Perdew John P, Ruzsinszky Adrienn, Sun Jianwei, Burke Kieron
Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA.
Department of Chemistry and Department of Physics, University of California, Irvine, California 92697, USA.
J Chem Phys. 2014 May 14;140(18):18A533. doi: 10.1063/1.4870763.
Approximations to the exact density functional for the exchange-correlation energy of a many-electron ground state can be constructed by satisfying constraints that are universal, i.e., valid for all electron densities. Gedanken densities are designed for the purpose of this construction, but need not be realistic. The uniform electron gas is an old gedanken density. Here, we propose a spherical two-electron gedanken density in which the dimensionless density gradient can be an arbitrary positive constant wherever the density is non-zero. The Lieb-Oxford lower bound on the exchange energy can be satisfied within a generalized gradient approximation (GGA) by bounding its enhancement factor or simplest GGA exchange-energy density. This enhancement-factor bound is well known to be sufficient, but our gedanken density shows that it is also necessary. The conventional exact exchange-energy density satisfies no such local bound, but energy densities are not unique, and the simplest GGA exchange-energy density is not an approximation to it. We further derive a strongly and optimally tightened bound on the exchange enhancement factor of a two-electron density, which is satisfied by the local density approximation but is violated by all published GGA's or meta-GGA's. Finally, some consequences of the non-uniform density-scaling behavior for the asymptotics of the exchange enhancement factor of a GGA or meta-GGA are given.
通过满足通用的约束条件（即对所有电子密度均有效），可以构建多电子基态交换关联能精确密度泛函的近似。为构建此泛函而设计的假想密度不一定需要符合实际情况。均匀电子气就是一种古老的假想密度。在此，我们提出一种球形双电子假想密度，其中无量纲密度梯度在密度非零处可为任意正的常数。通过限制其增强因子或最简单的广义梯度近似（GGA）交换能密度，在广义梯度近似下可以满足交换能的Lieb - Oxford下界。众所周知，这种增强因子界限是充分的，但我们的假想密度表明它也是必要的。传统的精确交换能密度不满足这种局部界限，但能量密度并非唯一，且最简单的GGA交换能密度并非对其的近似。我们进一步推导了双电子密度交换增强因子的一个强且最优收紧的界限，局域密度近似满足该界限，但所有已发表的GGA或meta - GGA均违反此界限。最后，给出了非均匀密度标度行为对GGA或meta - GGA交换增强因子渐近性的一些影响。
