Unified kernel function approach to two-center integrations in quantum-chemical calculations

Abstract

We present a unified kernel function treatment of the two-center integrals over atomic orbitals appearing in quantum chemical calculations on the basis of density functional theory. By using the Fourier transform method, we obtain analytical expressions for the kernel functions, from which actual integral values can be evaluated through twofold integration. As our only assumption is that the angular dependences of the atomic basis functions is given by spherical harmonics, our kernel functions are applicable to almost arbitrary types of analytically given or numerically tabulated atomic radial functions. In addition, we give some interesting new recurrence relations holding between parts of kernel functions, which enable an entirely recursive treatment of the angular momentum arguments. When applied to analytically given atomic basis functions, which contain a factor rν+l, a qualitatively new and general set of recurrence relations for auxiliary functions of two-center integrals and, in some cases, of the two-center integrals themselves, results. As an example, two-center overlap integrals over Slater-type orbitals are calculated recursively for higher angular momentum quantum numbers and compared with the results of a direct calculation. We expect the present recurrence relations to be superior to other existing recurrences with respect to angular momentum quantum numbers.