New Aspects of the Bipolar Expansion and Molecular Multicenter Integrals

Abstract

Standard bipolar expansions contain radial functions Jl1l2 L(r1, r2, R) which depend on three variables, r1, r2, R. Using representations of Bessel functions as series of Laguerre polynomials, the J's are expanded in terms of functions of the individual variables. As a result, the inverse distance between two points is brought into the form r12−1 = ∑ nlllml ∑ n2l2m2 Fn1l1m1, n2l2m2(R, θ, Ψ) Yl1m1(θ1, φ1)fn1l1(r1)Yl2m2(θ2, θ2)fn2l2(r2). This modified bipolar expansion is used to derive expressions for multicenter electron repulsion and nuclear attraction integrals. The method is particularly suitable for Gaussian orbitals expressed with spherical harmonics and yields compact expressions directly. For Slater-type orbitals, the multicenter energy integrals appear as series involving only integrals of the two-center overlap type. The one-center and multipole limits of the bipolar expansion are examined.