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 , 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 one-electron operators and two-electron operators . Using the Slater-Condon rules we can simply evaluate
which we recognize is simply an off-diagonal element of the Fock matrix . 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.