Origin of static and dynamic steps in exact Kohn-Sham potentials

Abstract

Knowledge of exact properties of the exchange-correlation (xc) functional is important for improving the approximations made within density functional theory. Features such as steps in the exact xc potential are known to be necessary for yielding accurate densities, yet little is understood regarding their shape, magnitude and location. We use systems of a few electrons, where the exact electron density is known, to demonstrate general properties of steps. We find that steps occur at points in the electron density where there is a change in the 'local effective ionization energy' of the electrons. We provide practical arguments, based on the electron density, for determining the position, shape and height of steps for ground-state systems, and extend the concepts to time-dependent systems. These arguments are intended to inform the development of approximate functionals, such as the mixed localization potential (MLP), which already demonstrate their capability to produce steps in the Kohn-Sham potential.PACS numbers: 31.15. 71.15.Mb, 31.15.A,

Orbital nodal surfaces: Topological challenges for density functionals

“…To achieve the cancellation, the asymptotic limits of t/s 2 and u/s 3 have to be equal, as they are in the spherical symmetric case and exactly on a pure nodal surface. However, for a fixed distance to the nodal surface z > 0 the limits of these semilocal ratios differ slightly as r → ∞, as a comparison of the second-order terms in the asymptotic relations (8) and (9) readily demonstrates. Consequently, the leading-order terms remain, i.e.,…”

“…This success is based on the favorable ratio of accuracy to computational cost that DFT offers, especially with semilocal approximations for the exchange-correlation (xc) energy E xc [n(r)]. However, while the low computational cost of semilocal functionals has very much contributed to the success of DFT because it enables access to large systems of practical relevance, the functional derivatives of typical semilocal functionals, i.e., their corresponding xc potentials, miss important features of the exact xc potential, in particular discontinuities [3,4] and step structures [5][6][7][8][9] that are relevant, e.g., in charge-transfer situations [10][11][12] and ionization processes [5,[13][14][15][16]. Many attempts have been made to incorporate some of the missing features into semilocal DFT [17][18][19][20][21][22][23][24][25][26][27].…”

Exact exchange-correlation potential of effectively interacting Kohn-Sham systems

“…In the past decades, various exchange-correlation functionals have been developed to realize accurate description of the electronic ground state such as local density approximations (LDA) [3,4], generalized gradient approximations (GGA) [5,6], meta-GGA [7][8][9], and hybrid functions [10][11][12]. Furthermore, detailed studies clarified several exact properties of the exact exchange-correlation functional and potential such as asymptotic behavior of the potential in Coulombic systems [5,13,14], and spiky features in the molecular dissociation [15][16][17][18][19]. However, the systematic improvement of the exchange-correlation functionals and potentials is still a non-trivial task due to highly nonlinear and nonlocal natures of the density functional [20,21].…”

Functional derivative of the zero-point-energy functional from the strong-interaction limit of density-functional theory

“…Besides the XC functional itself, another quantity that plays an important role in KS DFT is its functional derivative with respect to the density, which determines the XC potential entering in the KS equations. The exact (or very accurate) XC potential has been studied for small systems in several works, using various reverseengineering procedures [23][24][25]: these works have shown that for strongly-correlated systems the XC potential must display very peculiar features, such as "peaks" and "steps" [26][27][28]. While the functional derivative of the SCE leading term has been evaluated and used as an approximation for the XC potential in the self-consistent KS equations in various works [11][12][13]29], the potential associated to the next leading term has never been investigated in an exact manner (only very recently, a semi-local approximation for the ZPE has been used to look at KS potentials coming from functionals that interpolate between the weak-and strong-coupling limits of the XC functional [30]).…”

“…For this reason, our investigation starts from a simple, yet non trivial, case: two electrons confined in one dimension (1D). Similar 1D models have been widely used to investigate features in exact KS DFT, proving to provide a good qualitative description of the relevant features of their 3D counterparts [27,[31][32][33][34][35].…”