Theory of the Unrestricted Hartree-Fock Equation and Its Solutions. III: Classification of Instabilities and Interconnection Relation between the Eight Classes of UHF Solutions

Abstract

A general theory on classification of instability and interconnection relation between UHF solutions is developed. It is shown that instability of a UHF solution is splitted into subinstabilities irreducible to the invariance group of the UHF solution. All types of instability for the eight classes of UHF solutions are listed up and explicit conditions for the instabilities are derived. It is shown that the type of a UHF solution emerging from an instability is determined by the type of the instability. The invariance groups of the UHF solutions emerging from instability of a UHF solution are the subgroups of the invariance group of the initial UHF solution. From this, the interconnection relation between the eight classes of UHF solutions is determined. Expressions for HF energy for the eight classes of UHF solutions are derived.