发表机构
Tulane University; Cambridge University（杜兰大学; 剑桥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文论证电子交换能存在精确的二阶密度梯度展开，其梯度系数为 Sham 系数的 10/7，并讨论长程反常及对密度泛函近似的意义。
AI 中文摘要
基态密度泛函理论中一个古老但反复出现的问题，与密度泛函近似的构建相关，是关于交换能是否存在一个对空间缓慢变化的密度精确的二阶梯度展开。在此，我们将论证确实存在这样的展开，其梯度系数为 Sham 系数的 10/7，正如理论/数学推导以及对模型缓慢变化金属密度的数值计算所建议的那样，尽管一些标准推导并未完全捕捉到正确的梯度系数。我们还讨论了围绕电子的交换和关联空穴梯度展开中的长程反常。合理的可能是，正如先前和近期所建议的，标准推导对于交换和关联合并的二阶梯度展开系数仍然是正确的，在这种情况下，关联的正确梯度系数为 $C_{\mathrm{c}}$ = $C_{\mathrm{xc}}$ - $C_{\mathrm{x}}$。我们提出了对 $C_{\mathrm{xc}}$ 和 $C_{\mathrm{x}}$ 的可能数值评估方法，这些方法不受长程反常的影响，并讨论了它们对密度泛函近似的意义。Axel Becke 1988 年的广义梯度近似（B88 GGA）用于交换能，将密度泛函理论引入了化学领域，其交换梯度系数在 GGA 水平上对原子和分子是最优的，但对固体则过大。Meta-GGA 似乎对分子和固体都同样有效，但可能尚未完善。文章最后以第一作者与 Axel Becke 讨论的一些回忆作为结尾。
英文摘要
An old but recurring question in ground-state density functional theory, relevant to the construction of density functional approximations, is the existence of a second-order gradient expansion for the exchange energy that is exact for densities that vary slowly over space. Here we will argue that there is such an expansion, with a gradient coefficient 10/7 of Sham's, as suggested by theoretical/mathematical derivations and by numerical calculations on model slowly-varying metallic densities, although some standard derivations do not quite capture the correct gradient coefficient. We also discuss long-range anomalies in the gradient expansions for the exchange and correlation holes around an electron. It is plausible that, as suggested earlier and recently, standard derivations are still correct for the coefficient of the second-order gradient expansion for exchange and correlation together, in which case the correct gradient coefficient for correlation is $C_{\mathrm{c}}$ = $C_{\mathrm{xc}}$ - $C_{\mathrm{x}}$. We propose possible numerical evaluations of $C_{\mathrm{xc}}$ and $C_{\mathrm{x}}$ that are free from long-range anomalies, and discuss implications for density functional approximations. Axel Becke's 1988 generalized gradient approximation (the B88 GGA) for the exchange energy brought density functional theory into chemistry, with a gradient coefficient for exchange that is optimal at the GGA level for atoms and molecules but too large for solids. Meta-GGAs seem to work as well for molecules as for solids, but may not be perfected yet. The article concludes with a few memories that the first author has of discussions with Axel Becke.
