THE AVERAGE ENERGY APPROXIMATION FOR ELECTRONIC PERTURBATION PROBLEMS∗PROBLEMS^{*}

$^{*}$This research was supported in part by a grant extended to the University of Washington by the Office of Naval Research. $^{\dag}$In absentia from Department of Chemistry, University of Washington, Seattle 5, Washington. $^{\dag\dag}$dLNational Science Foundation Cooperative Fellow 1961-63.Author Institution: Department of Chemistry, University of California“Ordinary second order perturbation theory, when applied to an electronic perturbation ${P}$ [for convenience assume $(o|{P}|o) = o]$, yields an intractable sum over states $\Sigma_{i}’ (E_{o} - E_{e})^{-1}$. Expanding each denominator $E_{o}-E_{1}$, about an average energy $\epsilon_{av}$, leads to a power series in $(E_{o}-E_{i}-\epsilon_{nv})/\epsilon_{nv}$. The first term, after closure, is $/\epsilon_{nv}$ which is just the average energy approximation. Setting the second term equal to zero determines an $\epsilon_{nv}$ which is identical to the one obtained by minimizing the energy with, respect to a trial wavefunction of the form $\Psi_{trial} = \Psi_{o} + \epsilon_{av}^{-1} {P} \Psi_{o}$. Certain ${P}_{s}$, however, possess singularities which would give rise to a non-allowed $\Psi_{trial}$ (actually, $\epsilon_{{ av}}^{-1}$ is forced to be zero). This suggests choosing different $\epsilon_{av}’s$ for different regions in space so that singularities can be ignored. A general method for doing this has been developed which leads, ultimately, to replacement of the sum over states by an integral over regions in spare. Furthermore, $\Psi_{trial}$ can be correctly scaled by a suitable perturbation procedure. Applications to electric and magnetic perturbation problems will be discussed.