The Group $R_4$ in Atomic-Structure Theory: The Hydrogenic $R_4$ versus the Mathematical $R_4$ and the Coulomb Interaction in $2s^{m}2p^n$ and $3s^{m}3p^{n}3d^r$ Configurations

The rotation group in four dimensions $R_4$ is applied to the study of the $2s^{m}2p^n$ configurations of atoms. This Lie group is used in both the mathematical sense, describing the transformation properties of the angular parts of the $2s$ and $2p$ electrons, and in the more physical sense of an approximate symmetry group for the first-row atoms. The various states of each configuration are classified by means of $R_4$. By expressing the Coulomb interaction in terms of tensors, transforming according to irreducible representations of $R_4$, Coulomb and exchange integrals are evaluated group theoretically. The method is extended to $3s^{m}3p^{n}3d^{\mathcal{r}}$ configurations. The states are classified by means of $R_4$ and Coulomb and exchange integrals are approximately evaluated.