Numerical Solution of the ($1s1s$) and ($1s2s$) Hydrogenic Pair Equations

The pair functions which determine the exact first-order wave function for the ground state of the three-electron atom have been found with the matrix finite-difference method. The second- and third-order energies for the $\mathrm{}\left(1s1s\right)\mathrm{}{}_{}{}^1{}_{}{}^{}S$, $\mathrm{}\left(1s2s\right)\mathrm{}{}_{}{}^3{}_{}{}^{}S$, and $\mathrm{}\left(1s2s\right)\mathrm{}{}_{}{}^1{}_{}{}^{}S$ states of the two-electron atom are presented along with contour and perspective plots of the pair functions.