Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind
© 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...
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| Format: | Article |
| Language: | English |
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Springer Science and Business Media LLC
2022
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| Online Access: | https://hdl.handle.net/1721.1/143972 |
| _version_ | 1826217426186928128 |
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| author | Wang, Y Kong, J |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Wang, Y Kong, J |
| author_sort | Wang, Y |
| collection | MIT |
| description | © 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. |
| first_indexed | 2024-09-23T17:03:27Z |
| format | Article |
| id | mit-1721.1/143972 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T17:03:27Z |
| publishDate | 2022 |
| publisher | Springer Science and Business Media LLC |
| record_format | dspace |
| spelling | mit-1721.1/1439722023-01-10T17:01:22Z Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind Wang, Y Kong, J Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science © 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. 2022-07-22T15:30:10Z 2022-07-22T15:30:10Z 2021-02-01 2022-07-22T15:22:09Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/143972 Wang, Y and Kong, J. 2021. "Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind." Journal of Mathematical Chemistry, 59 (2). en 10.1007/s10910-020-01204-4 Journal of Mathematical Chemistry Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Springer Science and Business Media LLC DOE repository |
| spellingShingle | Wang, Y Kong, J Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind |
| title | Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind |
| title_full | Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind |
| title_fullStr | Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind |
| title_full_unstemmed | Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind |
| title_short | Efficient spherical surface integration of Gauss functions in three-dimensional spherical coordinates and the solution for the modified Bessel function of the first kind |
| title_sort | efficient spherical surface integration of gauss functions in three dimensional spherical coordinates and the solution for the modified bessel function of the first kind |
| url | https://hdl.handle.net/1721.1/143972 |
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