The refinement of atomic parameters by the technique known in X-ray crystallography as ‘the method of steepest descents’

Abstract

This paper is the first of a proposed set of three in which applications of the methods of ‘steepest descents’ are made to the problems of crystal structure analysis. The proposed programme is in three parts: (1) Working details of the method. (2) A discussion of the accuracy and convergence of the method. (3) Examples of the use of the method in actual structure determination. The present paper forms the first of this set and is complete in itself. A discussion of the normal methods of structure analysis results in a formulation of the problem as one of minimization, and leads naturally to the idea of optimum procedures for this purpose. Three techniques are derived from a unified presentation: ( a ) the ‘relaxation’ method, ( b ) the ‘least squares’ refinement, ( c ) the method of ‘steepest descents’. These are examined in detail and their virtues and defects noted. A complete discussion of the application of the steepest descent procedure to various types of structure analysis is given and techniques are derived for the cases of: refinement of a set of parameters already reasonably good; refinement of a structure where general configuration but not orientation is known; and refinement of structures given by electron densities on a lattice. Several variants of the process are given, in one of which only those planes of zero observed intensity being used; this gives a process which is independent of phase angle values and should be useful in the case of structures for which no chemical data is available.