Lesiuk Michał, Moszynski Robert
Faculty of Chemistry, University of Warsaw Pasteura 1, 02-093 Warsaw, Poland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Dec;90(6):063319. doi: 10.1103/PhysRevE.90.063319. Epub 2014 Dec 29.
In this paper we consider the calculation of two-center exchange integrals over Slater-type orbitals (STOs). We apply the Neumann expansion of the Coulomb interaction potential and consider calculation of all basic quantities which appear in the resulting expression. Analytical closed-form equations for all auxiliary quantities have already been known but they suffer from large digital erosion when some of the parameters are large or small. We derive two differential equations which are obeyed by the most difficult basic integrals. Taking them as a starting point, useful series expansions for small parameter values or asymptotic expansions for large parameter values are systematically derived. The resulting expansions replace the corresponding analytical expressions when the latter introduce significant cancellations. Additionally, we reconsider numerical integration of some necessary quantities and present a new way to calculate the integrand with a controlled precision. All proposed methods are combined to lead to a general, stable algorithm. We perform extensive numerical tests of the introduced expressions to verify their validity and usefulness. Advances reported here provide methodology to compute two-electron exchange integrals over STOs for a broad range of the nonlinear parameters and large angular momenta.
在本文中，我们考虑斯莱特型轨道（STO）上双中心交换积分的计算。我们应用库仑相互作用势的诺伊曼展开，并考虑对所得表达式中出现的所有基本量进行计算。所有辅助量的解析闭式方程早已为人所知，但当某些参数很大或很小时，它们会遭受严重的数字精度损失。我们推导了两个最难的基本积分所满足的微分方程。以它们为起点，系统地推导了小参数值时的有用级数展开式或大参数值时的渐近展开式。当相应的解析表达式引入显著的抵消项时，所得展开式可替代这些表达式。此外，我们重新考虑了一些必要量的数值积分，并提出了一种以可控精度计算被积函数的新方法。所有提出的方法相结合，形成了一种通用、稳定的算法。我们对引入的表达式进行了广泛的数值测试，以验证其有效性和实用性。本文所报告的进展为计算广泛的非线性参数和大角动量下STO上的双电子交换积分提供了方法。
