Theory of Electron-Pair Density Functions and Its Quantum Chemical Applications

电子对在原子和分子中的运动由电子对胞内(相对运动)和胞外(质心运动)密度函数来描述。对于原子系统，这些电子对密度用它们的球面平均值来表征。电子对密度是理解原子和分子中电子-电子相互作用的重要概率密度函数，但到目前为止，我们对其理论结构和在量子化学问题中的实际应用还缺乏足够的了解，本研究的目的是阐明电子对密度函数的理论结构，并将新的结果应用于各种化学和物理问题。本课题两年来的研究成果总结如下：1.从电子对密度的角度，我们证明了平均电子动量由低动量和高动量两部分组成。内径和外径之和分别构成了电子对相对半径和质心半径的上界和下界。发现电子动量密度由低动量密度和高动量密度组成。阐明了电子对密度的数学结构，推导了电子排斥能与电子角的关系的一般表达式。在同时考虑两个电子的基础上，我们证明了电子的径向密度严格地分为内密度和外密度。结果表明，亚壳平均动量可以分为低动量和高动量，它们的大小与双Zeta基函数的指数有关。

The motions of electron-pairs in atoms and molecules are described by the electron-pair intracule (relative motion) and extracule (center-of-mass motion) density functions. For atomic systems, these electron-pair densities are characterized by their spherical averages. The electron-pair densities are important probability density functions to understand the electron-electron interactions in atoms and molecules, but up to now we have neither sufficient knowledge about their theoretical structures nor practical applications to quantum chemical problems.The purpose of this research project has been to clarify the theoretical structure of the electron-pair density functions and to apply new results to various chemical and physical problems. The results of this research project during the two years are summarized as follows:1. From the viewpoint of electron-pair densities, we showed that the average electron momentum consists of two components, low momentum and high momentum.2. The sum and difference of the inner and outer radii were shown to constitute the upper and lower bounds, respectively, of the electron-pair relative and center-of-mass radii.3. We found that the electron momentum density consists of low momentum density and high momentum density.4. We clarified the mathematical structure of electron-pair densities and derived a general expression for the dependence of the electron repulsion energy on the interelectronic angle.5. Based on the simultaneous consideration of two electrons, we showed that the electron radial density is rigorously separated into inner and outer densities.6. We showed that the subshell average momentum can be separated into low and high momenta, and their values are related to the exponents of double-zeta basis functions.