Baerends E J
Section Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands.
Phys Chem Chem Phys. 2017 Jun 21;19(24):15639-15656. doi: 10.1039/c7cp02123b.
It is often stated that the Kohn-Sham occupied-unoccupied gap in both molecules and solids is "wrong". We argue that this is not a correct statement. The KS theory does not allow to interpret the exact KS HOMO-LUMO gap as the fundamental gap (difference (I - A) of electron affinity (A) and ionization energy (I), twice the chemical hardness), from which it indeed differs, strongly in molecules and moderately in solids. The exact Kohn-Sham HOMO-LUMO gap in molecules is much below the fundamental gap and very close to the much smaller optical gap (first excitation energy), and LDA/GGA yield very similar gaps. In solids the situation is different: the excitation energy to delocalized excited states and the fundamental gap (I - A) are very similar, not so disparate as in molecules. Again the Kohn-Sham and LDA/GGA band gaps do not represent (I - A) but are significantly smaller. However, the special properties of an extended system like a solid make it very easy to calculate the fundamental gap from the ground state (neutral system) band structure calculations entirely within a density functional framework. The correction Δ from the KS gap to the fundamental gap originates from the response part v of the exchange-correlation potential and can be calculated very simply using an approximation to v. This affords a calculation of the fundamental gap at the same level of accuracy as other properties of crystals at little extra cost beyond the ground state bandstructure calculation. The method is based on integer electron systems, fractional electron systems (an ensemble of N- and (N + 1)-electron systems) and the derivative discontinuity are not invoked.
人们常说分子和固体中的科恩 - 沈（Kohn - Sham）占据 - 未占据能隙是“错误的”。我们认为这种说法并不正确。科恩 - 沈理论不允许将精确的科恩 - 沈最高已占分子轨道 - 最低未占分子轨道（HOMO - LUMO）能隙解释为基本能隙（电子亲和能（A）与电离能（I）的差值（I - A），即化学硬度的两倍），实际上在分子中二者差异很大，在固体中差异适中。分子中精确的科恩 - 沈HOMO - LUMO能隙远低于基本能隙，且非常接近小得多的光学能隙（第一激发能），而局域密度近似（LDA）/广义梯度近似（GGA）给出的能隙非常相似。在固体中情况则不同：到离域激发态的激发能与基本能隙（I - A）非常相似，不像在分子中那样相差悬殊。同样，科恩 - 沈和LDA/GGA带隙并不代表（I - A），而是明显更小。然而，像固体这样的扩展系统的特殊性质使得在密度泛函框架内完全从基态（中性系统）能带结构计算中计算基本能隙变得非常容易。从科恩 - 沈能隙到基本能隙的修正Δ源自交换关联势的响应部分v，并且可以使用v的近似值非常简单地计算出来。这使得在几乎不增加除基态能带结构计算之外的额外成本的情况下，以与晶体其他性质相同的精度水平计算基本能隙成为可能。该方法基于整数电子系统，不涉及分数电子系统（N电子和（N + 1）电子系统的系综）和导数不连续性。
