How Interatomic Steps in the Exact Kohn–Sham Potential Relate to Derivative Discontinuities of the Energy

Abstract

Accurate density functional calculations hinge on reliable approximations to the unknown exchange-correlation (xc) potential. The most popular approximations usually lack features of the exact xc potential that are important for an accurate prediction of the fundamental gap and the distribution of charge in complex systems. Two principal features in this regard are the spatially uniform shift in the potential, as the number of electrons infinitesimally surpasses an integer, and the spatial steps that form, for example, between the atoms of stretched molecules. Although both aforementioned concepts are well known, the exact relationship between them remained unclear. Here we establish this relationship via an analytical derivation. We support our result by numerically solving the many-electron Schrödinger equation to extract the exact Kohn-Sham potential and directly observe its features. Spatial steps in the exact xc potential of a full configuration-interaction (FCI) calculation of a molecule are presented in three dimensions.

Ultranonlocality and accurate band gaps from a meta-generalized gradient approximation

“…In molecular physics and chemistry, missing field-counteracting terms have serious, detrimental consequences for the calculation of polarizabilities, hyperpolarizabilities, and charge transfer [13][14][15]. These problems originate from several fundamental shortcomings that are closely interconnected: a missing derivative discontinuity [16][17][18], a lack of step structures in the exchange correlation (xc) potential [19][20][21][22], and the delocalization error [23,24]. Density-driven errors [25,26] characterize many of these problems.…”

Sum-rules of the response potential in the strongly-interacting limit of DFT

“…This piece of the xc potential has been shown to be critical for the correct description of virtual KS orbitals' levels, needed for the calculation of molecular excitation energies in TDDFT, [38,39] as well as for the proper description of electron localization in a dissociating heteronuclear molecule [19][20][21][22]30,37,40] and for the construction of the Levy-Zahariev potential. [41] Here we show that, in cases in which the SCE limit can be solved exactly (one-dimensional and spherically symmetric systems), its response potential satisfies a simple sum rule, see Eqs.…”

Efficient implementation of the superposition of atomic potentials initial guess for electronic structure calculations in Gaussian basis sets

“…Although the exact exchange operator is linear in the density matrix, the corresponding local scalar potential may behave discontinuously in the number of electrons. 58 Because of this, the approaches can be better contrasted within a density functional approximation: SAD yields a local ex-change potential V SAD x (r) ∝ − A n A (r) 1/3 (17) whereas SAP yields…”