Kohn-Sham Exchange Potential Research Articles

Relationship between Slater and Kohn-Sham exchange potentials.

Relationship between Slater and Kohn-Sham exchange potentials.

The gradient expansion approximation for exchange: A physical prespective

In recent work, we have provided a rigorous physical interpretation for the exchange energy and potential (or functional derivative) as obtained within the local-density approximation via the Harbola–Sahni formulation of many-electron theory. In this article, we analyze the gradient-expansion approximation (GEA) for these properties from the same physical perspective. The source charge distribution in this approximation is the GEA Fermi hole to O(▽3). This charge distribution is unphysical, so that the resulting force field and work done cannot be defined in a physically meaningful manner, and the exchange energy is singular. Thus, when viewed from the perspective of a source charge, the existence of the gradient expansions for the potential and energy is questionable. We next discuss the conventional method of employing a screened-Coulomb interaction to eliminate the singularities due to the GEA source charge, and show that it leads to inconsistent results. These inconsistencies are also intrinsic to a proof of the inequivalence of the Harbola–Sahni and Kohn–Sham exchange potentials within the GEA. Thus, although the inequivalence of these potentials has been established by other analyses, this proof is shown not to be rigorous. Finally, we demonstrate that when the physics of the GEA exchange source charge is corrected by the satisfaction of sum rules, the modified charge distribution then leads to a well-behaved local exchange potential and exchange energy density, and to a finite exchange energy. The consequences of our analysis on the gradient expansions for the correlation and exchange-correlation potential and energy are also noted. © 1992 John Wiley & Sons, Inc.

THEOREMS FOR EXACT LOCAL EXCHANGE POTENTIAL

Theorems for the exact local Kohn-Sham exchange potential in density-functional theory are reviewed. Trial exchange potentials based on these theorems are illustrated.

Nonlinear dielectric response to a point-donor impurity of an electron-gas-model semiconductor that includes the effect of the Dirac-Slater exchange correlation.

The Thomas-Fermi statistical theory, including the Dirac-Slater local-density treatment of exchange correlation, has been applied to the problem of nonlinear screening of a donor point charge embedded in an electron-gas-model semiconductor. The nonlinear screening equation is solved numerically, giving spatial dielectric functions and screening radii with exchange-correlation strength and ion-charge state as parameters. Illustrations and tabulations of these results are given for five semiconductors, four ion charges, and two nonzero values of the exchange-correlation strength corresponding to the Kohn-Sham and Slater exchange potentials. A variational principle equivalent of the nonlinear equation leads to approximate analytical expressions for the spatial dielectric functions which are in close agreement with the exact results. Variational parameters are given for a subset of the semiconductors and charge states. Dielectric functions of silicon are used to illustrate typical comparisons between the two methods of solution.

Implication of nonintegral occupation number and Fermi-Dirac statistics in the local-spin-density approximation applied to finite systems.

In electronic-structure calculations for finite systems using the local-spin-density (LSD) approximation, it is assumed that the eigenvalues of the Kohn-Sham equation should obey Fermi-Dirac (FD) statistics. In order to comply with this assumption for some of the transition-metal atoms, a nonintegral occupation number is used which also minimizes the total energy. It is shown here that for finite systems it is not necessary that the eigenvalues of the Kohn-Sham equation obey FD statistics. It is also shown that the Kohn-Sham exchange potential used in all LSD models is correct only for integer occupation number. With a noninteger occupation number the LSD exchange potential will be smaller than that given by the Kohn-Sham potential. Ab initio self-consistent spin-polarized calculations have been performed numerically for the total energy of an iron atom. It is found that the ground state belongs to the 3d/sup 6/4s/sup 2/ configuration. The ionization potentials of all the Fe/sup n//sup +/ ions are reported and are in agreement with experiment.

Nonlocal Kohn-Sham exchange corrections to Si band gaps and binding energy.

We compare the binding energy and energy band gaps of Si calculated with the local-density approximation to the Kohn-Sham exchange potential and with the correction to that potential which is exact to first order in \ensuremath{\rho}(K)/${\ensuremath{\rho}}_{0}$. Although the correction is not small, running between 1 eV and -3.5 eV in the unit cell, its effect on both the band gaps and binding energy is very small. Thus we conclude that the discrepancy between calculated energy band gaps and experiment must be almost entirely due to the discontinuity in the zeroth Fourier component of the exchange-correlation potential.

First approach to the band structure ofα-rhombohedral boron

We report a calculation of the energy-band structure of $\ensuremath{\alpha}$-rhombohedral boron. The lack of theoretical studies of solid B (a technologically interesting material) can be explained by the difficulties of a quantitative calculation: twelve atoms in the cell and a very open structure. The "muffin-tin-orbitals" method for atomic spheres having a large overlap has been used. The total potential is a superposition of individual potentials of finite range, built from the atomic charge density. The Kohn-Sham exchange potential is correctly calculated using the superposition of atomic densities and then numerically fitted with a superposition of individual functions. The results show a nonmetallic behavior of $\ensuremath{\alpha}\ensuremath{-}\mathrm{B}$ with an indirect bandgap. But this agreement with experiment is not quantitative: The gap is significantly too large and the occupied band too narrow. The modifications required for improving this calculation are reviewed.

Fermi surface of YAl2

The energy bandstructure for the cubic Laves-phase intermetallic compound YAl2, which has recently attracted interest as a proper reference material for the study of anomalous physical properties of rare-earth dialuminides, is calculated self-consistently by a non-relativistic APW method with the Kohn-Sham exchange potential. The Fermi surface consists of various sheets, for which the extremal cross section and the cyclotron mass is carefully calculated. The eighth-band Fermi surface consists of small hole pockets around the Delta axes. The ninth-band Fermi surface is a multiply connected hole sheet, the topology of which is a junglegym running from an X point to a nearest-neighbour X point. Open orbits exist in (100) directions. The tenth-band Fermi surface is an electron sphere centred at the Gamma point, but has bumps in (100) directions. The calculated extremal cross sectional area of the tenth-band Fermi surface agrees within 10% with an experimental result of the recent de Haas-van Alphen effect measurement.

Band structure and x-ray photoelectron spectrum of ZrB2

The band structure and the density of states of Zr${\mathrm{B}}_{2}$ have been calculated by an augmented-plane-wave method and compared with the observed x-ray photoelectron spectrum of valence band. The theoretical energy-distribution curve obtained by using the Kohn-Sham exchange potential was in good agreement with the experimental one. The calculated density of states at the Fermi level and the interband transition energies at the $\ensuremath{\Gamma}$ point were consistent with the available experimental data of a low-temperature specific heat and a reflectance measurement. The band structure shows that the bonding nature of Zr${\mathrm{B}}_{2}$ can be explained by a combination of the graphitic bonding model of boron network and the hep-metal bonding model of zirconium.

Electronic-Structure Studies of Solids. II. "Exact" Hartree-Fock Calculations for Cubic Atomic-Hydrogen Crystals

A comprehensive description is given of rigorous Hartree-Fock calculations for cubic atomic-hydrogen crystals. All energy and overlap integrals are reduced to reciprocal-lattice sums. The exchange effects are treated exactly. The Bloch functions and Fermi surfaces are optimized. The near-Hartree-Fock results indicate essentially spherical Fermi surfaces and strong atomic-orbital behavior of the Bloch functions. The energetic effect associated with the atomlike inhomogeneities is about 15%. Good agreement is found between Kohn-Sham and Slater local-exchange potentials and exchange potentials deduced from our Hartree-Fock results.

Phonon Frequencies of Alkali Metals

Phonon frequencies of alkali metals are calculated using the single orthogonalized-plane-wave electron-ion matrix element and the dielectric functions due to Singwi et al. and to Geldart and Vosko. We have calculated the bare-ion potential by the Hartree-Fock-Slater method, where the Slater exchange potential is replaced by a modified Kohn-Sham exchange potential which approximates correlation effects according to the simple prescription of Robinson et al. The calculated phonon frequencies of Li, Na, K, and Rb are in good agreement with the neutron spectroscopy measurements.