Department of Chemistry, McGill University, 801 Sherbrooke St. West, Montreal QC H3A 0B8, Canada.
J Chem Theory Comput. 2021 Sep 14;17(9):5568-5581. doi: 10.1021/acs.jctc.1c00379. Epub 2021 Aug 9.
The key idea of the variable-metric approach to orbital localization is to allow nonorthogonality between orbitals while, at the same time, preventing them from becoming linearly dependent. The variable-metric localization has been shown to improve the locality of occupied nonorthogonal orbitals relative to their orthogonal counterparts. In this work, numerous localization algorithms are designed and tested to exploit the conceptual simplicity of the variable-metric approach with the goal of creating a straightforward and reliable localization procedure for virtual orbitals. The implemented algorithms include the steepest descent, conjugate gradient (CG), limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), and hybrid procedures as well as trust-region (TR) methods based on the CG and Cauchy-point subproblem solvers. Comparative analysis shows that the CG-based TR algorithm is the best overall method to obtain nonorthogonal localized molecular orbitals (NLMOs), occupied or virtual. The L-BFGS and CG algorithms can also be used to obtain NLMOs reliably but often at higher computational cost. Extensive tests demonstrate that the implemented methods allow us to obtain well-localized Boys-Foster (i.e., maximally localized Wannier functions) and Pipek-Mezey, orthogonal and nonorthogonal, and occupied and virtual orbitals for a variety of gas-phase molecules and periodic materials. The tests also show that virtual NLMOs, which have not been described before, are, on average, 13% (Boys-Foster) and 18% (Pipek-Mezey) more localized than their orthogonal counterparts.
变尺度方法在轨道定位中的关键思想是允许轨道之间存在非正交性，同时防止它们线性相关。变尺度定位已被证明可以提高占据非正交轨道的局部性，相对于它们的正交对应物。在这项工作中，设计并测试了许多定位算法，以利用变尺度方法的概念简单性，旨在为虚拟轨道创建一个简单可靠的定位过程。所实现的算法包括最速下降法、共轭梯度 (CG)、有限内存 Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) 和混合程序以及基于 CG 和 Cauchy 点子问题求解器的信赖域 (TR) 方法。比较分析表明，基于 CG 的 TR 算法是获得非正交定域分子轨道（NLMO）的最佳总体方法，无论是占据还是虚拟。L-BFGS 和 CG 算法也可以可靠地获得 NLMO，但通常计算成本更高。广泛的测试表明，所实现的方法允许我们获得良好定域的 Boys-Foster（即最大定域化的 Wannier 函数）和 Pipek-Mezey、正交和非正交以及占据和虚拟轨道，适用于各种气相分子和周期性材料。测试还表明，以前未描述的虚拟 NLMO 比其正交对应物平均更局部化 13%（Boys-Foster）和 18%（Pipek-Mezey）。
