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 of one-electron operators $h(1)=-{\frac {1}{2}}\nabla _{1}^{2}-\sum _{\alpha }{\frac {Z_{\alpha }}{r_{1\alpha }}}$ and the other of 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.

Brillouin's theorem

Proof

Further reading