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Brillouin's theorem

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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 | ψ 0 ⟩ , 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.

Proof

The electronic Hamiltonian of the system can be divided into two parts: one consisting of one-electron operators h ( 1 ) = − 1 2 ∇ 1 2 − ∑ α Z α r 1 α and the other of two-electron operators ∑ j | r 1 − r j | − 1 . Using the Slater-Condon rules we can simply evaluate

which we recognize is simply an off-diagonal element of the Fock matrix ⟨ χ a | f | χ r ⟩ . 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.

References

Brillouin's theorem Wikipedia
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