ZExpansion of Hartree-Fock Wave Functions

Abstract

Let a many-electron Hartree-Fock radial function ${\Phi }^{\mathrm{HF}}(nl; r)=\Pi \stackrel{N}{i=1\mathrm{}}\left[P^{\mathrm{HF}}\right(n^i{l}^i; r_i\left)\right]$ and the corresponding nonseparable variational function $\Phi (nl; r)$ be expanded in hydrogenic product functions ${\Phi }^{\mathrm{H}}(nl; r)$. The expansion coefficients $〈\mathrm{H}{n}^{\prime }l|nl〉$ of $\Phi$ are $O\left(Z^{-1\mathrm{}}\right)$ for $n^{\prime }\not\equiv n$; they vanish when the sets $n$, $n^{\prime }$ differ by more than two one-electron quantum numbers. It is shown that the expansion coefficients $〈\mathrm{H}{n}^{\prime }l|\mathrm{HF}n〉$ are $O\left(Z^{-2\mathrm{}}\right)$ when $\nu (n, n^{\prime })=N-\Sigma \stackrel{N}{i=1\mathrm{}}\delta \left(n^i{n}^{i\prime }\right)=2,$ i.e., when $n$ and $n^{\prime }$ differ in two places, and are $O\left(Z^{-1\mathrm{}}\right)$ when $\nu (n, n^{\prime })=1\mathrm{}$. In the second case $〈\mathrm{H}{n}^{\prime }l|\mathrm{HF}\mathrm{nl}〉=〈\mathrm{H}{n}^{\prime }l|nl〉+O\left(Z^{-2\mathrm{}}\right)$; the Hartree-Fock and the nonseparable expansion coefficients coincide to first order. This result holds only if the one-electron Hartree-Fock functions $P^{\mathrm{HF}}$ are not restricted by auxiliary conditions other than normalization. It does not hold, for example, if one requires the $P^{\mathrm{HF}}$ to satisfy the orthogonality condition

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