Nuclear Magnetic Shielding Constants with the Polarizable Density Embedding Model

We extend the polarizable density embedding (PDE) model to support the calculation of nuclear magnetic resonance (NMR) shielding constants using gauge-including atomic orbitals (GIAOs) within a density functional theory (DFT) framework. The PDE model divides the total system into fragments, describing some by quantum mechanics (QM) and the others through an embedding model. The PDE model uses anisotropic polarizabilities, inter-fragment two-electron Coulomb integrals, and a non-local repulsion operator to emulate the QM effects. The terms involving Coulomb integrals are straightforwardly extended with GIAOs. In contrast, we consider two approaches to handle the gauge dependency of the non-local operator, employing either simple symmetrization or a gauge transformation. We find the latter approach to be most stable with respect to increasing the basis set size of the QM region. We examine the accuracy of the PDE model for calculating NMR shielding constants on several solutes in a water solution. The performance is compared with the classical polarizable embedding (PE) model in addition to supermolecular reference calculations. Based on these systems, we address the basis set convergence characteristics and the QM region size requirements. Furthermore, we investigate the performance of the PDE model for a system with significant electron spill-out. In many cases, we find that the PDE model outperforms the PE model, especially regarding the accuracy of nuclear shielding constants when using small QM region sizes and in systems with significant electron spill-out.我们将可极化密度嵌入（polarizable density embedding, PDE）模型拓展至可在密度泛函理论（density functional theory, DFT）框架下，利用含规范原子轨道（gauge-including atomic orbitals, GIAOs）计算核磁共振（nuclear magnetic resonance, NMR）屏蔽常数。该模型将整体体系划分为若干片段，其中部分片段采用量子力学（quantum mechanics, QM）描述，其余片段则通过嵌入模型表征。PDE模型借助各向异性极化率、片段间双电子库仑积分与非局域排斥算符来模拟量子力学效应。其中涉及库仑积分的项可直接通过GIAOs进行拓展；与之相对，我们针对非局域算符的规范依赖性设计了两种处理方案：分别采用简单对称化操作或是规范变换（gauge transformation）。我们发现，采用规范变换的方案在增大QM区域基组尺寸时稳定性更佳。我们针对水溶液中的多种溶质开展了NMR屏蔽常数计算，以检验PDE模型的计算精度。除超分子参考计算外，我们还将该模型的性能与经典可极化嵌入（polarizable embedding, PE）模型进行了对比。基于上述测试体系，我们分析了基组收敛特性与QM区域尺寸的需求规律。此外，我们还针对存在显著电子溢出（electron spill-out）的体系，探究了PDE模型的计算表现。在多数测试案例中，PDE模型的性能均优于PE模型，尤其在采用较小QM区域尺寸时，以及针对存在显著电子溢出的体系，其核屏蔽常数的计算精度更为突出。