Theory of the Unrestricted Hartree-Fock Equation and Its Solutions. I

概要

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A general theory of the unrestricted Hartree-Fock equation and its solutions is developed. Two different but, of course, equivalent non-conventional formulations of the Hartree-Fock equation are presented. The first formulation is derived from the equation of the density matrix. It is shown that the Hartree-Fock equation is equivalent to an intersection problem of certain quadratic hypersurfaces. The second formulation is derived from a parametric representation of the density matrix and the orbitals. The Hartree-Fock equation is converted to a coupled non-linear equation for the parameter matrices. This form of the Hartree-Fock equation may be transformed into the form analogous to the Bogoliubov equation of superconductivity. Various representations of the Hartree-Fock energy are derived. A close correlation between the Hartree-Fock energy and the gaps of the orbital energies is indicated. An estimation for the upper and lower bounds of the number of different solutions of the Hartree-Fock equation is obtained. They are respectively 2^<(N^2-K)> and 2^K, where N is the dimension of the Hilbert space of the orbitals and K is the smaller number of n and N-n, n being the number of electrons in the system.
理論物理学刊行会の論文
1971-05-25