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 $|\psi _{0}\rangle$, 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.

$\langle \psi _{0}|{\hat {H}}|\psi _{a}^{r}\rangle =0$

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 $h(1)=-{\frac {1}{2}}\nabla _{1}^{2}-\sum _{\alpha }{\frac {Z_{\alpha }}{r_{1\alpha }}}$ and the other of two-electron operators $\sum _{j}|r_{1}-r_{j}|^{-1}$. Using the Slater–Condon rules we can simply evaluate

$\langle \psi _{0}|{\hat {H}}|\psi _{a}^{r}\rangle =\langle a|h|r\rangle +\sum _{b}\langle ab||rb\rangle =\langle a|h|r\rangle +\sum _{b}\left(\langle ab|rb\rangle -\langle ab|br\rangle \right)=\langle a|h|r\rangle +\sum _{b}\left(\langle a|2{\hat {J}}_{b}-{\hat {K}}_{b}|r\rangle \right)$

which we recognize is simply an off-diagonal element of the Fock matrix $\langle \chi _{a}|f|\chi _{r}\rangle$. 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. This is evident if we multiply the both sides of a Fock equation

${\hat {F}}\chi _{r}=\epsilon _{r}\chi _{r}$

by $\chi _{a}^{\ast }({\vec {r}})$ and integrate over the electronic coordinate:

$\int \limits _{-\infty }^{\infty }\chi _{a}^{\ast }({\vec {r}}){\hat {F}}\chi _{r}({\vec {r}})d^{3}{\vec {r}}=\epsilon _{r}\int \limits _{-\infty }^{\infty }\chi _{a}^{\ast }({\vec {r}})\chi _{r}({\vec {r}})d^{3}{\vec {r}}$

Since the wavefunction used as the ground-state determinant for the single-reference multielectron-basis set methods (CI, MPn, etc.) is the converged wavefunction of the Hartree–Fock problem, the Fock matrix has been already diagonalized and hence the states $\chi _{r}^{\ast }({\vec {r}})$ and $\chi _{a}({\vec {r}})$, being the eigenstates of the Fock operator, are orthogonal; thus their overlap is zero. It makes all the right-hand side of the equation zero:

$\int \limits _{-\infty }^{\infty }\chi _{a}^{\ast }({\vec {r}}){\hat {F}}\chi _{r}({\vec {r}})d^{3}{\vec {r}}=\langle \psi _{0}|{\hat {H}}|\psi _{a}^{r}\rangle =0$,

which proves the Brillouin theorem.

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.

The theorem had also been proven directly from the variational principle (by Mayer) and is essentially equivalent to the Hartree–Fock equations in general.

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.