The Brillouin Theorem

Abstract

Let 'PG be the Hartree-Fock ground-state wave function and 'PE a singly excited state which differs from 'P G by substituting an occupied spinorbital by a virtual one. Then the following statement holds (Brillouin theorem): (11.1) In words, singly excited states do not interact with the HF ground state: the corresponding matrix element of the Hamiltonian is zero. This famous theorem (Brillouin 1933) plays an important role in the Hartree-Fock theory as well as in more sophisticated methods based on a Hartree-Fock reference state. It can be shown that Eq. (11.1) is a necessary and sufficient condition for 'PG to be the exact Hartree-Fock wave function, and, in fact, the most general derivation of the Hartree-Fock equations is possible through the Brillouin theorem which can be proved directly from the variation principle (Mayer 1971, 1973, 1974). We shall not prove here the complete equivalence of the Hartree-Fock equations and Eq. (11.1), it will be shown only that the Brillouin theorem is fulfilled for the Hartree-Fock wave function. The proof will make use of second quantization which helps us to evaluate the matrix element easily. To this goal, Eq. (11.1) should be rewritten in the second quantized notation. The ground state is simply represented the Fermi vacuum: