Brillouin's theorem

In quantum chemistry, Brillouin's theorem, proposed by the French physicist Léon Brillouin in 1934, is a fundamental theorem that simplifies theoretical calculations of electronic structure. It states that within the common Hartree–Fock approximation, the electronic ground state does not directly mix or interact with electronic states where only a single electron has been promoted to a higher energy level (a "singly excited" state). The Hartree–Fock method is a foundational approach for approximating the wavefunction and energy of a quantum many-body system, such as the electrons in an atom or molecule.

The main consequence of the theorem arises when improving upon the Hartree-Fock approximation, a process known as including electronic correlation. Methods like configuration interaction (CI) build a more accurate wavefunction by combining the ground state with various excited states. Brillouin's theorem implies that when performing a CI calculation, all contributions from singly excited states will be zero. Therefore, to improve the Hartree-Fock energy, one only needs to consider states where two or more electrons have been excited, which significantly reduces the computational complexity. Mathematically, the theorem states that the matrix element of the Hamiltonian ${\hat {H}}$ between the ground state Hartree–Fock wavefunction $|\psi _{0}\rangle$ and a singly excited determinant $|\psi _{a}^{r}\rangle$ (i.e. one where an occupied orbital a is replaced by a virtual orbital r) is zero: $\langle \psi _{0}|{\hat {H}}|\psi _{a}^{r}\rangle =0$ 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.