Lesiuk Michał, Moszynski Robert
Faculty of Chemistry, University of Warsaw and Pasteura 1, 02-093 Warsaw, Poland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Dec;90(6):063318. doi: 10.1103/PhysRevE.90.063318. Epub 2014 Dec 29.
In this paper, which constitutes the first part of the series, we consider calculation of two-center Coulomb and hybrid integrals over Slater-type orbitals. General formulas for these integrals are derived with no restrictions on the values of the quantum numbers and nonlinear parameters. Direct integration over the coordinates of one of the electrons leaves us with the set of overlaplike integrals which are evaluated by using two distinct methods. The first one is based on the transformation to the ellipsoidal coordinates system and the second utilizes a recursive scheme for consecutive increase of the angular momenta in the integrand. In both methods simple one-dimensional numerical integrations are used in order to avoid severe digital erosion connected with the straightforward use of the alternative analytical formulas. It is discussed that the numerical integration does not introduce a large computational overhead since the integrands are well-behaved functions, calculated recursively with decent speed. Special attention is paid to the numerical stability of the algorithms. Applicability of the resulting scheme over a large range of the nonlinear parameters is tested on examples of the most difficult integrals appearing in the actual calculations including, at most, 7i-type functions (l=6).
在构成本系列第一部分的本文中，我们考虑斯莱特型轨道上双中心库仑积分和混合积分的计算。推导了这些积分的通用公式，对量子数和非线性参数的值没有限制。对其中一个电子的坐标进行直接积分后，我们得到了一组重叠型积分，这些积分通过两种不同的方法进行计算。第一种方法基于向椭球坐标系的变换，第二种方法利用一种递归方案来连续增加被积函数中的角动量。在这两种方法中，都使用简单的一维数值积分，以避免与直接使用替代解析公式相关的严重数字精度损失。文中讨论了数值积分不会带来很大的计算开销，因为被积函数是行为良好的函数，可以以相当不错的速度递归计算。特别关注了算法的数值稳定性。通过实际计算中出现的最困难积分的例子，包括最多到7i型函数（l = 6），测试了所得方案在大范围非线性参数上的适用性。
