– 6230,  DOI: 10.1021/jp9528020
      70
      Dynamical and Nondynamical Correlation
      Mok, Daniel K. W.; Neumann, Ralf; Handy, Nicholas C.
      Journal of Physical Chemistry (1996), 100 (15), 6225-30CODEN: JPCHAX; ISSN:0022-3654. (American Chemical Society)
      The variation is studied of correlation energies with bond distances of various first row diat. mols. SCF and complete active space SCF potential curves of these mols. are calcd. Exact potential energy curves are constructed from exptl. data using the Rydberg-Klein-Rees method. With appropriate definitions, the dynamical and nondynamical correlation energies are obtained and the variation of these with bond distance is calcd. Two definitions of nondynamical correlation are examd. Classifying the angular correlation as dynamical seems to be a better way to partition the correlation energy. The correlation functionals of d. functional theory, VWN, LYP, and P86, are also evaluated and compared with the ab initio dynamical correlation energies. LYP appears to give the closest agreement with the dynamical correlation energy.
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    71. 71
      Hollett, J. W.; Gill, P. M. The two faces of static correlation. J. Chem. Phys. 2011, 134, 114111,  DOI: 10.1063/1.3570574
      71
      The two faces of static correlation
      Hollett, Joshua W.; Gill, Peter M. W.
      Journal of Chemical Physics (2011), 134 (11), 114111/1-114111/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      RHF and UHF wavefunctions for beryllium-like ions with nuclear charge 3 ≤ Z ≤ 5 are found using a near-complete Slater basis set. The triplet (RHF → UHF) instability and correlation energy are investigated as a function of Z and we find that the instability vanishes for Z > 4.5. We reproduce this surprising behavior using a minimal-basis model and, by comparing with the stretched H2 mol., conclude that "static" (also known as nondynamical, near-degeneracy, first-order, or strong) correlation comes in two flavors: one that can be captured by UHF and another that cannot. In the former (Type A), there is an "abs. near-degeneracy"; in the latter (Type B), there is a "relative near-degeneracy." This dichotomy clarifies discussions of static correlation effects. (c) 2011 American Institute of Physics.
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    72. 72
      Benavides-Riveros, C. L.; Lathiotakis, N. N.; Marques, M. A. Towards a formal definition of static and dynamic electronic correlations. Phys. Chem. Chem. Phys. 2017, 19, 12655– 12664,  DOI: 10.1039/C7CP01137G
      72
      Towards a formal definition of static and dynamic electronic correlations
      Benavides-Riveros, Carlos L.; Lathiotakis, Nektarios N.; Marques, Miguel A. L.
      Physical Chemistry Chemical Physics (2017), 19 (20), 12655-12664CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      Some of the most spectacular failures of d.-functional and Hartree-Fock theories are related to an incorrect description of the so-called static electron correlation. Motivated by recent progress in the N-representability problem of the one-body d. matrix for pure states, we propose a method to quantify the static contribution to the electronic correlation. By studying several mol. systems we show that our proposal correlates well with our intuition of static and dynamic electron correlation. Our results bring out the paramount importance of the occupancy of the highest occupied natural spin-orbital in such quantification.
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    73. 73
      Bulik, I. W.; Henderson, T. M.; Scuseria, G. E. Can single-reference coupled cluster theory describe static correlation?. J. Chem. Theory Comput. 2015, 11, 3171– 3179,  DOI: 10.1021/acs.jctc.5b00422
      73
      Can Single-Reference Coupled Cluster Theory Describe Static Correlation?
      Bulik, Ireneusz W.; Henderson, Thomas M.; Scuseria, Gustavo E.
      Journal of Chemical Theory and Computation (2015), 11 (7), 3171-3179CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      While restricted single-ref. coupled cluster theory truncated to singles and doubles (CCSD) provides very accurate results for weakly correlated systems, it usually fails in the presence of static or strong correlation. This failure is generally attributed to the qual. breakdown of the ref., and can accordingly be cor. by using a multideterminant ref., including higher-body cluster operators in the ansatz, or allowing symmetry breaking in the ref. None of these solns. are ideal; multireference coupled cluster is not black box, including higher-body cluster operators is computationally demanding, and allowing symmetry breaking leads to the loss of good quantum nos. It has long been recognized that quasidegeneracies can instead be treated by modifying the coupled cluster ansatz. The recently introduced pair coupled cluster doubles (pCCD) approach is one such example which avoids catastrophic failures and accurately models strong correlations in a symmetry-adapted framework. Here, we generalize pCCD to a singlet-paired coupled cluster model (CCD0) intermediate between coupled cluster doubles and pCCD, yielding a method that possesses the invariances of the former and much of the stability of the latter. Moreover, CCD0 retains the full structure of coupled cluster theory, including a fermionic wave function, antisym. cluster amplitudes, and well-defined response equations and d. matrixes.
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    74. 74
      Karton, A.; Rabinovich, E.; Martin, J. M.; Ruscic, B. W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions. J. Chem. Phys. 2006, 125, 144108,  DOI: 10.1063/1.2348881
      74
      W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions
      Karton, Amir; Rabinovich, Elena; Martin, Jan M. L.; Ruscic, Branko
      Journal of Chemical Physics (2006), 125 (14), 144108/1-144108/17CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      In an attempt to improve on our earlier W3 theory [A. D. Boese et al., J. Chem. Phys. 120, 4129 (2004)] we consider such refinements as more accurate ests. for the contribution of connected quadruple excitations (T4), inclusion of connected quintuple excitations (T5), diagonal Born-Oppenheimer corrections (DBOC), and improved basis set extrapolation procedures. Revised exptl. data for validation purposes were obtained from the latest version of the Active Thermochem. Tables thermochem. network. The recent CCSDT(Q) method offers a cost-effective way of estg. T4, but is insufficient by itself if the mol. exhibits some nondynamical correlation. The latter considerably slows down basis set convergence for T4, and anomalous basis set convergence in highly polar systems makes two-point extrapolation procedures unusable. However, we found that the CCSDTQ-CCSDT(Q) difference converges quite rapidly with the basis set, and that the formula 1.10[CCSDT(Q)/cc-pVTZ + CCSDTQ/cc-pVDZ - CCSDT(Q)/cc-pVDZ] offers a very reliable as well as fairly cost-effective est. of the basis set limit T4 contribution. The T5 contribution converges very rapidly with the basis set, and even a simple double-zeta basis set appears to be adequate. The largest T5 contribution found in the present work is on the order of 0.5 kcal/mol (for ozone). DBOCs are significant at the 0.1 kcal/mol level in hydride systems. Post-CCSD(T) contributions to the core-valence correlation energy are only significant at that level in systems with severe nondynamical correlation effects. Based on the accumulated experience, a new computational thermochem. protocol for first- and second-row main-group systems, to be known as W4 theory, is proposed. Its computational cost is not insurmountably higher than that of the earlier W3 theory, while performance is markedly superior. Our W4 atomization energies for a no. of key species are in excellent agreement (better than 0.1 kcal/mol on av., 95% confidence intervals narrower than 1 kJ/mol) with the latest exptl. data obtained from Active Thermochem. Tables. Lower-cost variants are proposed: the sequence W1 → W2.2 → W3.2 → W4lite → W4 is proposed as a converging hierarchy of computational thermochem. methods. A simple a priori est. for the importance of post-CCSD(T) correlation contributions (and hence a pessimistic est. for the error in a W2-type calcn.) is proposed.
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    75. 75
      Hait, D.; Tubman, N. M.; Levine, D. S.; Whaley, K. B.; Head-Gordon, M. What levels of coupled cluster theory are appropriate for transition metal systems? A study using near-exact quantum chemical values for 3d transition metal binary compounds. J. Chem. Theory Comput. 2019, 15, 5370– 5385,  DOI: 10.1021/acs.jctc.9b00674
      75
      What Levels of Coupled Cluster Theory Are Appropriate for Transition Metal Systems? A Study Using Near-Exact Quantum Chemical Values for 3d Transition Metal Binary Compounds
      Hait, Diptarka; Tubman, Norman M.; Levine, Daniel S.; Whaley, K. Birgitta; Head-Gordon, Martin
      Journal of Chemical Theory and Computation (2019), 15 (10), 5370-5385CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Transition metal compds. are traditionally considered to be challenging for std. quantum chem. approxns. like coupled cluster (CC) theory, which are usually employed to validate lower level methods like d. functional theory (DFT). To explore this issue, we present a database of bond dissocn. energies (BDEs) for 74 spin states of 69 diat. species contg. a 3d transition metal atom and a main group element, in the moderately sized def2-SVP basis. The presented BDEs appear to have an (estd.) 3σ error less than 1 kJ/mol relative to the exact solns. to the nonrelativistic Born-Oppenheimer Hamiltonian. These benchmark values were used to assess the performance of a wide range of std. single ref. CC models, as the results should be beneficial for understanding the limitations of these models for transition metal systems. We find that interactions between metals and monovalent ligands like hydride and fluoride are well described by CCSDT. Similarly, CCSDTQ appears to be adequate for bonds between metals and nominally divalent ligands like oxide and sulfide. However, interactions with polyvalent ligands like nitride and carbide are more challenging, with even CCSDTQ(P)Λ yielding errors on the scale of a few kJ/mol. We also find that many perturbative and iterative approxns. to higher order terms either yield disappointing results or actually worsen the performance relative to the baseline low level CC method, indicating that complexity does not always guarantee accuracy.
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    76. 76
      Roos, B. O.; Taylor, P. R.; Sigbahn, P. E. A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys. 1980, 48, 157– 173,  DOI: 10.1016/0301-0104(80)80045-0
      76
      A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach
      Roos, Bjoern O.; Taylor, Peter R.; Siegbahn, E. M.
      Chemical Physics (1980), 48 (2), 157-73CODEN: CMPHC2; ISSN:0301-0104.
      A d. matrix formulation of the super-CI MCSCF method is presented. The MC expansion is assumed to be complete in an active subset of the orbital space, and the corresponding CI secular problem is solved by a direct scheme using the unitary group approach. With a d. matrix formulation the orbital optimization step becomes independent of the size of the CI expansion. It is possible to formulate the super-CI in terms of d. matrices defined only in the small active subspace; the doubly occupied orbitals (the inactive subspace) do not enter. Further, in the unitary group formalism it is straightforward and simple to obtain the necessary d. matrices from the symbolic formula list. It then becomes possible to treat very long MC expansions, the largest so far comprising 726 configurations. The method is demonstrated in a calcn. of the potential curves for the 3 lowest states (1.sum.g+, 3.sum.u+ and 3πg) of the N2 mol., using a medium-sized gaussian basis set. 7 Active orbitals were used yielding the following results: Dc:8.76(9.90), 2.43(3.68) and 3.39 (4.90) eV; rc:1.108 (1.098), 1.309(1.287) and 1.230 (1.213) Å; ωe: 2333 (2359), 1385 (1461) and 1680 (1733) cm-1, for the 3 states (exptl. values within parentheses). The results of these calcns. indicate that it is important to consider not only the dissocn. limit but also the united atom limit in partitioning the occupied orbital space into an active and an inactive part.
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    77. 77
      Chan, G. K.-L.; Sharma, S. The density matrix renormalization group in quantum chemistry. Annu. Rev. Phys. Chem. 2011, 62, 465– 481,  DOI: 10.1146/annurev-physchem-032210-103338
      77
      The density matrix renormalization group in quantum chemistry
      Chan, Garnet Kin-Lic; Sharma, Sandeep
      Annual Review of Physical Chemistry (2011), 62 (), 465-481CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)
      A review. The d. matrix renormalization group is a method that is useful for describing mols. that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the d. matrix renormalization group works, a survey of its existing applications to mol. problems, and some thoughts on the future of the method.
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    78. 78
      Mouesca, J.-M. Density functional theory-broken symmetry (DFT-BS) methodology applied to electronic and magnetic properties of bioinorganic prosthetic groups. In Metalloproteins; Springer: New York, 2014; pp 269– 296.
      There is no corresponding record for this reference.
    79. 79
      Scuseria, G. E.; Jiménez-Hoyos, C. A.; Henderson, T. M.; Samanta, K.; Ellis, J. K. Projected quasiparticle theory for molecular electronic structure. J. Chem. Phys. 2011, 135, 124108,  DOI: 10.1063/1.3643338
      79
      Projected quasiparticle theory for molecular electronic structure
      Scuseria, Gustavo E.; Jimenez-Hoyos, Carlos A.; Henderson, Thomas M.; Samanta, Kousik; Ellis, Jason K.
      Journal of Chemical Physics (2011), 135 (12), 124108/1-124108/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We derive and implement symmetry-projected Hartree-Fock-Bogoliubov (HFB) equations and apply them to the mol. electronic structure problem. All symmetries (particle no., spin, spatial, and complex conjugation) are deliberately broken and restored in a self-consistent variation-after-projection approach. We show that the resulting method yields a comprehensive black-box treatment of static correlations with effective one-electron (mean-field) computational cost. The ensuing wave function is of multireference character and permeates the entire Hilbert space of the problem. The energy expression is different from regular HFB theory but remains a functional of an independent quasiparticle d. matrix. All reduced d. matrixes are expressible as an integration of transition d. matrixes over a gauge grid. We present several proof-of-principle examples demonstrating the compelling power of projected quasiparticle theory for quantum chem. (c) 2011 American Institute of Physics.
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    80. 80
      Helgaker, T.; Jorgensen, P.; Olsen, J. Molecular electronic-structure theory; John Wiley & Sons: Chichester, 2000.
      There is no corresponding record for this reference.
    81. 81
      Shavitt, I.; Bartlett, R. J. Many-body methods in chemistry and physics: MBPT and coupled-cluster theory; Cambridge University Press: Cambridge, 2009.
      There is no corresponding record for this reference.
    82. 82
      Čížek, J. On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods. J. Chem. Phys. 1966, 45, 4256– 4266,  DOI: 10.1063/1.1727484
      82
      Correlation problem in atomic and molecular systems. Calculation of wave function components in ursell-type expansion using quantum-field theoretical methods
      Cizek, Jiri
      Journal of Chemical Physics (1966), 45 (11), 4256-66CODEN: JCPSA6; ISSN:0021-9606.
      A method is suggested for the calcn. of the matrix elements of the logarithm of an operator which gives the exact wave function when operating on the wave function in the 1-electron approxn. The method is based on the use of the creation and annihilation operators, hole-particle formalism, Wick's theorem, and the technique of Feynman-like diagrams. The connection of this method with the configuration interaction method as well as with the perturbation theory in the quantum-field theoretical form is discussed. The method is applied to the simple models of N and C6H6 mols. The results are compared with those obtained with the configuration-interaction method considering all possible configurations within the chosen basis of 1-electron functions.
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    83. 83
      Paldus, J.; Čížek, J.; Shavitt, I. Correlation Problems in Atomic and Molecular Systems. IV. Extended Coupled-Pair Many-Electron Theory and Its Application to the BH3 Molecule. Phys. Rev. A 1972, 5, 50– 67,  DOI: 10.1103/PhysRevA.5.50
      There is no corresponding record for this reference.
    84. 84
      Arponen, J. Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems. Ann. Phys. 1983, 151, 311– 382,  DOI: 10.1016/0003-4916(83)90284-1
      84
      Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems
      Arponen, Jouko
      Annals of Physics (San Diego, CA, United States) (1983), 151 (2), 311-82CODEN: APNYA6; ISSN:0003-4916.
      The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed by using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using 2 different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the av. values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes.
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    85. 85
      Faulstich, F. M.; Laestadius, A.; Legeza, O.; Schneider, R.; Kvaal, S. Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry. SIAM J. Numer. Anal. 2019, 57, 2579– 2607,  DOI: 10.1137/18M1171436
      There is no corresponding record for this reference.
    86. 86
      Csirik, M. A.; Laestadius, A. Coupled-cluster theory revisited. arXiv:2105.13134 2021, DOI: 10.48550/arXiv.2105.13134 .
      There is no corresponding record for this reference.
    87. 87
      Schütz, M.; Werner, H.-J. Low-order scaling local electron correlation methods. IV. Linear scaling local coupled-cluster (LCCSD). J. Chem. Phys. 2001, 114, 661– 681,  DOI: 10.1063/1.1330207
      87
      Low-order scaling local electron correlation methods. IV. Linear scaling local coupled-cluster (LCCSD)
      Schutz, Martin; Werner, Hans-Joachim
      Journal of Chemical Physics (2001), 114 (2), 661-681CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A new implementation of local coupled-cluster theory with single and double excitations (LCCSD) is presented for which asymptotically all computational resources (CPU, memory, and disk) scale only linearly with the mol. size. This is achieved by: (i) restricting the correlation space for each electron pair to domains that are independent of mol. size; (ii) classifying the pairs according to a distance criterion and treating only strong pairs at the highest level; (iii) using efficient pre-screening algorithms in the integral transformation and other integral-direct procedures; and (iv) neglect of small couplings of electron pairs that are far apart from each other. The errors caused by the various approxns. are negligible. LCCSD calcns. on mols. including up to 300 correlated electrons and over 1000 basis functions in C1 symmetry are reported, all carried out on a workstation.
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    88. 88
      Riplinger, C.; Neese, F. An efficient and near linear scaling pair natural orbital based local coupled cluster method. J. Chem. Phys. 2013, 138, 034106,  DOI: 10.1063/1.4773581
      88
      An efficient and near linear scaling pair natural orbital based local coupled cluster method
      Riplinger, Christoph; Neese, Frank
      Journal of Chemical Physics (2013), 138 (3), 034106/1-034106/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      In previous publications, it was shown that an efficient local coupled cluster method with single- and double excitations can be based on the concept of pair natural orbitals (PNOs) . The resulting local pair natural orbital-coupled-cluster single double (LPNO-CCSD) method has since been proven to be highly reliable and efficient. For large mols., the no. of amplitudes to be detd. is reduced by a factor of 105-106 relative to a canonical CCSD calcn. on the same system with the same basis set. In the original method, the PNOs were expanded in the set of canonical virtual orbitals and single excitations were not truncated. This led to a no. of fifth order scaling steps that eventually rendered the method computationally expensive for large mols. (e.g., >100 atoms). In the present work, these limitations are overcome by a complete redesign of the LPNO-CCSD method. The new method is based on the combination of the concepts of PNOs and projected AOs (PAOs). Thus, each PNO is expanded in a set of PAOs that in turn belong to a given electron pair specific domain. In this way, it is possible to fully exploit locality while maintaining the extremely high compactness of the original LPNO-CCSD wavefunction. No terms are dropped from the CCSD equations and domains are chosen conservatively. The correlation energy loss due to the domains remains below <0.05%, which implies typically 15-20 but occasionally up to 30 atoms per domain on av. The new method has been given the acronym DLPNO-CCSD ("domain based LPNO-CCSD"). The method is nearly linear scaling with respect to system size. The original LPNO-CCSD method had three adjustable truncation thresholds that were chosen conservatively and do not need to be changed for actual applications. In the present treatment, no addnl. truncation parameters have been introduced. Any addnl. truncation is performed on the basis of the three original thresholds. There are no real-space cutoffs. Single excitations are truncated using singles-specific natural orbitals. Pairs are prescreened according to a multipole expansion of a pair correlation energy est. based on local orbital specific virtual orbitals (LOSVs). Like its LPNO-CCSD predecessor, the method is completely of black box character and does not require any user adjustments. It is shown here that DLPNO-CCSD is as accurate as LPNO-CCSD while leading to computational savings exceeding one order of magnitude for larger systems. The largest calcns. reported here featured >8800 basis functions and >450 atoms. In all larger test calcns. done so far, the LPNO-CCSD step took less time than the preceding Hartree-Fock calcn., provided no approxns. have been introduced in the latter. Thus, based on the present development reliable CCSD calcns. on large mols. with unprecedented efficiency and accuracy are realized. (c) 2013 American Institute of Physics.
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    89. 89
      Shiozaki, T.; Valeev, E. F.; Hirata, S. Explicitly correlated coupled-cluster methods. In Annual Reports in Computational Chemistry; Elsevier: Amsterdam, 2009; Vol. 5, Chapter 6, pp 131– 148.
      There is no corresponding record for this reference.
    90. 90
      Izsák, R. Single-reference coupled cluster methods for computing excitation energies in large molecules: The efficiency and accuracy of approximations. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2020, 10, e1445,  DOI: 10.1002/wcms.1445
      90
      Single-reference coupled cluster methods for computing excitation energies in large molecules: The efficiency and accuracy of approximations
      Izsak, Robert
      Wiley Interdisciplinary Reviews: Computational Molecular Science (2020), 10 (3), e1445CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell)
      A review. While methodol. developments in the last decade made it possible to compute coupled cluster (CC) energies including excitations up to a perturbative triples correction for mols. contg. several hundred atoms, a similar breakthrough has not yet been reported for excited state computations. Accurate CC methods for excited states are still expensive, although some promising candidates for an efficient and accurate excited state CC method have emerged recently. This review examines the various approxn. schemes with particular emphasis on their performance for excitation energies and summarizes the best state-of-the-art results which may pave the way for a robust excited state method applicable to mols. of hundreds of atoms. Among these, special attention will be given to exploiting the techniques of similarity transformation, perturbative approxns. as well as integral decompn., local and embedding techniques within the equation of motion CC framework. This article is categorized under:Electronic Structure Theory > Ab Initio Electronic Structure Methods Structure and Mechanism > Mol. Structures.
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    91. 91
      Lyakh, D. I.; Musiał, M.; Lotrich, V. F.; Bartlett, R. J. Multireference nature of chemistry: The coupled-cluster view. Chem. Rev. 2012, 112, 182– 243,  DOI: 10.1021/cr2001417
      91
      Multireference Nature of Chemistry: The Coupled-Cluster View
      Lyakh, Dmitry I.; Musial, Monika; Lotrich, Victor F.; Bartlett, Rodney J.
      Chemical Reviews (Washington, DC, United States) (2012), 112 (1), 182-243CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
      A review. The following topics are discussed: Exponential era of electron correlation theory; Genuine MR CC theory in Hilbert space and in Fock space; Alternative MR CC methods. Numerical illustrations are presented.
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    92. 92
      Huron, B.; Malrieu, J. P.; Rancurel, P. Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions. J. Chem. Phys. 1973, 58, 5745– 5759,  DOI: 10.1063/1.1679199
      92
      Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions
      Huron, B.; Malrieu, J. P.; Rancurel, P.
      Journal of Chemical Physics (1973), 58 (12), 5745-59CODEN: JCPSA6; ISSN:0021-9606.
      A method is proposed for calcg. the effect of configuration interaction by a Rayleigh Schroedinger perturbation expansion when starting from a multiconfigurational wavefunction. A careless choice of H0 may lead to absurd transition energies between 2 states, at the 1st orders of the perturbation, even when the perturbation converges for both states. A barycentric definition of H0 is proposed, which ensures the cancellation of common diagrams in the calcd. transition energies. A practical iterative procedure is defined which allows a progressive improvement of the unperturbed wavefunction ψ0; the CI (configuration interaction) matrix restricted to a subspace S of strongly interacting determinants is diagonalized. The desired eigenvector ψ0 of this matrix is perturbed by the determinants which do not belong to S. The most important determinants in ψ1 are added to S, etc. The energy thus obtained after the 2nd-order correction is compared with the ordinary perturbation series where ψ0 is a single determinant. For the ground state, this procedure includes, besides the whole 2nd-order correction, the most important terms of the 3rd and 4th orders. The question of orthogonality of excited states is discussed. This technique was tested on the ground and several excited states of H2, Ne, and MgO, showing both a rapid convergence of the calcd. transition energy and the importance of correlation effects on transition energy.
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    93. 93
      Austin, B. M.; Zubarev, D. Y.; Lester Jr, W. A. Quantum Monte Carlo and related approaches. Chem. Rev. 2012, 112, 263– 288,  DOI: 10.1021/cr2001564
      93
      Quantum Monte Carlo and Related Approaches
      Austin, Brian M.; Zubarev, Dmitry Yu.; Lester, William A., Jr.
      Chemical Reviews (Washington, DC, United States) (2012), 112 (1), 263-288CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
      A review. The following topics are discussed: Variational Monte Carlo, Fixed-node diffusion Monte Carlo, Self-healing diffusion Monte Carlo, Auxiliary field quantum Monte Carlo, Reptation quantum Monte Carlo, Full CI quantum Monte Carlo, Time-dependent quantum Monte Carlo; Trial electronic wave functions (antisym., backflow transformed, Jastrow), Trial wave function optimization, Effective core potentials; Computational considerations (Linear scaling quantum Monte Carlo, Parallelization and hardware acceleration, Advances in algorithms and software); Applications.
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    94. 94
      Booth, G. H.; Thom, A. J.; Alavi, A. Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space. J. Chem. Phys. 2009, 131, 054106,  DOI: 10.1063/1.3193710
      94
      Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space
      Booth, George H.; Thom, Alex J. W.; Alavi, Ali
      Journal of Chemical Physics (2009), 131 (5), 054106/1-054106/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We have developed a new quantum Monte Carlo method for the simulation of correlated many-electron systems in full configuration-interaction (Slater determinant) spaces. The new method is a population dynamics of a set of walkers, and is designed to simulate the underlying imaginary-time Schrodinger equation of the interacting Hamiltonian. The walkers (which carry a pos. or neg. sign) inhabit Slater determinant space, and evolve according to a simple set of rules which include spawning, death and annihilation processes. We show that this method is capable of converging onto the full configuration-interaction (FCI) energy and wave function of the problem, without any a priori information regarding the nodal structure of the wave function being provided. Walker annihilation is shown to play a key role. The pattern of walker growth exhibits a characteristic plateau once a crit. (system-dependent) no. of walkers has been reached. At this point, the correlation energy can be measured using two independent methods-a projection formula and a energy shift; agreement between these provides a strong measure of confidence in the accuracy of the computed correlation energies. We have verified the method by performing calcns. on systems for which FCI calcns. already exist. In addn., we report on a no. of new systems, including CO, O2, CH4, and NaH-with FCI spaces ranging from 109 to 1014, whose FCI energies we compute using modest computational resources. (c) 2009 American Institute of Physics.
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    95. 95
      Petruzielo, F. R.; Holmes, A. A.; Changlani, H. J.; Nightingale, M. P.; Umrigar, C. J. Semistochastic Projector Monte Carlo Method. Phys. Rev. Lett. 2012, 109, 230201,  DOI: 10.1103/PhysRevLett.109.230201
      95
      Semistochastic projector monte carlo method
      Petruzielo, F. R.; Holmes, A. A.; Changlani, Hitesh J.; Nightingale, M. P.; Umrigar, C. J.
      Physical Review Letters (2012), 109 (23), 230201/1-230201/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We introduce a semistochastic implementation of the power method to compute, for very large matrixes, the dominant eigenvalue and expectation values involving the corresponding eigenvector. The method is semistochastic in that the matrix multiplication is partially implemented numerically exactly and partially stochastically with respect to expectation values only. Compared to a fully stochastic method, the semistochastic approach significantly reduces the computational time required to obtain the eigenvalue to a specified statistical uncertainty. This is demonstrated by the application of the semistochastic quantum Monte Carlo method to systems with a sign problem: the fermion Hubbard model and the carbon dimer.
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    96. 96
      Spencer, J. S.; Blunt, N. S.; Foulkes, W. M. The sign problem and population dynamics in the full configuration interaction quantum Monte Carlo method. J. Chem. Phys. 2012, 136, 054110,  DOI: 10.1063/1.3681396
      96
      The sign problem and population dynamics in the full configuration interaction quantum Monte Carlo method
      Spencer, J. S.; Blunt, N. S.; Foulkes, W. M. C.
      Journal of Chemical Physics (2012), 136 (5), 054110/1-054110/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The recently proposed full CI quantum Monte Carlo method allows access to essentially exact ground-state energies of systems of interacting fermions substantially larger than previously tractable without knowledge of the nodal structure of the ground-state wave function. We investigate the nature of the sign problem in this method and how its severity depends on the system studied. We explain how cancellation of the pos. and neg. particles sampling the wave function ensures convergence to a stochastic representation of the many-fermion ground state and accounts for the characteristic population dynamics obsd. in simulations. (c) 2012 American Institute of Physics.
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    97. 97
      Cleland, D.; Booth, G. H.; Alavi, A. Communications: Survival of the fittest: Accelerating convergence in full configuration-interaction quantum Monte Carlo. J. Chem. Phys. 2010, 132, 041103,  DOI: 10.1063/1.3302277
      97
      Communications: Survival of the fittest: Accelerating convergence in full configuration-interaction quantum Monte Carlo
      Cleland, Deidre; Booth, George H.; Alavi, Ali
      Journal of Chemical Physics (2010), 132 (4), 041103/1-041103/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We provide a very simple adaptation of our recently published quantum Monte Carlo algorithm in full configuration-interaction (Slater determinant) spaces which dramatically reduces the no. of walkers required to achieve convergence. A survival criterion is imposed for newly spawned walkers. We define a set of initiator determinants such that progeny of walkers spawned from such determinants onto unoccupied determinants are able to survive, while the progeny of walkers not in this set can survive only if they are spawned onto determinants which are already occupied. The set of initiators is originally defined to be all determinants constructable from a subset of orbitals, in analogy with complete-active spaces. This set is dynamically updated so that if a non-initiator determinant reaches an occupation larger than a preset limit, it becomes an initiator. The new algorithm allows sign-coherent sampling of the FCI space to be achieved with relatively few walkers. Using the N2 mol. as an illustration, we show that rather small initiator spaces and nos. of walkers can converge with submilli-Hartree accuracy to the known full configuration-interaction (FCI) energy (in the cc-pVDZ basis), in both the equil. geometry and the multiconfigurational stretched case. We use the same method to compute the energy with cc-pVTZ and cc-pVQZ basis sets, the latter having an FCI space of over 1015 with very modest computational resources. (c) 2010 American Institute of Physics.
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    98. 98
      Cleland, D. M.; Booth, G. H.; Alavi, A. A study of electron affinities using the initiator approach to full configuration interaction quantum Monte Carlo. J. Chem. Phys. 2011, 134, 024112,  DOI: 10.1063/1.3525712
      98
      A study of electron affinities using the initiator approach to full configuration interaction quantum Monte Carlo
      Cleland, D. M.; Booth, George H.; Alavi, Ali
      Journal of Chemical Physics (2011), 134 (2), 024112/1-024112/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      For the atoms with Z ≤ 11, energies obtained using the "initiator" extension to full CI quantum Monte Carlo (i-FCIQMC) come to within statistical errors of the FCIQMC results. As these FCIQMC values have been shown to converge onto FCI results, the i-FCIQMC method allows similar accuracy to be achieved while significantly reducing the scaling with the size of the Slater determinant space. The i-FCIQMC electron affinities of the Z ≤ 11 atoms in the aug-cc-pVXZ basis sets are presented here. In every case, values are obtained to well within chem. accuracy the mean abs. deviation (MAD) from the relativistically cor. exptl. values is 0.41 mEh, and significantly improve on coupled cluster with singles, doubles and perturbative triples CCSD(T) results. Since the only remaining source of error is basis set incompleteness, we have investigated using CCSD(T)-F12 contributions to correct the i-FCIQMC results. By doing so, much faster convergence with respect to basis set size may be achieved for both the electron affinities and the FCIQMC ionization potentials presented in a previous paper. With this F12 correction, the MAD can be further reduced to 0.13 mEh for the electron affinities and 0.31 mEh for the ionization potentials. (c) 2011 American Institute of Physics.
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    99. 99
      Eriksen, J. J. The shape of full configuration interaction to come. J. Phys. Chem. Lett. 2021, 12, 418– 432,  DOI: 10.1021/acs.jpclett.0c03225
      99
      The Shape of Full Configuration Interaction to Come
      Eriksen, Janus J.
      Journal of Physical Chemistry Letters (2021), 12 (1), 418-432CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      A review. We present a Perspective on what the future holds for full CI (FCI) theory, with an emphasis on conceptual rather than tech. details. Upon revisiting the early history of FCI, a no. of its key contemporary approxns. are compared on as equal a footing as possible, using a recent blind challenge on the benzene mol. as a testbed [Eriksen et al., J. Phys. Chem. Lett, 2020, 11, 8920]. In the process, we review the scope of applications for which FCI continues to prove indispensable, and the required traits in terms of robustness, efficacy, and reliability its modern approxns. must satisfy are discussed. We close by conveying a no. of general observations on the merits offered by the state-of-the-art alongside some of the challenges still faced to this day. While the field has altogether seen immense progress over the years-the past decade, in particular-it remains clear that our community as a whole has a substantial way to go in enhancing the overall applicability of near-exact electronic structure theory for systems of general compn. and increasing size.
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    100. 100
      Eriksen, J. J.; Gauss, J. Many-body expanded full configuration interaction. I. Weakly correlated regime. J. Chem. Theory Comput. 2018, 14, 5180– 5191,  DOI: 10.1021/acs.jctc.8b00680
      100
      Many-Body Expanded Full Configuration Interaction. I. Weakly Correlated Regime
      Eriksen, Janus J.; Gauss, Juergen
      Journal of Chemical Theory and Computation (2018), 14 (10), 5180-5191CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Over the course of the past few decades, the field of computational chem. has managed to manifest itself as a key complement to more traditional lab-oriented chem. This is particularly true in the wake of the recent renaissance of full CI (FCI)-level methodologies, albeit only if these can prove themselves sufficiently robust and versatile to be routinely applied to a variety of chem. problems of interest. In the present series of works, performance and feature enhancements of one such avenue toward FCI-level results for medium to large one-electron basis sets, the recently introduced many-body expanded full CI (MBE-FCI) formalism [J. Phys. Chem. Lett. 2017, 8, 4633], will be presented. Specifically, in this opening part of the series, the capabilities of the MBE-FCI method in producing near-exact ground state energies for weakly correlated mols. of any spin multiplicity will be demonstrated.
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    101. 101
      Eriksen, J. J.; Gauss, J. Many-body expanded full configuration interaction. II. Strongly correlated regime. J. Chem. Theory Comput. 2019, 15, 4873– 4884,  DOI: 10.1021/acs.jctc.9b00456
      101
      Many-Body Expanded Full Configuration Interaction. II. Strongly Correlated Regime
      Eriksen, Janus J.; Gauss, Juergen
      Journal of Chemical Theory and Computation (2019), 15 (9), 4873-4884CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      In this second part of our series on the recently proposed many-body expanded full CI (MBE-FCI) method, we introduce the concept of multideterminantal expansion refs. Through theor. arguments and numerical validations, the use of this class of starting points is shown to result in a focused compression of the MBE decompn. of the FCI energy, thus allowing chem. problems dominated by strong correlation to be addressed by the method. The general applicability and performance enhancements of MBE-FCI are verified for std. stress tests such as the bond dissocns. in H2O, N2, C2, and a linear H10 chain. Furthermore, the benefits of employing a multideterminantal expansion ref. in accelerating calcns. of high accuracy are discussed, with an emphasis on calcns. in extended basis sets. As an illustration of this latter quality of the MBE-FCI method, results for H2O and C2 in basis sets ranging from double- to pentuple-ζ quality are presented, demonstrating near-ideal parallel scaling on up to almost 25000 processing units.
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    102. 102
      White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 1992, 69, 2863– 2866,  DOI: 10.1103/PhysRevLett.69.2863
      102
      Density matrix formulation for quantum renormalization groups
      White
      Physical review letters (1992), 69 (19), 2863-2866 ISSN:.
      There is no expanded citation for this reference.
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    103. 103
      Chilkuri, V. G.; Neese, F. Comparison of many-particle representations for selected-CI I: A tree based approach. J. Comput. Chem. 2021, 42, 982– 1005,  DOI: 10.1002/jcc.26518
      103
      Comparison of many-particle representations for selected-CI I: A tree based approach
      Chilkuri, Vijay Gopal; Neese, Frank
      Journal of Computational Chemistry (2021), 42 (14), 982-1005CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      The full CI (FCI) method is only applicable to small mols. with few electrons in moderate size basis sets. One of the main alternatives to obtain approx. FCI energies for bigger mols. and larger basis sets is selected CI. However, due to: (a) the lack of a well-defined structure in a selected CI Hamiltonian, (b) the potentially large no. of electrons together with (c) potentially large orbital spaces, a computationally and memory efficient algorithm is difficult to construct. In the present series of papers, we describe our attempts to address these issues by exploring tree-based approaches. At the same time, we devote special attention to the issue of obtaining eigenfunctions of the total spin squared operator since this is of particular importance in tackling magnetic properties of complex open shell systems. Dedicated algorithms are designed to tackle the CI problem in terms of determinant, configuration (CFG) and configuration state function many-particle bases by effective use of the tree representation. In this paper we describe the underlying logic of our algorithm design and discuss the advantages and disadvantages of the different many particle bases. We demonstrate by the use of small examples how the use of the tree simplifies many key algorithms required for the design of an efficient selected CI program. Our selected CI algorithm, called the iterative configuration expansion, is presented in the penultimate part. Finally, we discuss the limitations and scaling characteristics of the present approach.
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    104. 104
      Li Manni, G.; Dobrautz, W.; Alavi, A. Compression of spin-adapted multiconfigurational wave functions in exchange-coupled polynuclear spin systems. J. Chem. Theory Comput. 2020, 16, 2202– 2215,  DOI: 10.1021/acs.jctc.9b01013
      104
      Compression of Spin-Adapted Multiconfigurational Wave Functions in Exchange-Coupled Polynuclear Spin Systems
      Li Manni, Giovanni; Dobrautz, Werner; Alavi, Ali
      Journal of Chemical Theory and Computation (2020), 16 (4), 2202-2215CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We present a protocol based on unitary transformations of MOs to reduce the no. of nonvanishing coeffs. of spin-adapted CI expansions. Methods that exploit the sparsity of the Hamiltonian matrix and compactness of its eigensolns., such as the full CI quantum Monte Carlo (FCIQMC) algorithm in its spin-adapted implementation, are well suited to this protocol. The wave function compression resulting from this approach is particularly attractive for antiferromagnetically coupled polynuclear spin systems, such as transition-metal cubanes in biocatalysis, and Mott and charge-transfer insulators in solid-state physics. Active space CI calcns. on N2 and CN- at various bond lengths, the stretched square N4 compds., the chromium dimer, and a [Fe2S2]2- model system are presented as a proof-of-concept. For the Cr2 case, large and intermediate bond distances are discussed, showing that the approach is effective in cases where static and dynamic correlations are equally important. The [Fe2S2]2- case shows the general applicability of the method.
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    105. 105
      Li Manni, G. Modeling magnetic interactions in high-valent trinuclear [Mn3(IV)O4]4+ complexes through highly compressed multi-configurational wave functions. Phys. Chem. Chem. Phys. 2021, 23, 19766– 19780,  DOI: 10.1039/D1CP03259C
      105
      Modeling magnetic interactions in high-valent trinuclear [Mn3(IV)O4]4+ complexes through highly compressed multi-configurational wave functions
      Li Manni, Giovanni
      Physical Chemistry Chemical Physics (2021), 23 (35), 19766-19780CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      In this work we apply a quantum chem. framework, recently designed in our labs., to rationalize the low-energy electronic spectrum and the magnetic properties of an homo-valent trinuclear [Mn3(IV)O4]4+ model of the oxygen-evolving center in photosystem II. The method is based on chem. motivated MO unitary transformations, and the optimization of spin-adapted many-body wave functions, both for ground- and excited-states, in the transformed MO basis. In this basis, the CI Hamiltonian matrix of exchange-coupled multi-center clusters is extremely sparse and characterized by a unique block diagonal structure. This property leads to highly compressed wave functions (oligo- or single-ref.) and crucially enables state-specific optimizations. This work is the first showing that compression and selective targeting of ground- and excited-states wave functions is possible for systems with three magnetic centers that are not exactly half-filled, and that potentially exhibit frustrated spin interactions. The reduced multi-ref. character of the wave function greatly simplifies the interpretation of the ground- and excited-state electronic structures, and provides a route for the direct rationalization of magnetic interactions in these compds., often considered a challenge in polynuclear transition-metal chem. In this study, strong electron correlation effects have explicitly been described by conventional and stochastic multiconfigurational methodologies, while dynamic correlation effects have been accounted for by multiconfigurational second order perturbation theory, CASPT2. Ab initio results for the [Mn3(IV)O4]4+ system have been mapped to a three-site Heisenberg model with two magnetic coupling consts. The magnetic coupling consts. and the temp. dependence of the effective magnetic moment predicted by the ab initio calcns. are in good agreement with the available exptl. data, and confirm the antiferromagnetic interaction among the three magnetic centers, while providing a simple and rigorous description of the noncollinearity of the local spins, that characterize most of the low-energy states for this system.
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    106. 106
      Li Manni, G.; Dobrautz, W.; Bogdanov, N. A.; Guther, K.; Alavi, A. Resolution of low-energy states in spin-exchange transition-metal clusters: Case study of singlet states in [Fe(III)4S4] cubanes. J. Phys. Chem. A 2021, 125, 4727– 4740,  DOI: 10.1021/acs.jpca.1c00397
      106
      Resolution of Low-Energy States in Spin-Exchange Transition-Metal Clusters: Case Study of Singlet States in [Fe(III)4S4] Cubanes
      Li Manni, Giovanni; Dobrautz, Werner; Bogdanov, Nikolay A.; Guther, Kai; Alavi, Ali
      Journal of Physical Chemistry A (2021), 125 (22), 4727-4740CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      Polynuclear transition-metal (PNTM) clusters owe their catalytic activity to numerous energetically low-lying spin states and stable oxidn. states. The characterization of their electronic structure represents one of the greatest challenges of modern chem. We propose a theor. framework that enables the resoln. of targeted electronic states with ease and apply it to two [Fe(III)4S4] cubanes. Through direct access to their many-body wave functions, we identify important correlation mechanisms and their interplay with the geometrical distortions obsd. in these clusters, which are core properties in understanding their catalytic activity. The simulated magnetic coupling consts. predicted by our strategy allow us to make qual. connections between spin interactions and geometrical distortions, demonstrating its predictive power. Moreover, despite its simplicity, the strategy provides magnetic coupling consts. in good agreement with the available exptl. ones. The complexes are intrinsically frustrated anti-ferromagnets, and the obtained spin structures together with the geometrical distortions represent two possible ways to release spin frustration (spin-driven Jahn-Teller distortion). Our paradigm provides a simple, yet rigorous, route to uncover the electronic structure of PNTM clusters and may be applied to a wide variety of such clusters.
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    107. 107
      Shee, J.; Loipersberger, M.; Hait, D.; Lee, J.; Head-Gordon, M. Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition. J. Chem. Phys. 2021, 154, 194109,  DOI: 10.1063/5.0047386
      107
      Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition
      Shee, James; Loipersberger, Matthias; Hait, Diptarka; Lee, Joonho; Head-Gordon, Martin
      Journal of Chemical Physics (2021), 154 (19), 194109CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      In this work, we provide a nuanced view of electron correlation in the context of transition metal complexes, reconciling computational characterization via spin and spatial symmetry breaking in single-ref. methods with qual. concepts from ligand-field and MO theories. These insights provide the tools to reliably diagnose the multi-ref. character, and our anal. reveals that while strong (i.e., static) correlation can be found in linear mols. (e.g., diatomics) and weakly bound and antiferromagnetically coupled (monometal-noninnocent ligand or multi-metal) complexes, it is rarely found in the ground-states of mono-transition-metal complexes. This leads to a picture of static correlation that is no more complex for transition metals than it is, e.g., for org. biradicaloids. In contrast, the ability of organometallic species to form more complex interactions, involving both ligand-to-metal σ-donation and metal-to-ligand π-backdonation, places a larger burden on a theory's treatment of dynamic correlation. We hypothesize that chem. bonds in which inter-electron pair correlation is non-negligible cannot be adequately described by theories using MP2 correlation energies and indeed find large errors vs expt. for carbonyl-dissocn. energies from double-hybrid d. functionals. A theory's description of dynamic correlation (and to a less important extent, delocalization error), which affects relative spin-state energetics and thus spin symmetry breaking, is found to govern the efficacy of its use to diagnose static correlation. (c) 2021 American Institute of Physics.
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    108. 108
      Benavides-Riveros, C. L.; Lathiotakis, N. N.; Schilling, C.; Marques, M. A. Relating correlation measures: The importance of the energy gap. Phys. Rev. A 2017, 95, 032507,  DOI: 10.1103/PhysRevA.95.032507
      108
      Relating correlation measures: the importance of the energy gap
      Benavides-Riveros, Carlos L.; Lathiotakis, Nektarios N.; Schillin, Christian; Marques, Miguel A. L.
      Physical Review A (2017), 95 (3), 032507/1-032507/6CODEN: PRAHC3; ISSN:2469-9934. (American Physical Society)
      The concept of correlation is central to all approaches that attempt the description of many-body effects in electronic systems. Multipartite correlation is a quantum information theor. property that is attributed to quantum states independent of the underlying physics. In quantum chem., however, the correlation energy (the energy not seized by the Hartree-Fock ansatz) plays a more prominent role.We show that these two different viewpoints on electron correlation are closely related. The key ingredient turns out to be the energy gap within the symmetry-adapted subspace. We then use a few-site Hubbard model and the stretched H2 to illustrate this connection and to show how the corresponding measures of correlation compare.
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    109. 109
      Jiang, W.; DeYonker, N. J.; Wilson, A. K. Multireference character for 3d transition-metal-containing molecules. J. Chem. Theory Comput. 2012, 8, 460– 468,  DOI: 10.1021/ct2006852
      109
      Multireference Character for 3d Transition-Metal-Containing Molecules
      Jiang, Wanyi; DeYonker, Nathan J.; Wilson, Angela K.
      Journal of Chemical Theory and Computation (2012), 8 (2), 460-468CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Coupled cluster and CI diagnostics have been examd. in order to assess the reliability of single ref. quantum methods for a series of 3d transition metal species including hydrides, nitrides, chalcogenides, halides, small clusters, coordination complexes, and metal dimers. Several means of diagnostics have been considered including T1 and D1 diagnostics (the Frobenius norm and matrix 2-norm of coupled cluster amplitudes for single excitations, resp.), C02 (the wt. of leading configuration of a complete active space wave function), and %TAE (percent total atomization energy). T1 and D1 diagnostics are strongly correlated for certain metal-ligand bonding types. The use of T1 and D1 together with %TAE can provide more reliable assessment of the severity of nondynamical correlation than a single indicator can provide. New criteria, namely T1 > 0.05, D1 > 0.15, and |%TAE| > 10, are suggested to identify inorg. species with substantial nondynamical correlation. For these systems, energies and spectroscopic properties computed using single ref. electronic correlation methods may suffer from large errors and unpredictable behavior. Conversely, a computation where a mol. is below one or more of these thresholds does not always imply domination by a single ref. Some historically pathol. mols. such as Mn2 and Cr2 show T1 < 0.05 and D1 < 0.15. Current implementations of coupled cluster diagnostics may still be insufficient for categorization of mols. that have pronounced nondynamical correlation.
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    110. 110
      Lee, T. J.; Rice, J. E.; Scuseria, G. E.; Schaefer, H. F. Theoretical investigations of molecules composed only of fluorine, oxygen and nitrogen: determination of the equilibrium structures of FOOF, (NO)2 and FNNF and the transition state structure for FNNF cis-trans isomerization. Theor. Chim. Acta 1989, 75, 81– 98,  DOI: 10.1007/BF00527711
      110
      Theoretical investigations of molecules composed only of fluorine, oxygen and nitrogen: determination of the equilibrium structures of difluoroperoxide (FOOF), nitric oxide dimer, and difluorodiazene (FNNF) and the transition state structure for difluorodiazene cis-trans
      Lee, Timothy J.; Rice, Julia E.; Scuseria, Gustavo E.; Schaefer, Henry F., III
      Theoretica Chimica Acta (1989), 75 (2), 81-98CODEN: TCHAAM; ISSN:0040-5744.
      The deficiencies of common ab initio methods were studied for the reliable prediction of the equil. structures of compds. composed of only the fluorine, oxygen and nitrogen atoms. Specifically, the importance of using large one-particle basis sets with multiple sets of polarization functions was studied. Addnl., the need for a set of f basis functions was investigated. Several different single-ref. electron correlation methods were tested to det. whether it is possible for a single-ref.-based method to be used routinely on such chem. systems. These electron-correlation methods include second-order Moeller-Plesset perturbation theory (MP2), singles and doubles CI (CISD), the coupled-pair-functional (CPF) approach, and singles and doubles coupled-cluster (CCSD) theory. The mol. systems studied included difluoroperoxide (FOOF), the cis form of the NO dimer, cis and trans difluorodiazene (FNNF), and the transition state to interconversion of the cis and trans isomers of FNNF. This is the first time that the cis-trans isomerization transition state has been reported. At the highest level of theory employed, the equil. structures of cis and trans FNNF agreed very well with the exptl. structures. However, the barrier to interconversion was predicted to be 65 kcal/mol. which is substantially higher than the exptl. activation energy of 32 kcal/mol. Potential sources of error are discussed. A new diagnostic method for detg. a priori the reliability of single-ref.-based electron correlation methods is suggested, and discussed.
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    111. 111
      Liakos, D. G.; Neese, F. Interplay of Correlation and Relativistic Effects in Correlated Calculations on Transition-Metal Complexes: The (Cu2O2)2+ Core Revisited. J. Chem. Theory Comput. 2011, 7, 1511– 1523,  DOI: 10.1021/ct1006949
      111
      Interplay of Correlation and Relativistic Effects in Correlated Calculations on Transition-Metal Complexes: The (Cu2O2)2+ Core Revisited
      Liakos, Dimitrios G.; Neese, Frank
      Journal of Chemical Theory and Computation (2011), 7 (5), 1511-1523CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Owing to the availability of large-scale computing facilities and the development of efficient new algorithms, wave function-based ab initio calcns. are becoming more common in bioinorg. chem. In principle they offer a systematic route toward high accuracy. However, these calcns. are by no means trivial. In this contribution we address some pertinent points through a systematic theor. study for the equil. between the peroxo- and bis-(μ-oxo) isomers of the [{Cu(C2H8N2)}2O2]2+ complex. While this system is often regarded as a prototypical multireference case, we treat it with the single ref. local-pair natural orbital coupled cluster method and reiterate that the multireference character in this system is very limited. A set of intermediate structures, for the interconversion between the two isomers, is calcd. through a relaxed surface scan thus allowing the calcn. of an energetic profile that cleanly connects the bis-(μ-oxo) and side-on peroxo min. on the ground-state potential energy surface. Only at the highest level of theory involving complete basis set extrapolation, triple excitation contributions as well as relativistic and solvent effects, the bis-(μ-oxo) isomer is found to be slightly more stable than the peroxo structure. This is in agreement with the exptl. findings. The effects of basis set, triples excitation, relativity, and solvent contribution have all been analyzed in detail. Finally, the ab initio results are compared with d. functional calcns. using various functionals. It is demonstrated that the largest part of the discrepancies of the results reported in the literature are due to an inconsistent handling of relativistic effects, which are large in both ab initio and d. functional theory calcns.
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    112. 112
      Head-Gordon, M. Characterizing unpaired electrons from the one-particle density matrix. Chem. Phys. Lett. 2003, 372, 508– 511,  DOI: 10.1016/S0009-2614(03)00422-6
      112
      Characterizing unpaired electrons from the one-particle density matrix
      Head-Gordon, Martin
      Chemical Physics Letters (2003), 372 (3,4), 508-511CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
      A new definition of the unpaired electrons in a mol. is proposed, which derives from the one-particle reduced d. matrix. It yields lower ests. of the no. of radical electrons than the widely discussed distribution of effectively unpaired electrons', with a max. possible difference of a factor of two. Unlike the existing definition, the new definition cannot yield nos. of unpaired electrons higher than the total no. of electrons, and also recovers the intuitively expected result for the dissocn. of O2.
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    113. 113
      Ramos-Cordoba, E.; Matito, E. Local Descriptors of Dynamic and Nondynamic Correlation. J. Chem. Theory Comput. 2017, 13, 2705– 2711,  DOI: 10.1021/acs.jctc.7b00293
      113
      Local Descriptors of Dynamic and Nondynamic Correlation
      Ramos-Cordoba, Eloy; Matito, Eduard
      Journal of Chemical Theory and Computation (2017), 13 (6), 2705-2711CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Quant. accurate electronic structure calcns. rely on the proper description of electron correlation. A judicious choice of the approx. quantum chem. method depends upon the importance of dynamic and nondynamic correlation, which is usually assesed by scalar measures. Existing measures of electron correlation do not consider sep. the regions of the Cartesian space where dynamic or nondynamic correlation are most important. We introduce real-space descriptors of dynamic and nondynamic electron correlation that admit orbital decompn. Integration of the local descriptors yields global nos. that can be used to quantify dynamic and nondynamic correlation. Illustrative examples over different chem. systems with varying electron correlation regimes are used to demonstrate the capabilities of the local descriptors. Since the expressions only require orbitals and occupation nos., they can be readily applied in the context of local correlation methods, hybrid methods, d. matrix functional theory, and fractional-occupancy d. functional theory.
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    114. 114
      Löwdin, P.-O. Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction. Phys. Rev. 1955, 97, 1474– 1489,  DOI: 10.1103/PhysRev.97.1474
      There is no corresponding record for this reference.
    115. 115
      Ramos-Cordoba, E.; Salvador, P.; Matito, E. Separation of dynamic and nondynamic correlation. Phys. Chem. Chem. Phys. 2016, 18, 24015– 24023,  DOI: 10.1039/C6CP03072F
      115
      Separation of dynamic and nondynamic correlation
      Ramos-Cordoba, Eloy; Salvador, Pedro; Matito, Eduard
      Physical Chemistry Chemical Physics (2016), 18 (34), 24015-24023CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      The account of electron correlation and its efficient sepn. into dynamic and nondynamic parts plays a key role in the development of computational methods. In this paper we suggest a phys.-sound matrix formulation to split electron correlation into dynamic and nondynamic parts using the two-particle cumulant matrix and a measure of the deviation from idempotency of the first-order d. matrix. These matrixes are applied to a two-electron model, giving rise to a simplified electron correlation index that (i) depends only on natural orbitals and their occupancies, (ii) can be straightforwardly decompd. into orbital contributions and (iii) splits into dynamic and nondynamic correlation parts that (iv) admit a local version. These expressions are shown to account for dynamic and nondynamic correlation in a variety of systems contg. different electron correlation regimes, thus providing the first sepn. of dynamic and nondynamic correlation using solely natural orbital occupancies.
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    116. 116
      Juhász, T.; Mazziotti, D. A. The cumulant two-particle reduced density matrix as a measure of electron correlation and entanglement. J. Chem. Phys. 2006, 125, 174105,  DOI: 10.1063/1.2378768
      116
      The cumulant two-particle reduced density matrix as a measure of electron correlation and entanglement
      Juhasz, Tamas; Mazziotti, David A.
      Journal of Chemical Physics (2006), 125 (17), 174105/1-174105/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Several measures of electron correlation are compared based on two criteria: (i) the presence of a unique mapping between the reduced variables in the measure and the many-electron wave function and (ii) the linear scaling of the measure and its variables with system size. We propose the squared Frobenius norm of the cumulant part of the two-particle reduced d. matrix (2-RDM) as a measure of electron correlation that satisfies these criteria. An advantage of this cumulant-based norm is its ability to measure the correlation from spin entanglement, which is not contained in the correlation energy. Alternative measures based on the 2-RDM, such as the von Neumann entropy, do not scale linearly with system size. Properties of the measures are demonstrated with Be, F2, HF, N2, and a hydrogen chain.
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    117. 117
      Crittenden, D. L. A Hierarchy of Static Correlation Models. J. Phys. Chem. A 2013, 117, 3852– 3860,  DOI: 10.1021/jp400669p
      117
      A Hierarchy of Static Correlation Models
      Crittenden, Deborah L.
      Journal of Physical Chemistry A (2013), 117 (18), 3852-3860CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      It is commonly accepted in the scientific literature that the static correlation energy, Estat, of a system can be defined as the exact correlation energy of its valence electrons in a minimal basis. Unfortunately, the computational cost of calcg. the exact correlation energy within a fully optimized minimal basis grows exponentially with system size, making such calcns. intractable for all but the smallest systems. However, analogous to single-ref. methods, it is possible to systematically approx. both the treatment of electron correlation and flexibility of the minimal basis to reduce computational cost. This yields a hierarchy of methods for calcg. Estat, ranging from coupled cluster methods in a minimal at. basis up to full valence complete active space methods with a minimal MO basis constructed from a near-complete AO basis. By examg. a variety of dissocg. diatomics, along with equil. and transition structures for polyat. systems, we show that std. coupled cluster models with minimal at. basis sets (e.g., STO-3G) offer a convenient and cost-effective hierarchy of black box ests. for Estat in small- to medium-sized systems near their equil. geometries. To properly describe homolytic bond dissocn., it is better to use a more flexible basis set expansion so that each AO can effectively adapt to its mol. environment.
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    118. 118
      Pauncz, R. Spin Eigenfunctions: Construction and Use; Plenum Press: New York, 1979.
      There is no corresponding record for this reference.
    119. 119
      Perdew, J. P.; Ruzsinszky, A.; Sun, J.; Nepal, N. K.; Kaplan, A. D. Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories. Proc. Nat. Acad. Sci. 2021, 118, e2017850118,  DOI: 10.1073/pnas.2017850118
      119
      Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories
      Perdew, John P.; Ruzsinszky, Adrienn; Sun, Jianwei; Nepal, Niraj K.; Kaplan, Aaron D.
      Proceedings of the National Academy of Sciences of the United States of America (2021), 118 (4), e2017850118CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      Strong correlations within a symmetry-unbroken ground-state wavefunction can show up in approx. d. functional theory as symmetry-broken spin densities or total densities, which are sometimes observable. They can arise from soft modes of fluctuations (sometimes collective excitations) such as spin-d. or charge-d. waves at nonzero wavevector. In this sense, an approx. d. functional for exchange and correlation that breaks symmetry can be more revealing (albeit less accurate) than an exact functional that does not. The examples discussed here include the stretched H2 mol., antiferromagnetic solids, and the static charge-d. wave/Wigner crystal phase of a low-d. jellium. Time-dependent d. functional theory is used to show quant. that the static charge-d. wave is a soft plasmon. More precisely, the frequency of a related d. fluctuation drops to zero, as found from the frequency moments of the spectral function, calcd. from a recent constraint-based wavevector- and frequency-dependent jellium exchange-correlation kernel.
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    120. 120
      Perdew, J. P.; Ruzsinszky, A.; Constantin, L. A.; Sun, J.; Csonka, G. I. Some fundamental issues in ground-state density functional theory: A guide for the perplexed. J. Chem. Theory Comput. 2009, 5, 902– 908,  DOI: 10.1021/ct800531s
      120
      Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for the Perplexed
      Perdew, John P.; Ruzsinszky, Adrienn; Constantin, Lucian A.; Sun, Jianwei; Csonka, Gabor I.
      Journal of Chemical Theory and Computation (2009), 5 (4), 902-908CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Some fundamental issues in ground-state d. functional theory are discussed without equations: (1) The std. Hohenberg-Kohn and Kohn-Sham theorems were proven for a Hamiltonian that is not quite exact for real atoms, mols., and solids. (2) The d. functional for the exchange-correlation energy, which must be approximated, arises from the tendency of electrons to avoid one another as they move through the electron d. (3) In the absence of a magnetic field, either spin densities or total electron d. can be used, although the former choice is better for approxns. (4) "Spin contamination" of the determinant of Kohn-Sham orbitals for an open-shell system is not wrong but right. (5) Only to the extent that symmetries of the interacting wave function are reflected in the spin densities should those symmetries be respected by the Kohn-Sham noninteracting or determinantal wave function. Functionals below the highest level of approxns. should however sometimes break even those symmetries, for good phys. reasons. (6) Simple and commonly used semilocal (lower-level) approxns. for the exchange-correlation energy as a functional of the d. can be accurate for closed systems near equil. and yet fail for open systems of fluctuating electron no. (7) The exact Kohn-Sham noninteracting state need not be a single determinant, but common approxns. can fail when it is not. (8) Over an open system of fluctuating electron no., connected to another such system by stretched bonds, semilocal approxns. make the exchange-correlation energy and hole-d. sum rule too neg. (9) The gap in the exact Kohn-Sham band structure of a crystal underestimates the real fundamental gap but may approx. the first exciton energy in the large-gap limit. (10) D. functional theory is not really a mean-field theory, although it looks like one. The exact functional includes strong correlation, and semilocal approxns. often overestimate the strength of static correlation through their semilocal exchange contributions. (11) Only under rare conditions can excited states arise directly from a ground-state theory.
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    121. 121
      Filatov, M.; Shaik, S. Spin-restricted density functional approach to the open-shell problem. Chem. Phys. Lett. 1998, 288, 689– 697,  DOI: 10.1016/S0009-2614(98)00364-9
      121
      Spin-restricted density functional approach to the open-shell problem
      Filatov, Michael; Shaik, Sason
      Chemical Physics Letters (1998), 288 (5,6), 689-697CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
      Open-shell one-electron equations are derived by application of Roothaan's coupling operator technique to the variational procedure of finding the Kohn-Sham orbitals and minimizing the energy of an open-shell system, represented within the d. functional vector coupling scheme. The final equations are presented in a form suitable for std. quantum-chem. codes using finite basis set Kohn-Sham calcns. Examples of multiplets for which the theory is applicable are discussed.
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    122. 122
      Gagliardi, L.; Truhlar, D. G.; Li Manni, G.; Carlson, R. K.; Hoyer, C. E.; Bao, J. L. Multiconfiguration pair-density functional theory: A new way to treat strongly correlated systems. Acc. Chem. Res. 2017, 50, 66– 73,  DOI: 10.1021/acs.accounts.6b00471
      122
      Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems
      Gagliardi, Laura; Truhlar, Donald G.; Li Manni, Giovanni; Carlson, Rebecca K.; Hoyer, Chad E.; Bao, Junwei Lucas
      Accounts of Chemical Research (2017), 50 (1), 66-73CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)
      A review. The electronic energy of a system provides the Born-Oppenheimer potential energy for internuclear motion and thus dets. mol. structure and spectra, bond energies, conformational energies, reaction barrier heights, and vibrational frequencies. The development of more efficient and more accurate ways to calc. the electronic energy of systems with inherently multiconfigurational electronic structure is essential for many applications, including transition metal and actinide chem., systems with partially broken bonds, many transition states, and most electronically excited states. Inherently multiconfigurational systems are called strongly correlated systems or multireference systems, where the latter name refers to the need for using more than one ("multiple") configuration state function to provide a good zero-order ref. wave function. The present account describes (MC-PDFT), which was developed as a way to combine the advantages of wave function theory (WFT) and d. functional theory (DFT) to provide a better treatment of strongly correlated systems. First we review background material: the widely used Kohn-Sham DFT (which uses only a single Slater determinant as ref. wave function), multiconfiguration WFT methods that treat inherently multiconfigurational systems based on an active space, and previous attempts to combine multiconfiguration WFT with DFT. Then we review the formulation of MC-PDFT. MC-PDFT is a generalization of Kohn-Sham DFT in that the electron kinetic energy and classical electrostatic energy are calcd. from a ref. wave function, with the rest of the energy obtained from a d. functional. However, there are two main differences: (i) The ref. wave function is multiconfigurational rather than being a single Slater determinant. (ii) The d. functional is a function of the total d. and the on-top pair d. rather than being a function of the spin-up and spin-down densities. In work carried out so far, the multiconfigurational wave function is a multiconfiguration self-consistent-field wave function. The new formulation has the advantage that the ref. wave function has the correct spatial and spin symmetry and can describe bond dissocn. (of both single and multiple bonds) and electronic excitations in a formally and phys. correct way. We then review the formulation of d. functionals in terms of the on-top pair d. Finally we review successful applications of the theory to bond energies and bond dissocn. potential energy curves of main-group and transition metal bonds, to barrier heights (including pericyclic reactions), to proton affinities, to the hydrogen bond energy of water dimer, to ground- and excited-state charge transfer, to valence and Rydberg excitations of mols., and to singlet-triplet splittings of radicals. We find that MC-PDFT can give accurate results not only with complete-active-space multiconfiguration wave functions, but also with generalized-active-space multiconfiguration wave functions, which are practical for larger nos. of active electrons and active orbitals than are complete-active-space wave functions. The sepd.-pair approxn., which is a special case of generalized active space self-consistent-field theory, is esp. promising. MC-PDFT, because it requires much less computer time and storage than previous WFT methods, has the potential to open larger and more complex strongly correlated systems to accurate simulation.
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    123. 123
      Cremer, D. Density functional theory: coverage of dynamic and non-dynamic electron correlation effects. Mol. Phys. 2001, 99, 1899– 1940,  DOI: 10.1080/00268970110083564
      123
      Density functional theory: coverage of dynamic and non-dynamic electron correlation effects
      Cremer, Dieter
      Molecular Physics (2001), 99 (23), 1899-1940CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
      The electron correlation effects covered by d. functional theory (DFT) can be assessed qual. by comparing DFT densities ρ(r) with suitable ref. densities obtained with wavefunction theory (WFT) methods that cover typical electron correlation effects. The anal. of difference densities ρ(DFT)-ρ(WFT) reveals that LDA and GGA exchange (X) functionals mimic non-dynamic correlation effects in an unspecified way. It is shown that these long range correlation effects are caused by the self-interaction error (SIE) of std. X functionals. Self-interaction cor. (SIC) DFT exchange gives, similar to exact exchange, for the bonding region a delocalized exchange hole, and does not cover any correlation effects. Hence, the exchange SIE is responsible for the fact that DFT densities often resemble MP4 or MP2 densities. The correlation functional changes X-only DFT densities in a manner obsd. when higher order coupling effects between lower order N-electron correlation effects are included. Hybrid functionals lead to changes in the d. similar to those caused by SIC-DFT, which simply reflects the fact that hybrid functionals have been developed to cover part of the SIE and its long range correlation effects in a balanced manner. In the case of spin-unrestricted DFT (UDFT), non-dynamic electron correlation effects enter the calcn. both via the X functional and via the wavefunction, which may cause a double-counting of correlation effects. The use of UDFT in the form of permuted orbital and broken-symmetry DFT (PO-UDFT, BS-UDFT) can lead to reasonable descriptions of multireference systems provided certain conditions are fulfilled. More reliable, however, is a combination of DFT and WFT methods, which makes the routine description of multireference systems possible. The development of such methods implies a sepn. of dynamic and non-dynamic correlation effects. Strategies for accomplishing this goal are discussed in general and tested in practice for CAS (complete active space)-DFT.
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    124. 124
      Ziegler, T.; Rauk, A.; Baerends, E. J. On the calculation of multiplet energies by the Hartree-Fock-Slater method. Theor. Chim. Acta 1977, 43, 261– 271,  DOI: 10.1007/BF00551551
      124
      On the calculation of multiplet energies by the Hartree-Fock-Slater method
      Ziegler, Tom; Rauk, Arvi; Baerends, Evert J.
      Theoretica Chimica Acta (1977), 43 (3), 261-71CODEN: TCHAAM; ISSN:0040-5744.
      A consistent application of the statistical-exchange approxn. (J. C. Slater, 1974; B., et at., 1973) of the Hartree-Fock-Slater method requires use of the sum method for calcn. of the energy Es1 of singlet excited states of closed-shell mols. Values of Es1 were calcd. in satisfactory agreement with the available exptl. data for a no. of mols. Multiplet splittings other than singlet-triplet were also calcd. with the Hartree-Fock-Slater method.
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    125. 125
      Noodleman, L. Valence bond description of antiferromagnetic coupling in transition metal dimers. J. Chem. Phys. 1981, 74, 5737– 5743,  DOI: 10.1063/1.440939
      125
      Valence bond description of antiferromagnetic coupling in transition metal dimers
      Noodleman, Louis
      Journal of Chemical Physics (1981), 74 (10), 5737-43CODEN: JCPSA6; ISSN:0021-9606.
      A single configuration model contg. nonorthogonal magnetic orbitals is developed to represent the important features of the antiferromagnetic state of a transition metal dimer. A state of mixed spin symmetry and lowered space symmetry is constructed which has both conceptual and practical computational value. Either UHF theory or spin polarized d. functional theory, e.g., Xα theory, can be used to generate the mixed spin state wave function. The most important consequence of the theory is that the Heisenberg exchange coupling const. J can be calcd. simply from the energies of the mixed spin state and the highest pure spin multiplet.
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    126. 126
      Daul, C. Density functional theory applied to the excited states of coordination compounds. Int. J. Quantum Chem. 1994, 52, 867– 877,  DOI: 10.1002/qua.560520414
      126
      Density functional theory applied to the excited states of coordination compounds
      Daul, Claude
      International Journal of Quantum Chemistry (1994), 52 (4), 867-77CODEN: IJQCB2; ISSN:0020-7608. (Wiley)
      Coordination compds. are usually sym. mols. with degenerate orbitals. Hence, the individual multiplet states arising from open-shell configurations can, in general, not be expressed by a single determinant. We have, therefore, exploited symmetry to the largest possible extent in order to simplify the relation between the multiplet splitting and single-determinant energies, and, thus, developed a new method based on vector coupling to keep the computational effort to a min. A system of computer programs, working on both mainframe and personal computers, was developed, carrying out (for any desired point group) the required group-theor. manipulations. The description of the method is illustrated by considering three practical examples.
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    127. 127
      Neese, F. Definition of corresponding orbitals and the diradical character in broken symmetry DFT calculations on spin coupled systems. J. Phys. Chem. Solids 2004, 65, 781– 785,  DOI: 10.1016/j.jpcs.2003.11.015
      127
      Definition of corresponding orbitals and the diradical character in broken symmetry DFT calculations on spin coupled systems
      Neese, Frank
      Journal of Physics and Chemistry of Solids (2004), 65 (4), 781-785CODEN: JPCSAW; ISSN:0022-3697. (Elsevier Science B.V.)
      The broken symmetry (BS) concept is an extremely useful tool for the prediction of exchange coupling consts. in mols. with interacting paramagnetic centers. An anal. of the BS wave functions is presented and the relationship between the overlap of magnetic orbitals and the exchange coupling is stressed. The corresponding orbital transformation is introduced as a useful tool in order to det. the non-orthogonal valence bond-like magnetic orbital pairs in many electron systems.
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    128. 128
      Gunnarsson, O.; Lundqvist, B. I. Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys. Rev. B 1976, 13, 4274– 4298,  DOI: 10.1103/PhysRevB.13.4274
      128
      Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism
      Gunnarsson, O.; Lundqvist, B. I.
      Physical Review B: Solid State (1976), 13 (10), 4274-98CODEN: PLRBAQ; ISSN:0556-2805.
      The spin-d.-functional (SDF) formalism (e.g., G., et al., 1974-5) was extended to apply to generalized Hamiltonians and to lowest excited states with different types of symmetry. A relation between the exchange-correlation functional and the pair-correlation function was derived, and was used to interpret approx. versions of the theory, esp. the local-spin-d. (LSD) approxn., which can be used in calcn. of the exchange-correlation energy (Exc) in rather inhomogeneous systems. Calcns. done on the homogeneous spin-polarized electron liq., where the charge-d. fluctuations were described by using a plasmon model, provide interpolation formulas for detg. Exc and the exchange-correlation potentials in the LSD approxn. Other properties calcd. for the electron liq. include: bulk modulus at const. magnetization, compressibility at const. magnetic field, and magnetic susceptibility. Applications of the SDF formalism in calcns. of the properties of atoms, mols., and metals are discussed.
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    129. 129
      Anderson, P. W. More is different: broken symmetry and the nature of the hierarchical structure of science. Science 1972, 177, 393– 396,  DOI: 10.1126/science.177.4047.393
      129
      More is different
      Anderson, P. W.
      Science (Washington, DC, United States) (1972), 177 (4047), 393-6CODEN: SCIEAS; ISSN:0036-8075.
      A review with 15 refs., on the general relations of broken symmetry and the phys. properties of inanimate and living many-body systems, includes discussion of: elec. dipole properties, supercond. and superfluidity, phase transitions, and temporal regularity in the biol. activities of living things.
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    130. 130
      Amos, A.; Hall, G. Single determinant wave functions. Proc. R. Soc. A 1961, 263, 483– 493,  DOI: 10.1098/rspa.1961.0175
      There is no corresponding record for this reference.
    131. 131
      Ladner, R. C.; Goddard III, W. A. Improved Quantum Theory of Many-Electron Systems. V. The Spin-Coupling Optimized GI Method. J. Chem. Phys. 1969, 51, 1073– 1087,  DOI: 10.1063/1.1672106
      131
      Improved quantum theory of many-electron systems. V. The spin-coupling optimized GI method
      Ladner, Robert C.; Goddard, William A., III
      Journal of Chemical Physics (1969), 51 (3), 1073-87CODEN: JCPSA6; ISSN:0021-9606.
      The previously developed GI (gauge invariant) methods have an arbitrary aspect since they are based on a particular representation of the symmetric group. Here, the authors remove this arbitrariness by optimizing the representation, that is, optimizing the spin-coupling scheme simultaneously with the optimization of the orbitals. The resulting wavefunctions, called the spin-coupling optimized GI or SOGI wavefunctions, have all of the general properties of GI wavefunctions including the independent particle interpretation and are found as the solns. to a set of coupled differential equations which differ from the GI equations only in that the equations are constructed from a different representation of the symmetric group. The authors have applied this method to the ground state and some excited states of Li, to the ground states of Be+ and B++ and to the ground state of LiH. In each of these cases, they found that the SOGI wavefunction was only slightly different from the GI wavefunction and led to very similar energies and other spatial properties. For the spin d. at the nucleus, however, SOGI led to much better results. To illustrate the effects of spatial symmetry on the SOGI orbitals, the authors examd. the lowest 1B1g, 3A2g, and 3Eu states of sq. H4 and the 2Σu+ state of linear sym. H3. They find that in 3 of these cases optimization of the spin representation is crucial to providing an adequate description of the state. To investigate how the SOGI method would describe chem. reactions, the SOGI wavefunctions were computed for several other nuclear configurations of the H3 system along the reaction path. These calcns. showed that the spin coupling changed significantly during the reaction H2 + H .dblharw. H + H2 and that the variation of the SOGI orbitals provides a clear description of the changes in bonding which occur during this reaction.
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    132. 132
      Bobrowicz, F. W.; Goddard, W. A. The Self-Consistent Field Equations for Generalized Valence Bond and Open-Shell Hartree─Fock Wave Functions. In Methods of Electronic Structure Theory; Springer: New York, 1977; Chapter 4, pp 79– 127.
      There is no corresponding record for this reference.
    133. 133
      Ghosh, P.; Bill, E.; Weyhermüller, T.; Neese, F.; Wieghardt, K. Noninnocence of the Ligand Glyoxal-bis (2-mercaptoanil). The Electronic Structures of [Fe(gma)]2,[Fe(gma)(py)].py,[Fe(gma)(CN)]1–/0,[Fe(gma)I], and [Fe(gma)(PR3)n] (n= 1, 2). Experimental and Theoretical Evidence for “Excited State” Coordination. J. Am. Chem. Soc. 2003, 125, 1293– 1308,  DOI: 10.1021/ja021123h
      133
      Noninnocence of the Ligand Glyoxal-bis(2-mercaptoanil). The Electronic Structures of [Fe(gma)]2, [Fe(gma)(py)]·py, [Fe(gma)(CN)]1-/0, [Fe(gma)I], and [Fe(gma)(PR3)n] (n = 1, 2). Experimental and Theoretical Evidence for "Excited State" Coordination
      Ghosh, Prasanta; Bill, Eckhard; Mueller, Thomas Weyherm; Neese, Frank; Wieghardt, Karl
      Journal of the American Chemical Society (2003), 125 (5), 1293-1308CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      The electronic structure of the known Fe complexes [Fe(gma)]2 (St = 0) (1) (D. Sellmann, 1992) and [Fe(gma)(py)]·py (St = 1) (2) (P. Karsten, 1997) (H2(gma) = glyoxal-bis(2-mercaptoanil)) was shown by x-ray crystallog., Mossbauer spectroscopy, and d. functional theory calcns. to be best described as ferric (SFe = 3/2) complexes contg. a coordinated open-shell π radical trianion (gma•)3- and not as previously reported as ferrous species with a coordinated closed-shell dianion (gma)2-. Compd. 1 (or 2) can be oxidized by I2 yielding [FeIII(gma)I] (St = 1/2) (3). With cyanide anions, complex 1 forms [Bu4N][FeIII(gma•)(CN)] (St = 1) (4), which can be 1-electron oxidized with I yielding the neutral species [FeIII(gma)(CN)] (St = 1/2) (5). With phosphines complex 1 also forms adducts7 of which [FeIII(gma•)(P(n-propyl)3)] (St = 1) (6) was isolated and characterized by x-ray crystallog. [FeII(gma)(P(n-propyl)3)2] (St = 0) (7) represents the only genuine ferrous species of the series. D. functional theory (DFT) calcns. at the BP86 and B3LYP levels were applied to calc. the structural as well as the EPR and Mossbauer spectroscopic parameters of the title compds. and of [Zn(gma)]0/- and [Ni(gma)]0/-. Overall, the calcns. give excellent agreement with the available spectroscopic information, thus lending support to the following electronic structure descriptions: The gma ligand features an unusually low lying LUMO, which readily accepts an electron to give (gma•)3-. The 1-electron redn. of [Zn(gma)] and [Ni(gma)] is strictly ligand centered and differences in the phys. properties of [Zn(gma•)]- and [Ni(gma•)]- are readily accounted for in terms of a model that features enhanced back-bonding from the metal to the gma LUMO in the case of [Ni(gma•)]-. In the case of [Fe(gma)(PH3)], [Fe(gma)(py)], and [Fe(gma)(CN)]- an electron transfer from the Fe to the gma LUMO takes place to give strong antiferromagnetic coupling between an intermediate spin Fe(III) (SFe = 3/2) and (gma•)3- (Sgma = 1/2), yielding a total spin St = 1. Broken symmetry DFT calcns. take properly account of this exptl. calibrated electronic structure description. By contrast, [Fe(gma)(PH3)2] and [Fe(PhBMA)] feature closed-shell ligands with a low-spin Fe(II) (SFe = St = 0) and an intermediate spin central Fe(II) (SFe = St = 1), resp. The most interesting case is provided by the 1-electron oxidized species [Fe(gma)(py)]+, [Fe(gma)I], and [Fe(gma)(CN)]. Here the combination of theory and expt. suggests the coupling of an intermediate spin Fe(III) (SFe = 3/2) to the dianionic ligand (gma)2- formally in its 1st excited triplet state (Sgma = 1) to give a resulting St = 1/2. All phys. properties are in accord with this interpretation. Probably this unique excited state coordination is energetically driven by the strong antiferromagnetic exchange interaction between the metal and the ligand, which cannot occur for the closed-shell form of the ligand.
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    134. 134
      Herebian, D.; Wieghardt, K. E.; Neese, F. Analysis and interpretation of metal-radical coupling in a series of square planar nickel complexes: correlated ab initio and density functional investigation of [Ni(LISQ)2](LISQ= 3,5-di-tert-butyl-o-diiminobenzosemiquinonate(1-)). J. Am. Chem. Soc. 2003, 125, 10997– 11005,  DOI: 10.1021/ja030124m
      134
      Analysis and Interpretation of Metal-Radical Coupling in a Series of Square Planar Nickel Complexes: Correlated Ab Initio and Density Functional Investigation of [Ni(LISQ)2] (LISQ=3,5-di-tert-butyl-o-diiminobenzosemiquinonate (1-))
      Herebian, Diran; Wieghardt, Karl E.; Neese, Frank
      Journal of the American Chemical Society (2003), 125 (36), 10997-11005CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      The author report a detailed theor. study of the interaction between a central low-spin d8 nickel ion and two N,N-coordinating diiminobenzosemiquinonate(1-) ligands in a square planar arrangement. Such complexes have recently attracted much attention due to their unusual bonding patterns, structures, optical, and magnetic properties. Geometry optimizations using various levels of d. functional theory (DFT) result in excellent agreement with the exptl. detd. structure and in particular reproduce the quinoidal distortions in the arom. rings well. A detailed anal. of the orbital structure reveals that the complex features essentially two strongly interacting ligand radicals which interact with each other via an efficient superexchange mechanism that is mediated by a back-bonding interaction to the central metal. An anal. of the broken symmetry DFT wave function is presented and a new index for the diradical character is proposed which shows that [Ni(LISQ)2] has a diradical character of ∼77%. These results are in full agreement with elaborate multireference post-Hartree-Fock ab initio calcns. for [Ni(LISQ)2] using the difference dedicated CI (DDCI) method as well as second-order multireference Moller-Plesset (MR-MP2) theory, which give diradical characters of 65-80%. On the basis of these calcns. our best est. for the singlet-triplet gap in this system is 3096 cm-1. This very large value results from an efficient mixing of the ionic configurations into the mainly singlet diradical ground state which is feasible because the semiquinonate SOMOs are delocalized and, therefore, have moderate on-site Coulomb repulsion parameters. As pointed out in the discussion, this represents an interesting difference to the case of magnetically interacting transition metal ions which typically show much smaller magnetic exchange couplings.
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    135. 135
      Chłopek, K.; Muresan, N.; Neese, F.; Wieghardt, K. Electronic Structures of Five-Coordinate Complexes of Iron Containing Zero, One, or Two π-Radical Ligands: A Broken-Symmetry Density Functional Theoretical Study. Chem. Eur. J. 2007, 13, 8390– 8403,  DOI: 10.1002/chem.200700897
      135
      Electronic structures of five-coordinate complexes of iron containing zero, one, or two π-radical ligands: a broken-symmetry density functional theoretical study
      Chlopek, Krzysztof; Muresan, Nicoleta; Neese, Frank; Wieghardt, Karl
      Chemistry - A European Journal (2007), 13 (30), 8390-8403CODEN: CEUJED; ISSN:0947-6539. (Wiley-VCH Verlag GmbH & Co. KGaA)
      The electronic structures of a series of five-coordinate complexes of iron contg. zero, one, or two bidentate, org. π-radical ligands and a monodentate ligand (pyridine, iodide) have been studied by broken-symmetry (BS) d. functional theor. (DFT) methods. By analyzing the set of corresponding orbitals (CO) a convenient division of the spin-up and spin-down orbitals into (1) essentially doubly-occupied MOs , (2) exactly singly-occupied MOs, (3) spin-coupled pairs, and (4) virtual orbitals can be achieved and a clear picture of the spin coupling between the ligands (non-innocence vs. innocence) and the central metal ion (dN configuration) can be generated. We have identified three classes of complexes which all contain a ferric ion (d5) with an intrinsic intermediate spin (SFe = 3/2) that yield (1) an St = 3/2 ground spin state if the two bidentate ligands are closed-shell species (innocent ligands); (2) if one π-radical ligand is present, an St = 1 ground state is obtained through intramol. antiferromagnetic coupling; (3) if two such radicals are present, an St = 1/2 ground state is obtained. We show unambiguously for the first time that the pentane-2,4-dione-bis(S-alkylisothiosemicarbazonato) ligand can bind as π-radical dianion (L•TSC)2- in [FeIII(L•TSC)I](St = 1); the description as [FeIV-(LTSC3-)I] is incorrect. Similarly, the diamagnetic monoanion in 14 must be described as [FeIII(CN)2(L•TSC)]-(St = 0) with a low-spin ferric ion (d5, SFe = 1/2) coupled antiferromagnetically to a π-radical ligand; [FeII(CN)2(LTSC-)]- is an incorrect description.
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    136. 136
      Neese, F. Prediction of molecular properties and molecular spectroscopy with density functional theory: From fundamental theory to exchange-coupling. Coord. Chem. Rev. 2009, 253, 526– 563,  DOI: 10.1016/j.ccr.2008.05.014
      136
      Prediction of molecular properties and molecular spectroscopy with density functional theory: From fundamental theory to exchange-coupling
      Neese, Frank
      Coordination Chemistry Reviews (2009), 253 (5+6), 526-563CODEN: CCHRAM; ISSN:0010-8545. (Elsevier B.V.)
      This review provides a detailed account of d. functional theory (DFT) and its application to the calcn. of mol. properties of inorg. compds. After introducing some fundamental quantum mech. concepts, the foundations of DFT and their realization in the framework of the Kohn-Sham construction are described. Following a brief exposition of the computational machinery required to carry out large-scale DFT calcns., the application of analytic deriv. theory to DFT is developed in some detail. The cases covered include geometric, elec., magnetic, and time-dependent perturbations. The developed theor. app. is then applied to the calcns. of mol. structures, vibrational energies as well as a wide variety of properties including absorption, CD, magnetic CD, resonance Raman, x-ray absorption, Moessbauer, and ESR spectroscopies. Finally, the important subjects of spin state energetics and exchange couplings in oligomeric transition metal clusters is discussed.
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    137. 137
      Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev. 1964, 136, B864– B871,  DOI: 10.1103/PhysRev.136.B864
      There is no corresponding record for this reference.
    138. 138
      Kutzelnigg, W. Density functional theory in terms of a Legendre transformation for beginners. J. Mol. Struct.: THEOCHEM 2006, 768, 163– 173,  DOI: 10.1016/j.theochem.2006.05.012
      138
      Density functional theory in terms of a Legendre transformation for beginners
      Kutzelnigg, Werner
      Journal of Molecular Structure: THEOCHEM (2006), 768 (1-3), 163-173CODEN: THEODJ; ISSN:0166-1280. (Elsevier B.V.)
      The derivation of the Hohenberg-Kohn (HK) theorem by means of a simplified version of Lieb's Legendre transformation is presented, with the stress on phys. problematic aspects, and caring about the definition of the domains of the resp. functionals only to the extent, that this is absolutely necessary. The take-home lesson consists in a crit. anal. of some statements often found in the DFT literature. As a simple illustration for a Legendre transformation we discuss that in parameter space for the family of n-electron isoelectronic at. ions, with the nuclear charge as parameter.
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    139. 139
      Lieb, E. H. Density functionals for Coulomb systems. Int. J. Quantum Chem. 1983, 24, 243– 277,  DOI: 10.1002/qua.560240302
      139
      Density functionals for Coulomb systems
      Lieb, Elliott H.
      International Journal of Quantum Chemistry (1983), 24 (3), 243-77CODEN: IJQCB2; ISSN:0020-7608.
      Some of the math. connections between N-particle wave functions and their single-particle densities ρ are discussed. The math. underpinnings of "universal d. functional" theory are given for the ground state energy. The Hohenberg-Kohn functional is not defined for all ρ. Several ways around this difficulty are given. Since the functional mentioned above is not computable, examples of explicit functionals that have the virtue of yielding rigorous bounds to the energy are reviewed.
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    140. 140
      Penz, M.; Tellgren, E. I.; Csirik, M. A.; Ruggenthaler, M.; Laestadius, A. structure of the density-potential mapping. Part I: Standard density-functional theory. arXiv:2211.16627 2022, DOI: 10.48550/arXiv.2211.16627 .
      There is no corresponding record for this reference.
    141. 141
      Casida, M. E.; Huix-Rotllant, M. Progress in time-dependent density-functional theory. Annu. Rev. Phys. Chem. 2012, 63, 287– 323,  DOI: 10.1146/annurev-physchem-032511-143803
      141
      Progress in time-dependent density-functional theory
      Casida, M. E.; Huix-Rotllant, M.
      Annual Review of Physical Chemistry (2012), 63 (), 287-323CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)
      The classic d.-functional theory (DFT) formalism introduced by Hohenberg, Kohn, and Sham in the mid-1960s is based on the idea that the complicated N-electron wave function can be replaced with the math. simpler 1-electron charge d. in electronic structure calcns. of the ground stationary state. As such, ordinary DFT cannot treat time-dependent (TD) problems nor describe excited electronic states. In 1984, Runge and Gross proved a theorem making TD-DFT formally exact. Information about electronic excited states may be obtained from this theory through the linear response (LR) theory formalism. Beginning in the mid-1990s, LR-TD-DFT became increasingly popular for calcg. absorption and other spectra of medium- and large-sized mols. Its ease of use and relatively good accuracy has now brought LR-TD-DFT to the forefront for this type of application. As the no. and the diversity of applications of TD-DFT have grown, so too has our understanding of the strengths and weaknesses of the approx. functionals commonly used for TD-DFT. The objective of this article is to continue where a previous review of TD-DFT in Vol. 55 of the Annual Review of Phys. Chem. left off and highlight some of the problems and solns. from the point of view of applied phys. chem. Because doubly-excited states have a particularly important role to play in bond dissocn. and formation in both thermal and photochem., particular emphasis is placed on the problem of going beyond or around the TD-DFT adiabatic approxn., which limits TD-DFT calcns. to nominally singly-excited states.
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    142. 142
      Kohn, W. Density Functional and Density Matrix Method Scaling Linearly with the Number of Atoms. Phys. Rev. Lett. 1996, 76, 3168– 3171,  DOI: 10.1103/PhysRevLett.76.3168
      142
      Density functional and density matrix method scaling linearly with the number of atoms
      Kohn, W.
      Physical Review Letters (1996), 76 (17), 3168-71CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      A widely applicable "nearsightedness" principle is first discussed as the phys. basis for the existence of computational methods scaling linearly with the no. of atoms. This principle applies to the one particle d. matrix n(r,r') but not to individual eigenfunctions. A variational principle for n(r,r') is derived in which, by the use of a penalty functional P[n(r,r')], the (difficult) idempotency of n(r,r') need not be assured in advance but is automatically achieved. The method applies to both insulators and metals.
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    143. 143
      Prodan, E.; Kohn, W. Nearsightedness of electronic matter. Proc. Nat. Acad. Sci. 2005, 102, 11635– 11638,  DOI: 10.1073/pnas.0505436102
      143
      Nearsightedness of electronic matter
      Prodan, E.; Kohn, W.
      Proceedings of the National Academy of Sciences of the United States of America (2005), 102 (33), 11635-11638CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      In an earlier paper, W. Kohn had qual. introduced the concept of "nearsightedness" of electrons in many-atom systems. It can be viewed as underlying such important ideas as Pauling's "chem. bond," "transferability," and Yang's computational principle of "divide and conquer.". It describes the fact that, for fixed chem. potential, local electronic properties, such as the d. n(r), depend significantly on the effective external potential only at nearby points. Changes of that potential, no matter how large, beyond a distance R have limited effects on local electronic properties, which rapidly tend to zero as a function of R. In the present paper, the concept is first sharpened for representative models of uncharged fermions moving in external potentials, and then the effects of electron-electron interactions and of perturbing external charges are discussed.
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    144. 144
      Li, L.; Burke, K. Recent developments in density functional approximations. In Handbook of Materials Modeling: Methods: Theory and Modeling; Springer: Cham, 2020; pp 213– 226.
      There is no corresponding record for this reference.
    145. 145
      Mayer, J. E. Electron Correlation. Phys. Rev. 1955, 100, 1579– 1586,  DOI: 10.1103/PhysRev.100.1579
      145
      Electron correlation
      Mayer, Joseph E.
      Physical Review (1955), 100 (), 1579-86CODEN: PHRVAO; ISSN:0031-899X.
      Math. The state of a real gas of electrons of uniform d. is examd. An equation for the co.ovrddot.ordinate representation of the d. matrix for 2 particles is found that satisfies the various necessary conditions and that gives a lower energy than the antisymmetrized single Slater determinant. The addnl. neg. correlation energy found is proportional to the 1/6-th power of the d. at high ds.
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    146. 146
      Bopp, F. Ableitung der Bindungsenergie vonN-Teilchen-Systemen aus 2-Teilchen-Dichtematrizen. Z. Phys. 1959, 156, 348– 359,  DOI: 10.1007/BF01461233
      There is no corresponding record for this reference.
    147. 147
      Mazziotti, D. A. Structure of Fermionic Density Matrices: Complete N-Representability Conditions. Phys. Rev. Lett. 2012, 108, 263002,  DOI: 10.1103/PhysRevLett.108.263002
      147
      Structure of fermionic density matrices: complete N-representability conditions
      Mazziotti, David A.
      Physical Review Letters (2012), 108 (26), 263002/1-263002/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We present a constructive soln. to the N-representability problem: a full characterization of the conditions for constraining the two-electron reduced d. matrix to represent an N-electron d. matrix. Previously known conditions, while rigorous, were incomplete. Here, we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompns. of model Hamiltonians. Existing conditions D, Q, G, T1, and T2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.
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    148. 148
      Gilbert, T. L. Hohenberg-Kohn theorem for nonlocal external potentials. Phys. Rev. B 1975, 12, 2111– 2120,  DOI: 10.1103/PhysRevB.12.2111
      There is no corresponding record for this reference.
    149. 149
      Müller, A. Explicit approximate relation between reduced two- and one-particle density matrices. Phys. Lett. A 1984, 105, 446– 452,  DOI: 10.1016/0375-9601(84)91034-X
      There is no corresponding record for this reference.
    150. 150
      Kutzelnigg, W. Density-cumulant functional theory. J. Chem. Phys. 2006, 125, 171101,  DOI: 10.1063/1.2387955
      150
      Density-cumulant functional theory
      Kutzelnigg, Werner
      Journal of Chemical Physics (2006), 125 (17), 171101/1-171101/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Starting point is the energy expectation value as a functional of the one-particle d. matrix γ and the two-particle d. cumulant λ2. We decomp. γ into a best idempotent approxn. κ and a correction τ, that is entirely expressible in terms of λ2. So we get the energy E as a functional of κ and λ2, which can be varied independently. Approx. n-representability conditions, derived by perturbation theory are imposed on the variation of λ2. A nonlinear system of equations satisfied by λ2 is derived, the linearized version of which turns out to be equiv. to the CEPA, variant zero. The start for κ is Hartree-Fock, but κ is then updated to become the best idempotent approxn. of γ. Relations to d. matrix functional theory and Kohn-Sham type d. functional theory are discussed.
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    151. 151
      Piris, M.; Ugalde, J. M. Perspective on natural orbital functional theory. Int. J. Quantum Chem. 2014, 114, 1169– 1175,  DOI: 10.1002/qua.24663
      151
      Perspective on natural orbital functional theory
      Piris, Mario; Ugalde, Jesus M.
      International Journal of Quantum Chemistry (2014), 114 (18), 1169-1175CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
      A review. The natural orbital functional (NOF) theory is briefly reviewed. The meaning of the top-down and bottom-up approaches for the construction of a NOF is analyzed. A particular reconstruction of the two-particle reduced d. matrix (2-RDM) based on the cumulant expansion is discussed. The cumulant is expressed by two auxiliary matrixes, which are constrained to certain bounds due to the N-representabilty conditions of the 2-RDM. Appropriate forms of these matrixes lead to different implementations known in the literature as PNOFi (i = 1-5). The strengths and weaknesses of PNOF5 are assessed. Its main strength is its ability to deal with the intrapair electron correlation at a reasonable computational cost. Its main limitation is the absence of the interpair electron correlation. The inclusion of the missing correlation via a multiconfigurational perturbation theory is shortly described. The growing interest in methods based on NOF theory points to a promising future in this field. © 2014 Wiley Periodicals, Inc.
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    152. 152
      Pernal, K., Giesbertz, K. J. H. Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time-Dependent RDMFT (TD-RDMFT). In Density-Functional Methods for Excited States; Springer: Cham, 2016; pp 125– 183.
      There is no corresponding record for this reference.
    153. 153
      Coleman, A. J. Structure of fermion density matrices. Rev. Mod. Phys. 1963, 35, 668– 686,  DOI: 10.1103/RevModPhys.35.668
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    154. 154
      Cioslowski, J.; Pernal, K.; Buchowiecki, M. Approximate one-matrix functionals for the electron–electron repulsion energy from geminal theories. J. Chem. Phys. 2003, 119, 6443– 6447,  DOI: 10.1063/1.1604375
      154
      Approximate one-matrix functionals for the electron-electron repulsion energy from geminal theories
      Cioslowski, Jerzy; Pernal, Katarzyna; Buchowiecki, Marcin
      Journal of Chemical Physics (2003), 119 (13), 6443-6447CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A simple extension of the antisymmetrized product of strongly orthogonal geminals theory produces a "JK-only" one-matrix functional for the electron-electron repulsion energy of a closed-shell system that is exact for two-electron singlet ground states, size-extensive, and incorporates some intergeminal correlation and thus dispersion effects. The functional is defined only for one-matrixes with occupation nos. that can be arranged into sets with elements that sum up to two. Its possible generalizations are discussed.
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    155. 155
      Piris, M. A natural orbital functional based on an explicit approach of the two-electron cumulant. Int. J. Quantum Chem. 2013, 113, 620– 630,  DOI: 10.1002/qua.24020
      155
      A natural orbital functional based on an explicit approach of the two-electron cumulant
      Piris, M.
      International Journal of Quantum Chemistry (2013), 113 (5), 620-630CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
      A review. The cumulant expansion gives rise to an useful decompn. of the two-matrix in which the pair correlated matrix (cumulant) is disconnected from the antisym. product of the one-matrixes. The cumulant can be approximated in terms of two matrixes, Δ and Π, which are explicit functions of the occupation nos. of the natural orbitals. It produces a natural orbital functional (NOF) that reduces to the exact expression for the total energy in two-electron systems. The N-representability positivity necessary conditions of the two-matrix impose several bounds on the matrixes Δ and Π. Appropriate forms of these matrixes lead to different implementations of the NOF known in the literature as PNOFi (i = 1-5). The basic features of these functionals are reviewed here. The strengths and weaknesses of the different PNOFs are assessed. © 2012 Wiley Periodicals, Inc.
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    156. 156
      Simmonett, A. C.; Wilke, J. J.; Schaefer, H. F., III; Kutzelnigg, W. Density cumulant functional theory: First implementation and benchmark results for the DCFT-06 model. J. Chem. Phys. 2010, 133, 174122,  DOI: 10.1063/1.3503657
      156
      Density cumulant functional theory: First implementation and benchmark results for the DCFT-06 model
      Simmonett, Andrew C.; Wilke, Jeremiah J.; Schaefer, Henry F., III; Kutzelnigg, Werner
      Journal of Chemical Physics (2010), 133 (17), 174122/1-174122/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      D. cumulant functional theory is implemented for the first time. Benchmark results are provided for atoms and diat. mols., demonstrating the performance of DCFT-06 for both nonbonded and bonded interactions. The results show that DCFT-06 appears to perform similarly to coupled cluster theory with single and double excitations (CCSD) in describing dispersion. For covalently bound systems, the phys. properties predicted by DCFT-06 appear to be at least of CCSD quality around equil. geometries. The computational scaling of both DCFT-06 and CCSD is O(N6), but the former has reduced nonlinearities among the variables and a Hermitian energy functional, making it an attractive alternative. (c) 2010 American Institute of Physics.
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    157. 157
      Kutzelnigg, W. Density Functional Theory (DFT) and ab-initio Quantum Chemistry (AIQC). Story of a difficult partnership. In Trends and Perspectives in Modern Computational Science; Brill: Leiden, 2006; pp 23– 62.
      There is no corresponding record for this reference.
    158. 158
      Copan, A. V.; Sokolov, A. Y. Linear-response density cumulant theory for excited electronic states. J. Chem. Theory Comput. 2018, 14, 4097– 4108,  DOI: 10.1021/acs.jctc.8b00326
      158
      Linear-Response Density Cumulant Theory for Excited Electronic States
      Copan, Andreas V.; Sokolov, Alexander Yu.
      Journal of Chemical Theory and Computation (2018), 14 (8), 4097-4108CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We present a linear-response formulation of d. cumulant theory (DCT) that provides a balanced and accurate description of many electronic states simultaneously. In the original DCT formulation, only information about a single electronic state (usually, the ground state) is obtained. We discuss the derivation of linear-response DCT, present its implementation for the ODC-12 method (LR-ODC-12), and benchmark its performance for excitation energies in small mols. (N2, CO, HCN, HNC, C2H2, and H2CO), as well as challenging excited states in ethylene, butadiene, and hexatriene. For small mols., LR-ODC-12 shows smaller mean abs. errors in excitation energies than equation-of-motion coupled cluster theory with single and double excitations (EOM-CCSD), relative to the ref. data from EOM-CCSDT. In a study of butadiene and hexatriene, LR-ODC-12 correctly describes the relative energies of the singly excited 11Bu and the doubly excited 21Ag states, in excellent agreement with highly accurate semistochastic heat-bath CI results, while EOM-CCSD overestimates the energy of the 21Ag state by almost 1 eV. Our results demonstrate that linear-response DCT is a promising theor. approach for excited states of mols.
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    159. 159
      Cioslowski, J., Ed. Many-electron densities and reduced density matrices; Springer Science & Business Media: New York, 2000.
      There is no corresponding record for this reference.
    160. 160
      Gidofalvi, G.; Mazziotti, D. A. Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the N-electron Hamiltonian. J. Chem. Phys. 2008, 129, 134108,  DOI: 10.1063/1.2983652
      160
      Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the N-electron Hamiltonian
      Gidofalvi, Gergely; Mazziotti, David A.
      Journal of Chemical Physics (2008), 129 (13), 134108/1-134108/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Mol. systems in chem. often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the no. N of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-d.-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calcn. where the 2-RDM is constrained by approx. N-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as ra6 where ra is the no. of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calcn. of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, H2O, and N2 as well as n-acene chains for n=2-8. When n°4, the acenes cannot be treated by conventional CASSCF methods; for example, when n=8, CASSCF requires optimization over approx. 1.47×1017 configuration state functions. The natural occupation nos. of the n-acenes show the emergence of bi- and polyradical character with increasing chain length. (c) 2008 American Institute of Physics.
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    161. 161
      Mullinax, J. W.; Sokolov, A. Y.; Schaefer, H. F., III Can density cumulant functional theory describe static correlation effects?. J. Chem. Theory Comput. 2015, 11, 2487– 2495,  DOI: 10.1021/acs.jctc.5b00346
      161
      Can Density Cumulant Functional Theory Describe Static Correlation Effects?
      Mullinax, J. Wayne; Sokolov, Alexander Yu.; Schaefer, Henry F.
      Journal of Chemical Theory and Computation (2015), 11 (6), 2487-2495CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We evaluate the performance of d. cumulant functional theory (DCT) for capturing static correlation effects. In particular, we examine systems with significant multideterminant character of the electronic wave function, such as the beryllium dimer, diat. carbon, m-benzyne, 2,6-pyridyne, twisted ethylene, as well as the barrier for double-bond migration in cyclobutadiene. We compute mol. properties of these systems using the ODC-12 and DC-12 variants of DCT and compare these results to multireference CI and multireference coupled-cluster theories, as well as single-ref. coupled-cluster theory with single, double (CCSD), and perturbative triple excitations [CCSD(T)]. For all systems the DCT methods show intermediate performance between that of CCSD and CCSD(T), with significant improvement over the former method. In particular, for the beryllium dimer, m-benzyne, and 2,6-pyridyne, the ODC-12 method along with CCSD(T) correctly predict the global min. structures, while CCSD predictions fail qual., underestimating the multireference effects. Our results suggest that the DC-12 and ODC-12 methods are capable of describing emerging static correlation effects but should be used cautiously when highly accurate results are required. Conveniently, the appearance of multireference effects in DCT can be diagnosed by analyzing the DCT natural orbital occupations, which are readily available at the end of the energy computation.
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    162. 162
      Lathiotakis, N. N.; Helbig, N.; Rubio, A.; Gidopoulos, N. I. Local reduced-density-matrix-functional theory: Incorporating static correlation effects in Kohn-Sham equations. Phys. Rev. A 2014, 90, 032511,  DOI: 10.1103/PhysRevA.90.032511
      162
      Local reduced-density-matrix-functional theory: incorporating static correlation effects in Kohn-Sham equations
      Lathiotakis, Nektarios N.; Helbig, Nicole; Rubio, Angel; Gidopoulos, Nikitas I.
      Physical Review A: Atomic, Molecular, and Optical Physics (2014), 90 (3-A), 032511/1-032511/8CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
      We propose a scheme to bring reduced-d.-matrix-functional theory into the realm of d. functional theory (DFT) that preserves the accurate d. functional description at equil., while incorporating accurately static and left-right correlation effects in mols. and keeping the good computational performance of DFT-based schemes. The key ingredient is to relax the requirement that the local potential is the functional deriv. of the energy with respect to the d. Instead, we propose to restrict the search for the approx. natural orbitals within a domain where these orbitals are eigenfunctions of a single-particle Hamiltonian with a local effective potential. In this way, fractional natural occupation nos. are accommodated into Kohn-Sham equations allowing for the description of mol. dissocn. without breaking spin symmetry. Addnl., our scheme provides a natural way to connect an energy eigenvalue spectrum to the approx. natural orbitals and this spectrum is found to represent accurately the ionization potentials of atoms and small mols.
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    163. 163
      Piris, M. Global Method for Electron Correlation. Phys. Rev. Lett. 2017, 119, 063002,  DOI: 10.1103/PhysRevLett.119.063002
      163
      Global method for electron correlation
      Piris, Mario
      Physical Review Letters (2017), 119 (6), 063002/1-063002/5CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
      A review. The current work presents a new single-ref. method for capturing at the same time the static and dynamic electron correlation. The starting point is a determinant wave function formed with natural orbitals obtained from a new interacting-pair model. The latter leads to a natural orbital functional (NOF) capable of recovering the complete intrapair, but only the static interpair correlation. Using the soln. of the NOF, two new energy functionals are defined for both dynamic (Edyn) and static (Esta) correlation. Edyn is derived from a modified second-order Moller-Plesset perturbation theory (MP2), while Esta is obtained from the static component of the new NOF. Double counting is avoided by introducing the amt. of static and dynamic correlation in each orbital as a function of its occupation. As a result, the total energy is represented by the sum EHF + Edyn + Esta, where EHF is the Hartree-Fock energy obtained with natural orbitals. The new procedure called NOF-MP2 scales formally as O(M5) (where M is the no. of basis functions), and is applied successfully to the homolytic dissocn. of a selected set of diat. mols., paradigmatic cases of near-degeneracy effects. The size consistency has been numerically demonstrated for singlets. The values obtained are in good agreement with the exptl. data.
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    164. 164
      Gibney, D.; Boyn, J.-N.; Mazziotti, D. A. Density Functional Theory Transformed into a One-Electron Reduced-Density-Matrix Functional Theory for the Capture of Static Correlation. J. Phys. Chem. Lett. 2022, 13, 1382– 1388,  DOI: 10.1021/acs.jpclett.2c00083
      164
      Density Functional Theory Transformed into a One-Electron Reduced-Density-Matrix Functional Theory for the Capture of Static Correlation
      Gibney, Daniel; Boyn, Jan-Niklas; Mazziotti, David A.
      Journal of Physical Chemistry Letters (2022), 13 (6), 1382-1388CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      D. Functional Theory (DFT), the most widely adopted method in modern computational chem., fails to describe accurately the electronic structure of strongly correlated systems. Here we show that DFT can be formally and practically transformed into a one-electron reduced-d.-matrix (1-RDM) functional theory, which can address the limitations of DFT while retaining favorable computational scaling compared to wave function based approaches. In addn. to relaxing the idempotency restriction on the 1-RDM in the kinetic energy term, we add a quadratic 1-RDM-based term to DFT's d.-based exchange-correlation functional. Our approach, which we implement by quadratic semidefinite programming at DFT's computational scaling of O(r3), yields substantial improvements over traditional DFT in the description of static correlation in chem. structures and processes such as singlet biradicals and bond dissocns.
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    165. 165
      Martin, P. C.; Schwinger, J. Theory of many-particle systems. I. Phys. Rev. 1959, 115, 1342– 1373,  DOI: 10.1103/PhysRev.115.1342
      There is no corresponding record for this reference.
    166. 166
      Luttinger, J. M.; Ward, J. C. Ground-state energy of a many-fermion system. II. Phys. Rev. 1960, 118, 1417– 1427,  DOI: 10.1103/PhysRev.118.1417
      There is no corresponding record for this reference.
    167. 167
      Baym, G.; Kadanoff, L. P. Conservation laws and correlation functions. Phys. Rev. 1961, 124, 287– 299,  DOI: 10.1103/PhysRev.124.287
      There is no corresponding record for this reference.
    168. 168
      Ziesche, P. Cumulant expansions of reduced densities, reduced density matrices, and Green’s functions. In Many-Electron Densities and Reduced Density Matrices; Springer: New York, 2000; Chapter 3, pp 33– 56.
      There is no corresponding record for this reference.
    169. 169
      Sun, Q.; Chan, G. K.-L. Quantum embedding theories. Acc. Chem. Res. 2016, 49, 2705– 2712,  DOI: 10.1021/acs.accounts.6b00356
      169
      Quantum Embedding Theories
      Sun, Qiming; Chan, Garnet Kin-Lic
      Accounts of Chemical Research (2016), 49 (12), 2705-2712CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)
      A review. In complex systems, it is often the case that the region of interest forms only one part of a much larger system. The idea of joining two different quantum simulations - a high level calcn. on the active region of interest, and a low level calcn. on its environment - formally defines a quantum embedding. While any combination of techniques constitutes an embedding, several rigorous formalisms have emerged that provide for exact feedback between the embedded system and its environment. These three formulations: d. functional embedding, Green's function embedding, and d. matrix embedding, resp., use the single-particle d., single-particle Green's function, and single-particle d. matrix as the quantum variables of interest. Many excellent reviews exist covering these methods individually. However, a unified presentation of the different formalisms is so far lacking. Indeed, the various languages commonly used, functional equations for d. functional embedding, diagrammatics for Green's function embedding, and entanglement arguments for d. matrix embedding, make the three formulations appear vastly different. In this Account, we introduce the basic equations of all three formulations in such a way as to highlight their many common intellectual strands. While we focus primarily on a straightforward theor. perspective, we also give a brief overview of recent applications and possible future developments. The first section starts with d. functional embedding, where we introduce the key embedding potential via the Euler equation. We then discuss recent work concerning the treatment of the nonadditive kinetic potential, before describing mean-field d. functional embedding and wave function in d. functional embedding. We finish the section with extensions to time-dependence and excited states. The second section is devoted to Green's function embedding. Here, we use the Dyson equation to obtain equations that parallel as closely as possible the d. functional embedding equations, with the hybridization playing the role of the embedding potential. Embedding a high-level self-energy within a low-level self-energy is treated analogously to wave function in d. functional embedding. The numerical computation of the high-level self-energy allows us to briefly introduce the bath representation in the quantum impurity problem. We then consider translationally invariant systems to bring in the important dynamical mean-field theory. Recent developments to incorporate screening and long-range interactions are discussed.The third section concerns d. matrix embedding. Here, we first highlight some math. complications assocd. with a simple Euler equation derivation, arising from the open nature of fragments. This motivates the d. matrix embedding theory, where we use the Schmidt decompn. to represent the entanglement through bath orbitals. The resulting impurity plus bath formulation resembles that of dynamical mean-field theory. We discuss the numerical self-consistency assocd. with using a high-level correlated wave function with a mean-field low-level treatment, and connect the resulting numerical inversion to that used in d. functional embedding. We finish with perspectives on the future of all three methods.
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    170. 170
      Kotliar, G.; Savrasov, S. Y.; Haule, K.; Oudovenko, V. S.; Parcollet, O.; Marianetti, C. Electronic structure calculations with dynamical mean-field theory. Rev. Mod. Phys. 2006, 78, 865– 951,  DOI: 10.1103/RevModPhys.78.865
      170
      Electronic structure calculations with dynamical mean-field theory
      Kotliar, G.; Savrasov, S. Y.; Haule, K.; Oudovenko, V. S.; Parcollet, O.; Marianetti, C. A.
      Reviews of Modern Physics (2006), 78 (3), 865-951CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
      A review of the basic ideas and techniques of the spectral d.-functional theory is presented. This method is currently used for electronic structure calcns. of strongly correlated materials where the one-electron description breaks down. The method is illustrated with several examples where interactions play a dominant role: systems near metal-insulator transitions, systems near vol. collapse transitions, and systems with local moments.
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    171. 171
      Lin, N.; Marianetti, C.; Millis, A. J.; Reichman, D. R. Dynamical mean-field theory for quantum chemistry. Phys. Rev. Lett. 2011, 106, 096402,  DOI: 10.1103/PhysRevLett.106.096402
      171
      Dynamical Mean-Field Theory for Quantum Chemistry
      Lin, Nan; Marianetti, C. A.; Millis, Andrew J.; Reichman, David R.
      Physical Review Letters (2011), 106 (9), 096402/1-096402/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The dynamical mean-field concept of approximating an unsolvable many-body problem in terms of the soln. of an auxiliary quantum impurity problem, introduced to study bulk materials with a continuous energy spectrum, is here extended to mols., i.e., finite systems with a discrete energy spectrum. The application to small clusters of hydrogen atoms yields ground state energies which are competitive with leading quantum chem. approaches at intermediate and large interat. distances as well as good approxns. to the excitation spectrum.
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    172. 172
      Zgid, D.; Chan, G. K.-L. Dynamical mean-field theory from a quantum chemical perspective. J. Chem. Phys. 2011, 134, 094115,  DOI: 10.1063/1.3556707
      172
      Dynamical mean-field theory from a quantum chemical perspective
      Zgid, Dominika; Chan, Garnet Kin-Lic
      Journal of Chemical Physics (2011), 134 (9), 094115/1-094115/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We investigate the dynamical mean-field theory (DMFT) from a quantum chem. perspective. Dynamical mean-field theory offers a formalism to extend quantum chem. methods for finite systems to infinite periodic problems within a local correlation approxn. In addn., quantum chem. techniques can be used to construct new ab initio Hamiltonians and impurity solvers for DMFT. Here, we explore some ways in which these things may be achieved. First, we present an informal overview of dynamical mean-field theory to connect to quantum chem. language. Next, we describe an implementation of dynamical mean-field theory where we start from an ab initio Hartree-Fock Hamiltonian that avoids double counting issues present in many applications of DMFT. We then explore the use of the CI hierarchy in DMFT as an approx. solver for the impurity problem. We also investigate some numerical issues of convergence within DMFT. Our studies are carried out in the context of the cubic hydrogen model, a simple but challenging test for correlation methods. Finally, we finish with some conclusions for future directions. (c) 2011 American Institute of Physics.
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    173. 173
      Hedin, L. New method for calculating the one-particle Green’s function with application to the electron-gas problem. Phys. Rev. 1965, 139, A796– A823,  DOI: 10.1103/PhysRev.139.A796
      173
      New method for calculating the one-particle Green's function with application to the electron-gas problem
      Hedin, Lars
      Physical Review (1965), 139 (3A), 796-823CODEN: PHRVAO; ISSN:0031-899X.
      A set of successively more accurate self-consistent equations for the 1-electron Green's function were derived. They correspond to an expansion in a screened potential rather than the bare Coulomb potential. The 1st equation is adequate for many purposes. Each equation follows from the demand that a corresponding expression for the total energy be stationary with respect to variations in the Green's function. The main information to be obtained, besides the total energy, is 1-particle-like excitation spectra, i.e., spectra characterized by the quantum nos. of a single particle. This includes the low-excitation spectra in metals as well as configurations in atoms, mols., and solids with one electron outside or one electron missing from a closed-shell structure. In the latter cases, an approx. description is obtained by a modified Hartree-Fock equation involving a "Coulomb hole" and a static screened potential in the exchange term. As an example, spectra of some atoms are discussed. To investigate the convergence of successive approxn. for the Green's function, extensive calcns. were made for the electron gas at a range of metallic ds. The results are expressed in terms of quasiparticle energies Ε(k) and quasiparticle interactions f(k,k'). The very 1st approxn. gives a good value for the magnitude of Ε(k.). To est. the deriv. of Ε(k), both the 1st- and the 2nd-order terms are needed. The derivative, and thus the sp. heat, differs from the free-particle value by only a few percent. The correction to the sp. heat keeps the same sign down to the lowest alkalimetal ds., and is smaller than those obtained recently by Silverstein (CA 59, 144d) and by Rice (CA 62, 7218f). The results for the paramagnetic susceptibility are unreliable in the alkalimetal-d.-region owing to poor convergence of the expansion for f. Besides the proof of a modified Luttinger-Ward-Klein variational principle and a related self-consistency idea, there is not much new in principle but emphasis is on the development of a numerically manageable approxn. scheme.
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    174. 174
      Aryasetiawan, F.; Gunnarsson, O. The GW method. Rep. Prog. Phys. 1998, 61, 237– 312,  DOI: 10.1088/0034-4885/61/3/002
      174
      The GW method
      Aryasetiawan, F.; Gunnarsson, O.
      Reports on Progress in Physics (1998), 61 (3), 237-312CODEN: RPPHAG; ISSN:0034-4885. (Institute of Physics Publishing)
      A review with many refs. Calcns. of ground-state and excited-state properties of materials have been one of the major goals of condensed matter physics. Ground-state properties of solids have been extensively investigated for several decades within the std. d. functional theory. Excited-state properties, on the other hand, were relatively unexplored in ab initio calcns. until a decade ago. The most suitable approach up to now for studying excited-state properties of extended systems is the Green function method. To calc. the Green function one requires the self-energy operator which is non-local and energy dependent. In this article we describe the GW approxn. which has turned out to be a fruitful approxn. to the self-energy. The Green function theory, numerical methods for carrying out the self-energy calcns., simplified schemes, and applications to various systems are described. Self-consistency issues and new developments beyond the GW approxn. are also discussed as well as the success and shortcomings of the GW approxn.
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    175. 175
      Reining, L. The GW approximation: content, successes and limitations. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2018, 8, e1344,  DOI: 10.1002/wcms.1344
      There is no corresponding record for this reference.
    176. 176
      Phillips, J. J.; Zgid, D. Communication: The description of strong correlation within self-consistent Green’s function second-order perturbation theory. J. Chem. Phys. 2014, 140, 241101,  DOI: 10.1063/1.4884951
      176
      Communication: The description of strong correlation within self-consistent Green's function second-order perturbation theory
      Phillips, Jordan J.; Zgid, Dominika
      Journal of Chemical Physics (2014), 140 (24), 241101/1-241101/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We report an implementation of self-consistent Green's function many-body theory within a second-order approxn. (GF2) for application with mol. systems. This is done by iterative soln. of the Dyson equation expressed in matrix form in an AO basis, where the Green's function and self-energy are built on the imaginary frequency and imaginary time domain, resp., and fast Fourier transform is used to efficiently transform these quantities as needed. We apply this method to several archetypical examples of strong correlation, such as a H32 finite lattice that displays a highly multireference electronic ground state even at equil. lattice spacing. In all cases, GF2 gives a phys. meaningful description of the metal to insulator transition in these systems, without resorting to spin-symmetry breaking. Our results show that self-consistent Green's function many-body theory offers a viable route to describing strong correlations while remaining within a computationally tractable single-particle formalism. (c) 2014 American Institute of Physics.
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    177. 177
      Pokhilko, P.; Zgid, D. Interpretation of multiple solutions in fully iterative GF2 and GW schemes using local analysis of two-particle density matrices. J. Chem. Phys. 2021, 155, 024101,  DOI: 10.1063/5.0055191
      177
      Interpretation of multiple solutions in fully iterative GF2 and GW schemes using local analysis of two-particle density matrices
      Pokhilko, Pavel; Zgid, Dominika
      Journal of Chemical Physics (2021), 155 (2), 024101CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Due to the presence of non-linear equations, iterative Green's function methods can result in multiple different solns. even for simple mol. systems. In contrast to the wave-function methods, a detailed and careful anal. of such mol. solns. was not performed before. In this work, we use two-particle d. matrixes to investigate local spin and charge correlators that quantify the charge resonance and covalent characters of these solns. When applied within the unrestricted orbital set, spin correlators elucidate the broken symmetry of the solns., contg. necessary information for building effective magnetic Hamiltonians. Based on GW and GF2 calcns. of simple mols. and transition metal complexes, we construct Heisenberg Hamiltonians, four-spin-four-center corrections, and biquadratic spin-spin interactions. These Hamiltonian parameterizations are compared to previous wave-function calcns. (c) 2021 American Institute of Physics.
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    178. 178
      Blase, X.; Duchemin, I.; Jacquemin, D. The Bethe–Salpeter equation in chemistry: relations with TD-DFT, applications and challenges. Chem. Soc. Rev. 2018, 47, 1022– 1043,  DOI: 10.1039/C7CS00049A
      178
      The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges
      Blase, Xavier; Duchemin, Ivan; Jacquemin, Denis
      Chemical Society Reviews (2018), 47 (3), 1022-1043CODEN: CSRVBR; ISSN:0306-0012. (Royal Society of Chemistry)
      We review the many-body Green's function Bethe-Salpeter equation (BSE) formalism that is rapidly gaining importance for the study of the optical properties of mol. org. systems. We emphasize in particular its similarities and differences with time-dependent d. functional theory (TD-DFT), both methods sharing the same formal O(N4) computing time scaling with system size. By comparison with higher level wavefunction based methods and exptl. results, the advantages of BSE over TD-DFT are presented, including an accurate description of charge-transfer states and an improved accuracy for the challenging cyanine dyes. We further discuss the models that have been developed for including environmental effects. Finally, we summarize the challenges to be faced so that BSE reaches the same popularity as TD-DFT.
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    179. 179
      Pokhilko, P.; Iskakov, S.; Yeh, C.-N.; Zgid, D. Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green’s function methods: Theory and application to GW and GF2. J. Chem. Phys. 2021, 155, 024119,  DOI: 10.1063/5.0054661
      179
      Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green's function methods: Theory and application to GW and GF2
      Pokhilko, Pavel; Iskakov, Sergei; Yeh, Chia-Nan; Zgid, Dominika
      Journal of Chemical Physics (2021), 155 (2), 024119CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      One-particle Green's function methods can model mol. and solid spectra at zero or non-zero temps. One-particle Green's functions directly provide electronic energies and one-particle properties, such as dipole moment. However, the evaluation of two-particle properties, such as 〈S2〉 and 〈N2〉, can be challenging because they require a soln. of the computationally expensive Bethe-Salpeter equation to find two-particle Green's functions. We demonstrate that the soln. of the Bethe-Salpeter equation can be completely avoided. Applying the thermodn. Hellmann-Feynman theorem to self-consistent one-particle Green's function methods, we derive expressions for two-particle d. matrixes in a general case and provide explicit expressions for GF2 and GW methods. Such d. matrixes can be decompd. into an antisymmetrized product of correlated one-electron d. matrixes and the two-particle electronic cumulant of the d. matrix. Cumulant expressions reveal a deviation from ensemble representability for GW, explaining its known deficiencies. We analyze the temp. dependence of 〈S2〉 and 〈N2〉 for a set of small closed-shell systems. Interestingly, both GF2 and GW show a non-zero spin contamination and a non-zero fluctuation of the no. of particles for closed-shell systems at the zero-temp. limit. (c) 2021 American Institute of Physics.
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    180. 180
      Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Fractional spins and static correlation error in density functional theory. J. Chem. Phys. 2008, 129, 121104,  DOI: 10.1063/1.2987202
      180
      Fractional spins and static correlation error in density functional theory
      Cohen, Aron J.; Mori-Sanchez, Paula; Yang, Weitao
      Journal of Chemical Physics (2008), 129 (12), 121104/1-121104/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Electronic states with fractional spins arise in systems with large static correlation (strongly correlated systems). Such fractional-spin states are shown to be ensembles of degenerate ground states with normal spins. It is proven here that the energy of the exact functional for fractional-spin states is a const., equal to the energy of the comprising degenerate pure-spin states. Dramatic deviations from this exact constancy condition exist with all approx. functionals, leading to large static correlation errors for strongly correlated systems, such as chem. bond dissocn. and band structure of Mott insulators. This is demonstrated with numerical calcns. for several mol. systems. Approximating the constancy behavior for fractional spins should be a major aim in functional constructions and should open the frontier for d. functional theory to describe strongly correlated systems. The key results are also shown to apply in reduced d.-matrix functional theory. (c) 2008 American Institute of Physics.
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    181. 181
      Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Insights into current limitations of density functional theory. Science 2008, 321, 792– 794,  DOI: 10.1126/science.1158722
      181
      Insights into Current Limitations of Density Functional Theory
      Cohen, Aron J.; Mori-Sanchez, Paula; Yang, Weitao
      Science (Washington, DC, United States) (2008), 321 (5890), 792-794CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
      A review. D. functional theory of electronic structure is widely and successfully applied in simulations throughout engineering and sciences. However, for many predicted properties, there are spectacular failures that can be traced to the delocalization error and static correlation error of commonly used approxns. These errors can be characterized and understood through the perspective of fractional charges and fractional spins introduced recently. Reducing these errors will open new frontiers for applications of d. functional theory.
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    182. 182
      Grimme, S.; Hansen, A. A practicable real-space measure and visualization of static electron-correlation effects. Angew. Chem., Int. Ed. 2015, 54, 12308– 12313,  DOI: 10.1002/anie.201501887
      182
      A Practicable Real-Space Measure and Visualization of Static Electron-Correlation Effects
      Grimme, Stefan; Hansen, Andreas
      Angewandte Chemie, International Edition (2015), 54 (42), 12308-12313CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
      The inclusion of dynamical and static electron correlation (SEC) is mandatory for accurate quantum chem. (QC). SEC is particularly difficult to calc. and hence a qual. understanding is important to judge the applicability of approx. QC methods. Existing scalar SEC diagnostics, however, lack the important information where the SEC effects occur in a mol. We introduce an anal. tool based on a fractional occupation no. weighted electron d. (ρFOD) that is plotted in 3D for a pre-defined contour surface value. The scalar field is obtained by finite-temp. DFT calcns. with pre-defined electronic temp. (e.g. TPSS at 5000 K). FOD plots only show the contribution of the "hot" (strongly correlated) electrons. We discuss illustrative plots for a broad range of chem. systems from small mols. to large conjugated mols. with polyradicaloid character. Spatial integration yields a single no. which can be used to globally quantify SEC.
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    183. 183
      Muechler, L.; Badrtdinov, D. I.; Hampel, A.; Cano, J.; Rösner, M.; Dreyer, C. E. Quantum embedding methods for correlated excited states of point defects: Case studies and challenges. Phys. Rev. B 2022, 105, 235104,  DOI: 10.1103/PhysRevB.105.235104
      183
      Quantum embedding methods for correlated excited states of point defects: Case studies and challenges
      Muechler, Lukas; Badrtdinov, Danis I.; Hampel, Alexander; Cano, Jennifer; Rosner, Malte; Dreyer, Cyrus E.
      Physical Review B (2022), 105 (23), 235104CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      A quant. description of the excited electronic states of point defects and impurities is crucial for understanding materials properties, and possible applications of defects in quantum technologies. This is a considerable challenge for computational methods, since Kohn-Sham d. functional theory (DFT) is inherently a ground-state theory, while higher-level methods are often too computationally expensive for defect systems. Recently, embedding approaches have been applied that treat defect states with many-body methods, while using DFT to describe the bulk host material. We implement such an embedding method, based on Wannierization of defect orbitals and the constrained RPA approach, and perform systematic characterization of the method for three distinct systems with current technol. relevance: a carbon dimer replacing a B and N pair in bulk hexagonal BN (CBCN), the neg. charged nitrogen-vacancy center in diamond (NV-), and an Fe impurity on the Al site in wurtzite AlN (FeAl). In the context of these test-case defects, we demonstrate that crucial considerations of the methodol. include convergence of the bulk screening of the active-space Coulomb interaction, the choice of exchange-correlation functional for the initial DFT calcn., and the treatment of the "double-counting" correction. For CBCN we show that the embedding approach gives many-body states in agreement with anal. results on the Hubbard dimer model, which allows us to elucidate the effects of the DFT functional and double-counting correction. For the NV- center, our method demonstrates good quant. agreement with expts. for the zero-phonon line of the triplet-triplet transition. Finally, we illustrate challenges assocd. with this method for detg. the energies and orderings of the complex spin multiplets in FeAl.
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    184. 184
      Nielsen, M. A.; Chuang, I. L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, 2010.
      There is no corresponding record for this reference.
    185. 185
      Almeida, M.; Omar, Y.; Rocha Vieira, V. Introduction to entanglement and applications to the simulation of many-body quantum systems. In Strongly Correlated Systems, Coherence And Entanglement; World Scientific: Singapore, 2007; Chapter 19, pp 525– 547.
      There is no corresponding record for this reference.
    186. 186
      Ding, L.; Knecht, S.; Zimborás, Z.; Schilling, C. Quantum Correlations in Molecules: from Quantum Resourcing to Chemical Bonding. Quantum Sci. Technol. 2023, 8, 015015,  DOI: 10.1088/2058-9565/aca4ee
      There is no corresponding record for this reference.
    187. 187
      Omar, Y. Particle statistics in quantum information processing. Int. J. Quantum Inf. 2005, 3, 201– 205,  DOI: 10.1142/S021974990500075X
      There is no corresponding record for this reference.
    188. 188
      Benatti, F.; Floreanini, R.; Franchini, F.; Marzolino, U. Entanglement in indistinguishable particle systems. Phys. Rep. 2020, 878, 1– 27,  DOI: 10.1016/j.physrep.2020.07.003
      There is no corresponding record for this reference.
    189. 189
      Henderson, T. M.; Bulik, I. W.; Stein, T.; Scuseria, G. E. Seniority-based coupled cluster theory. J. Chem. Phys. 2014, 141, 244104,  DOI: 10.1063/1.4904384
      189
      Seniority-based coupled cluster theory
      Henderson, Thomas M.; Bulik, Ireneusz W.; Stein, Tamar; Scuseria, Gustavo E.
      Journal of Chemical Physics (2014), 141 (24), 244104/1-244104/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Doubly occupied CI (DOCI) with optimized orbitals often accurately describes strong correlations while working in a Hilbert space much smaller than that needed for full CI. However, the scaling of such calcns. remains combinatorial with system size. Pair coupled cluster doubles (pCCD) is very successful in reproducing DOCI energetically, but can do so with low polynomial scaling (N3, disregarding the two-electron integral transformation from at. to MOs). We show here several examples illustrating the success of pCCD in reproducing both the DOCI energy and wave function and show how this success frequently comes about. What DOCI and pCCD lack are an effective treatment of dynamic correlations, which we here add by including higher-seniority cluster amplitudes which are excluded from pCCD. This frozen pair coupled cluster approach is comparable in cost to traditional closed-shell coupled cluster methods with results that are competitive for weakly correlated systems and often superior for the description of strongly correlated systems. (c) 2014 American Institute of Physics.
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    190. 190
      Boguslawski, K.; Tecmer, P. Orbital entanglement in quantum chemistry. Int. J. Quantum Chem. 2015, 115, 1289– 1295,  DOI: 10.1002/qua.24832
      190
      Orbital entanglement in quantum chemistry
      Boguslawski, Katharina; Tecmer, Pawel
      International Journal of Quantum Chemistry (2015), 115 (19), 1289-1295CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)
      The basic concepts of orbital entanglement and its application to chem. are briefly reviewed. The calcn. of orbital entanglement measures from correlated wavefunctions is discussed in terms of reduced n-particle d. matrixes. Possible simplifications in their evaluation are highlighted in case of seniority-zero wavefunctions. Specifically, orbital entanglement allows us to dissect electron correlation effects in its strong and weak contributions, to det. bond orders, to assess the quality and stability of active space calcns., to monitor chem. reactions, and to identify points along the reaction coordinate where electronic wavefunctions change drastically. Thus, orbital entanglement represents a useful and intuitive tool to interpret complex electronic wavefunctions and to facilitate a qual. understanding of electronic structure and how it changes in chem. processes. © 2014 Wiley Periodicals, Inc.
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    191. 191
      Ding, L.; Zimboras, Z.; Schilling, C. Quantifying Electron Entanglement Faithfully. arXiv:2207.03377 2022, DOI: 10.48550/arXiv.2207.03377 .
      There is no corresponding record for this reference.
    192. 192
      Legeza, Ö.; Sólyom, J. Optimizing the density-matrix renormalization group method using quantum information entropy. Phys. Rev. B 2003, 68, 195116,  DOI: 10.1103/PhysRevB.68.195116
      192
      Optimizing the density-matrix renormalization group method using quantum information entropy
      Legeza, O.; Solyom, J.
      Physical Review B: Condensed Matter and Materials Physics (2003), 68 (19), 195116/1-195116/19CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      In order to optimize the ordering of the lattice sites in the momentum space and quantum chem. versions of the d.-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various mols. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chem. DMRG to a great extent. The effect of lattice site ordering on the no. of block states to be kept during the RG procedure is also investigated.
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    193. 193
      Stein, C. J.; Reiher, M. Measuring multi-configurational character by orbital entanglement. Mol. Phys. 2017, 115, 2110– 2119,  DOI: 10.1080/00268976.2017.1288934
      193
      Measuring multi-configurational character by orbital entanglement
      Stein, Christopher J.; Reiher, Markus
      Molecular Physics (2017), 115 (17-18), 2110-2119CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
      One of the most crit. tasks at the very beginning of a quantum chem. investigation is the choice of either a multi- or single-configurational method. Naturally, many proposals exist to define a suitable diagnostic of the multi-configurational character for various types of wave functions in order to assist this crucial decision. Here, we present a new orbital-entanglement-based multi-configurational diagnostic termed Zs(1). The correspondence of orbital entanglement and static (or non-dynamic) electron correlation permits the definition of such a diagnostic. We chose our diagnostic to meet important requirements such as well-defined limits for pure single-configurational and multi-configurational wave functions. The Zs(1) diagnostic can be evaluated from a partially converged, but qual. correct, and therefore inexpensive d. matrix renormalization group wave function as in our recently presented automated active orbital selection protocol. Its robustness and the fact that it can be evaluated at low cost make this diagnostic a practical tool for routine applications.
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    194. 194
      Stein, C. J.; Reiher, M. Automated selection of active orbital spaces. J. Chem. Theory Comput. 2016, 12, 1760– 1771,  DOI: 10.1021/acs.jctc.6b00156
      194
      Automated Selection of Active Orbital Spaces
      Stein, Christopher J.; Reiher, Markus
      Journal of Chemical Theory and Computation (2016), 12 (4), 1760-1771CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      One of the key challenges of quantum-chem. multi-configuration methods is the necessity to manually select orbitals for the active space. This selection requires both expertise and experience and can therefore impose severe limitations on the applicability of this most general class of ab initio methods. A poor choice of the active orbital space may yield even qual. wrong results. This is obviously a severe problem, esp. for wave function methods that are designed to be systematically improvable. Here, we show how the iterative nature of the d. matrix renormalization group combined with its capability to include up to about 100 orbitals in the active space can be exploited for a systematic assessment and selection of active orbitals. These benefits allow us to implement an automated approach for active orbital space selection, which can turn multi-configuration models into black box approaches.
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    195. 195
      Boguslawski, K.; Tecmer, P.; Barcza, G.; Legeza, O.; Reiher, M. Orbital entanglement in bond-formation processes. J. Chem. Theory Comput. 2013, 9, 2959– 2973,  DOI: 10.1021/ct400247p
      195
      Orbital Entanglement in Bond-Formation Processes
      Boguslawski, Katharina; Tecmer, Pawel; Barcza, Gergely; Legeza, Ors; Reiher, Markus
      Journal of Chemical Theory and Computation (2013), 9 (7), 2959-2973CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      The accurate calcn. of the (differential) correlation energy is central to the quantum chem. description of bond-formation and bond-dissocn. processes. In order to est. the quality of single- and multireference approaches for this purpose, various diagnostic tools have been developed. In this work, we elaborate on our previous observation that one- and two-orbital-based entanglement measures provide quant. means for the assessment and classification of electron correlation effects among MOs. The dissocn. behavior of some prototypical diat. mols. features all types of correlation effects relevant for chem. bonding. We demonstrate that our entanglement anal. is convenient to dissect these electron correlation effects and to provide a conceptual understanding of bond-forming and bond-breaking processes from the point of view of quantum information theory.
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    196. 196
      Huang, Z.; Kais, S. Entanglement as measure of electron–electron correlation in quantum chemistry calculations. Chem. Phys. Lett. 2005, 413, 1– 5,  DOI: 10.1016/j.cplett.2005.07.045
      196
      Entanglement as measure of electron-electron correlation in quantum chemistry calculations
      Huang, Zhen; Kais, Sabre
      Chemical Physics Letters (2005), 413 (1-3), 1-5CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)
      In quantum chem. calcns., the correlation energy is defined as the difference between the Hartree-Fock limit energy and the exact soln. of the nonrelativistic Schroedinger equation. With this definition, the electron correlation effects are not directly observable. In this report, we show that the entanglement can be used as an alternative measure of the electron correlation in quantum chem. calcns. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics. As an example we calc. the entanglement for He atom and H2 mol. with different basis sets.
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    197. 197
      Boguslawski, K.; Tecmer, P.; Legeza, O.; Reiher, M. Entanglement measures for single-and multireference correlation effects. J. Phys. Chem. Lett. 2012, 3, 3129– 3135,  DOI: 10.1021/jz301319v
      197
      Entanglement Measures for Single- and Multireference Correlation Effects
      Boguslawski, Katharina; Tecmer, Pawel; Legeza, Ors; Reiher, Markus
      Journal of Physical Chemistry Letters (2012), 3 (21), 3129-3135CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      Electron correlation effects are essential for an accurate ab initio description of mols. A quant. a priori knowledge of the single- or multireference nature of electronic structures as well as of the dominant contributions to the correlation energy can facilitate the decision regarding the optimum quantum chem. method of choice. We propose concepts from quantum information theory as orbital entanglement measures that allow us to evaluate the single- and multireference character of any mol. structure in a given orbital basis set. By studying these measures we can detect possible artifacts of small active spaces.
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    198. 198
      Ziesche, P. Correlation strength and information entropy. Int. J. Quantum Chem. 1995, 56, 363– 369,  DOI: 10.1002/qua.560560422
      198
      Correlation strength and information entropy
      Ziesche, Paul
      International Journal of Quantum Chemistry (1995), 56 (4), 363-69CODEN: IJQCB2; ISSN:0020-7608. (Wiley)
      The correlation present in the nondegenerate ground state of an interacting Fermi system is discussed in terms of reduced d. matrixes and their cumulant expansion. By generalizing a result obtained for the interacting uniform electron gas (correlation induced exchange-hole narrowing), possible measures of the correlation strength in terms of natural occupation nos. (the eigenvalues of the true one-particle d. matrix) are introduced. These quantities, the ν-order nonidempotency and the information entropy of the natural occupation nos., result from the correlated many-body wave function and characterize the ground-state correlation in addn. to the usual correlation energy. The uniform electron gas serves as a first illustrative example.
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    199. 199
      Ghosh, K. J.; Kais, S.; Herschbach, D. R. Geometrical picture of the electron–electron correlation at the large-D limit. Phys. Chem. Chem. Phys. 2022, 24, 9298– 9307,  DOI: 10.1039/D2CP00438K
      199
      Geometrical picture of the electron-electron correlation at the large-D limit
      Ghosh, Kumar J. B.; Kais, Sabre; Herschbach, Dudley R.
      Physical Chemistry Chemical Physics (2022), 24 (16), 9298-9307CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      In electronic structure calcns., the correlation energy is defined as the difference between the mean field and the exact soln. of the non relativistic Schroedinger equation. Such an error in the different calcns. is not directly observable as there is no simple quantum mech. operator, apart from correlation functions, that correspond to such quantity. Here, we use the dimensional scaling approach, in which the electrons are localized at the large-dimensional scaled space, to describe a geometric picture of the electronic correlation. Both, the mean field, and the exact solns. at the large-D limit have distinct geometries. Thus, the difference might be used to describe the correlation effect. Moreover, correlations can be also described and quantified by the entanglement between the electrons, which is a strong correlation without a classical analog. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics and bounded by the area law for local gapped Hamiltonians of interacting many-body systems. This study opens up the possibility of presenting a geometrical picture of the electron-electron correlations and might give a bound on the correlation energy. The results at the large-D limit and at D = 3 indicate the feasibility of using the geometrical picture to get a bound on the electron-electron correlations.
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    200. 200
      Rissler, J.; Noack, R. M.; White, S. R. Measuring orbital interaction using quantum information theory. Chem. Phys. 2006, 323, 519– 531,  DOI: 10.1016/j.chemphys.2005.10.018
      200
      Measuring orbital interaction using quantum information theory
      Rissler, Joerg; Noack, Reinhard M.; White, Steven R.
      Chemical Physics (2006), 323 (2-3), 519-531CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)
      Quantum information theory gives rise to a straightforward definition of the interaction of electrons Ip,q in two orbitals p,q for a given many-body wave function. A convenient way to calc. the von Neumann entropies needed is presented in this work, and the orbital interaction Ip,q is successfully tested for different types of chem. bonds. As an example of an application of Ip,q beyond the interpretation of wave functions, Ip,q is then used to investigate the ordering problem in the d.-matrix renormalization group.
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    201. 201
      Cao, Y.; Romero, J.; Olson, J. P.; Degroote, M.; Johnson, P. D.; Kieferová, M.; Kivlichan, I. D.; Menke, T.; Peropadre, B.; Sawaya, N. P. D.; Sim, S.; Veis, L.; Aspuru-Guzik, A. Quantum Chemistry in the Age of Quantum Computing. Chem. Rev. 2019, 119, 10856– 10915,  DOI: 10.1021/acs.chemrev.8b00803
      201
      Quantum Chemistry in the Age of Quantum Computing
      Cao, Yudong; Romero, Jonathan; Olson, Jonathan P.; Degroote, Matthias; Johnson, Peter D.; Kieferova, Maria; Kivlichan, Ian D.; Menke, Tim; Peropadre, Borja; Sawaya, Nicolas P. D.; Sim, Sukin; Veis, Libor; Aspuru-Guzik, Alan
      Chemical Reviews (Washington, DC, United States) (2019), 119 (19), 10856-10915CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
      A review. Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chem. communities over the past century. Although many approxn. methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging complexity landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chem. such as the electronic structure of mols. In the past two decades significant advances have been made in developing algorithms and phys. hardware for quantum computing, heralding a revolution in simulation of quantum systems. This article is an overview of the algorithms and results that are relevant for quantum chem. The intended audience is both quantum chemists who seek to learn more about quantum computing, and quantum computing researchers who would like to explore applications in quantum chem.
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    202. 202
      Reiher, M.; Wiebe, N.; Svore, K. M.; Wecker, D.; Troyer, M. Elucidating reaction mechanisms on quantum computers. Proc. Nat. Acad. Sci. 2017, 114, 7555– 7560,  DOI: 10.1073/pnas.1619152114
      202
      Elucidating reaction mechanisms on quantum computers
      Reiher, Markus; Wiebe, Nathan; Svore, Krysta M.; Wecker, Dave; Troyer, Matthias
      Proceedings of the National Academy of Sciences of the United States of America (2017), 114 (29), 7555-7560CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      With rapid recent advances in quantum technol., we are close to the threshold of quantum devices whose computational powers can exceed those of classical supercomputers. Here, we show that a quantum computer can be used to elucidate reaction mechanisms in complex chem. systems, using the open problem of biol. nitrogen fixation in nitrogenase as an example. We discuss how quantum computers can augment classical computer simulations used to probe these reaction mechanisms, to significantly increase their accuracy and enable hitherto intractable simulations. Our resource ests. show that, even when taking into account the substantial overhead of quantum error correction, and the need to compile into discrete gate sets, the necessary computations can be performed in reasonable time on small quantum computers. Our results demonstrate that quantum computers will be able to tackle important problems in chem. without requiring exorbitant resources.
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    203. 203
      Tubman, N. M.; Mejuto-Zaera, C.; Epstein, J. M.; Hait, D.; Levine, D. S.; Huggins, W.; Jiang, Z.; McClean, J. R.; Babbush, R.; Head-Gordon, M.; Whaley, K. B. Postponing the orthogonality catastrophe: efficient state preparation for electronic structure simulations on quantum devices. arXiv:1809.05523 2018, DOI: 10.48550/arXiv.1809.05523 .
      There is no corresponding record for this reference.
    204. 204
      Carbone, A.; Galli, D. E.; Motta, M.; Jones, B. Quantum circuits for the preparation of spin eigenfunctions on quantum computers. Symmetry 2022, 14, 624,  DOI: 10.3390/sym14030624
      204
      Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers
      Carbone, Alessandro; Galli, Davide Emilio; Motta, Mario; Jones, Barbara
      Symmetry (2022), 14 (3), 624CODEN: SYMMAM; ISSN:2073-8994. (MDPI AG)
      The application of quantum algorithms to the study of many-particle quantum systems requires the ability to prep. wave functions that are relevant in the behavior of the system under study. Hamiltonian symmetries are important instruments used to classify relevant many-particle wave functions and to improve the efficiency of numerical simulations. In this work, quantum circuits for the exact and approx. prepn. of total spin eigenfunctions on quantum computers are presented. Two different strategies are discussed and compared: exact recursive construction of total spin eigenfunctions based on the addn. theorem of angular momentum, and heuristic approxn. of total spin eigenfunctions based on the variational optimization of a suitable cost function. The construction of these quantum circuits is illustrated in detail, and the prepn. of total spin eigenfunctions is demonstrated on IBM quantum devices, focusing on three- and five-spin systems on graphs with triangle connectivity.
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    205. 205
      Lacroix, D.; Ruiz Guzman, E. A.; Siwach, P. Symmetry breaking/symmetry preserving circuits and symmetry restoration on quantum computers. Eur. Phys. J. A 2023, 59, 3,  DOI: 10.1140/epja/s10050-022-00911-7
      205
      Symmetry breaking/symmetry preserving circuits and symmetry restoration on quantum computers - A quantum many-body perspective
      Lacroix, Denis; Ruiz Guzman, Edgar Andres; Siwach, Pooja
      European Physical Journal A: Hadrons and Nuclei (2023), 59 (1), 3CODEN: EPJAFV; ISSN:1434-601X. (Springer International Publishing AG)
      Abstr.: We discuss here some aspects related to the symmetries of a quantum many-body problem when trying to treat it on a quantum computer. Several features related to symmetry conservation, symmetry breaking, and possible symmetry restoration are reviewed. After briefly discussing some of the std. symmetries relevant for many-particle systems, we discuss the advantage of encoding some symmetries directly in quantum ansatze, esp. to reduce the quantum register size. It is, however, well-known that the use of symmetry-breaking states can also be a unique way to incorporate specific internal correlations when a spontaneous symmetry breaking occurs. These aspects are discussed in the quantum computing context. Ultimately, an accurate description of quantum systems can be achieved only when the initially broken symmetries are properly restored. We review several methods explored previously to perform symmetry restoration on a quantum computer, for instance, the ones based on symmetry filtering by quantum phase estn. and by an iterative independent set of Hadamard tests. We propose novel methods that pave the new directions to perform symmetry restoration, like those based on the purifn. of the state employing the linear combination of unitaries (LCU) approach.
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    206. 206
      Lee, S.; Lee, J.; Zhai, H.; Tong, Y.; Dalzell, A. M.; Kumar, A.; Helms, P.; Gray, J.; Cui, Z.-H.; Liu, W.; Kastoryano, M.; Babbush, R.; Preskill, J.; Reichman, D. R.; Campbell, E. T.; Valeev, E. F.; Lin, L.; Chan, G. K.-L. Is there evidence for exponential quantum advantage in quantum chemistry. arXiv:2208.02199 2022, DOI: 10.48550/arXiv.2208.02199 .
      There is no corresponding record for this reference.
    207. 207
      Tazhigulov, R. N.; Sun, S.-N.; Haghshenas, R.; Zhai, H.; Tan, A. T.; Rubin, N. C.; Babbush, R.; Minnich, A. J.; Chan, G. K.-L. Simulating models of challenging correlated molecules and materials on the Sycamore quantum processor. PRX Quantum 2022, 3, 040318,  DOI: 10.1103/PRXQuantum.3.040318
      There is no corresponding record for this reference.
    208. 208
      Amsler, M.; Deglmann, P.; Degroote, M.; Kaicher, M. P.; Kiser, M.; Kühn, M.; Kumar, C.; Maier, A.; Samsonidze, G.; Schroeder, A.; Streif, M.; Vodola, D.; Wever, C. Quantum-enhanced quantum Monte Carlo: an industrial view. arXiv:2301.11838 2023, DOI: 10.48550/arXiv.2301.11838 .
      There is no corresponding record for this reference.
    209. 209
      Rubin, N. C.; Berry, D. W.; Malone, F. D.; White, A. F.; Khattar, T.; Sicolo, A. E. D.; Kühn, S.; Kaicher, M.; Lee, M.; Babbush, J. Fault-tolerant quantum simulation of materials using Bloch orbitals. arXiv:2302.05531 2023, DOI: 10.48550/arXiv.2302.05531 .
      There is no corresponding record for this reference.
    210. 210
      Lykos, P.; Pratt, G. W. Discussion on The Hartree-Fock Approximation. Rev. Mod. Phys. 1963, 35, 496– 501,  DOI: 10.1103/RevModPhys.35.496
      There is no corresponding record for this reference.
    211. 211
      Slater, J. C. The Theory of Complex Spectra. Phys. Rev. 1929, 34, 1293– 1322,  DOI: 10.1103/PhysRev.34.1293
      211
      The theory of complex spectra
      Slater, J. C.
      Physical Review (1929), 34 (), 1293-1323CODEN: PHRVAO; ISSN:0031-899X.
      At. multiplets are treated by wave mechanics. The first part deals with the derivation of Hund's scheme for multiplet classification directly from theory. The second part deals with the computation of energy distances between multiplets and their comparison with exptl. values for some examples.
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    212. 212
      Slater, J. C. Solid-state and molecular theory: a scientific biography; Wiley-Interscience: New York, 1975.
      There is no corresponding record for this reference.
    213. 213
      Matsen, F. Spin-free quantum chemistry. In Adv. Quantum Chem.; Interscience: New York, 1964; Vol. 1, pp 59– 114.
      There is no corresponding record for this reference.
    214. 214
      Paldus, J. The Unitary Group for the Evaluation of Electronic Energy Matrix Elements. In The Unitary Group for the Evaluation of Electronic Energy Matrix Elements; Springer: Berlin, 1981; pp 1– 50.
      There is no corresponding record for this reference.
    215. 215
      Shavitt, I. The Graphical Unitary Group Approach and its Application to Direct Configuration Interaction Calculations. In The Unitary Group for the Evaluation of Electronic Energy Matrix Elements; Springer: Berlin, 1981; pp 51– 99.
      There is no corresponding record for this reference.
    216. 216
      Dobrautz, W.; Smart, S. D.; Alavi, A. Efficient formulation of full configuration interaction quantum Monte Carlo in a spin eigenbasis via the graphical unitary group approach. J. Chem. Phys. 2019, 151, 094104,  DOI: 10.1063/1.5108908
      216
      Efficient formulation of full configuration interaction quantum Monte Carlo in a spin eigenbasis via the graphical unitary group approach
      Dobrautz, Werner; Smart, Simon D.; Alavi, Ali
      Journal of Chemical Physics (2019), 151 (9), 094104/1-094104/33CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We provide a spin-adapted formulation of the Full CI Quantum Monte Carlo (FCIQMC) algorithm, based on the Graphical Unitary Group Approach (GUGA), which enables the exploitation of SU(2) symmetry within this stochastic framework. Random excitation generation and matrix element calcn. on the Shavitt graph of GUGA can be efficiently implemented via a biasing procedure on the branching diagram. The use of a spin-pure basis explicitly resolves the different spin-sectors and ensures that the stochastically sampled wavefunction is an eigenfunction of the total spin operator ̂S2. The method allows for the calcn. of states with low or intermediate spin in systems dominated by Hund's first rule, which are otherwise generally inaccessible. Furthermore, in systems with small spin gaps, the new methodol. enables much more rapid convergence with respect to walker no. and simulation time. Some illustrative applications of the GUGA-FCIQMC method are provided: computation of the 2F - 4F spin gap of the cobalt atom in large basis sets, achieving chem. accuracy to expt., and the 1Σg+, 3Σg+, 5Σg+, and 7Σg+ spin-gaps of the stretched N2 mol., an archetypal strongly correlated system. (c) 2019 American Institute of Physics.
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    217. 217
      McLean, A. D.; Lengsfield, B. H.; Pacansky, J.; Ellinger, Y. Symmetry breaking in molecular calculations and the reliable prediction of equilibrium geometries. The formyloxyl radical as an example. J. Chem. Phys. 1985, 83, 3567– 3576,  DOI: 10.1063/1.449162
      217
      Symmetry breaking in molecular calculations and the reliable prediction of equilibrium geometries. The formyloxyl radical as an example
      McLean, A. D.; Lengsfield, B. H., III; Pacansky, J.; Ellinger, Y.
      Journal of Chemical Physics (1985), 83 (7), 3567-76CODEN: JCPSA6; ISSN:0021-9606.
      A systematic approach is given to symmetry breaking in mol. calcns., based on MCSCF and multiref. CI (MRCI) wave functions. A series of MCSCF expansions is generated by successively incorporating resonance effects and size effects into the wave functions. The character of the potential surface obtained at each level is analyzed. As an example, the potential energy curves of the ground state (σ) and the 1st excited state (π) of the HCO2 radical are characterized. The σ and π equil. structures are sym., with an adiabatic σ-π excitation energy of 9.2 kcal/mol. Unlike earlier theor. studies, present MCSCF model produces a qual. correct potential surface. Reliable vibrational frequencies are calcd. from the MRCI potential surface.
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    218. 218
      Ayala, P. Y.; Schlegel, H. B. A nonorthogonal CI treatment of symmetry breaking in sigma formyloxyl radical. J. Chem. Phys. 1998, 108, 7560– 7567,  DOI: 10.1063/1.476190
      218
      A nonorthogonal CI treatment of symmetry breaking in sigma formyloxyl radical
      Ayala, Philippe Y.; Schlegel, H. Bernhard
      Journal of Chemical Physics (1998), 108 (18), 7560-7567CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Spatial symmetry breaking can occur in Hartree-Fock (HF) wavefunctions when there are ≥2 close-lying configurations that can mix strongly, such as in HCO2•, NO2 and allyl radical. Like spin contamination, spatial symmetry breaking can cause sizeable errors when perturbation theory is used to est. the correlation energy. With conventional methodol., very large MCSCF and MRCI calcns. are necessary to overcome the spatial symmetry-breaking problem. This paper explores an alternative approach in which a 2×2 nonorthogonal CI is used to recombine the 2 symmetry broken HF determinants. The necessary matrix elements closely resemble those used in spin-projection calcns. Second-order perturbation theory is used to include electron correlation energy in this approach. With perturbative corrections for correlation energy, this approach predicts that the 2B2 structure is a min., in agreement with the best available calcns.
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    219. 219
      Manne, R. Brillouin’s theorem in Roothaan’s open-shell SCF method. Mol. Phys. 1972, 24, 935– 944,  DOI: 10.1080/00268977200102061
      219
      Brillouin's theorem in Roothaan's open-shell SCF method
      Manne, Rolf
      Molecular Physics (1972), 24 (5), 935-44CODEN: MOPHAM; ISSN:0026-8976.
      The underlying assumptions of Roothaan's symmetry-restricted SCF method for open-shell systems are considered. A restricted Brillouin theorem is formulated and applied in a discussion of total energy discontinuities arising in restricted SCF calcns. of systems which exhibit Jahn-Teller instability, such as the tetrahedral 2T2 state of CH4+.
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    220. 220
      Davidson, E. R.; Borden, W. T. Symmetry breaking in polyatomic molecules: real and artifactual. J. Phys. Chem. 1983, 87, 4783– 4790,  DOI: 10.1021/j150642a005
      220
      Symmetry breaking in polyatomic molecules: real and artifactual
      Davidson, Ernest R.; Borden, Weston Thatcher
      Journal of Physical Chemistry (1983), 87 (24), 4783-90CODEN: JPCHAX; ISSN:0022-3654.
      A review is presented with over 40 refs. of recent work by the authors and their collaborators on mols. with broken symmetry.
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    221. 221
      Čížek, J.; Paldus, J. Stability Conditions for the Solutions of the Hartree–Fock Equations for Atomic and Molecular Systems. Application to the Pi-Electron Model of Cyclic Polyenes. J. Chem. Phys. 1967, 47, 3976– 3985,  DOI: 10.1063/1.1701562
      221
      Stability conditions for the solutions of the Hartree-Fock equations for atomic and molecular systems. Application to the π-electron model of cyclic polyenes
      Cizek, Jiri; Paldus, Josef
      Journal of Chemical Physics (1967), 47 (10), 3976-85CODEN: JCPSA6; ISSN:0021-9606.
      The stability conditions which ensure that the Hartree-Fock determinant minimizes the energy expectation value are rederived by using the language familiar in quantum chemistry. These stability conditions are then specified for the case of closed-shell electronic systems which allow addnl. simplification of the conditions as well as a certain classification of the instabilities. Examples of the instabilities of different types are presented and the case of the so-called singlet instabilities (most interesting from the phys. point of view) is studied in detail for the π-electron model of cyclic polyenes.
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    222. 222
      Čížek, J.; Paldus, J. Stability Conditions for the Solutions of the Hartree–Fock Equations for Atomic and Molecular Systems. III. Rules for the Singlet Stability of Hartree–Fock Solutions of π-Electronic Systems. J. Chem. Phys. 1970, 53, 821– 829,  DOI: 10.1063/1.1674065
      222
      Stability conditions for the solutions of the Hartree-Fock equations for atomic and molecular systems. III. Rules for the singlet stability of Hartree-Fock solutions of π-electronic systems
      Cizek, Jiri; Paldus, Josef
      Journal of Chemical Physics (1970), 53 (2), 821-9CODEN: JCPSA6; ISSN:0021-9606.
      The singlet stability conditions for closed-shell electronic systems, which ensure that the Hartree-Fock (H.F.) determinant with doubly occupied orbitals minimizes the energy expectation value, are applied to the symmetry adapted H.F. solns. of linear polyacenes, by using the Pariser-Parr-Pople-type semiempirical Hamiltonian. The symmetry adapted H.F. solns. for linear polyacenes contg. an even no. of benzene rings are always singlet stable, while the H.F. solns. for linear polyacenes having an odd no. of benzene rings may exhibit singlet instability if the coupling const. is large enough. For cases where singlet instability was found, new H.F. solns. having lower energy than the symmetry-adapted H.F. solns. were calcd. These new H.F. solns. violate the space symmetry conservation laws as usual. Furthermore, the qual. rules for the existence of singlet stability of the symmetry adapted H.F. soln. of π-electronic systems with conjugated double bonds are derived. These rules are formulated through the simple symmetry properties of possible Kekule structures of the system. These rules are used to explain the results of stability calcns. for linear polyacenes as well as further illustrated on other examples.
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    223. 223
      Paldus, J.; Čížek, J. Stability Conditions for the Solutions of the Hartree-Fock Equations for Atomic and Molecular Systems. VI. Singlet-Type Instabilities and Charge-Density-Wave Hartree-Fock Solutions for Cyclic Polyenes. Phys. Rev. A 1970, 2, 2268– 2283,  DOI: 10.1103/PhysRevA.2.2268
      There is no corresponding record for this reference.
    224. 224
      Davidson, E. R. Spin-restricted open-shell self-consistent-field theory. Chem. Phys. Lett. 1973, 21, 565– 567,  DOI: 10.1016/0009-2614(73)80309-4
      224
      Spin-restricted open-shell self-consistent-field theory
      Davidson, Ernest R.
      Chemical Physics Letters (1973), 21 (3), 565-7CODEN: CHPLBC; ISSN:0009-2614.
      A method is given for eliminating the off-diagonal Lagrangian multipliers which appear in open-shell SCF theory. This leads to a set of coupled eigenville equations which is easily solved for a new guess to the SCF orbitals. This procedure has proven more convenient than many others now in use.
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    225. 225
      Edwards, W. D.; Zerner, M. C. A generalized restricted open-shell Fock operator. Theor. Chim. Acta 1987, 72, 347– 361,  DOI: 10.1007/BF01192227
      225
      A generalized restricted open-shell Fock operator
      Edwards, W. Daniel; Zerner, Michael C.
      Theoretica Chimica Acta (1987), 72 (5-6), 347-61CODEN: TCHAAM; ISSN:0040-5744.
      The open shell RHF theory is reexamd. and Fock-like operators are developed, that are general and easy to implement on a computer. A table of "vector coupling coeffs." that define this operator for most of the cases that commonly arise is presented. The form of this operator is compared with that suggested by others, and the orbitals obtained by this procedure are discussed with respect to the generalized Brillouin's theorem; the orbital energies are discussed with respect to Koopmans' approxn.
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    226. 226
      Löwdin, P.-O. Quantum Theory of Many-Particle Systems. III. Extension of the Hartree-Fock Scheme to Include Degenerate Systems and Correlation Effects. Phys. Rev. 1955, 97, 1509– 1520,  DOI: 10.1103/PhysRev.97.1509
      There is no corresponding record for this reference.
    227. 227
      Mayer, I. The spin-projected extended Hartree-Fock method. In Adv. Quantum Chem.; Academic Press: New York, 1980; Vol. 12, pp 189– 262.
      There is no corresponding record for this reference.
    228. 228
      Tsuchimochi, T.; Scuseria, G. E. Communication: ROHF theory made simple. J. Chem. Phys. 2010, 133, 141102,  DOI: 10.1063/1.3503173
      228
      Communication: ROHF theory made simple
      Tsuchimochi, Takashi; Scuseria, Gustavo E.
      Journal of Chemical Physics (2010), 133 (14), 141102/1-141102/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Restricted open-shell Hartree-Fock (ROHF) theory is formulated as a projected self-consistent unrestricted HF (UHF) model by math. constraining spin d. eigenvalues. This constrained UHF (CUHF) wave function is identical to that obtained from Roothaan's effective Fock operator. The α and β CUHF Fock operators are parameter-free and have eigenvalues (orbital energies) that are phys. meaningful as in UHF, except for eliminating spin contamination. This new way of solving ROHF leads to orbitals that turn out to be identical to semicanonical orbitals. The present approach removes ambiguities in ROHF orbital energies. (c) 2010 American Institute of Physics.
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    229. 229
      Rittby, M.; Bartlett, R. J. An open-shell spin-restricted coupled cluster method: application to ionization potentials in nitrogen. J. Phys. Chem. 1988, 92, 3033– 3036,  DOI: 10.1021/j100322a004
      229
      An open-shell spin-restricted coupled cluster method: application to ionization potentials in nitrogen
      Rittby, Magnus; Bartlett, Rodney J.
      Journal of Physical Chemistry (1988), 92 (11), 3033-6CODEN: JPCHAX; ISSN:0022-3654.
      To circumvent the problem of spin contamination in UHF based coupled cluster calcns., a new method of calcn. is given for certain classes of open-shell systems. The approach ensures that the proper spin component of the resulting correlated wave function is projected out in the energy evaluation by the use of a ref. function constructed from suitably chosen restricted open-shell Hartree-Fock or closed-shell Hartree-Fock orbitals. This single-ref. open-shell spin-restricted CC method is applied to the calcn. of ionization potentials in the N2 mol., and it is shown that highly accurate results can be obtained in a 5s4p1d basis. The mean error for all the principal ionization potentials of N2 compared to expt. is 0.45%.
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    230. 230
      Neese, F. Importance of direct spin-spin coupling and spin-flip excitations for the zero-field splittings of transition metal complexes: A case study. J. Am. Chem. Soc. 2006, 128, 10213– 10222,  DOI: 10.1021/ja061798a
      230
      Importance of Direct Spin-Spin Coupling and Spin-Flip Excitations for the Zero-Field Splittings of Transition Metal Complexes: A Case Study
      Neese, Frank
      Journal of the American Chemical Society (2006), 128 (31), 10213-10222CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      This work reports the evaluation of several theor. approaches to the zero-field splitting (ZFS) in transition metal complexes. The exptl. well-known complex [Mn(acac)3] is taken as an example. The direct spin-spin contributions to the ZFS have been calcd. on the basis of d. functional theory (DFT) or complete active space SCF (CASSCF) wave functions and have been found to be much more important than previously assumed. The contributions of the direct term may exceed ∼1 cm-1 in magnitude and therefore cannot be neglected in any treatment that aims at a realistic quant. modeling of the ZFS. In the DFT framework, two different variants to treat the spin-orbit coupling (SOC) term have been evaluated. The first approach is based on previous work by Pederson, Khanna, and Kortus, and the second is based on a "quasi-restricted" DFT treatment which is rooted in our previous work on ZFS. Both approaches provide very similar results and underestimate the SOC contribution to the ZFS by a factor of 2 or more. The SOC is represented by an accurate multicenter spin-orbit mean-field (SOMF) approxn. which is compared to the popular effective DFT potential-derived SOC operator. In addn. to the DFT results, direct "infinite order" ab initio calcns. of the SOC contribution to the ZFS based on CASSCF wave functions, the spectroscopy-oriented CI (SORCI), and the difference-dedicated CI (DDCI) approach are reported. In general, the multireference ab initio results provide a more realistic description of the ZFS in [Mn(acac)3]. The conclusions likely carry over to many other systems. This is attributed to the explicit treatment of the multiplet effects which are of dominant importance, since the calcns. demonstrate that, even in the high-spin d4 system Mn(III), the spin-flip excitations make the largest contribution to the SOC. It is demonstrated that the ab initio methods can be used even for somewhat larger mols. (the present calcns. were done with more than 500 basis functions) in a reasonable time frame. Much more economical but still fairly reasonable results have been achieved with the INDO/S treatment based on CASSCF and SOC-CI wave functions.
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    231. 231
      Casanova, D.; Krylov, A. I. Spin-flip methods in quantum chemistry. Phys. Chem. Chem. Phys. 2020, 22, 4326– 4342,  DOI: 10.1039/C9CP06507E
      231
      Spin-flip methods in quantum chemistry
      Casanova, David; Krylov, Anna I.
      Physical Chemistry Chemical Physics (2020), 22 (8), 4326-4342CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      A review. This Perspective discusses salient features of the spin-flip approach to strong correlation and describes different methods that sprung from this idea. The spin-flip treatment exploits the different physics of low-spin and high-spin states and is based on the observation that correlation is small for same-spin electrons. By using a well-behaved high-spin state as a ref., one can access problematic low-spin states by deploying the same formal tools as in the excited-state treatments (i.e., linear response, propagator, or equation-of-motion theories). The Perspective reviews applications of this strategy within wave function and d. functional theory frameworks as well as the extensions for mol. properties and spectroscopy. The utility of spin-flip methods is illustrated by examples. Limitations and proposed future directions are also discussed.
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    232. 232
      Roemelt, M.; Neese, F. Excited states of large open-shell molecules: an efficient, general, and spin-adapted approach based on a restricted open-shell ground state wave function. J. Phys. Chem. A 2013, 117, 3069– 3083,  DOI: 10.1021/jp3126126
      232
      Excited States of Large Open-Shell Molecules: An Efficient, General, and Spin-Adapted Approach Based on a Restricted Open-Shell Ground State Wave function
      Roemelt, Michael; Neese, Frank
      Journal of Physical Chemistry A (2013), 117 (14), 3069-3083CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      A spin-adapted CI with singles method that is based on a restricted open-shell ref. function (ROCIS) with general total spin S is presented. All excited configuration state functions (CSFs) are generated with the aid of a spin-free second quantization formalism that only leads to CSFs within the first order interacting space. By virtue of the CSF construction, the formalism involves higher than singly excited determinants but not higher than singly excited configurations. Matrix elements between CSFs are evaluated on the basis of commutator relationships using a symbolic algebra program. The final equations were, however, hand-coded in order to maximize performance. The method can be applied to fairly large systems with more than 100 atoms in reasonable wall-clock times and also parallelizes well. Test calcns. demonstrate that the approach is far superior to UHF-based CI with single excitations but necessarily falls somewhat short of quant. accuracy due to the lack of dynamic correlation contributions. In order to implicitly account for dynamic correlation in a crude way, the program optionally allows for the use of Kohn-Sham orbitals in combination with a modest downscaling of two-electron integrals (DFT/ROCIS). All two-electron integrals of Kohn-Sham orbitals that appear in the Hamiltonian matrix are reduced by a total of three scaling parameters that are suitable for a wide range of mols. Test calcns. on open-shell org. radicals as well as transition metal complexes demonstrate the wide applicability of the method and its ability to calc. the electronic spectra of large mol. systems.
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    233. 233
      Izsák, R. Second quantisation for unrestricted references: formalism and quasi-spin-adaptation of excitation and spin-flip operators. Mol. Phys. 2022, e2126802,  DOI: 10.1080/00268976.2022.2126802
      There is no corresponding record for this reference.
    234. 234
      Veis, L.; Antalik, A.; Legeza, O.; Alavi, A.; Pittner, J. The intricate case of tetramethyleneethane: A full configuration interaction quantum Monte Carlo benchmark and multireference coupled cluster studies. J. Chem. Theory Comput. 2018, 14, 2439– 2445,  DOI: 10.1021/acs.jctc.8b00022
      234
      The Intricate Case of Tetramethyleneethane: A Full Configuration Interaction Quantum Monte Carlo Benchmark and Multireference Coupled Cluster Studies
      Veis, Libor; Antalik, Andrej; Legeza, Ors; Alavi, Ali; Pittner, Jiri
      Journal of Chemical Theory and Computation (2018), 14 (5), 2439-2445CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We have performed a full CI (FCI) quality benchmark calcn. for the tetramethyleneethane mol. in the cc-pVTZ basis set employing a subset of complete active space second order perturbation theory, CASPT2(6,6), natural orbitals for the FCI quantum Monte Carlo calcn. The results are in an excellent agreement with the previous large scale diffusion Monte Carlo calcns. by Pozun et al. and available exptl. results. Our computations verified that there is a max. on the potential energy surface (PES) of the ground singlet state (1A) 45° torsional angle, and the corresponding vertical singlet-triplet energy gap is 0.01 eV. We have employed this benchmark for the assessment of the accuracy of Mukherjee's coupled clusters with up to triple excitations (MkCCSDT) and CCSD tailored by the d. matrix renormalization group method (DMRG). Multireference MkCCSDT with CAS(2,2) model space, though giving good values for the singlet-triplet energy gap, is not able to properly describe the shape of the multireference singlet PES. Similarly, DMRG(24,25) is not able to correctly capture the shape of the singlet surface, due to the missing dynamic correlation. On the other hand, the DMRG-tailored CCSD method describes the shape of the ground singlet state with excellent accuracy but for the correct ordering requires computation of the zero-spin-projection component of the triplet state (3B1).
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    235. 235
      Sears, J. S.; Sherrill, C. D. Assessing the Performance of Density Functional Theory for the Electronic Structure of Metal-Salens: The 3d0-Metals. J. Phys. Chem. A 2008, 112, 3466– 3477,  DOI: 10.1021/jp711595w
      235
      Assessing the Performance of Density Functional Theory for the Electronic Structure of Metal-Salens: The 3d0-Metals
      Sears, John S.; Sherrill, C. David
      Journal of Physical Chemistry A (2008), 112 (15), 3466-3477CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      A series of metal-salen complexes of the 3d0 metals Sc(III), Ti(IV), V(V), Cr(VI), and Mn(VII) have been explored using high-level electronic structure methods including coupled-cluster theory with singles, doubles, and perturbative triples as well as complete active-space third-order perturbation theory. The performance of three common d. functional theory approaches has been assessed for both the geometries and the relative energies of the low-lying electronic states. The nondynamical correlation effects are demonstrated to be extremely large in all of the systems examd. Although d. functional theory provides reasonable results for some of the systems, the overall agreement is quite poor. This said, the d. functional theory approaches are shown to outperform the single-ref. perturbation theory and coupled-cluster theory approaches for cases of strong nondynamical correlation.
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    236. 236
      Sears, J. S.; Sherrill, C. D. Assessing the performance of Density Functional Theory for the electronic structure of metal-salens: the d2-metals. J. Phys. Chem. A 2008, 112, 6741– 6752,  DOI: 10.1021/jp802249n
      236
      Assessing the Performance of Density Functional Theory for the Electronic Structure of Metal-Salens: The d2-Metals
      Sears, John S.; Sherrill, C. David
      Journal of Physical Chemistry A (2008), 112 (29), 6741-6752CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      The performance of three common combinations of d. functional theory has been evaluated for the geometries and relative energies of a commonly-employed model complex of the salen ligand [salen = bis(salicylaldehydo)ethylenediamine] with the d2-metals Ti(II), V(III), Cr(IV), Zr(II), Nb(III), and Mo(IV). High-level ab initio methods including complete active-space third-order perturbation theory have been employed both as benchmarks for the d. functional theory results and to examine the multireference character of the low-lying electronic states in these systems. The strong multireference character of the systems has been clearly demonstrated. All of the functionals examd. provide geometries that are typically within 0.2 Å least root mean square deviation from the benchmark geometries. The performance of the d. functionals for the relative energies of the low-lying electronic states is significantly worse, providing qual. different descriptions in some instances. Of the systems explored, no significant difference is obsd. in the multireference character or in the reliability of the d. functional results when comparing 3d vs 4d transition-metal systems.
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    237. 237
      Olivares-Amaya, R.; Hu, W.; Nakatani, N.; Sharma, S.; Yang, J.; Chan, G. K.-L. The ab-initio density matrix renormalization group in practice. J. Chem. Phys. 2015, 142, 034102,  DOI: 10.1063/1.4905329
      237
      The ab-initio density matrix renormalization group in practice
      Olivares-Amaya, Roberto; Hu, Weifeng; Nakatani, Naoki; Sharma, Sandeep; Yang, Jun; Chan, Garnet Kin-Lic
      Journal of Chemical Physics (2015), 142 (3), 034102/1-034102/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The ab-initio d. matrix renormalization group (DMRG) is a tool that can be applied to a wide variety of interesting problems in quantum chem. Here, we examine the d. matrix renormalization group from the vantage point of the quantum chem. user. What kinds of problems is the DMRG well-suited to. What are the largest systems that can be treated at practical cost. What sort of accuracies can be obtained, and how do we reason about the computational difficulty in different mols.. By examg. a diverse benchmark set of mols.: π-electron systems, benchmark main-group and transition metal dimers, and the Mn-oxo-salen and Fe-porphine organometallic compds., we provide some answers to these questions, and show how the d. matrix renormalization group is used in practice. (c) 2015 American Institute of Physics.
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    238. 238
      Pandharkar, R.; Hermes, M. R.; Cramer, C. J.; Gagliardi, L. Localized Active Space-State Interaction: a Multireference Method for Chemical Insight. J. Chem. Theory Comput. 2022, 18, 6557– 6566,  DOI: 10.1021/acs.jctc.2c00536
      238
      Localized Active Space-State Interaction: a Multireference Method for Chemical Insight
      Pandharkar, Riddhish; Hermes, Matthew R.; Cramer, Christopher J.; Gagliardi, Laura
      Journal of Chemical Theory and Computation (2022), 18 (11), 6557-6566CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Multireference electronic structure methods, like the complete active space (CAS) SCF model, have long been used to characterize chem. interesting processes. Important work has been done in recent years to develop modifications having a lower computational cost than CAS, but typically these methods offer no more chem. insight than that from the CAS soln. being approximated. In this paper, we present the localized active space-state interaction (LASSI) method that can be used not only to lower the intrinsic cost of the multireference calcn. but also to improve interpretability. The localized active space (LAS) approach utilizes the local nature of the electron-electron correlation to express a composite wave function as an antisymmetrized product of unentangled wave functions in local active subspaces. LASSI then uses these LAS states as a basis from which to express complete mol. wave functions. This not only makes the mol. wave function more compact but also permits flexibility in choosing those states to be included in the basis. Such selective inclusion of states translates to the selective inclusion of specific types of interactions, thereby allowing a quant. anal. of these interactions. We demonstrate the use of LASSI to study charge migration and spin-flip excitations in multireference org. mols. We also compute the J coupling parameter for a bimetallic compd. using various LAS bases to construct the Hamiltonian to provide insights into the coupling mechanism.
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    239. 239
      Máté, M.; Petrov, K.; Szalay, S.; Legeza, Ö. Compressing multireference character of wave functions via fermionic mode optimization. J. Math. Chem. 2023, 61, 362– 375,  DOI: 10.1007/s10910-022-01379-y
      239
      Compressing multireference character of wave functions via fermionic mode optimization
      Mate, Mihaly; Petrov, Klara; Szalay, Szilard; Legeza, Ors
      Journal of Mathematical Chemistry (2023), 61 (2), 362-375CODEN: JMCHEG; ISSN:0259-9791. (Springer)
      Abstr.: In this work, we present a brief overview of the fermionic mode optimization within the framework of tensor network state methods (Krumnow et al. in Phys Rev Lett 117:210402, 2016, https://doi.org/10.1103/PhysRevLett.117.210402), and demonstrate that it has the potential to compress the multireference character of the wave functions after finding optimal MOs (modes), based on entanglement minimization. Numerical simulations have been performed for the nitrogen dimer in the cc-pVDZ basis for the equil. and for stretched geometries.
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    240. 240
      Smith, J. E. T.; Mussard, B.; Holmes, A. A.; Sharma, S. Cheap and Near Exact CASSCF with Large Active Spaces. J. Chem. Theory Comput. 2017, 13, 5468– 5478,  DOI: 10.1021/acs.jctc.7b00900
      240
      Cheap and Near Exact CASSCF with Large Active Spaces
      Smith, James E. T.; Mussard, Bastien; Holmes, Adam A.; Sharma, Sandeep
      Journal of Chemical Theory and Computation (2017), 13 (11), 5468-5478CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We use the recently-developed Heat-bath CI (HCI) algorithm as an efficient active-space solver to perform multi-configuration SCF calcns. (HCISCF) with large active spaces. We give a detailed derivation of the theory and show that difficulties assocd. with non-variationality of the HCI procedure can be overcome by making use of the Lagrangian formulation to calc. the HCI relaxed two body reduced d. matrix. HCISCF is then used to study the electronic structure of butadiene, pentacene, and Fe-porphyrin. One of the most striking results of our work is that the converged active space orbitals obtained from HCISCF are relatively insensitive to the accuracy of the HCI calcn. This allows us to obtain nearly converged CASSCF energies with an estd. error of less than 1 mHa using the orbitals obtained from the HCISCF procedure in which the integral transformation is the dominant cost. For example, an HCISCF calcn. on Fe-Porphyrin model complex with an active space of (44e, 44o) took only 412 s per iteration on a single node contg. 28 cores, out of which 185 s were spent in the HCI calcn. and the remaining 227 s were mainly used for integral transformation. Finally, we also show that active-space orbitals can be optimized using HCISCF to substantially speed up the convergence of the HCI energy to the Full CI limit because HCI is not invariant to unitary transformations within the active space.
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    241. 241
      Dobrautz, W.; Weser, O.; Bogdanov, N. A.; Alavi, A.; Li Manni, G. Spin-Pure Stochastic-CASSCF via GUGA-FCIQMC Applied to Iron–Sulfur Clusters. J. Chem. Theory Comput. 2021, 17, 5684– 5703,  DOI: 10.1021/acs.jctc.1c00589
      241
      Spin-Pure Stochastic-CASSCF via GUGA-FCIQMC Applied to Iron-Sulfur Clusters
      Dobrautz, Werner; Weser, Oskar; Bogdanov, Nikolay A.; Alavi, Ali; Li Manni, Giovanni
      Journal of Chemical Theory and Computation (2021), 17 (9), 5684-5703CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      In this work, we demonstrate how to efficiently compute the one- and two-body reduced d. matrixes within the spin-adapted full CI quantum Monte Carlo (FCIQMC) method, which is based on the graphical unitary group approach (GUGA). This allows us to use GUGA-FCIQMC as a spin-pure CI eigensolver within the complete active space SCF (CASSCF) procedure and hence to stochastically treat active spaces far larger than conventional CI solvers while variationally relaxing orbitals for specific spin-pure states. We apply the method to investigate the spin ladder in iron-sulfur dimer and tetramer model systems. We demonstrate the importance of the orbital relaxation by comparing the Heisenberg model magnetic coupling parameters from the CASSCF procedure to those from a CI-only (CASCI) procedure based on restricted open-shell Hartree-Fock orbitals. We show that the orbital relaxation differentially stabilizes the lower-spin states, thus enlarging the coupling parameters with respect to the values predicted by ignoring orbital relaxation effects. Moreover, we find that, while CASCI results are well fit by a simple bilinear Heisenberg Hamiltonian, the CASSCF eigenvalues exhibit deviations that necessitate the inclusion of biquadratic terms in the model Hamiltonian.
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    242. 242
      Dobrautz, W.; Katukuri, V. M.; Bogdanov, N. A.; Kats, D.; Li Manni, G.; Alavi, A. Combined unitary and symmetric group approach applied to low-dimensional Heisenberg spin systems. Phys. Rev. B 2022, 105, 195123,  DOI: 10.1103/PhysRevB.105.195123
      242
      Combined unitary and symmetric group approach applied to low-dimensional Heisenberg spin systems
      Dobrautz, Werner; Katukuri, Vamshi M.; Bogdanov, Nikolay A.; Kats, Daniel; Li Manni, Giovanni; Alavi, Ali
      Physical Review B (2022), 105 (19), 195123CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      A novel combined unitary and sym. group approach is used to study the spin-12 Heisenberg model and related Fermionic systems in a total spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground-state wave function representation-indicated by a larger leading ground-state coeff.-is obtained when combining the sym. group Sn, in the form of permutations of the underlying lattice site ordering, with the cumulative spin coupling based on the unitary group U(n). In one-dimensional systems the obsd. compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full CI quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest wt. already captures with high accuracy the spin-spin correlation function of the exact ground-state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chem. models, exemplified by one-dimensional hydrogen chains. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equiv. to the optimal ordering obtained for methods based on matrix-product states, such as the d.-matrix renormalization group approach.
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    243. 243
      Li Manni, G.; Kats, D.; Liebermann, N. Resolution of Electronic States in Heisenberg Cluster Models within the Unitary Group Approach. ChemRxiv 2022, DOI: 10.26434/chemrxiv-2022-rfmhk-v2 .
      There is no corresponding record for this reference.
    244. 244
      Pipek, J.; Mezey, P. G. A fast intrinsic localization procedure applicable for abinitio and semiempirical linear combination of atomic orbital wave functions. J. Chem. Phys. 1989, 90, 4916– 4926,  DOI: 10.1063/1.456588
      244
      A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions
      Pipek, Janos; Mezey, Paul G.
      Journal of Chemical Physics (1989), 90 (9), 4916-26CODEN: JCPSA6; ISSN:0021-9606.
      A new intrinsic localization algorithm is suggested based on a recently developed math. measure of localization. No external criteria are used to define a priori bonds, lone pairs, and core orbitals. The method similarly to Edmiston-Ruedenberg's localization prefers the well established chem. concept of σ-π sepn., while on the other hand, works as economically as Boys' procedure. For the applications of the new localization algorithm, no addnl. quantities are to be calcd., the knowledge of at. overlap integrals is sufficient. This feature allows a unique formulation of the theory, adaptable for both ab initio and semiempirical methods, even in those cases where the exact form of the at. basis functions is not defined (line in the EHT and PPP calcns). The implementation of the procedure in already existing program systems is particularly easy. The Emiston-Ruedenberg and Boys localized orbitals are compared with those calcd. by the method suggested here, within both the CNDO/2 and ab initio frameworks (using STO-3G and 6-31G** basis sets) for several mols. (CO, H2CO, B2H6, and N2O4).
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    245. 245
      Malrieu, J.-P.; Guihéry, N.; Calzado, C. J.; Angeli, C. Bond electron pair: Its relevance and analysis from the quantum chemistry point of view. J. Comput. Chem. 2007, 28, 35– 50,  DOI: 10.1002/jcc.20546
      245
      Bond electron pair: its relevance and analysis from the quantum chemistry point of view
      Malrieu, Jean-Paul; Guihery, Nathalie; Calzado, Carmen Jimenez; Angeli, Celestino
      Journal of Computational Chemistry (2007), 28 (1), 35-50CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      A review. This paper first comments on the surprisingly poor status that Quantum Chem. has offered to the fantastic intuition of Lewis concerning the distribution of the electrons in the mol. Then, it advocates in favor of a hierarchical description of the mol. wave-function, distinguishing the physics taking place in the valence space (in the bond and between the bonds), and the dynamical correlation effects. It is argued that the clearest pictures of the valence electronic population combine two localized views, namely the bond (and lone pair) MOs and the Valence Bond decompn. of the wave-function, preferably in the orthogonal version directly accessible from the complete active space self consistent field method. Such a reading of the wave function enables one to understand the work of the nondynamical correlation as an enhancement of the wt. of the low-energy VB components, i.e. as a better compromise between the electronic delocalization and the energetic preferences of the atoms. It is suggested that regarding the bond building, the leading dynamical correlation effect may be the dynamical polarization phenomenon. It is shown that most correlation effects do not destroy the bond electron pairs and remain compatible with Lewis' vision. A certain no. of free epistemol. considerations have been introduced in the development of the argument.
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    246. 246
      Lewis, G. N. The atom and the molecule. J. Am. Chem. Soc. 1916, 38, 762– 785,  DOI: 10.1021/ja02261a002
      246
      The atom and the molecule
      Lewis, G. N.
      Journal of the American Chemical Society (1916), 38 (), 762-85CODEN: JACSAT; ISSN:0002-7863.
      cf. C. A. 71 3865 and Bray and Branch, C. A. 7, 3865. Compds. should be classed as polar and nonpolar rather than inorg. and org. These classes are roughly the same. A nonpolar mol. is one in which the electrons belonging to the individual atom are held by such restraints that they do not move far from their normal positions, while in the polar mols. the electrons, being more mobile, so move as to sep. the mol. into positive and negative parts. In an extremely polar mol. such as NaCl it is probable that in the great majority of the mols. the Cl atom has acquired a unit negative charge and therefore the Na atom a unit positive charge, and the process of ionization probably consists only in a further sepn. of these charged parts. If a weakly polar mol. comes into the neighborhood of a more polar one it becomes itself more polar. In this process the weaker bipole stretches and its moment increases. A "cubical atom" is proposed as a basis of a new theory of atomic structure. Thus Li is a cube with a single electron on one corner, Be has 2 electrons, B 3, C 4, N 5, O 6, and F 7. This view is in harmony with the theory developed by Parson, C. A. 10, 406. An atom is considered as having an unalterable kernel which possesses an excess of positive charges corresponding in number to the ordinal number of the group in the periodic table to which the element belongs (cf. Thomson, C. A. 8, 824). There is a shell of electrons around the kernel which, in the case of a neutral atom, contains negative electrons equal in number to the excess of positive charges of the kernel, but the number of electrons in the shell may vary during chem. changes between zero and 8. The atom tends to hold an even number of electrons in the shell (especially 8 at the corners of the cube) but the electrons may ordinarily pass from one position to another in this shell. Two atomic shells are mutually interpenetrable. The paper is a discussion of these ideas applied to the structure of atoms and compds.
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    247. 247
      Chen, H.; Lai, W.; Shaik, S. Multireference and Multiconfiguration Ab Initio Methods in Heme-Related Systems: What Have We Learned So Far?. J. Phys. Chem. B 2011, 115, 1727– 1742,  DOI: 10.1021/jp110016u
      247
      Multireference and Multiconfiguration Ab Initio Methods in Heme-Related Systems: What Have We Learned So Far?
      Chen, Hui; Lai, Wenzhen; Shaik, Sason
      Journal of Physical Chemistry B (2011), 115 (8), 1727-1742CODEN: JPCBFK; ISSN:1520-5207. (American Chemical Society)
      A review. This work reviews the recent applications of ab initio multireference/multiconfiguration (MR/MC) electronic structure methods to heme-related systems, involving tetra-, penta-, and hexa-coordinate species, as well as the high-valent iron-oxo species. The current accuracy of these methods in the various systems is discussed, with special attention to potential sources of systematic errors. Thus, the review summarizes and tries to rationalize the key elements of MR/MC calcns., namely, the choice of the employed active space, esp. the so-called double-shell effect that has already been recognized to be important in transition-metal-contg. systems, and the impact of these elements on the spin-state energetics of heme species, as well as on the bonding mechanism of small mols. to the heme. It is shown that expansion of the MC wave function into one based on localized orbitals provides a compact and insightful view on some otherwise complex electronic structures. The effects of protein environment on the MR/MC results are summarized for the few available quantum mech./mol. mech. (QM/MM) studies. Comparisons with corresponding DFT results are also made wherever available. Potential future directions are proposed.
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    248. 248
      Li, Z.; Guo, S.; Sun, Q.; Chan, G. K.-L. Electronic landscape of the P-cluster of nitrogenase as revealed through many-electron quantum wavefunction simulations. Nat. Chem. 2019, 11, 1026– 1033,  DOI: 10.1038/s41557-019-0337-3
      248
      Electronic landscape of the P-cluster of nitrogenase as revealed through many-electron quantum wavefunction simulations
      Li, Zhendong; Guo, Sheng; Sun, Qiming; Chan, Garnet Kin-Lic
      Nature Chemistry (2019), 11 (11), 1026-1033CODEN: NCAHBB; ISSN:1755-4330. (Nature Research)
      The electronic structure of the nitrogenase metal cofactors is central to nitrogen fixation. However, the P-cluster and FeMo cofactor, each contg. eight Fe atoms, have eluded detailed characterization of their electronic properties. We report on the low-energy electronic states of the P-cluster in three oxidn. states through exhaustive many-electron wavefunction simulations enabled by new theor. methods. The energy scales of orbital and spin excitations overlap, yielding a dense spectrum with features that we trace to the underlying at. states and recouplings. The clusters exist in superpositions of spin configurations with non-classical spin correlations, complicating interpretation of magnetic spectroscopies, whereas the charges are mostly localized from reorganization of the cluster and its surroundings. On oxidn., the opening of the P-cluster substantially increases the d. of states, which is intriguing given its proposed role in electron transfer. These results demonstrate that many-electron simulations stand to provide new insights into the electronic structure of the nitrogenase cofactors.
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      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhvVOlur7K&md5=1b466a893a07cf4448f8e705e2f86f27
    249. 249
      Khedkar, A.; Roemelt, M. Modern multireference methods and their application in transition metal chemistry. Phys. Chem. Chem. Phys. 2021, 23, 17097– 17112,  DOI: 10.1039/D1CP02640B
      249
      Modern multireference methods and their application in transition metal chemistry
      Khedkar, Abhishek; Roemelt, Michael
      Physical Chemistry Chemical Physics (2021), 23 (32), 17097-17112CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      A review. Transition metal chem. is a challenging playground for quantum chem. methods owing to the simultaneous presence of static and dynamic electron correlation effects in many systems. Wavefunction based multireference (MR) methods constitute a phys. sound and systematically improvable Ansatz to deal with this complexity but suffer from some conceptual difficulties and high computational costs. The latter problem partially arises from the unfavorable scaling of the Full CI (Full-CI) problem which in the majority of MR methods is solved for a subset of the MO space, the so-called active space. In the last years multiple methods such as modern variants of selected CI, Full-CI Quantum Monte Carlo (FCIQMC) and the d. matrix renormalization group (DMRG) have been developed that solve the Full-CI problem approx. for a fraction of the computational cost required by conventional techniques thereby significantly extending the range of applicability of modern MR methods. This perspective review outlines recent advancements in the field of MR electronic structure methods together with the resulting chances and challenges for theor. studies in the field of transition metal chem. In light of its emerging importance a special focus is put on the selection of adequate active spaces and the concomitant development of numerous selection aides in recent years.
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    250. 250
      Tarrago, M.; Römelt, C.; Nehrkorn, J.; Schnegg, A.; Neese, F.; Bill, E.; Ye, S. Experimental and Theoretical Evidence for an Unusual Almost Triply Degenerate Electronic Ground State of Ferrous Tetraphenylporphyrin. Inorg. Chem. 2021, 60, 4966– 4985,  DOI: 10.1021/acs.inorgchem.1c00031
      250
      Experimental and Theoretical Evidence for an Unusual Almost Triply Degenerate Electronic Ground State of Ferrous Tetraphenylporphyrin
      Tarrago, Maxime; Roemelt, Christina; Nehrkorn, Joscha; Schnegg, Alexander; Neese, Frank; Bill, Eckhard; Ye, Shengfa
      Inorganic Chemistry (2021), 60 (7), 4966-4985CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      Iron porphyrins exhibit unrivalled catalytic activity for electrochem. CO2-to-CO conversion. Despite intensive exptl. and computational studies in the last four decades, the exact nature of the prototypical square-planar [FeII(TPP)] complex (1; TPP2- = tetraphenylporphyrinate dianion) remained highly debated. Specifically, its intermediate spin (S = 1) ground state was contradictorily assigned to either a nondegenerate 3A2g state with (dxy)2(dz2)2(dxz,yz)2 configuration or a degenerate 3Eθg state with (dxy)2(dxz,yz)3(dz2)1/(dz2)2(dxy)1(dxz,yz)3 configuration. To address this question, we present herein a comprehensive, spectroscopy-based theor. and exptl. electronic-structure investigation on complex 1. Highly correlated wave function-based computations predicted that 3A2g and 3Egθ are well-isolated from other triplet states by ca. 4000 cm-1, whereas their splitting ΔA-E is on par with the effective spin-orbit coupling (SOC) const. of iron(II) (≈ 400 cm-1). In order to model the entire manifold of the nine magnetic sublevels arising from SOC between the 3A2g and 3Eθg states explicitly, we invoked an effective Hamiltonian (EH) operating on the corresponding nine-dimensional Hilbert space. This approach enabled us to successfully simulate all spectroscopic data of 1 obtained by variable temp. and variable field magnetization, applied-field 57Fe Mossbauer, and THz-EPR measurements. Remarkably, the EH contains only three adjustable parameters, namely, the energy gap without SOC, ΔA-E, an angle θ that describes the mixing of (dxy)2(dxz,yz)3(dz2)1 and (dz2)2(dxy)1(dxz,yz)3 configurations, and the 〈rd-3〉 expectation value of the iron d-orbitals that is necessary to est. the 57Fe magnetic hyperfine coupling tensor. The simulations revealed ΔA-E = +950 cm-1, rendering 3Eθg slightly above 3A2g in energy. Hence, 1 has a triplet ground state with substantial parentage of both 3A2g ( < 88%) and 3Eθg ( > 12%). Thus, the electronic ground state cannot simply be interpreted as either 3A2g or 3Eθg, but is genuinely multiconfigurational, arising from accidental near-triple degeneracy. Consequently, although this low-lying triplet is isolated from other states by ca. 900 cm-1, the magnetic properties of 1 cannot be adequately understood by the conventional S = 1 spin Hamiltonian (SH), which is valid only for orbitally nondegenerate states. Instead, the EH treatment easily explains the obsd. huge effective magnetic moment (4.2μB at 300 K), strong temp.-independent paramagnetism and large and pos. axial zero-field splitting within the triplet, giving rise to a nondegenerate magnetic sublevel being lowest in energy. Application of an external magnetic field demonstrates that the three magnetic sublevels carry substantial orbital angular momentum in the xy plane. This results in a large magnetization and a large and pos. internal field at the 57Fe nucleus aligned in the xy plane. (In the alternative SH description, the magnetic anisotropy manifests itself in an unusually large g anisotropy for an S = 1 system, with g.perp. ≈ 3 and g|| vbr ≈ 1.7). Further in-depth analyses suggested that such strong easy-plane anisotropy is a general spectroscopic signature of near-triple orbital degeneracy with more than half filled pseudodegenerate orbital sets. Implications of the unusual electronic structure of 1 for its CO2 redn. reactivity are discussed.
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    251. 251
      Han, R.; Luber, S.; Li Manni, G. Magnetic Interactions in a [Co(II)3Er(III)(OR)4] Model Cubane Through Forefront Multiconfigurational Methods. ChemRxiv 2023, DOI: 10.26434/chemrxiv-2023-xd0wv .
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