Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind
Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind
复制标题
三维球坐标下高斯函数的高效球面积分及第一类修正贝塞尔函数的解
DOI：
10.1007/s10910-020-01204-4
复制
发表时间：
2021
影响因子：
1.7
通讯作者：
Kong, Jing
中科院分区：
文献类型：
--
作者：
Wang, Yiting;Kong, Jing
An efficient solution of calculating the spherical surface integral of a Gauss function defined as $$h\left( {s,{\mathbf{Q}}} \right) = \int_{0}^{2\pi } {\int_{0}^{\pi } {\left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{x}^{i} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{y}^{j} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{z}^{k} e^{{ - \gamma \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)^{2} }} } } \sin \theta d\theta d\varphi$$ is provided, where, andi,j,kare 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 forincludes a number of cases of elementary integrals, the most difficult of which is, with a nonnegative integernand positiveμ. This integral can be formed by linearly combining modified Bessel functions of the first kind, with a nonnegative integernand negativeμ. 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 forn= 0,1,2. The relative error of asymptotic function is in the order of 10−16with the first five terms of the asymptotic expansion. At last, we give a new asymptotic function ofbased on the expression forwhennis an integer andμis real and large in absolute value.
An efficient solution of calculating the spherical surface integral of a Gauss function defined as $$h\left( {s,{\mathbf{Q}}} \right) = \int_{0}^{2\pi } {\int_{0}^{\pi } {\left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{x}^{i} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{y}^{j} \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)_{z}^{k} e^{{ - \gamma \left( {{\mathbf{s}} + {\mathbf{Q}}} \right)^{2} }} } } \sin \theta d\theta d\varphi$$ is provided, where, andi,j,kare 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 forincludes a number of cases of elementary integrals, the most difficult of which is, with a nonnegative integernand positiveμ. This integral can be formed by linearly combining modified Bessel functions of the first kind, with a nonnegative integernand negativeμ. 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 forn= 0,1,2. The relative error of asymptotic function is in the order of 10−16with the first five terms of the asymptotic expansion. At last, we give a new asymptotic function ofbased on the expression forwhennis an integer andμis real and large in absolute value.