Fine Structure of the2P3and3P3States of Helium

Using an extension of the method of Pekeris for $S$ states, eigenvalues and eigenfunctions have been obtained for the $2{}_{}{}^1{}_{}{}^{}P$, $2{}_{}{}^3{}_{}{}^{}P$, $3{}_{}{}^1{}_{}{}^{}P$, and $3{}_{}{}^3{}_{}{}^{}P$ states of helium. The fine-structure splittings of the $2{}_{}{}^3{}_{}{}^{}P$ and $3{}_{}{}^3{}_{}{}^{}P$ states have been computed, including the ${\alpha }^3$ quantum electrodynamic correction and the singlet-triplet correction. Determinants up to order $n=220\mathrm{}$ were solved, and when the results were extrapolated to $n\to \infty$ good agreement was obtained with recent accurate measurements of the fine-structure splittings, substantiating the correctness of the ${\alpha }^3$ terms.