Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind

Abstract

© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature. An efficient solution of calculating the spherical surface integral of a Gauss function defined as h(s,Q)=∫02π∫0π(s+Q)xi(s+Q)yj(s+Q)zke-γ(s+Q)2sinθdθdφ is provided, where γ≥ 0 , and i, j, k are nonnegative integers. A computationally concise algorithm is proposed for obtaining the expansion coefficients of polynomial terms when the coordinate system is transformed from cartesian to spherical. The resulting expression for h(s, Q) includes a number of cases of elementary integrals, the most difficult of which is II(n,μ)=∫0πcosnθe-μcosθdθ, with a nonnegative integer n and positive μ. This integral can be formed by linearly combining modified Bessel functions of the first kind B(n,μ)=1π∫0πeμcosθcos(nθ)dθ, with a nonnegative integer n and negative μ. Direct applications of the standard approach using Mathematica and GSL are found to be inefficient and limited in the range of the parameters for the Bessel function. We propose an asymptotic function for this expression for n = 0,1,2. The relative error of asymptotic function is in the order of 10−16 with the first five terms of the asymptotic expansion. At last, we give a new asymptotic function of B(n, μ) based on the expression for e-μII(n, μ) when n is an integer and μ is real and large in absolute value.