Jin Ye, Su Neil Qiang, Chen Zehua, Yang Weitao
Department of Chemistry, Duke University, Durham, NC 27708, USA.
Faraday Discuss. 2020 Dec 4;224(0):9-26. doi: 10.1039/d0fd00102c.
A major challenge in density functional theory (DFT) is the development of density functional approximations (DFAs) to overcome errors in existing DFAs, leading to more complex functionals. For such functionals, we consider roles of the noninteracting reference systems. The electron density of the Kohn-Sham (KS) reference with a local potential has been traditionally defined as being equal to the electron density of the physical system. This key idea has been applied in two ways: the inverse calculation of such a local KS potential for the reference from a given density and the direct calculation of density and energy based on given DFAs. By construction, the inverse calculation can yield a KS reference with the density equal to the input density of the physical system. In application of DFT, however, it is the direct calculation of density and energy from a DFA that plays a central role. For direct calculations, we find that the self-consistent density of the KS reference defined by the optimized effective potential (OEP), is not the density of the physical system, when the DFA is dependent on the external potential. This inequality holds also for the density of generalized KS (GKS) or generalized OEP reference, which allows a nonlocal potential, when the DFA is dependent on the external potential. Instead, the density of the physical system, consistent with a given DFA, is given by the linear response of the total energy with respect to the variation of the external potential. This is a paradigm shift in DFT on the use of noninteracting references: the noninteracting KS or GKS references represent the explicit computational variables for energy minimization, but not the density of the physical system for external potential-dependent DFAs. We develop the expressions for the electron density so defined through the linear response for general DFAs, demonstrate the results for orbital functionals and for many-body perturbation theory within the second-order and the random-phase approximation, and explore the connections to developments in DFT.
密度泛函理论（DFT）中的一个主要挑战是开发密度泛函近似（DFA），以克服现有DFA中的误差，从而产生更复杂的泛函。对于此类泛函，我们考虑非相互作用参考系统的作用。传统上，具有局部势的Kohn-Sham（KS）参考的电子密度被定义为等于物理系统的电子密度。这一关键思想已通过两种方式应用：从给定密度对该参考的此类局部KS势进行逆计算，以及基于给定DFA直接计算密度和能量。通过构造，逆计算可以得到一个KS参考，其密度等于物理系统的输入密度。然而，在DFT的应用中，从DFA直接计算密度和能量起着核心作用。对于直接计算，我们发现，当DFA依赖于外部势时，由优化有效势（OEP）定义的KS参考的自洽密度不是物理系统的密度。当DFA依赖于外部势时，对于允许非局部势的广义KS（GKS）或广义OEP参考的密度，这种不等式也成立。相反，与给定DFA一致的物理系统的密度由总能量相对于外部势变化的线性响应给出。这是DFT在使用非相互作用参考方面的一个范式转变：非相互作用的KS或GKS参考代表能量最小化的显式计算变量，但对于依赖于外部势的DFA，不是物理系统的密度。我们通过对一般DFA的线性响应来推导如此定义的电子密度的表达式，展示轨道泛函以及二阶和随机相位近似内的多体微扰理论的结果，并探索与DFT发展的联系。
