Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. IV. An improved algorithm for three-electron kinetic energy integrals

An improved algorithm to evaluate the nonrelativistic three-electron Hylleraas-Configuration Interaction (Hy-CI) kinetic energy integrals over Slater orbitals and the Hamiltonian in Hylleraas coordinates is shown. The resulting analytical expressions are general for all quantum numbers of the orbitals. From there, the restriction of employing orbitals with quantum numbers $l\le 2$ of the above algorithm presented in paper I of this series has been removed. With the new algorithm it is possible, in the direct integration method described in this series, to carry out Hy-CI atomic structure calculations including $f$-, $g$-, $\dots$ , $l$ and higher angular-momentum Slater orbitals and to determine $F$, $G$, $\dots$ , $L$ and states of higher order symmetry.