Long-range behavior of electronic wave functions. Generalized Carlson-Keller expansion

The long-range behavior of symmetry-restricted multiconfigurational self-consistent-field orbitals is given by ${{χ}}_{i{α}a}⁽¹⁾{~}({k}_{i{α}a}+{r}^{{-}1}{A}_{i{α}a}) {r}^{{{η}}_{{α}}}{e}^{{-}{ζ}r}$, where the common orbital exponent ${ζ}$ comes from the eigenvalue ${{{ζ}}²}{2}$ of an "ionization potential" matrix ${I}^{{{α}}}$, which may be thought of as being calculated with mixed-symmetry ($N{-}1$)-electron functions extracted from the $N$-electron wave function. Here ${α}$ is a "symmetry" index, $a$ is a "component" index, and $i$ is a "serial" index. The ${k}_{i{{α}}a}$ and ${A}_{i{{α}}a}$ from the "key" orbitals ${{χ}}_{i{{α}}a}⁽¹⁾$ are derived from eigenvectors of ${I}^{{{α}}}$. The corresponding key ${{η}}_{{{α}}}$ is obtained from the screened nuclear charge and from ${ζ}$ by ${{η}}_{{{α}}}={(Z{-}N+1{-}{ζ})}{{ζ}}$, while the other ${{η}}_{{α}}$ satisfy ${{η}}_{{α}}{≤}{{η}}_{{{α}}}{-}{l}GT({{α}},{α}){-}1$, where ${l}GT({{α}},{α})$ is the smallest $l$ such that ${Y}ₗᵐ({θ},{φ})$ couples ${{α}}$ to ${α}$. The Carlson-Keller "theorem"---the diagonal expansion of the $N$-electron wave function on its $p$- and ($N{-}p$)-electron natural eigenfunctions---is generalized for symmetry-restricted natural eigenfunctions. Distinct symmetry-coupled parts of the wave function can separately but not simultaneously be put in diagonal form.