Application of the GroupR4to Energy-Level Separations and the Evaluation of Correlation Energy in Atoms

The group $R_4$ is applied to the study of energy-level separations and correlation energies in the ground-state configurations of first-row atoms. The separation between two nearly degenerate levels, e.g., $2s^2{}_{}{}^1{}_{}{}^{}S$ and $2p^2{}_{}{}^1{}_{}{}^{}S$, considering the $\Sigma {i>j}^{}\frac{1}{{\mathcal{r}}_{\mathrm{ij}}}$ part of the Hamiltonian only, can then be determined by means of group theory, eliminating the need of solving the secular equation arising from the usual configuration-interaction procedure. Extending this work to the one-electron part of the Hamiltonian enables us to estimate the actual splittings and the $\epsilon \left(2\mathrm{}{s}^2\right)$ nondynamical (near degeneracy) correlation energy for the first-row atoms. The $Z$ and $N$ dependence of this type of correlation is seen to be closely related to the group theoretical treatment. Finally this method is extended to a treatment of the nondynamical correlation in second-row atoms.