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)

${\displaystyle \langle \psi _{0}|{\hat {H}}|\psi _{a}^{r}\rangle =0}$

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 ${\displaystyle h(1)=-{\frac {1}{2}}\nabla _{1}^{2}-\sum _{\alpha }{\frac {Z_{\alpha }}{r_{1\alpha }}}}$ and the other of two-electron operators ${\displaystyle \sum _{j}|r_{1}-r_{j}|^{-1}}$. Using the Slater-Condon rules we can simply evaluate

${\displaystyle \langle \psi _{0}|{\hat {H}}|\psi _{a}^{r}\rangle =\langle a|h|r\rangle +\sum _{b}\langle ab||rb\rangle }$

which we recognize is simply an off-diagonal element of the Fock matrix ${\displaystyle \langle \chi _{a}|f|\chi _{r}\rangle }$. 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.

Further reading

• Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd. pp. 207–211. ISBN 0-471-48552-7.

• Szabo, Attila; Neil S. Ostlund (1996). Modern Quantum Chemistry. Mineola, New York: Dover Publications, Inc. pp. 350–353. ISBN 0-486-69186-1.