Perturbation theory in1Zfor atoms: First-order pair functions in anl-separated Hylleraas basis set

For the $\frac{1}{Z}$ perturbation theory of atoms, a partial-wave method is presented for determining first-order pair wave functions. It rests on the fact that the Hylleraas variational principle decouples for the individual partial waves ($l=0,1,2,\dots$) and that all partial waves for $l\gg 1$ are easily representable (Schwartz, 1962). The $l\mathrm{th}$ partial wave is approximated in basis functions obtained by projecting the well-known Hylleraas functions (containing the powers of $r_{12}$ onto $P_l\left(cos{\theta }_{12}\right)$. Results for the ${1s}^2$ ground state show rapid convergence. The variational value for the (total) second-order ${1s}^2$ energy, which would be provided by 45 Hylleraas functions, is achieved with 10, 12, 9, 8, 5, and 5 basis functions for $l=0,1,2,3,4\mathrm{}$, and $5$, respectively. For any $1\ge 6$, one function is sufficient. Also good convergence is found for three-electron integrals (parts of the second-order lithium energy).