Title: The Convexity Condition of Density-Functional Theory Show Chinese title

Title: 密度泛函理论的凸性条件

Abstract: It has long been postulated that within density-functional theory (DFT) the total energy of a finite electronic system is convex with respect to electron count, so that 2 E_v[N_0] <= E_v[N_0 - 1] + E_v[N_0 + 1]. Using the infinite-separation-limit technique, this article proves the convexity condition for any formulation of DFT that is (1) exact for all v-representable densities, (2) size-consistent, and (3) translationally invariant. An analogous result is also proven for one-body reduced density matrix functional theory. While there are known DFT formulations in which the ground state is not always accessible, indicating that convexity does not hold in such cases, this proof nonetheless confirms a stringent constraint on the exact exchange-correlation functional. We also provide sufficient conditions for convexity in approximate DFT, which could aid in the development of density-functional approximations. This result lifts a standing assumption in the proof of the piecewise linearity condition with respect to electron count, which has proven central to understanding the Kohn-Sham band-gap and the exchange-correlation derivative discontinuity of DFT. Show Chinese abstract

Abstract: 长期以来，人们一直假设在密度泛函理论（DFT）中，有限电子系统的总能量相对于电子数是凸的，因此 2 E_v[N_0] <= E_v[N_0 - 1] + E_v[N_0 + 1]。 使用无限分离极限技术，本文证明了对于任何满足以下条件的DFT表述，凸性条件成立：(1) 对所有v可表示密度都是精确的，(2) 尺寸一致，(3) 平移不变。 还证明了一个类似的结果适用于单体约化密度矩阵泛函理论。 虽然已知某些DFT表述中基态并不总是可访问的，这表明在这些情况下凸性不成立，但此证明仍然确认了对精确交换-相关泛函的严格约束。 我们还提供了近似DFT中凸性的充分条件，这可能有助于密度泛函近似的开发。 这一结果消除了在关于电子数的分段线性条件证明中的一个既定假设，该假设已被证明对于理解Kohn-Sham带隙和DFT的交换-相关导数不连续性至关重要。