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.
翻译:许多电子地面状态的交换-关系能量的精确密度功能相近。 许多电子地面状态的交换-关系能量的精确密度功能,可以通过满足通用的限制,即所有电子密度都适用的限制来构建。 Gedanken 密度的设计是为了这一构造的目的,但并不现实。 统一的电子气体是一种古老的斜体密度。 在这里, 我们提议一种球形的双电子斜体密度, 无尺寸密度梯度在密度不为零的地方可以成为任意的正常数。 在通用的梯度近似(GGAA)范围内可以满足对交换能量的较低约束。 通过约束其增强系数或最简单的GGGA交换- 能量密度, 这个增强系数是众所周知的足够,但我们的斜体密度表明它也是必要的。 常规的精确交换-能量密度不具有这种约束性,但最简单的GGA交换-GA交换- 温度密度不是接近的。我们进一步从强烈和最优化的GA- GA 升级的精确度, 其最终的加速度是GA- check- gA 的精确度的精确度。