Numerical calculation of overlap and kinetic integrals in prolate spheroidal coordinates

Abstract

ABSTRACT:The efficient algorithm calculating the overlap and the kinetic integrals for the numerical atomic orbitals is presented. The described algorithm exploits the properties of the prolate spheroidal coordinates. The overlap and the kinetic integrals in R 3 are reduced to the integrals over the rectangular domain in R 2 , what substantially reduces the complexity of the problem. We prove that the integrand over the rectangular domain is continuous and does not have any slope singularities. For calculation of the integral over the rectangle any adaptive algorithm can be applied. The exemplary results were obtained by application of the adaptive Gauss quadrature.

Numerical calculation of overlap and kinetic integrals in prolate spheroidal coordinates. II

“…We showed in Ref. 1 that, by application of prolate coordinate system with foci at the points A and B , the overlap integral I o ( q ) is given by (the † denotes the conjugate complex): where δ is Kronecker delta and A l,m is the normalization factor of spherical harmonic 2: The function I o ,1 ( q ) is defined by the two‐dimensional integral over the rectangle Ω = [−1, 1] × [1, ξ * ] with ξ * = 2 max{ r , r }/ q + 1: where P l,m ( x ) denotes the associated Legendre polynomial. Moreover, the kinetic integral, I k ( q ), of f a ( r ) and f b ( r ) is equal to the overlap integral of f a ( r ) and g b ( r ) = R̃ b ( r b ) Y (θ b , ϕ b ): where we have: Thus, the calculation of both I o ( q ) and I k ( q ) reduces to integration over the rectangle Ω = [−1, 1] × [1, ξ * ].…”

“…In the previous paper 1 we introduced the algorithm for numerical evaluation of the overlap and the kinetic integrals. Our algorithm uses prolate coordinate systems 2, 3 and reduces the problem from ℝ 3 to ℝ 2 .…”

“…The algorithm is applicable to atomic orbitals with finite support, i.e., atomic orbitals which vanish outside the sphere of given radius. It was shown 1 that for atomic orbitals with the finite support, the evaluation of the overlap and the kinetic integrals reduces to integration over the rectangle Ω = [−1, 1] × [1, ξ * ], where ξ * depends on the support of the atomic orbitals and the inter‐center distance.…”

Analytical evaluation of relativistic molecular integrals. II: Method of computation for molecular auxiliary functions involved

“…The domain of s 6 sum index is dependent to upper limit of summation which must covers all the terms in Eq. (40). For vectorized forms of the equations presented in this section see Appendix B.…”

“…Benchmark result obtained via global-adaptive method with Gauss-Kronrod extension. Results obtained via Eq (40)…”