Chowdhury Shah Tanvir Ur Rahman, Perdew John P
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA.
J Chem Phys. 2021 Dec 21;155(23):234110. doi: 10.1063/5.0072020.
The atomization energies of molecules from first-principles density functional approximations improve from the local spin-density approximation to the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) to the strongly constrained and appropriately normed (SCAN) meta-GGA, and their sensitivities to non-spherical components of the density increase in the same order. Thus, these functional advances increase density sensitivity and imitate the exact constrained search over correlated wavefunctions better than that over ensembles. The diatomic molecules studied here, singlet C and F plus triplet B and O, have cylindrically symmetric densities. Because the densities of the corresponding atoms are non-spherical, the approximate Kohn-Sham potentials for the atoms have a lower symmetry than that of the external (nuclear) potential so that the non-interacting wavefunctions are not eigenstates of the square of total orbital angular momentum, breaking a symmetry that yields a feature of the exact ground-state density. That spatial symmetry can be preserved by a non-self-consistent approach in which a self-consistent equilibrium-ensemble calculation is followed by integer re-occupation of the Kohn-Sham orbitals as the first of several steps. The symmetry-preserving approach is different from symmetry restoration based on projection. First-step space- (and space-spin-) symmetry preservation in atoms is shown to have a small effect on the atomization energies of molecules, quantifying earlier observations by Fertig and Kohn. Thus, the standard Kohn-Sham way of calculating atomization energies, with self-consistent symmetry breaking to minimize the energy, is justified at least for the common cases where the molecules cannot break symmetry. Unless symmetry breaking is allowed in the molecule, SCAN strongly underestimates the atomization energy of strongly correlated singlet C.
从第一性原理密度泛函近似得到的分子原子化能，从局域自旋密度近似到佩德韦-伯克-恩泽霍夫（PBE）广义梯度近似（GGA）再到强约束且适当归一化（SCAN）元GGA不断提高，并且它们对密度非球对称分量的敏感度也按相同顺序增加。因此，这些泛函的改进提高了密度敏感度，并且比系综更好地模拟了对相关波函数的精确约束搜索。这里研究的双原子分子，单重态C和F以及三重态B和O，具有圆柱对称密度。由于相应原子的密度是非球对称的，原子的近似科恩-沙姆势的对称性低于外部（核）势场，使得非相互作用波函数不是总轨道角动量平方的本征态，从而打破了产生精确基态密度特征的一种对称性。这种空间对称性可以通过一种非自洽方法来保持，即在几个步骤中的第一步，在自洽平衡系综计算之后对科恩-沙姆轨道进行整数重新占据。这种保持对称性的方法不同于基于投影的对称性恢复。原子中第一步的空间（以及空间自旋）对称性保持对分子的原子化能影响较小，这量化了费蒂格和科恩早期的观察结果。因此，至少对于分子不能打破对称性的常见情况，通过自洽对称性破缺来最小化能量的标准科恩-沙姆计算原子化能的方法是合理且有效的。除非分子允许对称性破缺，否则SCAN会严重低估强相关单重态C的原子化能。
