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

Abstract

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 ≤ 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-, . . . , l and higher angular-momentum Slater orbitals and to determine F , G, . . . , L and states of higher order

Hylleraas–Configuration Interaction calculations on the 11S ground state of helium atom

“…Moreover the calculations of the ground state of helium atom have proven to be computationally faster when using the Kolos and Roothaan algorithm. Nevertheless it is important to test the direct method of evaluation, because this is the one which has to be used for the three-electron kinetic energy integrals when using the Hamiltonian in Hylleraas coordinates, since in the three-electron case the Kolos and Roothaan transformation cannot be used [35].…”

Analytic evaluation of some three- and four- electron atomic integrals involving s STO's and exponential correlation with unlinked $r_{ij}$'s

“…Some three-electron and four-electron integrals have been evaluated by Ruiz [44,45] by integration over the coordinates of one electron and calculated to a high degree of accuracy. Same method of evaluation was employed by Ruiz [46,47] to calculate the two-electron kinetic energy and the three-electron kinetic energy. Frolov et al [48] have investigated bound state spectra and properties of the doublet states in Li atom and some Li-like ions employing Hy-CI and CI methods.…”

“…It is observed that, if we make either a change (2 ⇆ 3) in equation( 44) or a change (3 ⇆ 1) in equation( 33), we get equation (47). Hence, by inspection, the following closed-form expression for I g 33 is obtained either from equation( 45) by performing a change (2 ⇆ 3) or from equation( 35) by making a change (3 ⇆ 1) :…”