Section 16.1.3.2. `Multisolution' methods and trial structures

Successful crystal structure determination requires that sufficient phases be found such that a Fourier map computed using the corresponding structure factors will reveal the atomic positions. It is particularly important that the biggest terms (i.e., largest ) be included in the Fourier series. Thus, the first step in the phasing process is to sort the reflections in decreasing order according to their values and to choose the number of large reflections that are to be phased. The second step is to generate the possible invariants involving these intense reflections and then to sort them in decreasing order according to their values. Those invariants with the largest values are retained in sufficient number to achieve the desired overdetermination. Ab initio phase determination by direct methods requires not only a set of invariants, the average values of the cosines of which are presumed to be known, but also a set of starting phases. Therefore, the third step in the phasing process is the assignment of initial phase values. If enough pairs of phases, and , are known, the structure invariants can then be used to generate further phases which, in turn, can be used to evaluate still more phases. Repeated iterations will permit most reflections with large to be phased.

Depending on the space group, a small number of phases can be assigned arbitrarily in order to fix the origin position and, in noncentrosymmetric space groups, the enantiomorph. However, except for the simplest structures, these reflections provide an inadequate foundation for further phase development. Consequently, a `multisolution' or multi-trial approach (Germain & Woolfson, 1968) is normally taken in which other reflections are each assigned many different starting values in the hope that one or more of the resultant phase combinations will lead to a solution. Solutions, if they occur, must be identified on the basis of some suitable figure of merit. Although phases can be evaluated sequentially, the order determined by a so-called convergence map (Germain et al., 1970), it has become standard in recent years to use a random-number generator to assign initial values to all available phases from the outset (Baggio et al., 1978; Yao, 1981). A variant of this procedure is to use the random-number generator to assign initial coordinates to the atoms in the trial structures and then to obtain initial phases from a structure-factor calculation.