Generalized formulation of quantum and classical crystallography using Green's functions

Quantum crystallography is a methodology by which structural information about a crystalline material obtained from X-ray crystallography is combined with quantum mechanical methods. The objective is to enhance the data obtained from the X-ray diffraction experiment, which are related to the atomic structure of the crystal, and to predict the properties and efficacy of those chemical compounds from which the crystals are derived. One approach in quantum crystallography is to use a projector matrix with a normalized trace. In this approach, quantum mechanical parameters in the projector matrix are fit into crystallographic data. During this fitting, the properties of the projector matrix called idempotency and normalization are used. Throughout this implementation procedure, Clinton's iteration scheme has been used in addition to the least-squares technique. The purpose of the present study is to generalize Clinton's iterative equations in quantum crystallography by means of single-particle Green's functions with the aid of the equal atoms model in the theory of direct methods. Convergency characters of the novel iterative equations are discussed by the steepest descent procedure. Furthermore, whether the calculations are valid in nonorthogonal bases was also examined. The iteration schemes widely used in quantum crystallography have been generalized but, in addition, the generalized expressions relating to the phase determination procedure and the probabilities of the sign relations between the structure factors are obtained and discussed comprehensively. The phrase order of crystallography has been put forward as a new concept. (C) 2005 Wiley Periodicals, Inc.