O(4)andU(3)Symmetry Breaking in the2s−2pShell

A group-theoretical analysis of the states of first-row atoms and their internal electrostatic interactions is developed. The Hamiltonian is found to be completely expressible as a linear combination of operators that are diagonal in $U\left(3\right)\mathrm{}$, operators diagonal in $O\left(4\right)\mathrm{}$, and operators diagonal in both. This makes possible a simple and uniform treatment of the energy levels of first-row atoms. We use it here to analyze configuration mixing between the $2s^{2}2p^x$ and $2p^{x+2\mathrm{}}$ configurations, and determine the contribution of configurations of $O\left(4\right)\mathrm{}$ and $U\left(3\right)\mathrm{}$ symmetry in several types of mixed-configuration wave functions.