Density-functional treatment of an exactly solvable semiconductor model.

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

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 investigate the exact Kohn-Sham exchange-correlation potential at semiconductor interfaces, including Schottky barriers, heterojunctions, and semiconductor surfaces. By considering the electron density at the interface, we deduce the way in which the exact exchange-correlation potential differs from its bulk counterpart. The potential has a slow spatial variation related to the discontinuity, \ensuremath{\Delta}, that occurs on addition of an electron to the bulk semiconductor. This variation, which corresponds to an ultra-non-local ``vertex correction'' in the Kohn-Sham formulation of the dielectric response of the semiconductor, results in correction terms for Schottky barrier heights and band offsets calculated using Kohn-Sham orbital energies. The effect is exhibited numerically for a model semiconductor.

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.

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.

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 paper focuses on the numerical aspects of the atomistic modelling of materials in the Density Functional formalism. As usual in such a modelling, calculations can be run at various levels of theory and using different numerical options. After a review of the approximations underlying the Local Density Approximation (LDA), the Local Spin Density Approximation (LSDA), etc. the numerical options regarding the size of the basis functions, the use of pseudo-potentials or the selection of k-points are examined in details as well as the subsequent numerical tricks occuring in the calculations, e.g. in geometry optimizations. Finally, results regarding an apatitic mineral are reported and orders of magnitudE' of the expected accuracy are provided.

We show that electronic phase transitions in zinc-blende semimetals with quadratic band touching (QBT) at the center of the Brillouin zone, like GaBi, InBi, or HgTe, can occur exclusively due to a change of the electronic temperature without the need to involve structural transformations or electron-phonon coupling. The commonly used Kohn-Sham density-functional methods based on local and semilocal density functionals employing the local density approximation (LDA) or generalized gradient approximations (GGAs), however, are not capable of describing such phenomena because they lack an intrinsic temperature dependence and account for temperature only via the occupation of bands, which essentially leads only to a shift of the Fermi level without changing the shape or topology of bands. Kohn-Sham methods using the exact temperature-dependent exchange potential, not to be confused with the Hartree-Fock exchange potential, on the other hand, describe such phase transitions. A simple modeling of correlation effects can be achieved by screening of the exchange. In the considered zinc-blende compounds the QBT is unstable at low temperatures and a transition to electronic states without QBT takes place. In the case of HgTe and GaBi Weyl points of type I and type II, respectively, emerge during the transitions. This demonstrates that Kohn-Sham methods can describe such topological phase transitions provided they are based on functionals more accurate than those within the LDA or GGA. Moreover, the electronic temperature is identified as a handle to tune topological materials.

The ground state of an externally confined one-component Yukawa plasma is derived analytically using the local density approximation (LDA). In particular, the radial density profile is computed. The results are compared with the recently obtained mean-field (MF) density profile [Henning et al., Phys. Rev. E 74, 056403 (2006)]. While the MF results are more accurate for weak screening, the LDA with correlations included yields the proper description for large screening. By comparison with first-principles simulations for three-dimensional spherical Yukawa crystals, we demonstrate that the two approximations complement each other. Together they accurately describe the density profile in the full range of screening parameters.

The electronic structures of α-BiFeO3 are calculated by using a full-potential linearized-augmented-plane-wave method. We employed the local-density approximation (LDA) with the modified Becke-Johnson (mBJ) exchange potential and the LDA + U method. The indirect (direct) bandgap of 2.24 (2.44) eV obtained by LDA + U method is in good agreement with an experiment, while the mBJ potential produces the indirect bandgap of 2.55 eV, and the direct bandgap is slightly larger than the indirect one. The discrepancy between the experimental x-ray spectra and the calculated Fe-3d and O-2p density of states were revealed to be due to the effects of the core hole. The core-hole effects are also responsible for the smaller bandgap in x-ray spectroscopy than the optical spectroscopy. The calculated valence-charge density and the bonding character obtained by LDA + U method also provides the stronger ionic character of the compound than the mBJ potential. Although the mBJ method is very efficient one, it is still very time consuming compared to the LDA + U method. The most suitable exchange-correlation potential for α-BiFeO3 is the LDA + U. Therefore, it is better to use the LDA + U method for the electronic-structure calculations of BiFeO3 compound not only for reducing the calculational time but also for better description of bandgaps and some physical properties. From the similar calculations carried out for transition-metal monoxide system it was found that the inadequacy of using the mBJ potential for the description of the localized 3d-states is rather universal.

We assess the validity of various exchange-correlation functionals for computing the structural, vibrational, dielectric, and thermodynamical properties of materials in the framework of density-functional perturbation theory (DFPT). We consider five generalized-gradient approximation (GGA) functionals (PBE, PBEsol, WC, AM05, and HTBS) as well as the local density approximation (LDA) functional. We investigate a wide variety of materials including a semiconductor (silicon), a metal (copper), and various insulators (SiO${}_{2}$ $\ensuremath{\alpha}$-quartz and stishovite, ZrSiO${}_{4}$ zircon, and MgO periclase). For the structural properties, we find that PBEsol and WC are the closest to the experiments and AM05 performs only slightly worse. All three functionals actually improve over LDA and PBE in contrast with HTBS, which is shown to fail dramatically for $\ensuremath{\alpha}$-quartz. For the vibrational and thermodynamical properties, LDA performs surprisingly very well. In the majority of the test cases, it outperforms PBE significantly and also the WC, PBEsol and AM05 functionals though by a smaller margin (and to the detriment of structural parameters). On the other hand, HTBS performs also poorly for vibrational quantities. For the dielectric properties, none of the functionals can be put forward. They all (i) fail to reproduce the electronic dielectric constant due to the well-known band gap problem and (ii) tend to overestimate the oscillator strengths (and hence the static dielectric constant).

We report the results of a computational study of the dimerization of the all-trans-polyacetylene chain within the local-density-func- tional formalism employing the G\'asp\'ar-Kohn-Sham exchange-correlation potential. Our calculated bond alternation at equilibrium is less than experimentally implied by roughly a factor of 3. Evidence is presented that suggests that this underestimation of dimerization is closely connected with the much discussed band-gap problem in local-density-functional theory.

The local many-body potential of density-functional theory is thus far understood in its mathematical context as the functional derivative of the exchange-correlation energy functional of the density. In recent work we have attempted to provide a physical interpretation for this potential. We interpret it as the work required to move an electron against the electric field of its Fermi–Coulomb hole charge distribution. Implicit in this interpretation is that the potential is path-independent. For symmetric systems this is rigorously the case. For systems where this may not be the case, the potential may be derived from an effective charge distribution given by the divergence of the field, thus ensuring its path independence. Also implicit as a consequence of the total Coulomb hole charge being zero is that the asymptotic structure of the potential is entirely due to the Fermi hole charge distribution, and thus known precisely. The potential lies explicitly within the rubric of density-functional theory in that within the exchange-only approximation it satisfies the exchange energy virial theorem sum rule and all scaling properties that the exact exchange potential must satisfy. The potential does not satisfy the virial theorem sum rule for the correlation energy, and consequently a term proportional to the difference between the interacting and noninteracting system kinetic energies must be added for the sum rule to be satisfied exactly. The formalism differs from density-functional theory in that it is not derived from the variational principle for the energy, thus obviating the requirement of determining functional derivatives, as well as allowing for the study of excited states. The interpretation also leads to insights into the exact Slater exchange potential, and other approximation schemes such as the Xα method, and local density and gradient expansion approximations. The results of application to few-electron atomic and many-electron metallic surface inhomogeneous electronic systems are remarkably accurate when compared with other theoretical calculations and experiment.

First principles study of structural, bonding and vibrational properties of PtCoO2, PdCoO2 and PdRhO2 metallic delafossites

Study of the linear and nonlinear response properties of the zinc-blende AlP, GaP and their AlxGa1-xP ternary alloys using first principle calculations