SADDLE-POINT CHARACTER OF HARTREE-FOCK ENERGIES

We consider the effect of replacing a doubly filled Hartree-Fock (HF) orbital {θiθi} by the symmetrized open-shell form {uivi + viui} (suppressing spin terms) in a Hartree-Fock wave function which consists of a linear combination of Slater determinants. We suppose that ui and vi differ from θi by small amounts. It is shown that by making certain restrictions on ui and vi, the change in energy may be negative, to second order. Since the HF orbitals minimize the HF energy, we have demonstrated the 'saddle-point character' of the HF energy. For those systems whose HF wave function is a single determinant, we use the criterion of energy minimization in the neighbourhood of the HF point (in Hilbert space) to obtain a method by which ui and vi can be determined approximately.