Fuemmeler Eric G, Damle Anil, DiStasio Robert A
Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, United States.
Department of Computer Science, Cornell University, Ithaca, New York 14853, United States.
J Chem Theory Comput. 2023 Dec 12;19(23):8572-8586. doi: 10.1021/acs.jctc.1c00801. Epub 2023 Nov 9.
In this work, we extend the selected columns of the density matrix (SCDM) methodology [ 1463-1469]─a non-iterative and real-space procedure for generating localized occupied orbitals for condensed-phase systems─to the construction of local molecular orbitals (LMOs) in systems described using non-orthogonal atomic orbital (AO) basis sets. In particular, we introduce three different theoretical and algorithmic variants of SCDM (referred to as SCDM-M, SCDM-L, and SCDM-G) that can be used in conjunction with the AO basis sets used in standard quantum chemistry codebases. The SCDM-M and SCDM-L variants are based on a pivoted QR factorization of the Mulliken and Löwdin representations of the density matrix and are tantamount to selecting a well-conditioned set of projected atomic orbitals (PAOs) and projected (symmetrically-) orthogonalized atomic orbitals, respectively, as proto-LMOs that can be orthogonalized to exactly span the occupied space. The SCDM-G variant is based on a real-space (grid) representation of the wavefunction, and therefore has the added flexibility of considering a large number of grid points (or δ functions) when selecting a set of well-conditioned proto-LMOs. A detailed comparative analysis across molecular systems of varying size, dimensionality, and saturation level reveals that the LMOs generated by these three non-iterative/direct SCDM variants are robust, comparable in orbital locality to those produced with the iterative Boys or Pipek-Mezey (PM) localization schemes, and completely agnostic toward any single orbital locality metric. Although all three SCDM variants are based on the density matrix, we find that the character of the generated LMOs can differ significantly between SCDM-M, SCDM-L, and SCDM-G. In this regard, only the grid-based SCDM-G procedure (like PM) generates LMOs that qualitatively preserve σ-π symmetry (in systems such as s-trans alkenes), and are well-aligned with chemical (, Lewis structure) intuition. While the direct and standalone use of SCDM-generated LMOs should suffice for most chemical applications, our findings also suggest that the use of these orbitals as an unbiased and cost-effective (initial) guess also has the potential to improve the convergence of iterative orbital localization schemes, in particular for large-scale and/or pathological molecular systems.
在本工作中，我们将密度矩阵选定列（SCDM）方法[1463 - 1469]（一种用于为凝聚相系统生成定域占据轨道的非迭代实空间程序）扩展到使用非正交原子轨道（AO）基组描述的系统中的局域分子轨道（LMO）构建。具体而言，我们引入了SCDM的三种不同理论和算法变体（称为SCDM - M、SCDM - L和SCDM - G），它们可与标准量子化学代码库中使用的AO基组结合使用。SCDM - M和SCDM - L变体基于密度矩阵的Mulliken和Löwdin表示的枢轴QR分解，分别相当于选择一组条件良好的投影原子轨道（PAO）和投影（对称）正交化原子轨道作为原LMO，这些原LMO可以正交化以精确跨越占据空间。SCDM - G变体基于波函数的实空间（网格）表示，因此在选择一组条件良好的原LMO时具有考虑大量网格点（或δ函数）的额外灵活性。对不同大小、维度和饱和度水平的分子系统进行的详细比较分析表明，这三种非迭代/直接SCDM变体生成的LMO是稳健的，其轨道局域性与通过迭代的Boys或Pipek - Mezey（PM）局域化方案生成的LMO相当，并且对任何单个轨道局域性度量完全不敏感。尽管所有三种SCDM变体都基于密度矩阵，但我们发现SCDM - M、SCDM - L和SCDM - G生成的LMO的性质可能有显著差异。在这方面，只有基于网格的SCDM - G程序（如PM）生成的LMO在定性上保留σ - π对称性（在诸如s - 反式烯烃等系统中），并且与化学（如Lewis结构）直觉良好吻合。虽然直接单独使用SCDM生成的LMO对于大多数化学应用应该足够，但我们的研究结果还表明，将这些轨道用作无偏且经济高效的（初始）猜测也有可能改善迭代轨道局域化方案的收敛性，特别是对于大规模和/或病态分子系统。
