Nonlocal density functional theory of solids: Applications of the weighted density approximation to silicon

Using the density functional formalism together with the weighted density approximation (WDA), we have carried out selfconsistent bandstructure studies of bulk Si in order to identify the influences of different WDA potentials and different pair correlation functions on the electronic spectrum. Improvements of band energies due to consideration of nonlocalities in exchange and correlation energy depend very sensitively on the particular correlation function used. For the prototype semiconductor Si, only one of the six WDA versions tested seems useful.     

Meta-generalized gradient approximation: non-empirical construction and performance of a density functional

The local ingredients of a meta-generalized gradient approximation (meta-GGA) include the electron density, its gradient, and the Kohn–Sham orbital kinetic energy density. We discuss the strategy of constructing a successful meta-GGA density functional for the exchange-correlation energy that satisfies exact constraints without empirical parameters. The new feature of this functional is that it simultaneously respects the two paradigms of electronic structure theory: one- or two-electron densities and slowly-varying densities, and so is uniformly accurate for atoms, molecules and solids. Results of extensive numerical tests of the new functional are summarized and evaluated.     

Random-phase approximation correlation methods for molecules and solids

Random-phase approximation (RPA) correlation methods based on Kohn–Sham density-functional theory and Hartree–Fock are derived using the adiabatic-connection fluctuation dissipation theorem. It is shown that the correlation energy within the adiabatic-connection fluctuation-dissipation theorem is exact in a Kohn–Sham framework while for Hartree–Fock reference states this is not the case. This shows that Kohn–Sham reference states are probably better suited to describe electron correlation for use in RPA methods than Hartree–Fock reference states. Both, Kohn–Sham and Hartree–Fock RPA methods are related to each other both by comparing the underlying correlation functionals and numerically through the comparison of total energies and reaction energies for a set of small organic molecules.     

Improving band gap prediction in density functional theory from molecules to solids

A novel nonempirical scaling correction method is developed to tackle the challenge of band gap prediction in density functional theory. For finite systems the scaling correction largely restores the straight-line behavior of electronic energy at fractional electron numbers. The scaling correction can be generally applied to a variety of mainstream density functional approximations, leading to significant improvement in the band gap prediction. In particular, the scaled version of a modified local density approximation predicts band gaps with an accuracy consistent for systems of all sizes, ranging from atoms and molecules to solids. The scaled modified local density approximation thus provides a useful tool to quantitatively characterize the size-dependent effect on the energy gaps of nanostructures.     

Density functional approximation for van der Waals fluids： based on hard sphere density functional approximation

A universal theoretical approach is proposed which enables all hard sphere density functional approximations (DFAs) applicable to van der Waals fluids. The resultant DFA obtained by combining the universal theoretical approach with any hard sphere DFAs only needs as input a second-order direct correlation function (DCF) of a coexistence bulk fluid, and is applicable in both supercritical and subcritical temperature regions. The associated effective hard sphere density can be specified by a hard wall sum rule. It is indicated that the value of the effective hard sphere density so determined can be universal, i.e. can be applied to any external potentials different from the single hard wall. As an illustrating example, the universal theoretical approach is combined with a hard sphere bridge DFA to predict the density profile of a hard core attractive Yukawa model fluid influenced by diverse external fields; agreement between the present formalism's predictions and the corresponding simulation data is good or at least comparable to several previous DFT approaches. The primary advantage of the present theoretical approach combined with other hard sphere DFAs is discussed.     

A BBGKY-based density gradient approximation of interparticle forces: Application for discrete Boltzmann methods

The force term accounting for interparticle interactions, with application to discrete Boltzmann simulation methods, is derived using a density gradient expansion of the BBGKY collision operator. It is shown that previous calculations, based on essentially the same mean-field-theory philosophy, do not apply the density gradient approximation in a self consistent fashion. Thus these previous models have errors in the second virial coefficient as well as coefficients associated with gradient terms in the pressure tensor. This new treatment corrects these shortcomings.     

The gradient approximation to the exchange-correlation energy functional: A generalization that works

A complete wave vector analysis has been made of the 2nd order gradient expansion of the exchange-correlation energy for a non-uniform electronic system. When applied to a typical surface, the method shows that the gradient expansion disagrees markedly with the exact small k form, and that the substantial contribution to it comes from the universál region where the wavelength is much longer than the surface healing length. This verifies the substantial correctness of Rasolt and Geldart's RPA calculation for the gradient coefficient. However, it also shows beyond any doubt that the gradient expansion itself is incorrect for a metallic surface. We propose an approximate method, based on an “average gradient” rather than the “local gradient”. When applied to a surface, we find rough agreement with our earlier cruder interpolation scheme, and good agreement with exact results where they exist.     

Method of the spin density functional and the ground state of solids

By using the property of universality of the spin density functional, quasiparticles of the ground state of an electron fluid in a solid are introduced. A model is proposed for the spin density functional of the ground state of a solid.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 76–81, October, 1977.     

Local approximation of the correlation energy functional in the density matrix functional theory

A local approximation formula of the correlation energy functional E(c) in terms of the first-order reduced density matrix (1-RDM) is presented. With the contracted Schr?dinger equation the principal dependence of E(c) on the natural occupation numbers n(i) is identified. Using the effective mass theory, E(c) is expressed as a functional of the local density and the local variable, J = SUM (i)[square root of (n(i)(1-n(i))] /phi(i)/(2), where phi(i) are the natural spin orbitals. This local approximation satisfies the homogeneous coordinate scaling relation, gives the exact result for a one-electron system, and is almost free from the exchange energy error. It reproduced about 90% of the correlation energies of atoms and molecules.     

Double fluctuations on the attractive Hubbard model: ladder approximation

We explore, for the first time, the effect of double fluctuations on both the diagonal and off-diagonal selfenergy. We use the T-Matrix equations below T _c , developed recently by the Zürich group (M.H. Pedersen et al) for the local pair attraction Hamiltonian. Here, we include as well the effect of fluctuations on the order parameter (beyond the BCS solution) up to second order in U/t. This is equivalent to approximating the effective interaction by U in the off-diagonal self-energy. For U/t = ?6.0, T/t = 0.05, μ/t = ?5.5 and Δ/t = 1.5, we find four peaks both for the diagonal, A(n(π/16, π/16), ω), the far left peak has a vanishing small weight; (b) in B(n(π/16, π/16), ω) the far left and far right peaks have very small weights. The physical picture is, then, that the pair physics in the normal phase (T > T _c is still valid below T _c . However, the condensation of the e-h pairs produces an additional gap around the chemical potential as in BCS, in other words, superconductivity opens a gap in the lower branch of a Hubbard-type-I solution.     

Bridge density functional approximation for non-uniform hard core repulsive Yukawa fluid

In this work, a bridge density functional approximation （BDFA） （J. Chem. Phys. 112, 8079 （2000）） for a nonuniform hard-sphere fluid is extended to a non-uniform hard-core repulsive Yukawa （HCRY） fluid. It is found that the choice of a bulk bridge functional approximation is crucial for both a uniform HCRY fluid and a non-uniform HCRY fluid. A new bridge functional approximation is proposed, which can accurately predict the radial distribution function of the bulk HCRY fluid. With the new bridge functional approximation and its associated bulk second order direct correlation function as input, the BDFA can be used to well calculate the density profile of the HCRY fluid subjected to the influence of varying external fields, and the theoretical predictions are in good agreement with the corresponding simulation data. The calculated results indicate that the present BDFA captures quantitatively the phenomena such as the coexistence of solid-like high density phase and low density gas phase, and the adsorption properties of the HCRY fluid, which qualitatively differ from those of the fluids combining both hard-core repulsion and an attractive tail.     

A perturbation density functional theory for hydrogen bonding cyclic molecules

Using the framework of Wertheim's thermodynamic perturbation theory, a new polyatomic density functional theory is developed to account for the intermolecular association of cyclic molecules in interfacial systems. To test the theory, Monte Carlo simulations in the canonical ensemble were performed for the specific case of an associating triatomic ring with one association site next to a hard wall. The theory and simulation results were found to be in good agreement.     

Global dynamical correlation energies in covariant density functional theory: Cranking approximation

The global dynamical correlation energies for 575 even-even nuclei with proton numbers ranging from Z = 8 to Z = 108 calculated with the covariant density functional theory using the PC-PK1 parametrization are presented. The dynamical correlation energies include the rotational correction energies obtained with the cranking approximation and the quadrupole vibrational correction energies. The systematic behavior of the present correlation energies is in good agreement with that obtained from the projected generator coordinate method using the SLy4 Skyrme force although our values are systematically smaller. After including the dynamical correlation energies, the root-mean-square deviation predicted by the PC-PK1 for the 575 even-even nuclei masses is reduced from 2.58 MeV to 1.24 MeV.     

The polarisability of atoms and molecules: a comparison between a conceptual density functional theory approach and time-dependent density functional theory

We investigate the local polarisability or polarisability density using both a conceptual density functional theory approach based on the linear response function and time-dependent density functional theory. Using a zero frequency in the latter, we can immediately compare both approaches. Using an analytical expression for the linear response kernel, we are able to systematically analyse α(r) throughout the periodic table. An extension to molecules is also made with a study of the CO molecule retrieving the connection between local softness and local polarisability.     

Method of the spin density functional and the ground state of solids. II

A model is proposed for the ground-state quasi-particles of an electron fluid in a solid state. Possible types of quasi-particles are found, and their internal structure is investigated, A general expression is written for the kinetic and exchange-correlation energy of the valence electrons.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 54–59, March, 1978.     

The ladder approximation in quantum optics and the Wigner-Weisskopf theory

It is shown that the Wigner-Weisskopf theory is an example of what Walls has called the ladder approximation in quantum optics.