Variational Calculation of the Multiplet Spacings in the(3d)2Electron Configurations

The effect of spatial correlation of the two valence electrons on the multiplet spacings has been investigated for atoms with the ${\mathrm{}\left(1s\right)\mathrm{}}^2{\mathrm{}\left(2s\right)\mathrm{}}^2{\mathrm{}\left(2p\right)\mathrm{}}^6{\mathrm{}\left(3s\right)\mathrm{}}^2{\mathrm{}\left(3p\right)\mathrm{}}^6{\mathrm{}\left(3d\right)\mathrm{}}^2$ ground configuration. The atom is taken as a two-electron system with each electron moving in a Hartree-Fock effective potential. The zeroth-order wave function is taken to be a Clebsch-Gordan combination of products of one-electron Hartree-Fock orbitals. A variational function of the form $1+c{\mathcal{r}}_{12}+c^{\prime }{({\mathcal{r}}_1-{\mathcal{r}}_2)}^2$ is inserted into this wave function for each of the multiplet members, and $c$ and $c^{\prime }$ are determined by the variational method. When unrestricted Hartree-Fock functions are used as the zeroth-order basis set, good results which are insensitive to $c^{\prime }$ are obtained for the multiplet spacings. The ${}_{}{}^1{}_{}{}^{}G-{}_{}{}^3{}_{}{}^{}F$ spacing is overcorrected by about - 12% compared to +23% without correlation, while the ${}_{}{}^1{}_{}{}^{}D-{}_{}{}^3{}_{}{}^{}F$ spacing is overcorrected by - 3% compared to +34% without correlation.