The evaluation of some four-electron correlated integrals with a Slater basis arising in linear and nonlinear r12 theories

This study considers the analytic evaluation of correlated integrals involving explicit inter-electron separation distances using a basis of standard Slater-type functions. These basis functions are employed in Hylleraas-CI calculations, Hylleraas calculations on atomic four-electron systems, and in non-Born-Oppenheimer calculations on small molecular systems. A Fourier transform approach allows a principal auxiliary integral to be evaluated analytically, permitting a number of four-electron integrals to be reduced to various one-dimensional Cauchy-Frullani integrals, which can be evaluated in closed form. The stability of some of the formulas with respect to numerical evaluation, particularly for the high-order derivative cases, is discussed in detail. Numerical values for a selection of test integrals are provided.