标题： Coulomb积分对于1、2和3电子距离算符的数值评估，$R_{C1}^{-n}R_{D1}^{-m}$、$R_{C1}^{-n}r_{12}^{-m}$和$r_{12}^{-n}r_{13}^{-m}$，具有实数$(n,m$以及3维笛卡尔乘积的常见密度泛函数值积分方案 显示英文标题

标题： Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, $R_{C1}^{-n}R_{D1}^{-m}$, $R_{C1}^{-n}r_{12}^{-m}$ and $r_{12}^{-n}r_{13}^{-m}$ with real $(n,m$) and the Descartes product of 3 dim. common density functional numerical integration scheme

摘要： 解析解对于积分来说比数值解更有用，然而在许多情况下前者并不可用。 我们对标题中提到的积分进行数值计算，这些积分在量子化学的一些方法中是必需的。 在标题中，R 表示核-电子距离，r 表示电子-电子距离，n, m= 0 的情况是平凡的，(n, m)= (1,0) 或 (0,1) 的情况是众所周知的，在积分中是一个基本的里程碑，并且在计算量子化学中广泛使用，如果使用高斯函数，解析积分也是可能的。 对于其余的情况，解析解受到限制，但已经为一些情况推导出来了，例如 对于 n, m= 0,1,2 使用高斯函数的情况。 在本工作中，我们将 Becke-Lebedev-Voronoi 三维数值积分方案（在密度泛函理论中常用）通过笛卡尔积推广到 6 维和 9 维，以评估标题中提到的积分，并对其进行测试。 这种数值方法 （包括带有种子 exp(-|r1|2) 的高斯被积函数，以及正负实数 n 和 m 值）对于处理电子间距离的高阶矩很有用，例如在相关计算中；此外，我们的数值方案也适用于具有种子 exp(-|r1|) 的斯莱特型函数。 显示英文摘要

摘要： Analytical solutions to integrals are far more useful than numeric, however, the former is not available in many cases. We evaluate integrals indicated in the title numerically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the n, m= 0 case is trivial, the (n, m)= (1,0) or (0,1) cases are well known, a fundamental milestone in the integration and widely used in computational quantum chemistry, as well as analytical integration is possible if Gaussian functions are used. For the rest of the cases the analytical solutions are restricted, but worked out for some, e.g. for n, m= 0,1,2 with Gaussians. In this work we generalize the Becke-Lebedev-Voronoi 3 dimensions numerical integration scheme (commonly used in density functional theory) to 6 and 9 dimensions via Descartes product to evaluate integrals indicated in the title, and test it. This numerical recipe (up to Gaussian integrands with seed exp(-|r1|2), as well as positive and negative real n and m values) is useful for manipulation with higher moments of inter-electronic distances, for example, in correlation calculations; more, our numerical scheme works for Slaterian type functions with seed exp(-|r1|) as well.