Scaling limits of the electronic Schrödinger equation fordiatomic molecules: Asymptotic prediction of correlation structure,potential energy curves, and symmetry quantum numbers

电子Schrödinger方程在分子物理和量子化学中起着核心作用，因为它提供了分子电子结构的定量和化学特定描述。然而，迄今为止，关于这个方程的严格的数学工作主要集中在定性的、普遍的方面。在这里，我们将开发在自然尺度极限下严格提取定量和化学特定预测的方法，(i)固定电子数下的高核电荷极限和（ii）高核电荷和大原子间距离的双极限。得到了同核二聚体系列H_2、He_2、Li_2、Be_2、B_2、C_2、N_2、O_2、F_2、Ne_2的相关结构、势能曲线和对称量子数的显式渐近结果。这需要将Friesecke和Goddard[1,2]的泛函数解析和对称约简方法从原子推广到二聚体，仔细使用双原子分子的单电子Schrödinger方程理论，并利用数学等符号计算机软件包来处理具有六个或更多电子的情况。渐近结果将与实验数据、hundn - mulliken分子轨道理论的半经验图像（我们期望，基于原子的类似发现[1,2]，状态（ii）中的渐近基态与此图像相似）和最先进的计算方法（波函数和密度泛函类型）进行比较。设想的结果将首次在Schrödinger方程的定量精度和化学键的半经验概念之间建立严格的联系，从数学上阐明导致化学键形成的高化学特异性的机制，并为计算方法的设计和验证提供罕见的基准相关多电子数据。

The electronic Schrödinger equation plays a central role in molecular physics and quantum chemistry as it provides a quantitative and chemically specific description of a molecule's electronic structure. However, rigorous mathematical work on this equation has hitherto focused on qualitative, universal aspects. Here we will develop methods for the rigorous extraction of quantitative and chemically specific predictions in natural scaling limits, (i) the limit of high nuclear charge at fixed electron number and (ii) the double-limit of high nuclear charge and large interatomic distance. Explicit asymptotic results on correlation structure, potential energy curves, and symmetry quantum numbers will be worked out for the homonuclear dimer series H_2, He_2, Li_2, Be_2, B_2, C_2, N_2, O_2, F_2, Ne_2. This will require generalizing the functional-analytic and symmetry-reduction methods of Friesecke and Goddard [1, 2] from atoms to dimers, careful use of the theory of the single-electron Schrödinger equation for diatomic molecules, and utilization of symbolic computer packages such as Mathematics to handle cases with six or more electrons.The asymptotic results will be compared to experimental data, to the semi-empirical picture of Hund-Mulliken molecular orbital theory (we expect, based on analogous findings for atoms [1, 2], that asymptotic ground states in regime (ii) are similar to this picture), and state-of-the-art computational methods (both of wave function and density functional type). The envisioned results will establish, for the first time, a rigorous link between the quantitative precision of the Schrödinger equation and semi-empirical concepts of chemical bonding, elucidate mathematically the mechanisms leading to the high chemical specificity of bond formation, and provide rare benchmark correlated many-electron data for the design and validation of computational methods.