Radial Electron–Electron Distributions and the Coulomb Hole for Be

Abstract

Individual radial electron–electron distribution functions, $P_i{\mathrm{(r}}_{12}\mathrm{)\text{'}}\mathrm{s}$ , are defined in terms of the eigen‐functions (the natural geminals, NG's) of the spin‐free 2‐matrix, $P^{\text{(2)}}({\mathbf{r}}_1,{\mathbf{r}}_2\mathrm{\hspace{0.17em}|\hspace{0.17em}}{\mathbf{r}}_1\mathrm{\prime ,}{\mathbf{r}}_2\mathrm{\prime )}$ . The total radial electron–electron distribution function, $P_0{\mathrm{(r}}_{12})$ , is defined in terms of $P^{\text{(2)}}$ . It is demonstrated that the structure of individual two‐particle eigenfunctions can easily be determined from the corresponding $P_i{\mathrm{(r}}_{12})$ . Distributions are obtained for the ${}^{1}S$ ground state of the Be atom by numerical Fourier transformation of x‐ray intensity values. The natural spin geminal (NSG) analysis by Barnett and Shull of Weiss' configuration interaction (CI) wavefunction is used for the determination of correlated $P_i{\mathrm{(r}}_{12}\mathrm{)\text{'}}\mathrm{s}$ , while the 2‐matrix analysis of Clementi's Restricted Hartree–Fock (RHF) function is employed in the evaluation of independent particle model (IPM) $P_i{\mathrm{(r}}_{12}\mathrm{)\text{'}}\mathrm{s}$ . The correlated and RHF $P_0{\mathrm{(r}}_{12}\mathrm{)\text{'}}\mathrm{s}$ are compared with that obtained from a two‐configuration wavefunction of Watson, which takes into account the near degeneracy of $\text{2s}$ and $\text{2p}$ orbitals. By subtracting the RHF from the Weiss $P_0{\mathrm{(r}}_{12})$ , a description of the Coulomb hole function, ${\mathrm{\Delta (r}}_{12})$ , is obtained for the ${}^{1}S$ Be ground state. A measure of the Fermi correlation may be obtained with $P_i{\mathrm{(r}}_{12}\mathrm{)\text{'}}\mathrm{s}$ constructed from those ${}^{1}S$ and ${}^{3}S$ RHF geminals which involve the spatially symmetric or the spatially antisymmetric combination of $\text{1s}$ and $\text{2s}$ orbitals.