Kvaal Simen, Ekström Ulf, Teale Andrew M, Helgaker Trygve
Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
J Chem Phys. 2014 May 14;140(18):18A518. doi: 10.1063/1.4867005.
The universal density functional F of density-functional theory is a complicated and ill-behaved function of the density-in particular, F is not differentiable, making many formal manipulations more complicated. While F has been well characterized in terms of convex analysis as forming a conjugate pair (E, F) with the ground-state energy E via the Hohenberg-Kohn and Lieb variation principles, F is nondifferentiable and subdifferentiable only on a small (but dense) subset of its domain. In this article, we apply a tool from convex analysis, Moreau-Yosida regularization, to construct, for any ε > 0, pairs of conjugate functionals ((ε)E, (ε)F) that converge to (E, F) pointwise everywhere as ε → 0(+), and such that (ε)F is (Fréchet) differentiable. For technical reasons, we limit our attention to molecular electronic systems in a finite but large box. It is noteworthy that no information is lost in the Moreau-Yosida regularization: the physical ground-state energy E(v) is exactly recoverable from the regularized ground-state energy (ε)E(v) in a simple way. All concepts and results pertaining to the original (E, F) pair have direct counterparts in results for ((ε)E, (ε)F). The Moreau-Yosida regularization therefore allows for an exact, differentiable formulation of density-functional theory. In particular, taking advantage of the differentiability of (ε)F, a rigorous formulation of Kohn-Sham theory is presented that does not suffer from the noninteracting representability problem in standard Kohn-Sham theory.
密度泛函理论中的通用密度泛函F是密度的一个复杂且性质不良的函数——特别是，F不可微，这使得许多形式上的操作更加复杂。虽然根据凸分析，F已被很好地表征为通过 Hohenberg-Kohn 和 Lieb 变分原理与基态能量E形成共轭对(E, F)，但F仅在其定义域的一个小（但稠密）子集上不可微且次可微。在本文中，我们应用凸分析中的一个工具——Moreau-Yosida 正则化，来构造对于任何ε>0的共轭泛函对((ε)E, (ε)F)，当ε→0(+)时，它们在各处逐点收敛到(E, F)，并且使得(ε)F是（Fréchet）可微的。出于技术原因，我们将注意力限制在有限但很大的盒子中的分子电子系统上。值得注意的是，在 Moreau-Yosida 正则化中没有信息丢失：物理基态能量E(v)可以通过一种简单的方式从正则化基态能量(ε)E(v)中精确恢复。所有与原始(E, F)对相关的概念和结果在((ε)E, (ε)F)的结果中都有直接对应物。因此，Moreau-Yosida 正则化允许对密度泛函理论进行精确的、可微的表述。特别是，利用(ε)F的可微性，提出了一种严格的 Kohn-Sham 理论表述，该表述不存在标准 Kohn-Sham 理论中的非相互作用可表示性问题。
