密度泛函理论Density functional theory

1964 年霍恩伯格-科恩定理证明：基态电子密度唯一决定外势，因而决定体系所有性质，并且真实基态密度使能量泛函取最小值。1965 年科恩-沈方法引入一组无相互作用的虚拟电子，其密度与真实体系相同，把多体问题转化为求解单电子科恩-沈方程。所有未知的多体效应都归入交换-相关泛函 E_xc[ρ]，它的精确形式未知，需要近似：局域密度近似（LDA）、广义梯度近似（GGA，如 PBE）、杂化泛函（如 B3LYP）等，构成所谓“雅各布天梯”。

In 1964 the Hohenberg–Kohn theorems showed that the ground-state density uniquely determines the external potential and hence all properties, and that the true density minimises an energy functional. In 1965 the Kohn–Sham method introduced fictitious non-interacting electrons with the same density as the real system, turning the many-body problem into one-electron Kohn–Sham equations. All unknown many-body effects go into the exchange–correlation functional E_xc[ρ], whose exact form is unknown and must be approximated: the local density approximation (LDA), generalised gradient approximations (GGA, such as PBE), hybrid functionals (such as B3LYP) and others, forming 'Jacob's ladder'.

1927 年托马斯和费米提出用电子密度描述原子的模型，但精度很差。1964 年霍恩伯格和科恩、1965 年科恩和沈吕九奠定了现代 DFT。1980 年代末贝克和李-杨-帕尔发展的泛函（组合成 B3LYP，1993 年）使 DFT 在化学中迅速普及。科恩因发展密度泛函理论获 1998 年诺贝尔化学奖。DFT 论文是有史以来引用最多的科学论文之一。

Thomas and Fermi proposed describing atoms by electron density in 1927, but their model was crude. Hohenberg and Kohn (1964) and Kohn and Lu Jeu Sham (1965) founded modern DFT. Functionals developed by Becke and by Lee, Yang and Parr in the late 1980s, combined as B3LYP in 1993, made DFT spread rapidly through chemistry. Kohn received the 1998 Nobel Prize in Chemistry for developing density functional theory, and DFT papers are among the most cited in all of science.

DFT 用于计算分子结构、反应能垒、催化机理、能带结构、材料的力学和磁学性质，可处理数百到上千个原子。局限包括：常用泛函对范德华色散作用描述不足（需加色散校正，如 DFT-D3）、自相互作用误差导致带隙低估、对强关联体系和激发态（需含时 DFT）处理困难。

DFT is used for molecular structures, reaction barriers, catalytic mechanisms, band structures and mechanical and magnetic properties of materials, handling hundreds to thousands of atoms. Limitations include poor treatment of van der Waals dispersion by common functionals (needing corrections such as DFT-D3), self-interaction error that underestimates band gaps, and difficulty with strongly correlated systems and excited states (which require time-dependent DFT).