Exchange-correlation potentials at semiconductor interfaces.

Starting from exceedingly accurate many-body wavefunctions, we have constructed essentially exact densities, exchange-correlation potentials and components of the total energy for helium and twoelectron ions. These density functional results are compared to the corresponding quantities obtained from a variety of commonly used approximate density functionals, namely the local density approximation and various generalized gradient approximations in order to test the accuracy of the approximate functionals. Although the generalized gradient approximations yield improved energies compared to the local density approximation, the exchange and correlation potentials (especially the latter) obtained from the generalized gradient approximations are in poor agreement with the corresponding exact potentials. The large-distance asymptotic behavior of the exact exchange-correlation potential to O(1/r4) is found to agree with theoretical predictions. The short range behaviour of the exchangecorrelation potential is very close to quadratic. The prospects for the improved generalized gradient approximations are discussed. Keywords: density functional theory, exchange-correlation potential, generalized gradient approximation

An exact expression in terms of density matrices (DM) is derived for \ensuremath{\delta}F[n]/\ensuremath{\delta}n(r), the functional derivative of the Hohenberg-Kohn functional. The derivation starts from the differential form of the virial theorem, obtained here for an electron system with arbitrary interactions, and leads to an expression taking the form of an integral over a path that can be chosen arbitrarily. After applying this approach to the equivalent system of noninteracting electrons (Slater-Kohn-Sham scheme) and combining the corresponding result with the previous one, an exact expression for the exchange-correlation potential ${\mathit{v}}_{\mathrm{xc}}$(r) is obtained which is analogous in character to that for \ensuremath{\delta}F[n]/\ensuremath{\delta}n(r), but involving, besides the interacting-system DMs, also the noninteracitng DMs. Equating the former DMs to the latter ones, we reduce the result for the exact ${\mathit{v}}_{\mathrm{xc}}$(r) to that for an approximate exchange-only potential ${\mathit{v}}_{\mathit{x}}$(r). This leads naturally to the Harbola-Sahni exchange-only potential.

Time-dependent density-functional methods are used to compute excitation energies and, via the adiabatic-connection fluctuation-dissipation theorem, ground-state correlation energies of atoms, ions, and the H${}_{2}$ molecule at various bond lengths. Various exchange-correlation potentials ${v}_{xc}$ and exchange-correlation kernels ${f}_{xc}$ are tested. Accurate exchange-correlation potentials are found to be essential for getting accurate energies. Methods employing in the Kohn-Sham self-consistency process the exact local Kohn-Sham exchange potential while neglecting completely the correlation potential lead to better excitation and correlation energies than methods with exchange-correlation potentials within the local density approximation or the generalized gradient approximation. Taking into account the exact exchange-correlation potential and thus the exact Kohn-Sham potential further improves excitation and correlation energies.

We consider an exactly soluble model consisting of two electrons attracted to a common center by harmonic-oscillator forces with mutual Coulomb repulsion between the electrons. The exact wave function and energy for the singlet ground state is obtained by separating the Schr\"odinger equation in center-of-mass and relative coordinates and numerically integrating the relative-coordinate eigenvalue equation. The exact Kohn-Sham (KS) orbital for the two equivalent electrons with opposite spins is constructed from the exact density, and the Kohn-Sham equation is inverted, yielding the exact exchange-correlation potential as a function of position.The exact correlation potential, as a function of position, is then obtained by subtracting the exact exchange potential from the exact exchange-correlation potential. The exact ground-state energy, exchange potential, correlation potential, KS single-particle energy eigenvalues ${\ensuremath{\epsilon}}_{\mathrm{KS}}$, exchange energy, and correlation energy are compared to the results given by density-functional theory employing the local-density approximation (LDA) plus correction terms and the exact KS orbitals for values of the harmonic-oscillator spring constant that vary over 4 orders of magnitude. We find that although total energies are accurately given by LDA plus correction terms, the values of ${\ensuremath{\epsilon}}_{\mathrm{KS}}$ are significantly in error. We show that this is mainly due to errors in the exchange potential.In addition we show analytically that in a singlet ground state of a two-electron system, the exact expectation value of the exchange potential equals the exact exchange energy, whereas the LDA expectation value of the exchange potential is only (2/3) the LDA exchange energy. Furthermore, we show that although the gradient expansion approximations for the exchange energy are significant improvements over the LDA, neither of these approximations significantly decreases the error in the LDA for the expectation value of the exchange potential. Both the correlation energy and the expectation value of the correlation potential in the LDA plus corrections are significantly in error when compared with the results of the exact calculation, the most accurate results being given by the LDA with self-interaction correction.

We present local hybrid functionals that incorporate a position-dependent admixture of short-range (screened) nonlocal exact [Hartree-Fock-type (HF)] exchange. We test two limiting cases: screened local hybrids with no long-range HF exchange and long-range-corrected local hybrids with 100% long-range HF exchange. Long-range-corrected local hybrids provide the exact asymptotic exchange-correlation potential in finite systems, while screened local hybrids avoid the problems inherent to long-range HF exchange in metals and small-bandgap systems. We treat these functionals self-consistently using the nonlocal exchange potential constructed from Kohn-Sham orbital derivatives. Generalized Kohn-Sham calculations with screened and long-range-corrected local hybrids can provide accurate molecular thermochemistry and kinetics, comparable to existing local hybrids of full-range exchange. Generalized Kohn-Sham calculations with existing full-range local hybrids provide results consistent with previous non-self-consistent and "localized local hybrid" calculations. These new functionals appear to provide a promising extension of existing local and range-separated hybrids.

The band-gap problem is studied for an exactly solvable semiconductor model. The exact exchange-correlation potential is constructed and its discontinuity is studied. Depending on the parameter range, a large or a small discontinuity is obtained. The "scissor-operator" method is compared with the local-density approximation for an impurity calculation. In general, the local-density approximation is closer to the exact solution. Some general features of the density-functional formalism are illustrated.

Electron density profiles at charged metal surfaces in the weighted density approximation

Traditionally quantum chemical methods use the Hartree-Fock model as a starting point and explicitly or implicitly employ expansions of the wavefunction in determinants that are generated by orbital substitutions in the HF determinant. Increasingly sophisticated methods for the selection of determinants have been developed, such as threshold-based methods (MRDCI), direct-CI methods and coupled-cluster approaches. The development is driven by both advances in mathematical technique (algebraic as well as diagrammatical) and by computational convenience and feasibility. In contrast, density functional theory (DFT) concentrates on the form of the exchange-correlation part of the total energy expression and on the Kohn-Sham one-electron potential that enters the effective one-electron equations. The present interest in DFT from the (quantum)chemistry community stems from the notable improvement that is obtained over Hartree-Fock, in particular if also non-local corrections are included (cf. other papers in this volume). Evidently, it is desirable to understand precisely which effects of correlation are build into the exchange correlation potential and energy of present day DF appraoches. For two-electron systems such an analysis has been carried out for the exact Kohn-Sham potential by the present authors1. It has been shown that in this case V xc consists of three contributing terms, V cond , V kin and V N−1, that each have a clear physical meaning1. V cond , for instance, is the potential due to the conditional electron density that will be defined below (eq. 1.6): the density of the remaining electrons when one electron (the reference electron) is known to be at a given position r1· The conditional density thus incorporates exactly all effects of correlation (Fermi and Coulomb holes).

We use diffusion quantum Monte Carlo (DQMC) techniques to obtain accurate estimates of the components of the Kohn-Sham effective potential of density-functional theory for a model semiconductor, and the corresponding components of the energy functional, by determining the local potential which, when filled with noninteracting electrons, reproduces the DQMC electron density. The results are compared with the widely used local-density approximation. There is a large deviation in the exchange-correlation potential, a slight deviation in the electron density, but a very small deviation in the total energy and its components. We also use DQMC techniques to calculate the quasiparticle band gap, and hence the discontinuity \ensuremath{\Delta} in the exchange-correlation potential on the addition of an electron.

The exact static and time-dependent Kohn-Sham (KS) exchange-correlation (xc) potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalised by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.

Density scaling considerations are used to derive an exchange-correlation explicit density functional that is appropriate for the electron deficient side of the integer and which recovers the exact r → ∞ asymptotic behaviour of the exchange-correlation potential. The functional has an unconventional mathematical form with parameters that are system-dependent; the parameters for an N-electron system are determined in advance from generalised gradient approximation (GGA) calculations on the N- and (N - 1)-electron systems. Compared to GGA results, the functional yields similar exchange-correlation energies, but HOMO energies that are an order of magnitude closer to the negative of the vertical ionisation potential; for anions, the HOMO energies are negative, as required. Rydberg excitation energies are also notably improved and the exchange-correlation potential is visibly lowered towards the near-exact potential. Further development is required to improve valence excitations, static isotropic polarisabilities, and the shape of the potential in non-asymptotic regions. The functional is fundamentally different to conventional approximations.

In this work, nonadiabatic exchange-correlation (XC) potentials for time-dependent density-functional theory (TDDFT) for strongly correlated materials are derived in the limits of strong and weak correlations. After summarizing some essentials of the available dynamical mean-field theory (DMFT) XC potentials valid for these systems, we present details of the Sham–Schluter equation approach that we use to obtain, in principle, an exact XC potential from a many-body theory solution for the nonequilibrium electron self-energy. We derive the XC potentials for the one-band Hubbard model in the limits of weak and strong on-site Coulomb repulsion. To test the accuracy of the obtained potentials, we compare the TDDFT results obtained with these potentials with the corresponding nonequilibrium DMFT solution for the one-band Hubbard model and find that the agreement between the solutions is rather good. We also discuss possible directions to obtain a universal XC potential that would be appropriate for the case of intermediate interaction strengths, i.e., a nonadiabatic potential that can be used to perform TDDFT analysis of nonequilibrium phenomena, such as transport and other ultrafast properties of materials with any strength of electron correlation at any value in the applied perturbing field.

Density, density matrix, or propagator

Analysis of exchange and exchange-correlation potentials with non-zero constant asymptotics

We demonstrate by specific examples that if a Kohn-Sham exchange-correlation potential is given explicitly in terms of the electron density and its derivatives, then one can easily reconstruct the parent density functional by evaluating analytically (or numerically with one-dimensional quadratures) the van Leeuwen-Baerends line integral (Phys. Rev. A 1995, 51, 170-178) along a path of (coordinate)-scaled densities. The choice of a density scaling path amounts to defining the gauge of the resultant exchange-correlation energy density. The well-known Levy-Perdew virial relation for exchange potentials can be viewed as an analytical line integral along the electron-number-conserving uniform density scaling path. Energies obtained from model exchange-correlation potentials should be interpreted with caution because the reconstructed density functional is unique (up to a gauge transformation) only if the model Kohn-Sham potential is a functional derivative.