Gegenbauer expansions for three‐electron integrals

Abstract

An arbitrary power of ͉r i Ϫ r j ͉ can be expanded in terms of the magnitudes of r i and r j and Gegenbauer polynomials whose argument is the cosine of the angle between these two vectors. The Gegenbauer expansion has seen little use in the evaluation of three-electron integrals because the Gegenbauer polynomials are not orthogonal when integrated over the angular variables of a spherical coordinate system. It is shown here that this disadvantage is easily overcome and that the resulting formulas are not only simple and compact but also particularly suitable for the application of truncation and/or convergence acceleration schemes.

Explicitly Correlated Electrons in Molecules

“…Two- and three-electron integrals occur over the following operators (where k is 1 or 2) together with four-electron integrals over the operators Further integrals are required for the kinetic energy and nuclear attraction parts of the Hamiltonian. Progress on evaluating all of these integrals has been reported over a long period of time. − In particular, the “fully linked” three-electon integral r 12 r 13 / r 23 requires special attention, and important progress toward its analytical and efficient evaluation has been reported in the last few decades. − ,, Despite the progress in (atomic) many-electron integral evaluation, benchmark calculations using Hylleraas-type wave functions or the Hy-CI method have thus far been restricted to atoms not larger than Be (Clary and Handy performed Hy-CI calculations on the Ne atom in 1976, but they obtained only ca. 73.5% of the correlation energy due to computational constraints).…”