The aim of this article is to propose a new reduced-order modelling approach for parametric eigenvalue problems arising in electronic structure calculations. Namely, we develop nonlinear reduced basis techniques for the approximation of parametric eigenvalue problems inspired from quantum chemistry applications. More precisely, we consider here a one-dimensional model which is a toy model for the computation of the electronic ground state wavefunction of a system of electrons within a molecule, solution to the many-body electronic Schr\"odinger equation, where the varying parameters are the positions of the nuclei in the molecule. We estimate the decay rate of the Kolmogorov n-width of the set of solutions for this parametric problem in several settings, including the standard L2-norm as well as with distances based on optimal transport. The fact that the latter decays much faster than in the traditional L2-norm setting motivates us to propose a practical nonlinear reduced basis method, which is based on an offline greedy algorithm, and an efficient stochastic energy minimization in the online phase. We finally provide numerical results illustrating the capabilities of the method and good approximation properties, both in the offline and the online phase.

翻译：本文旨在提出一种适用于电子结构计算中参数化本征值问题的新型约化阶建模方法。具体而言，我们针对量子化学应用启发的参数化本征值问题，发展了非线性约化基技术。更精确地说，我们考虑了一个一维模型，该模型是计算分子内电子系统基态波函数的简化模型，其解满足多体电子薛定谔方程，其中变化参数为分子内原子核的位置。我们估算了该参数问题解集的Kolmogorov n-width衰减率，涵盖标准L2范数以及基于最优输运的距离度量。由于后者比传统L2范数框架下的衰减速度快得多，这促使我们提出一种实用的非线性约化基方法，该方法基于离线贪婪算法，并在在线阶段采用高效的随机能量最小化。最后，我们提供数值结果，展示该方法在离线与在线阶段均具有优良的逼近性能。