Section Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands.
J Chem Phys. 2018 Aug 7;149(5):054105. doi: 10.1063/1.5026951.
The relation of Kohn-Sham (KS) orbital energies to ionization energies and electron affinities is different in molecules and solids. In molecules, the local density approximation (LDA) and generalized gradient approximations (GGA) approximate the exact ionization energy (I) and affinity (A) rather well with self-consistently calculated (total energy based) I and A, respectively. The highest occupied molecular orbital (HOMO) energy and lowest unoccupied molecular orbital (LUMO) energy, however, differ significantly (by typically 4-6 eV) from these quantities, ϵ(mol)>-I(mol)≈-I(mol), ϵ(mol)<-A(mol)≈-A(mol). In solids, these relations are very different, due to two effects. The (almost) infinite extent of a solid makes the difference of orbital energies and (L)DFA calculated ionization energy and affinity disappear: in the solid state limit, ϵ(solid)=-I(solid) and ϵ(solid)=-A(solid). Slater's relation ∂E/∂n = ϵ for local density functional approximations (LDFAs) [and Hartree-Fock (HF) and hybrids] is useful to prove these relations. The equality of LDFA orbital energies and LDFA calculated -I and -A in solids does not mean that they are good approximations to the exact quantities. The LDFA total energies of the ions with a delocalized charge are too low, hence I(solid) < I and A(solid) > A, due to the local-approximation error, also denoted delocalization error, of LDFAs in extended systems. These errors combine to make the LDFA orbital energy band gap considerably smaller than the exact fundamental gap, ϵ(solid)-ϵ(solid)=I(solid)-A(solid)<I-A (the LDFA band gap problem). These results for density functional approximations are compared to exact KS and to HF and hybrids. For the exact KS HOMO energy, one has ϵ=-I. The exact KS LUMO energy does not approximate the experimental -A (neither in molecules nor in solids), but is considerably below, which is the main reason for the exact KS HOMO-LUMO energy gap being considerably below the fundamental gap I - A (the exact KS band gap problem).
Kohn-Sham（KS）轨道能与电离能和电子亲和能的关系在分子和固体中是不同的。在分子中，局域密度近似（LDA）和广义梯度近似（GGA）分别通过自洽计算的（基于总能量的）I 和 A 很好地近似了精确的电离能（I）和亲和能（A）。然而，最高占据分子轨道（HOMO）能量和最低未占据分子轨道（LUMO）能量与这些量有显著差异（典型地相差 4-6 eV），即 ϵ(mol)>-I(mol)≈-I(mol)，ϵ(mol)<-A(mol)≈-A(mol)。在固体中，由于两个效应，这些关系非常不同。固体的（几乎）无限延伸使得轨道能的差异和（L）DFA 计算的电离能和亲和能消失：在固体状态极限下，ϵ(solid)=-I(solid) 和 ϵ(solid)=-A(solid)。对于局域密度泛函近似（LDFA）[和 Hartree-Fock（HF）和杂化]，Slater 关系 ∂E/∂n = ϵ 可用于证明这些关系。LDFA 轨道能与固体中 LDFA 计算的 -I 和 -A 相等并不意味着它们是精确量的良好近似。由于 LDFA 在扩展系统中的局域近似误差（也称为离域误差），具有离域电荷的离子的 LDFA 总能量太低，因此 I(solid) < I 和 A(solid) > A。这些误差结合起来使得 LDFA 轨道能带隙明显小于精确的基本带隙，ϵ(solid)-ϵ(solid)=I(solid)-A(solid)<I-A（LDFA 能带隙问题）。这些密度泛函近似的结果与精确的 KS 和 HF 以及杂化进行了比较。对于精确的 KS HOMO 能量，有 ϵ=-I。精确的 KS LUMO 能量不近似于实验上的 -A（无论是在分子中还是在固体中），而是远低于它，这是精确的 KS HOMO-LUMO 能隙明显低于基本隙 I - A（精确的 KS 能带隙问题）的主要原因。
