With triple-phase relationships treated as linear equations it is possible to refine a set of phases from given initial values. Phases so obtained are better than those found by refining to self-consistency with the tangent formula. An investigation of the radius of convergence of the least-squares refinement process showed that a substantially correct solution may often be found even starting with random phases. Systems containing up to 300 phases have been investigated and the results and their implications are discussed. It is concluded that the random approach can, at the very least, be used to obtain 70--100 phases as a good starting point for phase development. There is also the possibility of obtaining a sufficient number of phases directly to define a reasonably complex structure, especially with a computer augmented by an array processor. A problem which can arise with linear equations, as with the tangent formula, is that the phases obtained do not adequately define the enantiomorph and give an E map with a pseudo centre of symmetry. Two methods of overcoming this problem are described.

For space groups with translational elements of symmetry it is possible to find relationships between the phases of pairs of reflexions which are of the same parity group. When the reflexions have one index in common then many indications for a given relationship may appear. A rigorous theory is described which enables the variance of the relationships to be derived although, in practice, a less rigorous approach is preferred which is more amenable to numerical work. The phase-coincidence procedure is being incorporated as an option in the MULTAN computer package for solving non-centrosymmetric structures.

The number of triple-phase relationships interrelating a set of phases, Q, may greatly exceed the number of unknown phases, M. Consequently it is usually possible to find a linearly independent set of M triple-phase invariants and the remaining Q-M may be expressed as linear combinations of these. From the expected distribution in the values of all the invariants and the interrelationships between them it is possible to find values for the independent set of relationships which are closer to their true values than zero, the ab initio expectation value of each of them. Ways of using the calculated values of the invariants are described and results are given for various trial structures. Possible improvements in the invariant-determining process are discussed.

Abstract In recent years there have been a number of developments in direct methods involving refinement processes applied to initially random sets of phases. Procedures which have been used for refinement include least-squares and gradient methods applied to triple-phase relationships expressed as linear equations and also the tangent formula. In the present investigation seven functions are investigated for the refinement of random phases; because of the awkward form of these functions the refinement process used is based on a parameter-shift algorithm. Some of the functions appear to be more effective than others but the most effective one was discovered through making a mistake with one of the others and no rational explanation for its efficacy can be given. Trials have been made with known structures and with three unknown structures which were originally solved by the processes described in the paper.

The application of the condition that a true set of phases for a substantial subset of normalized structure factors should satisfy Sayre's equation leads to a phase-refining equation called the Sayre tangent formula. Phases refined by this formula tend to satisfy Sayre's equation for a subset of E's containing some of large magnitude and some of small (ideally zero) magnitude. Trials indicate that the new formula, incorporated into a computer program SAYTAN, is more effective than MULTAN80, especially for symmorphic structures.

The Sayre-equation tangent formula (SETF) develops sets of phases tending to satisfy Sayre's equations for both large and small normalized structure factors. There are two components in the SETF, corresponding to contributions from phase triplets and quartets respectively. The development of objective algorithms for properly weighting these components and for gradually building up the quartet contribution has enabled the SETF, within the procedure SAYTAN, to be incorporated into MULTAN87, the latest version of the package. Examples of tests of MULTAN87 and its use in solving unknown structures are given.

In modern direct methods a very powerful multisolution approach involves the refinement by some means of initially random phases. In exploring a number of refinement functions Debaerdemaeker & Woolfson [Acta Cryst. (1983), A39, 193-196] found by accident a function of phases, the maximization of which was very effective in obtaining substantially correct phase sets. This has led to the XMY method which works well although no completely rational explanation can be offered for how it does so. Examples of its use in the ab initio solution of unknown structures are given and tests are described indicating its usefulness as a means of carrying out multisolution fragment development.

A magic-integer approach, called the P-S set method is described. A primary (P) set of reflexions contains some which fix the origin and enantiomorph and others expressed symbolically in magic-integer form. Probable phases for a secondary (S) set of reflexions are derived, also in symbolic form, from single triple-phase relationships containing a pair of P reflexions. Relationships which link the combined P and S sets give rise to the terms of a Fourier map, the peaks of which indicate likely sets of phases for all the reflexions under consideration. These sets of phases are used as starting points for the computer program MULTAN. The process is completely automated and is illustrated by the solution of the structure of cephalotaxine, C₁₈H₂₁O₄N, the space group of which is C2 with two molecules in the asymmetric unit.

Two new multisolution direct-methods procedures are described: MAGIC, which employs the magic-integer concept and YZARC which refines initially random sets of phases by a least-squares approach. Each procedure produces several sets of phases for a number of reflexions, usually in the range 35-100. These are then extended by the tangent formula but with the constraint that the basis phases are not allowed to change until the final cycle. It is shown that for difficult structures these methods, which deal simultaneously with many phase relationships, may have intrinsic advantages over the MUL TAN procedure. Examples of their use are given.

Multisolution direct methods for solving crystal structures lead to many plausible sets of phases and some means of determining the correct set is necessary. For centrosymmetric structures, figures of merit are usually quite discriminating and the examination of only one or two E maps is necessary. For noncentrosymmetric structures, figures of merit are unreliable and the necessity of examining a large number of E maps can sometime prove to be an almost insuperable obstacle to finding the correct structure. A procedure is described for overcoming this difficulty. The Cooley-Tukey fast-Fourier-transform technique is used to compute E maps and all peaks greater than a certain height are located. A selection of the highest of these peaks, whose number is chosen by the program user, is then analysed with respect to bond lengths and angles. Favourable projections of coherent groups of peaks are output on the line printer in the form of integers representing the ranking order of the peaks and in positions which represent an undistorted projection of the group. Computing time is of the order of one minute per set of phases and the examination of a set of 32 maps and finding the correct solution takes about 30 minutes.