Kraisler Eli, Kronik Leeor
Department of Materials and Interfaces, Weizmann Institute of Science, Rehovoth 76100, Israel.
J Chem Phys. 2014 May 14;140(18):18A540. doi: 10.1063/1.4871462.
The fundamental gap is a central quantity in the electronic structure of matter. Unfortunately, the fundamental gap is not generally equal to the Kohn-Sham gap of density functional theory (DFT), even in principle. The two gaps differ precisely by the derivative discontinuity, namely, an abrupt change in slope of the exchange-correlation energy as a function of electron number, expected across an integer-electron point. Popular approximate functionals are thought to be devoid of a derivative discontinuity, strongly compromising their performance for prediction of spectroscopic properties. Here we show that, in fact, all exchange-correlation functionals possess a derivative discontinuity, which arises naturally from the application of ensemble considerations within DFT, without any empiricism. This derivative discontinuity can be expressed in closed form using only quantities obtained in the course of a standard DFT calculation of the neutral system. For small, finite systems, addition of this derivative discontinuity indeed results in a greatly improved prediction for the fundamental gap, even when based on the most simple approximate exchange-correlation density functional--the local density approximation (LDA). For solids, the same scheme is exact in principle, but when applied to LDA it results in a vanishing derivative discontinuity correction. This failure is shown to be directly related to the failure of LDA in predicting fundamental gaps from total energy differences in extended systems.
基本能隙是物质电子结构中的一个核心量。不幸的是，即使在原则上，基本能隙通常也不等于密度泛函理论（DFT）中的科恩 - 沈能隙。这两个能隙的差异恰好在于导数不连续性，即在整数电子点处，交换关联能随电子数变化的斜率会发生突然变化。人们认为流行的近似泛函没有导数不连续性，这严重损害了它们预测光谱性质的性能。在此我们表明，事实上，所有的交换关联泛函都具有导数不连续性，它自然地源于在DFT中应用系综考虑，而无需任何经验主义。这种导数不连续性可以仅用中性系统标准DFT计算过程中得到的量以封闭形式表示。对于小的有限体系，即使基于最简单的近似交换关联密度泛函——局域密度近似（LDA），加上这种导数不连续性确实能大大改进对基本能隙的预测。对于固体，同样的方案原则上是精确的，但应用于LDA时，它会导致导数不连续性修正消失。这种失败被证明与LDA在从扩展体系的总能差预测基本能隙时的失败直接相关。
