Brillouin's theorem

In quantum chemistry, Brillouin's theorem, proposed by the French physicist Léon Brillouin in 1934, states that given a self-consistent optimized Hartree-Fock wavefunction $|\psi_0\rangle$, the matrix element of the Hamiltonian between the ground state and a single excited determinant (i.e. one where an occupied orbital a is replaced by a virtual orbital r)

$\langle \psi_0|\hat{H} |\psi_a^r \rangle=0$

This theorem is important in constructing a configuration interaction method, among other applications.

Proof

The electronic Hamiltonian of the system can be divided into two parts: one consisting one-electron operators $h(1)=-\frac{1}{2}\nabla^2_1 - \sum_{\alpha} \frac{Z_\alpha}{r_{1\alpha}}$ and two-electron operators $\sum_{j} |r_1-r_j|^{-1}$. Using the Slater-Condon rules we can simply evaluate

$\langle \psi_0|\hat{H} |\psi_a^r \rangle=\langle a|h|r\rangle + \sum_b \langle ab || rb\rangle$

which we recognize is simply an off-diagonal element of the Fock matrix $\langle \chi_a|f|\chi_r \rangle$. But the whole point of the SCF procedure was to diagonalize the Fock matrix and hence for an optimized wavefunction this quantity must be zero.