Remarks on Mayer's Reduced Density Matrix Method

Abstract

Mayer has proposed a variation method of finding the two-body reduced density matrix p~ for the ground state of a many fermion system. In this paper it is shown explicitly that his variation method is imperfect and sometimes leads to quite unsatisfactory results. Further, his trial form of p~ for electron gas is translated into that for the 77: electron system of benzene molecule and it is shown that such a trial form is unsatisfactory in several respects. The rigorous analysis of an assembly of mutually interacting particles by means of quantum theory constitutes a very difficult problem ; therefore, we must resort to some approximate methods. One of the most useful approximate methods is the well-known variation method based on the Schrodinger variation principle. In this method an ap· propriate trial form is assumed for the total wave function and the best is sought of all the wave functions of this type. The configuration interaction (CI) procedure in orbital theories is a typical example of this method. It is, however, usually necessary to take into account a very large number of configurations in order to obtain satisfactory results by means of CI procedure. Accordingly, it cannot be said to be a very useful procedure. In order to avoid this difficulty, <mother procedure has been proposed by Mott,ll Feynman2J and others. They assumed a single Slater determinant multiplied by a certain trial function which represents the correlation among particles. However, such treatments are. also intractable, whenever the density of particles is large, because of the considerable computational difficulty involved in deriving the functional forms of one- and two-body density matrices from the total wave functions given by such trial forms. Consequently, it is desirable to find a variation method in which a trial form is assumed directly for the two-body density matrix of assembly itself. Such an investigation was done by ]. E. Mayer3l for the ground state of electron gas. His method, however, seems to be open to several difficulties. In this paper, therefore, we want to make some remarks on its validity.