On the density functional theory

Based on the theoretical analysis of the Kohn Nobel lecture, three important analytical observations regarding the fundamental statements of the density functional theory are presented. It is also noted that the Kohn-Sham equation formally coincides with the Hartree-Fock-Slater equation: both equations have a single-particle character and differ from each other only by additions to the Hartree potential.     

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.     

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.     

Defect energy levels in density functional calculations: alignment and band gap problem

For materials of varying band gap, we compare energy levels of atomically localized defects calculated within a semilocal and a hybrid density-functional scheme. Since the latter scheme partially relieves the band gap problem, our study describes how calculated defect levels shift when the band gap approaches the experimental value. When suitably aligned, defect levels obtained from total-energy differences correspond closely, showing average shifts of at most 0.2 eV irrespective of band gap. Systematic deviations from ideal alignment increase with the extent of the defect wave function. A guideline for comparing calculated and experimental defect levels is provided.     

Band gap engineering in silicene: A theoretical study of density functional tight-binding theory

In this work, we performed first principles calculations based on self-consistent charge density functional tight-binding to investigate different mechanisms of band gap tuning of silicene. We optimized structures of silicene sheet, functionalized silicene with H, CH₃ and F groups and nanoribbons with the edge of zigzag and armchair. Then we calculated electronic properties of silicene, functionalized silicene under uniaxial elastic strain, silicene nanoribbons and silicene under external electrical fields. It is found that the bond length and buckling value for relaxed silicene is agreeable with experimental and other theoretical values. Our results show that the band gap opens by functionalization of silicene. Also, we found that the direct band gap at K point for silicene changed to the direct band gap at the gamma point. Also, the functionalized silicene band gap decrease with increasing of the strain. For all sizes of the zigzag silicene nanoribbons, the band gap is near zero, while an oscillating decay occurs for the band gap of the armchair nanoribbons with increasing the nanoribbons width. At finally, it can be seen that the external electric field can open the band gap of silicene. We found that by increasing the electric field magnitude the band gap increases.     

On the validity of the inverse conjecture in classical density functional theory

It is shown that the basic assumptions of the classical density functional approach are rigorously correct forH-stable systems in the grand canonical ensemble. Moreover, it is established that the set of all single-particle densities is convex. These results are derived by providing necessary and sufficient conditions for the solution of the classical inverse problem for single-particle densities. Analogous results are obtained for the solution of the higher-order correlation inverse problem, and the ramifications of these results for the validity of two-body decomposition of forces are discussed.Research supported in part by the National Science Foundation under grants PHY-8116101 A01 (J.T.C.), PHY-8301493 (L.C.) and PHY-8203669 (J.T.C. and L.C.).     

Nuclear density functional theory

An understanding of atomic nuclei is crucial for a complete nuclear theory, for the nuclear astrophysics, for performing new experimental tasks, and for various other applications. Within a density functional theory, the total binding energy of the nucleus is given by a functional of the nuclear density matrices and their derivatives. The variation of the energy density functional with respect to particle and pairing densities leads to the Hartree-Fock-Bogoliubov equations. The “Universal Nuclear Energy Density Functional” (UNEDF) SciDAC project to develop and optimize the energy density functional for atomic nuclei using state-of-the-art computational infrastructure, is briefly described. The ultimate goal is to replace current phenomenological models of the nucleus with a well-founded microscopic theory with minimal uncertainties, capable of describing nuclear data and extrapolating to unknown regions.     

Developments in excited-state density functional theory

This article discusses the reasons behind the apparent lack of success of density functional theory (DFT), during the past three decades, with excited states of many-electron systems. It describes various variational and non-variational approaches developed so far for dealing with this problem. Those include Theophilou’s equiensemble approach, extended to unequally weighted ensembles by Gross et al., Fritsche’s wavefunction partitioning approach, local scaling transformation theory by Kryachko et al., the work-function formalism developed by Harbola and Sahni, etc. Through intimate connections between time-dependence and excited states, under a perturbation, various time-dependent (TD) DFT approaches to excited states, e.g., a quantum fluid dynamical approach, a TD density-functional response theory and a TD optimized effective potential approach, are also reviewed.     

Memory in time-dependent density functional theory

Exact time-dependent density functionals remember both the entire history of the density and the initial wave function. We show that the two effects are intimately related, and all history dependence can be written as initial-state dependence, including that of the exchange-correlation kernel. For states that can be evolved from a ground state, all initial-state dependence is a dependence on a pseudo-prehistory, providing a route to excited-state densities from time-dependent density functional theory.     

Derivative discontinuities in density functional theory

Fifty years after the original formulation of density functional theory (DFT), subtle consequences of the mathematical mappings underlying its formalism continue to merit new views. In this article, we discuss the origin, the importance, and the challenges associated with finding the derivative discontinuity of the exchange-correlation (XC) energy of DFT at integer–electron numbers. We show how even the energy of a quantum electron gas with finite volume and number of electrons displays such derivative discontinuities, but continuous density functional approximations to the XC functional miss them entirely. We discuss some of the practical problems that arise due to this lack of derivative discontinuities in standard functionals, and explain new ways to recover them.     

The problem of the universal density functional and the density matrix functional theory

The analysis in this paper shows that the Hohenberg-Kohn theorem is the constellation of two statements: (i) the mathematically rigorous Hohenberg-Kohn lemma, which demonstrates that the same ground-state density cannot correspond to two different potentials of an external field, and (ii) the hypothesis of the existence of the universal density functional. Based on the obtained explicit expression for the nonrel-ativistic particle energy in a local external field, we prove that the energy of the system of more than two non-interacting electrons cannot be a functional of the inhomogeneous density. This result is generalized to the system of interacting electrons. It means that the Hohenberg-Kohn lemma cannot provide justification of the universal density functional for fermions. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect. In the framework of the density matrix functional theory, the hypothesis of the existence of the universal density matrix functional corresponds to the cases of noninteracting particles and to interaction in the Hartree-Fock approximation.     

Density functional theory for nonuniform polymer melts: Based on the universality of the free energy density functional

A density functional theory is proposed for nonuniform freely jointed tangential hard sphere polymer melts in which the bonding interaction is treated on the basis of the properties of the Dirac δ-function, thus avoiding the use of the single chain simulation in the theory. The excess free energy is treated by making use of the universality of the free energy density functional and the Verlet-modified (VM) bridge function. To proceed numerically, one of the input parameters, the second-order direct correlation function of a uniform polymer melt is obtained by solving numerically the Polymer-RISM integral equation with the Percus-Yevick (PY) closure. The predictions of the present theory for the site density distribution, the partition coefficient and the adsorption isotherm, near a hard wall or between two hard walls are compared with computer simulation results and with those of previous theories. Comparison indicates that the present approach is more accurate than the previous integral equation theory and the most accurate Monte Carlo density functional theories. The predicted oscillations of the medium-induced force between two hard walls immersed in polymer melts are consistent with the experimental results available in the literature. Received 18 April 2000     

Non-empirical pairing energy density functional

We perform systematic calculations of pairing gaps in semi-magic nuclei across the nuclear chart using the Energy Density Functional method and a non-empirical pairing functional derived, without further approximation, at lowest order in the two-nucleon vacuum interaction, including the Coulomb force. The correlated single-particle motion is accounted for by the SLy4 semi-empirical functional. Rather unexpectedly, both neutron and proton pairing gaps thus generated are systematically close to experimental data. Such a result further suggests that missing effects, i.e. higher partial waves of the NN interaction, the NNN interaction and the coupling to collective fluctuations, provide an overall contribution that is sub-leading as for generating pairing gaps in nuclei. We find that including the Coulomb interaction is essential as it reduces proton pairing gaps by up to 40%.     

Kinetic component of the correlation energy density functional: a quantitative description from information theory

In the framework of information theory, a new method to determine T _c , the kinetic energy component of the correlation energy density functional for atoms, is presented. This approach is based on Shannon entropy and information energy that are obtained by fractional occupation probabilities of natural atomic orbitals. It is indicated that the calculated Shannon entropy using discrete probabilities is an increasing function while information energy is a decreasing function of the number of electrons. An expression is proposed with explicit dependence on the Shannon entropy or information energy and atomic number for the purpose. Applications of formulas for estimation of T _c values for neutral atoms up to Xe and their first positive and negative ions are then examined and validity of the proposed approach is numerically verified.     

One-dimensional hypervirial theorems in density functional theory

For a restricted class of operators hypervirial theorems are established involving wavefunctions ψ_i for N fermions which move independently in a common external one-dimensional potential V(x). Using this class it is possible to perform the summation over states ψ_i yielding exact relations between ground state density ? and kinetic energy density ?_k and potential V(x). It is checked to what extent the Thomas-Fermi expressions for ? and ?_k satisfy these relations.     

Resonant raman scattering at the E1 energy gap of semiconductor crystals

We present a discussion of resonant Raman scattering by optical phonons at the E₁ energy gap of group IV and groups III–V compound semiconductor crystals (e.g., Ge and InSb). For allowed scattering by TO and LO phonons, the q-dependent “double resonant” two-band calculation of the Raman tensor may display destructive interference effects when the intermediate electron-hole pairs are uncorrelated. We also discuss the Franz-Keldysh mechanism of resonant electric field induced Raman scattering by LO phonons. The double resonance terms due to this mechanism will, for large electric fields, broaden and have its largest resonance enhancement at the energy gap.     

Single-particle energies and density of states in density functional theory

Time-dependent density functional theory (TD-DFT) is commonly used as the foundation to obtain neutral excited states and transition weights in DFT, but does not allow direct access to density of states and single-particle energies, i.e. ionisation energies and electron affinities. Here we show that by extending TD-DFT to a superfluid formulation, which involves operators that break particle-number symmetry, we can obtain the density of states and single-particle energies from the poles of an appropriate superfluid response function. The standard Kohn– Sham eigenvalues emerge as the adiabatic limit of the superfluid response under the assumption that the exchange– correlation functional has no dependence on the superfluid density. The Kohn– Sham eigenvalues can thus be interpreted as approximations to the ionisation energies and electron affinities. Beyond this approximation, the formalism provides an incentive for creating a new class of density functionals specifically targeted at accurate single-particle eigenvalues and bandgaps.     

On the optical energy gap of SmN

The fundamental absorption edge of SmN is measured by the optical reflection and the transmission technique using a FTIR spectrometer. The MgF₂ passivated thin films of SmN were grown by thermal evaporation. The optical spectra was collated in the energy range of 0.5 to 5.0 eV, and the optical energy gap is measured at 1.2 eV, the same as that of DyN. The measured value of the onset of the absorption in SmN does agree with that theoretically calculated, if the spin–orbit coupling is accounted for.