Brillouin's theorem

In quantum chemistry, Brillouin's theorem, proposed by the French physicist Léon Brillouin in 1934, relates to Hartree–Fock wavefunctions. Hartree–Fock, or the self-consistent field method, is a non-relativistic method of generating approximate wavefunctions for a many-bodied quantum system, based on the assumption that each electron is exposed to an average of the positions of all other electrons, and that the solution is a linear combination of pre-specified basis functions.

The theorem states that given a self-consistent optimized Hartree–Fock wavefunction ${\displaystyle |\psi _{0}\rangle }$, 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) must be zero. $${\displaystyle \langle \psi _{0}|{\hat {H}}|\psi _{a}^{r}\rangle =0}$$ This theorem is important in constructing a configuration interaction method, among other applications.

Another interpretation of the theorem is that the ground electronic states solved by one-particle methods (such as HF or DFT) already imply configuration interaction of the ground-state configuration with the singly excited ones. That renders their further inclusion into the CI expansion redundant.