Closed-form representations of the Coulomb integral over hydrogenic orbitals

IF 1.7 3区化学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Journal of Mathematical Chemistry Pub Date : 2025-05-07 DOI:10.1007/s10910-025-01734-9

Balakrishnan Viswanathan, Darien DeWolf

引用次数: 0

Abstract

The theory of atomic structure is conceptually built on hydrogen-like orbitals and computed using Slater or Gaussian orbitals, owing to the relative difficulty of computing integrals concerning the hydrogenic orbitals. The optimal set of hydrogenic orbitals in an atom is obtained by minimizing the energy with respect to the orbitals. The Coulomb integral is difficult to compute due to the inverse distance relationship. In this paper, we evaluate the Coulomb integral and its derivative using two expressions for the inverse distance: the Laplace expression and the Legendre expression. The two expressions for inverse distance are similar and yield different integral forms. The Laplace expression yields the Coulomb integral as a sum of hypergeometric functions while the Legendre expression yields a compact polynomial form. The derivative of the Coulomb integral (computed using both forms) with respect to the decay constant is also provided.

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氢轨道上库仑积分的封闭形式表示

由于计算氢轨道的积分相对困难，原子结构理论在概念上建立在类氢轨道上，并使用斯莱特轨道或高斯轨道进行计算。一个原子中最优的氢轨道集是通过最小化相对于轨道的能量得到的。由于距离的反比关系，库仑积分很难计算。本文用拉普拉斯表达式和勒让德表达式求出了逆距离的库仑积分及其导数。逆距离的两个表达式相似，但产生不同的积分形式。拉普拉斯表达式得到的库仑积分是超几何函数的和，而勒让德表达式得到的是紧多项式形式。还提供了库仑积分（使用两种形式计算）对衰减常数的导数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Mathematical Chemistry 化学-化学综合

CiteScore

3.70

自引率

17.60%

发文量

105

审稿时长

6 months

期刊介绍： The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches. Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.