Low-temperature (118 K) X-ray data on pyridinium-1-dicyanomethylide have been refined with the aspherical-atom formalism of Hirshfeld [Isr. J. Chem. (1977), 16, 226-230] and the related spherical harmonic expansion proposed by Stewart [Acta Cryst. (1976), A32, 565-575] as modified by Hansen & Coppens [Acta Cryst. (1978), A34, 909-921]. The results of the two methods are very similar, except when pseudostatic deformation maps are compared. The triple bond is clearly differentiated from the other bonds, while the ylide carbon atom is neutral. The molecular dipole moment from the multipole formalism is in close agreement with the moment obtained by direct-space integration of the electron density and with the value from solution measurements. On the other hand, the refinement with the Hirshfeld functions leads to a much larger molecular dipole moment. The difference between the two methods in this respect appears to be related to the presence of diffuse monopolar functions in the Hirshfeld expansion.

The implications of quantum-chemical concepts for the simultaneous interpretation of charge and spin density data are discussed. It is proposed that the scattering of the electrons involved in the metal-ligand interaction, on which both X-ray and polarized neutron information is available, be expressed in terms of the wavefunction, while the remainder of the electron distribution be described in terms of the multipole formalism. The discussion is based on a three-electron subsystem for metal-ligand bonding. At the restricted molecular orbital level it is shown that the magnitude of the overlap spin density is much larger than that of the overlap charge density, which may be close to zero when the electronegativity difference between metal and ligand is considerable. Spin polarization is introduced at the unrestricted molecular orbital level and implies that different κ parameters should be applied to the α and β electrons. Its effect on the spin and charge distribution is of first and second order respectively. The effect of correlation, described by the mixing of two or more configurations, leads to an apparent increase in covalency. The formalisms discussed may be applied in a stepwise manner, first at the spin-restricted level and subsequently with the inclusion of spin polarization and correlation effects.

The Coulomb potential in a crystal is discussed. It is shown that its Fourier series expansion has a singularity for the V(0, 0, 0) component, which is important when comparing different compounds, or when using the Coulomb potential as a probe for reactivity. Methods to calculate this term are discussed. Sum rules for multipolar moments of crystals in terms of structure factors are derived, which are of interest for the comparison of microscopic and macroscopic dielectric properties.

A theory of extinction is derived which is valid within the limit of the Darwin intensity transfer equations. An expression describing the effect of n-fold rescattering within an ideal crystallite is derived, which differs from the equation given by Zachariasen because independent coordinates x₁ and x₂ based on an external coordinate system have been used, rather than the coordinates t₁ and t₂ which are only mutually independent if the crystal is a parallelepiped with faces parallel to the incident and diffracted beams. Furthermore, the derivation of the earlier expressions is based on a generally unjustifiable reversal of the direction of the diffracted ray (interchange of t₂ and t'₂). An exact expression is derived for the diffraction cross section σ(ε₁) in the perfect crystallite, which contains a factor sin 2θ neglected in the earlier work. As a result, the previously used classification of crystals into type I and type II becomes less well defined because at very small Bragg angles particle size always becomes the dominant effect. It is shown that the extinction factor y_p (p = primary), for a perfect spherical crystallite, calculated with the present theory, is in good agreement with calculations based on the dynamical theory. Furthermore, the limiting behavior of the expressions at 2θ = 0 and π justifies some of the mathematical approximations made. For a mosaic crystal the extinction coefficient y is written as y_p . y_s (s = secondary), y_p is evaluated numerically from the expressions derived. An analytical expression for y_p is obtained by least-squares fit to the numerical values. A similar procedure is followed for y_s, in the case of a Gaussian, Lorentzian and Fresnellian distributions of the crystallites and a spherical mosaic crystal. Analysis of the results shows that the Zachariasen expression can be used for small extinction (y > 0.8), provided the θ dependent factor is properly introduced for particle-size-affected extinction. Allowance for polarization effects in the X-ray case is discussed. Absorption effects cannot be treated separately from extinction for all but small values of 1 -y. Coefficients of the analytical extinction expressions are given for absorbing spherical crystals with μR values ≤ 4. Application of the expressions and extension to non-spherical geometries will be treated in following publications.

The extinction-correction formalisms derived in a previous article have been applied to two sets of diffraction data on spherical crystals. Application to the neutron data on SrF₂, collected by Cooper & Rouse at three different wavelengths, shows that the theory gives a slightly better fit than a special empirical formula used previously to allow for extinction in this data set. A simultaneous refinement on all the data varying both particle size and mosaic spread and allowing for primary extinction leads to physically reasonable results. It is found that Lorentzian or Fresnellian mosaic-spread distributions fit the data considerably better than a Gaussian model. A similar refinement on the two-wavelength data on LiF confirms Lawrence's original conclusions that the extinction is mainly of the primary type. However, the results are in sharp disagreement with a treatment of the same data by Killean and co-workers, in which only secondary extinction was considered. The physical upper limit for the particle size in the sample is found to be 1.9 × 10^-4 cm.

Previously derived formalisms for extinction are extended to include crystals of non-spherical shape and anisotropy of mosaic spread and particle size. Expressions derived for extinction in an ellipsoidal crystal are compared with numerical calculations on a polyhedral specimen. A pseudo-spherical approximation for polyhedral crystals is described which is accurate to within 2% of the extinction factor y for crystals whose ratio of maximum and minimum dimensions is less than two. Anisotropy of mosaic spread is introduced in both the Coppens-Hamilton (C.H.) and Thornley-Nelmes (T.N.) descriptions, with both a Lorentzian or a Gaussian distribution function. The formalisms are applied to neutron data sets on LiTbF₄ (100°K and 300°K), tetracyanoethylene and LiOH. H₂O, and an X-ray data set on α-deutero oxalic acid dihydrate. The distinction between type I and type II crystals is quite clear on the basis of a comparison of R values. Only for LiF, which was studied earlier, was extinction dominated by particle size. In all other cases the best fit corresponds to mosaic-spread-dominated extinction, with a Lorentzian shape of the distribution function. This is especially clear when partial R values summed over the severely extinction-affected reflections are compared. The new formalisms are further supported by the consistency of the final parameters among various refinements in which the most severely extinction-affected reflections are eliminated. Simultaneous refinement on both the particle size and the mosaic spread was only successful in the earlier studied case of SrF₂. The T.N. description of anisotropy leads to lower partial R values, in agreement with physical arguments supporting the validity of this distribution.

The advantages of the step-scan mode over a continuous scan in X-ray data collection are discussed. Computer analysis of the recorded profile allows corrections for effects such as non-uniform background, β-filter cut-off, thermal diffuse scattering, and counter dead time, as well as a reduction in the experimental standard deviations. It is concluded that the continuous scan mode should be avoided in careful crystallographic studies.

Imaging-plate synchrotron data have been applied in the charge-density analyses of sodium nitroprusside and hexaamminechromium(III) hexacyanochromate(lll), collected at 100 and 50 K, respectively, and photons of wavelengths 0.656 and 0.394 Å at the SUNY X3 beamline at NSLS. The electron-density maps show good agreement between chemically equivalent sections, while the multipole aspherical atom refinements lead to chemically reasonable population parameters, with trends reproduced in the available theoretical calculations. The results indicate that the time required for charge-density mapping with diffraction data can be greatly reduced by the application of the new technology.