Nonlocal Exchange Energy Research Articles

The importance of fully quantitative non-local treatments of exchange energy in Density Functional Theory of atoms, molecules and clusters, characterised by strongly inhomogeneous electron liquids

After a brief summary of some recent work on non-local approximations to the exchange energy and potential in density functional theory, a table is presented for atoms from Be to Ar, with an even number of electrons, having large electron density gradients, which highlights the importance of exchange. Until non-local exchange energy is truly quantitative, refinements of correlation energy may be submerged through ‘noisy’ exchange.

Investigations of the Nonlocal Exchange Energy Model on a Single Body of Ferromagnetic Material

In this paper we do computations on a single spherical body of uniaxial ferro-magnetic material in a varying applied magnetic field. Our computations are based on a relaxed model with a nonlocal exchange energy. The model is based on Young-measures, but we will make use of recent calculations which indicate that our computations can be based on only the first and second moments. We extend the result of a single uniaxial sphere in a uniform applied magnetic field parallel to the easy direction of magnetization to uniform applied magnetic fields that are skew to the easy direction of magnetization. We show that when the exchange energy is zero, we duplicate the wide range of measured-value minimizers of De Simone's model. As the exchange energy grows, our model stabilizes at the saturated solutions of the Stoner-Wohlfarth model where the measured-valued minimizers are eliminated.

Weighted-density exchange and local-density Coulomb correlation energy functionals for finite systems: Application to atoms.

A nonlocal exchange energy functional developed within the weighted spin-density approximation (WSDA) and a local Coulomb correlation energy functional are combined within the density-functional scheme for the calculation of the electronic structure of many-electron systems. Both functionals include alternative forms of the pair-correlation functions governing the size and shape of the Fermi and Coulomb holes. Self-consistent calculations have been performed for all atoms from He to Rn within the exchange-only WSDA scheme. These have been compared to the results of similar calculations using Becke's exchange functional. Besides, the results of calculations for all atoms from He to Ar, including both exchange and correlation, are compared with those of the self-interaction-corrected method. In addition to obtaining accurate total, exchange, and correlation energies, we perform a successful calculation of the s\ensuremath{\rightarrow}d interconfigurational energies (ICE's) for transition-metal atoms. The average error of the calculated ICE values for 3d atoms is only 0.2 eV.

Non-local exchange and local Coulomb correlation energy density functionals for finite many-electron systems

A new non-local exchange energy functional E x[ n] is developed within the weighted density approximation (WDA) and a local Coulomb correlation energy functional E c[ n] is developed within the local density approximation (LDA) for the calculation of ground state energies. Both functionals include novel forms of the pair-correlation functions governing the size and shape of the Fermi and Coulomb holes. Self-consistent calculations for noble gas atoms show that the present approach provides a substantial improvement compared to the commonly used LDA functionals for finite systems, and is promising for molecules and clusters.

Non‐local relation between kinetic and exchange energy densities in Hartree–Fock theory

AbstractA non‐local generalization K(r, r') of the kinetic energy t(r) such that t(r) = ∫K(r, r') dr' is defined using the idempotency property of the Hartree–Fock first‐order density matrix. This is, in turn, related by means of an explicit differential equation to the non‐local exchange energy density X(r, r'). The relationship is illustrated for a couple of examples: with the Fermi‐hole in a uniform electron gas, of importance in the local density version of density functional theory, and with inhomogeneous electron systems.

Self-consistent implementation of nonlocal exchange and correlation in a Gaussian density-functional method.

The gradient-expansion approximation (GEA) and the generalized gradient approximation (GGA) to nonlocal exchange energy in concert with the nonlocal correlation energy functional of Perdew [Phys. Rev. B 33, 8822 (1986)] are analyzed when implemented in a fully self-consistent way in conjunction with the Vosko-Wilk-Nusair parametrization for the local exchange-correlation energy. It is shown that the lowest-order gradient expansion, even with corrected asymptotic behavior in the large-density-gradient limit, is still unsatisfactory in the chemically important region of electron densities where the basic assumption of the GEA (\ensuremath{\Vert}\ensuremath{\nabla}n\ensuremath{\Vert}/2${\mathit{k}}_{\mathit{F}}$n1) breaks down. In contrast, the GGA expansion behaves better. A shift by a constant additive term of an effective one-body Kohn-Sham potential in concert with the GGA nonlocal functional provides, within the framework of density-functional theory, a way of interpreting excitation energies. The nonlocal functionals significantly improve binding energies. The resulting nonlocal exchange-correlation potential is state independent; thus the present method is convenient from the computational point of view. Applications are presented for a number of atoms and small molecules, including ${\mathrm{O}}_{2}$, ${\mathrm{Mg}}_{2}$, ${\mathrm{CH}}_{2}$, and for a transition-metal cluster, ${\mathrm{Ni}}_{4}$.

Gradient correction in Thomas-Fermi theory.

A new derivation of the Weizsacker-type gradient corrections to Thomas-Fermi (TF) kinetic energy functional is presented. The development is based on the first-order reduced density matrix as obtained from the one-body Green's function in the mean-path approximation devised for the purpose, using the Feynman path-integral approach; the mean-path approximation turns out to be essentially equivalent to the eikonal approximation used in quantum collision theory for high-energy collisions. This derivation agrees with the conventional gradient expansion truncated at second order, in that it gives the kinetic energy functional of the TF-(1/9)W model, that is, the sum of the original TF kinetic energy and (1/9) of the Weizsacker gradient correction. However, in the present derivation, TF-(1/9)W results from a reduced density matrix of closed form; the original TF local relation between particle, density, and one-body potential is preserved; and the kinetic energy density contains a Laplacian of particle density with a factor half of that from the gradient expansion. Most significantly, the TF-(1/9)W kinetic energy functional is the consequence of representing both the diagonal and off-diagonal elements of the density matrix correctly to zero order through the mean-path approximation to the one-body Green's function, whereas in the conventional TF approximation, the zero order of the gradient expansion, off-diagonal elements are not correct to the same order. Other results of the present approach include a nonlocal exchange energy functional of density, a one-body effective potential that contains a contribution from the kinetic energy functional derivative, and the construction of closed-form density matrices that give various kinetic energy functionals of TF-\ensuremath{\lambda}W form (justifying various existing empirical \ensuremath{\lambda} values). Also presented are the results of numerical calculation for rare-gas atoms of TFD-\ensuremath{\lambda}W models (TFD denotes Thomas-Fermi-Dirac) with \ensuremath{\lambda}=(1/3), 0.186, (1/6), and (1/9). .AE

Nonlocal exchange energy of many-fermion systems of arbitrary dimensionality and interparticle interaction.

Nonlocal exchange energy of many-fermion systems of arbitrary dimensionality and interparticle interaction.

The use of floating 1-s gaussians for energy band calculation: Ne and Ar results

A method is presented for the study of the conduction bands of insulators based on the systematic use of floating 1-s gaussian basis functions. The use of this type of basis has two major advantages. One is the flexibility of varying the position of the gaussian inside the unit cell to best optimise the wavefunction. The other is the ease with which a set of floating 1-s gaussians can represent states of low symmetry, thereby eliminating the use of higher angular momentum spherical harmonics. In addition, various terms of the Hamiltonian matrix elements, including the non-local exchange energy, can be efficiently evaluated by means of a set of simple analytic interpolation formulae. The method is applied to calculate the conduction band structure of solid Ne and Ar, giving very good agreement with earlier Hartree-Fock works. The most remarkable feature is that a single optimised Bloch basis, constructed of a floating gaussian positioned at the interstitial site, gives the lowest conduction band energies which are within about 0.01 Hartree from the final results obtained using larger basis. The additional basis functions are only necessary to produce higher excited states of the same symmetry.

Hartree-Fock theory of the inhomogeneous electron gas at a jellium metal surface:Rigorousupper bounds to the surface energy and accurate work functions

The inhomogeneous electron gas at a jellium metal surface is studied in the Hartree-Fock approximation by Kohn-Sham density functional theory. Rigorous upper bounds to the surface energy are derived by application of the Rayleigh-Ritz variational principle for the energy, the surface kinetic, electrostatic, and nonlocal exchange energy functionals being determined exactly for the accurate linear-potential model electronic wave functions. The densities obtained by the energy minimization constraint are then employed to determine work-function results via the variationally accurate "displaced-profile change-in-self-consistent-field" expression. The theoretical basis of this non-self-consistent procedure and its demonstrated accuracy for the fully correlated system (as treated within the local-density approximation for exchange and correlation) leads us to conclude these results for the surface energies and work functions to be essentially exact. Work-function values are also determined by the Koopmans'-theorem expression, both for these densities as well as for those obtained by satisfaction of the constraint set on the electrostatic potential by the Budd-Vannimenus theorem. The use of the Hartree-Fock results in the accurate estimation of correlation-effect contributions to these surface properties of the nonuniform electron gas is also indicated. In addition, the original work and approximations made by Bardeen in this attempt at a solution of the Hartree-Fock problem are briefly reviewed in order to contrast with the present work.

Study of the exchange energy of an inhomogeneous electron gas at a surface

In this work we have studied the exchange energy property of an inhomogeneous electron gas at a jellium surface. The nonlocal surface exchange energy is determined exactly for the accurate set of single-particle wave functions generated within the linear-potential approximation to the effective potential at a surface, and written in terms of a universal function of the field strength. It is shown, that over the wide range of density profiles considered, this universal function increases by a factor of 6 as the profile is changed from rapidly to slowly varying. For a Hartree-Fock gas with bulk density such that $5>~{r}_{s}>~1$, the corresponding increase in the surface exchange energy is three orders of magnitude. The local density approximation (LDA) functional of the surface exchange energy is also determined for the same set of wave functions, and for the first time its convergence to the exact results as the density profile is made slowly varying demonstrated. It is shown that this LDA functional is a poor approximation for metallic densities, overstimating the exact results by 17-55%. The results of a previous calculation within the step-model approximation are also reinterpreted, and similar conclusions with regard to the inaccuracy of the LDA functional arrived at. The applicability of the nonlocal calculation of this work to the evaluation of other approximate nonlocal exchange energy functionals, and to the study of the inhomogeneous Hartree-Fock and fully correlated systems is also indicated.