Brillouin 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 ${\displaystyle |\psi _{0}\rangle }$, 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)

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

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

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