太大，太小，还是刚刚好？密度泛函理论预测小化学体系电子密度空间范围的基准评估
Too big, too small or just right? A benchmark assessment of density functional theory for predicting the spatial extent of the electron density of small chemical systems
摘要：
多极矩是能量对电场强度的空间导数的一阶响应。因此，分子多极矩的密度泛函理论（DFT）预测的质量表征了模拟电子密度本身的误差，以及描述分子与外部电场相互作用的性能。然而，只有最低的非零矩是几何不变的，使得高阶矩依赖于原点。因此，代替使用$3 \times 3$四极矩矩阵，我们利用电子密度的二阶累积量（或空间方差）的几何不变的$3 \times 3$矩阵作为感兴趣的量（由$\mathcal{K}$表示）。${\mathcal{K}}$的主成分是沿每个轴的电子密度的空间范围的平方。在完全基组（CBS）极限下，建立了100个小分子的${\mathcal{K}}$主成分的基准数据集，得到了213个独立的${\mathcal{K}}$主成分. 47流行的和最近的密度泛函的性能进行评估，对这个Var213数据集。几个泛函，特别是双杂交，以及SCAN和SCAN0产生可靠的第二累积量，虽然一些现代的，经验参数化的泛函产生更令人失望的性能。特别是H和Be原子对几乎所有方法都具有挑战性，这表明未来的功能开发可以受益于将其密度信息纳入训练或测试方案。
英文摘要：
Multipole moments are the first order responses of the energy to spatial derivatives of the electric field strength. The quality of density functional theory (DFT) prediction of molecular multipole moments thus characterizes errors in modeling the electron density itself, as well as the performance in describing molecules interacting with external electric fields. However, only the lowest non-zero moment is translationally invariant, making the higher order moments origin-dependent. Therefore, instead of using the $3 \times 3$ quadrupole moment matrix, we utilize the translationally invariant $3 \times 3$ matrix of second cumulants (or spatial variances) of the electron density as the quantity of interest (denoted by $\mathcal{K}$). The principal components of ${\mathcal{K}}$ are the square of the spatial extent of the electron density along each axis. A benchmark dataset of the prinicpal components of ${\mathcal{K}}$ for 100 small molecules at the coupled cluster singles and doubles with perturbative triples (CCSD(T)) at the complete basis set (CBS) limit is developed, resulting in 213 independent ${\mathcal{K}}$ components. The performance of 47 popular and recent density functionals is assessed against this Var213 dataset. Several functionals, especially double hybrids, and also SCAN and SCAN0 yield reliable second cumulants, although some modern, empirically parameterized functionals yield more disappointing performance. The H and Be atoms in particular are challenging for nearly all methods, indicating that future functional development could benefit from inclusion of their density information in training or testing protocols.
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