5 J ul 2 00 6 1 - Density Operators and Algebraic Version of The Hohenberg - Kohn Theorem

Interrelation of the Coleman's representabilty theory for 1-density operators and abstract algebraic form of the Hohenberg-Kohn theorem is studied in detail. Convenient realization of the Hohenberg-Kohn set of classes of 1-electron operators and the Coleman's set of ensemble representable 1-density operators is presented. Dependence of the Hohenberg-Kohn class structure on the boundary properties of the ground state 1-density operator is established and is illustrated on concrete simple examples. Algorithm of restoration of many electron determinant ensembles from a given 1-density diagonal is described. Complete description of the combinatorial structure of Coleman's polyhedrons is obtained. I. Introduction In the middle of the twentieth century there appeared two papers that became contemporary classics and called into being a new branch of quantum chemistry. The first paper, dedicated, at first glance, to solution of a very abstract problem of representability of 1-density operators by ensembles of many electron states, was written by Coleman and published in 1963 [1]. One year later, in 1964, Hohenberg and Kohn published their theorem [2] about correspondence between representable densities and external potentials. From mathematical point of view both results belong to the functional analysis and have close interrelation. Coleman's theorem gives explicit analytic description of the set of all ensemble representable 1-density operators and Hohenberg-Kohn theorem suggests implicit way of parametrization of representable 1-density functions by certain classes of 1-electron operators (potentials). Further development of Coleman's approach was mainly concentrated on attempts to get the necessary and sufficient conditions of ensemble representability for density operators of higher order, especially for 2-electron density operators (see [3, 4] and references therein). It turned out to be a very complicated mathematical problem and to the best of our knowledge the constructive analytic description of the set of all representable 2-density