Electron Density Errors and Density-Driven Exchange-Correlation Energy Errors in Approximate Density Functional Calculations

Since its formal introduction, density
functional theory has achieved
many successes in the fields of molecular and solid-state chemistry.
According to its central theorems, the ground state of a many-electron
system is fully described by its electron density, and the exact functional
minimizes the energy at the exact electron density. For many years
of density functional development, it was assumed that the improvements
in the energy are accompanied by the improvements in the density,
and the approximations approach the exact functional. In a recent
analysis (Medvedev et al. Science 2017, 355, 49−52.), it has been pointed
out for 14 first row (Be–Ne) atoms and cations with 2, 4, or
10 electrons that the nowadays popular flexible but physically less
rigorous approximate density functionals may provide large errors
in the calculated electron densities despite the accurate energies.
Although far-reaching conclusions have been drawn in this work, the
methodology used by the authors may need improvements. Most importantly,
their benchmark set was biased toward small atomic cations with compressed,
high electron densities. In our paper, we construct a molecular test
set with chemically relevant densities and analyze the performance
of several density functional approximations including the less-investigated
double hybrids. We apply an intensive error measure for the density,
its gradient, and its Laplacian and examine how the errors in the
density propagate into the semilocal exchange-correlation energy.
While we have confirmed the broad conclusions of Medvedev et al.,
our different way of analyzing the data has led to conclusions that
differ in detail. Finally, seeking for a rationale behind the global
hybrid or double hybrid methods from the density’s point of
view, we also analyze the role of the exact exchange and second-order
perturbative correlation mixing in PBE-based global hybrid and double
hybrid functional forms.

自正式问世以来，密度泛函理论（density functional theory）已在分子化学与固态化学领域取得诸多重要成果。根据其核心定理，多电子体系的基态完全由其电子密度所描述，且精确泛函可在对应精确电子密度下使体系能量取极小值。在密度泛函的长期发展历程中，学界曾长期假定：能量计算精度的提升会伴随电子密度计算精度的优化，且近似泛函将不断趋近于精确泛函。2017年，Medvedev等人于《Science》期刊发表的一项分析研究（Medvedev et al. Science 2017, 355, 49−52.）中指出：针对14种第一周期（铍至氖）原子及含2、4或10个电子的阳离子，当下流行的柔性但物理严谨性较弱的近似密度泛函，即便在能量计算结果较为精准的情况下，也可能在电子密度的计算中产生显著误差。尽管该工作得出了影响深远的结论，但作者所采用的方法论仍存在改进空间。其中最为关键的一点是，其基准数据集存在偏向性，仅涵盖了具有压缩高电子密度的小型原子阳离子。在本文中，我们构建了一套包含化学相关电子密度的分子测试集，并对多种密度泛函近似方法的性能展开系统性分析，其中包括研究相对较少的双杂化泛函。我们针对电子密度、其梯度及拉普拉斯量（Laplacian）设计了精细化的误差度量方案，并探究了电子密度中的误差是如何传播至半局域交换关联能中的。我们虽验证了Medvedev等人结论的普适性，但由于采用了差异化的数据分析思路，我们的结论在细节层面存在差异。最后，为从电子密度的视角阐释全局杂化泛函或双杂化泛函方法的内在原理，我们还分析了精确交换与二阶微扰相关混合在基于PBE的全局杂化及双杂化泛函形式中所发挥的作用。