The importance of Brillouin's Theorem

The Hartree-Fock solution for $1s2s\;^3\!S$ is a much better approximation than the one for $1s2s\;^1\!S$. This is directly related to Brillouin's Theorem. Let the approximate wave function be a linear expansion

$$\mbox{$\Psi\left(\mbox{$\gamma_{}\, LS$ }\right)$ } = \mbox{$......{\gamma^*}c_{\gamma^*} \mbox{$\Phi\left(\gamma^*\,LS\right)$ }$$ (42)

where $\mbox{$\Phi\left(\gamma^*\,LS\right)$ }$ is a mono-excited CSF. Then, when Brillouin's theorem holds, the first row/column of the interaction matrix will be zero and the Hartree-Fock energy will be an eigenvalue of the interaction matrix. Thus, the Hartree-Fock approximation already included the effect of the singly excited states in the approximate wave function. The same is not true when Brillouin's theorem does not hold for all the singly excited states.
 

2001-01-09