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,

By propagating the many-body Schrödinger equation, we determine the exact time-dependent Kohn-Sham potential for a system of strongly correlated electrons which undergo field-induced tunneling. Numerous features are entirely absent from the approximations commonly used in time-dependent density-functional theory. The self-interaction correction is strong and time dependent, owing to electron localization, and prominent dynamic spatial potential steps arise from minima in the charge density, as modified by the Coulomb interaction experienced by the partially tunneled electron.

We calculate the exact Kohn-Sham potential that describes, within time-dependent densityfunctional theory, the propagation of an electron quasiparticle wavepacket of non-zero crystal momentum added to a ground-state model semiconductor. The potential is observed to have a highly nonlocal functional dependence on the charge density, in both space and time, giving rise to features entirely lacking in local or adiabatic approximations. The dependence of the non-equilibrium part of the Kohn-Sham electric field on the local current and charge density is identified as a key element of the correct Kohn-Sham functional.PACS numbers: 71.15. Mb, 85.35.Be Density functional theory (DFT) in the local density approximation (LDA) has proven highly effective in the ground-state calculation of electronic structures and energies of a diverse range of systems [1]. It was natural therefore to extend the approximation to the adiabatic LDA (ALDA) within a time-dependent densityfunctional-theory (TDDFT) description of excited matter. However, while the ALDA is effective in certain classes of excited systems [2], it is shown to break down in a large number of physical scenarios of universal interest, for instance the absorption spectra of semiconductors and insulators [3], the polarization of systems subject to an external electric field [4,5], and the prediction of correlation energies [6] and double excitation spectra [7].One important class of non-equilibrium problems for which accurate TDDFT functionals are needed is the quantum transport of charge through matter. Historically, quantum treatments of electronic transport reduce the current response for an applied potential across a device to a transmission problem, and employ noninteracting electrons. More accurate calculations make use of effective potential fields including Hartree, exchange and approximate exchange-correlation terms. A natural step, then, is a Kohn-Sham (KS) TDDFT approach in which the exact potential is a functional of the initial state and the time-dependent density. The need to better understand Kohn-Sham functionals, including those depending explicitly on the current density, beyond the adiabatic and exchange-only regimes in order to describe quantum transport has been previously noted [8]. Maitra et al. [9] have also highlighted the importance of including the initial-state dependence, absent in adiabatic approximations, of the time-dependent Kohn-Sham potential in the desired functional.To advise the construction of appropriate functionals, we believe it is important to study the exact KohnSham potential itself. The reverse-engineering of groundstate KS potentials via direct numerical optimization of characteristic parameters has provided insight into the nature of nonlocal functional dependence of, for instance, polarized materials [4,5], while in the timedependent regime there have been calculations of exact time-dependent exchange-correlation potentials and kernels for non-equilibrium systems such as doubly-excited helium [10] and small Hubbard cha...

We introduce a new functional for simulating ground-state and time-dependent electronic systems within density-functional theory. The functional combines an expression for the exact Kohn-Sham (KS) potential in the limit of complete electron localization with a measure of the actual localization. We find accurate self-consistent charge densities, even for systems where the exact exchangecorrelation potential exhibits non-local dependence on the density, such as potential steps. We compare our results to the exact KS potential for each system. The self-interaction correction is accurately described, avoiding the need for orbital-dependent potentials.PACS numbers: 71.15. Mb, 73.63.Nm, Density-functional theory (DFT) [1] is the most widely used tool for the simulation of many-electron systems in numerous fields of physics, chemistry and materials science. Its success hinges on approximations [2,3] to the exchange-correlation (xc) part of the Kohn-Sham (KS) functional, which perform well across a range of groundstate systems. However, these approximations become much less secure in the presence of strong correlation [4,5] and/or current flow [6][7][8][9][10]. Particular attention has been given to improving the time-dependent xc potential, used within time-dependent DFT (TDDFT) [11], where the use of adiabatic functionals of the electron density ignores the role of currents and memory effects.In this Letter we demonstrate that electron localization, driven by the Coulomb interaction and the Pauli principle, can form a powerful ingredient in approximations for the KS potential. The electron localization function (ELF), L(x), as in Ref. 12, provides a useful indicator of localization: L = 1 is complete localization, i.e. the chance of finding one electron in the vicinity of another is zero. L(x) ranges from 0 to 1, and a homogeneous electron gas (HEG) has L = 0.5.Our starting point is the KS potential of Refs. 13 and 14, originally derived for a system of two spinful electrons in their spin-zero ground state. We observe that the logic applies exactly to any one-electron system, and, indeed, in a general system, to all regions of space where the electron density is dominated by any one Kohn-Sham orbital [15]. For such a region the KS equations may be approximated as [16] − 1 2 2 + V KS √ n = ε k √ n for the dominant orbital φ k , where n ≈ |φ k | 2 in the region, yielding the ground-state KS potential, which we term the single orbital approximation (SOA),(Here the zero of energy in the KS system is at ε k .) We begin by considering Eq. 1 as an approximation to the universal KS functional. We find that the SOA not only works well for the strongly localized orbital regions, but also accounts for non-local features and corrects selfinteraction in the KS potential in regions of low localization. We compare the SOA to the exact KS potential for a variety of ground-state and time-dependent systems that exhibit non-local behavior in the xc potential. We then extend our approach by combining the SOA with a potential suited to d...

The local density approximation (LDA) constructed through quantum Monte Carlo calculations of the homogeneous electron gas (HEG) is the most common approximation to the exchange-correlation functional in density functional theory. We introduce an alternative set of LDAs constructed from slablike systems of one, two and three electrons that resemble the HEG within a finite region, and illustrate the concept in one dimension. Comparing with the exact densities and Kohn-Sham potentials for various test systems, we find that the LDAs give a good account of the self-interaction correction, but are less reliable when correlation is stronger or currents flow.