Stability Conditions for the Solutions of the Hartree–Fock Equations for Atomic and Molecular Systems. II. Simple Open-Shell Case

The stability conditions for the solutions of the Hartree–Fock equations for the simple open-shell case, i.e., closed shell with one extra electron, are derived. It is shown that only “doublet stability” is relevant is this simple open-shell case, the solutions being always “nondoublet unstable.” The doublet stability conditions are then derived using the mathematical methods of quantum field theory, namely, occupation number representation, Wick's theorem, and Feynman-like diagrams. In order to familiarize the reader with the use of these concepts they are first used to rederive the singlet and nonsinglet stability conditions for the closed-shell case. A general method of finding new Hartree–Fock solutions, in the case that the symmetry adapted Hartree–Fock solutions are unstable, is briefly discussed. The implications of the instability on the ground-state correlation energy calculations and on the excitation energy calculations using time-dependent Hartree–Fock theory are considered.