The direct method based on a fitting of distributions of semi-invariants

Abstract

A new direct method (called TRYMIN) for phaseproblem solution is described. At first a methodical procedure is presented for the construction of a direct method. The aim of the calculation is formulated as a consistency between theoretical and calculated distributions of invariants. A minimized function is obtained and an algorithm for its minimization is proposed. The algorithm is based on the partial decomposition of phases into three subsets and on the cyclical improvement of the estimate of local minima. The efficiency of the method has been tested on 23 structures and a short evaluation of the results of computer experiments with test structures is presented.0108-7673/89/070456-08503.00 Definition of the problemThe direct method for the solution of the phase problem is based on a theory that gives for a great number of functions Y, of phases ~ a forecast of their values for the correct phases ~*. For arbitrary values l~, u~ the theory affords the probability that the theoretically correct value Y* = Yi(~0*) satisfies l, < Y* < u,.Implementing this theory we can formulate the following task: to construct an algorithm for the generation of a limited number of sets q3 for which the values 17"~ = Y~(~) will satisfy inequalities l; < ~' < u~ (for a priori given l~, u~) with frequency in correspondence with the theory.O 1989 International Union of Crystallography VACLAV Kl~i2; 457We can presume that some of the sets ~3 will be close to the correct value ~*. This assumption is acceptable, because the power of direct methods resides in the fact that the problem is overdetermined. For some number of unknown phases the theory gives typically a twenty times greater number of functions Y~.We shall use only functions Y~ of the form Y~(~) = rsd (~ ku~j-P,), where ko are equal to 1, -1 or 0, and most of them are zero; p~ is chosen so that the theoretical probability density function of the absolute values of Y~ has a maximum at zero; and rsd is the real function of the real argument defined as rsd (x) = z if and only if -Tr -z < 7r, and an integer value j(x) exists that satisfies x--z + 2rrj(x).We shall use the symbol K for the matrix (ku), n for the number of functions Y~ and rn for the number of unknown phases. Phases are chosen to fix the origin and their valuesprojected into terms pi. Now we can suppose that K'K is a non-singular matrix.In the examples we shall use only the triple-phase relationships as they are generated by the MUL-TAN80 SIGMA2 routine (Main, Fiske, Hull, Lessinger, Germain, Declercq & Woolfson, 1980) for phase determination. We shall use the symbol K~ for the parameter of their distribution functions. If Y~ is derived from triple-phase relationships then at most three coefficients k~ for each i = 1,..., m are non-zero and p~ contains the phase shift and the values of the phases fixing the origin. Fitting the distributionThe principle of crystal structure solution using distribution fitting methods was used by Ha~ek (1974). However, a general description of these methods was presented later (...