Test of density-functional approximations in an exactly soluble 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

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.

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.

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 band-gap problem is studied for an exactly solvable semiconductor model. The exact exchange-correlation potential is constructed and its discontinuity is studied. Depending on the parameter range, a large or a small discontinuity is obtained. The "scissor-operator" method is compared with the local-density approximation for an impurity calculation. In general, the local-density approximation is closer to the exact solution. Some general features of the density-functional formalism are illustrated.

Ab-initio investigation of structural, electronic, magnetic, and thermodynamic properties of XPt3 (X=V, Cr, Mn, and Fe) intermetallic compounds

The exchange and correlation energies and potentials, occurring in various density functional theory approaches and schemes are reviewed and their definitions compared. The ways of their determination and their long-range properties are discussed. General expressions for the exchange and correlation energy and the long-range asymptotic form of the exchange potential are obtained for mixed-state systems. Line-integral expressions for the exchange and correlation potentials, valid both for pure-state and mixed-state systems are derived. Approximation to the electrostatic-plus-exchange energy and corresponding potential for arbitrary mixed-state systems and approximation to the correlation energy and potential for a specific class of mixed-state systems are proposed. They are expressible in terms of any approximate functional of the density for the exchange energy and correlation energy, respectively, known for pure-state systems.

In this paper we assess the accuracy of the local density approximation (LDA) based expressions for exchange and correlation functionals associated with a two-dimensional, spin-polarized, dipolar Fermi gas. For this purpose we consider and study an exactly solvable system of two interacting spin-polarized dipolar fermions confined in a two-dimensional harmonic oscillator potential. We determine the exact triplet ground state wavefunction, energy and the density by separating the time-independent Schrodinger equation in center-of-mass and relative coordinates and numerically solving the eigenvalue equation in relative coordinate. The exact Kohn–Sham orbitals, corresponding energies and exact Kohn–Sham potential are obtained from the exact density by inverting the Kohn–Sham equation. These results serve as benchmark data for assessing the accuracies of the LDA based expressions for a two-dimensional, spin-polarized, dipolar Fermi gas. We find that LDA results for total energy, interaction energy, and Kohn–Sham eigenvalues match well with the exact results for smaller values of dipole–dipole interaction strength. For higher values of interaction strength, LDA results for the total energy are underestimated. On the other hand, the LDA based interaction potentials overestimate the repulsion between the dipoles and their spatial profiles differ considerably from their exact counterparts and the discrepancy between them is higher for stronger interaction.

We intended to discuss in this chapter, TBM (tight binding method), APW (augmentedplane-wave), OPW (orthogonalized -plane-wave) methods and corresponding theoretical concepts. In particulars, we pay a great attention to the theory of CNT (Carbon Nano Tube), but discuss in less details some conventional band structure models, unless nearly electron approximation (NFA), TBM, APW and OPW models have been used for determining the electron energy band structure of solids. In fact, this chapter is partly based on the many – electron description of nano transistor – CNTFET (carbon nano tube field effect transistor), which was done with a number of MSC and PhD students for a number of years at university of Mazandaran in Iran (See our published papers [1-7] for more details). We hope this chapter can complete the present book and be of interest for researchers whom work in the nano technology and for beginners. Some part of the material may be used in lection course for students. There are actually two different approaches for studying the band spectrum of CNT. In the first view, some researchers believe that carbon atoms are as isolated atoms and consider the CNT potential of neighbor's atoms as a perturbation and neglect the intra atomic potential. The second approach is about the density functional theory (DFT), in that the exact exchange energy (EXX) instead of the exchange energy given by the local – density approximation (LDA). The EXX energy, which corresponds to the Fock term in the HartreeFock scheme, is treated as a function of electron densities via the eigenfunctions of the Kohn-Sham KS equations [8]. This approach cannot satisfy the electron behavior in CNT due to its self-interaction-free in its construction. Indeed, this chapter discusses about electronic band energy. It is an energy interval in which electronic states exist in the CNT. This energy structure has been usually obtained by solving the Schrodinger equation for electrons in the CNT. As usual, the electronic wave functions depend on both the wave vector and the spatial coordinates. The eigenvalues and eigenvectors have been determined by Fourier – transforming the differential equation into an algebraic equation. The solution of this equation can be used for some special cases with some reasonable approximation, such as NFE and TB methods. However, these approaches cannot be used for samples with critical dimensions of less than 100 nm due to overlap integrals in nano scale samples. The reason is that carbon atoms are not in fact stationary, but continually undergo vibrations (like thermal vibrations of ions in a crystal) about their positions, in where, the

Interaction energies for a representative sample of 39 intermolecular complexes are calculated using two computational approaches based on the subsystem formulation of density functional theory introduced by Cortona (Phys. Rev. B 44:8454, 1991), adopted for studies of intermolecular complexes (Wesolowski and Weber in Chem. Phys. Lett. 248:71, 1996). The energy components (exchange-correlation and non-additive kinetic) expressed as explicit density functionals are approximated by means of gradient-free- (local density approximation) of gradient-dependent- (generalized gradient approximation) approximations. The sample of the considered intermolecular complexes was used previously by Zhao and Truhlar to compare the interaction energies derived using various methods based on the Kohn-Sham equations with high-level quantum chemistry results considered as the reference. It stretches from rare gas dimers up to strong hydrogen bonds. Our results indicate that the subsystem-based methods provide an interesting alternative to that based on the Kohn-Sham equations. Local density approximation, which is the simplest approximation for the relevant density functionals and which does not rely on any empirical data, leads to a computational approach comparing favorably with more than twenty methods based on the Kohn-Sham equations including the ones, which use extensively empirical parameterizations. For various types of non-bonding interactions, the strengths and weaknesses of gradient-free and gradient-dependent approximations to exchange-correlation and non-additive kinetic energy density functionals are discussed in detail.

An accurate spin-polarized exchange-only Kohn-Sham (KS) potential is constructed from a consideration of the optimized-effective-potential (OEP) method. A detailed analysis of the OEP integral equation for the exchange-only case results in a set of conditions which are manifestly satisfied by the exact OEP; these conditions are employed to construct an approximate OEP, ${\mathit{V}}_{\mathit{x}\mathrm{\ensuremath{\sigma}}}$, and therefore an approximate KS exchange-only potential as a functional of KS orbitals. Further, it is shown that this ${\mathit{V}}_{\mathit{x}\mathrm{\ensuremath{\sigma}}}$ can be derived analytically based on a simple approximation of the Green's functions in the OEP integral equation. The constructed potential, although approximate, contains many of the key analytic features of the exact KS potential: it reduces to the exact KS result in the homogeneous-electron-gas limit, approaches -1/r as r\ensuremath{\rightarrow}\ensuremath{\infty}, yields highest occupied-orbital energy eigenvalues ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}\mathrm{\ensuremath{\sigma}}}$ that satisfy Koopmans's theorem, and exhibits an integer discontinuity when considered as a function of fractional occupancy of the highest-energy occupied single-particle state of a given spin projection \ensuremath{\sigma}. In addition ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}\mathrm{\ensuremath{\sigma}}}$ nearly exactly satisfies Janak's theorem. The approximate OEP is a simple but remarkably accurate representation of the exact, numerically derived exchange-only OEP.Detailed numerical results obtained by employing ${\mathit{V}}_{\mathit{x}\mathrm{\ensuremath{\sigma}}}$ as the exchange-only potential for ten atoms with closed subshells yield total energies, Hartree potentials, single-particle expectation values, and ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}}$ which are in excellent agreement with both exact OEP and Hartree-Fock (HF) results and represent a significant improvement over the results obtained by employing other exchange-only potentials. Similarly, the properties of alkali-metal atoms are calculated including the separate spin-up and spin-down densities to obtain results in excellent agreement with those of spin-unrestricted OEP and HF methods. Finally, we demonstrate the accuracy of ${\mathit{V}}_{\mathit{x}\mathrm{\ensuremath{\sigma}}}$ by calculating the total energy, ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}\mathrm{\ensuremath{\uparrow}}}$, and ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}\mathrm{\ensuremath{\downarrow}}}$ as a function of fractional filling f, of the highest occupied single-particle orbital for the magnesium atom (Z=12) from N=9--12 electrons and find excellent agreement with both spin-unrestricted OEP and HF results even when ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{m}\mathrm{\ensuremath{\sigma}}}$ is strongly dependent on f. In addition we display the integer discontinuity in ${\mathit{V}}_{\mathit{x}\mathrm{\ensuremath{\sigma}}}$ when the highest-energy spin subshell begins to be filled.

We present electronic band-structure calculations of simple metals (Li, Na, K, Rb, and Ca), where we use the exact Kohn-Sham density-functional exchange (EXX) potential combined with the LDA (local-density approximation) correlation potential. The method is the same as one exploited in the previous papers [Phys. Rev. B 50, 14 816 (1994) ; Phys. Rev. Lett. 74, 2989 (1995)]. Surprisingly, in contrast to the case of semiconductors and insulators, the resulting eigenvalues of valence bands are mostly in agreement with the corresponding LDA eigenvalues. In the case of Ca, where the empty d band exists above the Fermi energy, however, the EXX calculation modifies the position of the d bands relative to the s bands considerably, implying the importance of treating the correlation energy on the same footing as the exchange energy.

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.

Chapter 3 Density functional theory of the ground-state magnetic properties of rare earths and actinides

Extensive benchmarks and reviews of time-dependent density functional theory (TDDFT) have been published, covering at least 50 functionals. Here, we do not use one of the (meta)GGA or hybrid functionals but highlight the particular TDDFT method that does not use the Kohn-Sham (KS) potential of some exchange-correlation functional, but uses the exact Kohn-Sham potential or a close approximation to it in the SCF calculations. Such KS-potential-based TDDFT results have been proven to yield excellent results. For routine application, it is required that a computationally simple approximation (as a density functional) to the exact KS potential is available. In this paper, we benchmark TDDFT calculations with a recently developed model KS potential and compare them to advanced quantum chemical methods and experimental data. The target systems here are medium-sized molecules. These TDDFT calculations based on good KS potentials prove to be competitive in accuracy with sophisticated ab initio methods. An advantage is the possibility to use large basis sets (also for large molecules), enabling a description of valence excitations and Rydberg (or mixed valence-Rydberg) excitations on the same footing. This is necessary for high accuracy; a sophisticated but expensive method that cannot handle large basis sets cannot achieve high accuracy. An advantage of the use of a (close to) exact Kohn-Sham potential is the realistic nature (shape and energy) of both the occupied and the virtual KS orbitals, affording an interpretation of excitations in terms of one or a few single orbital-to-orbital transitions. This obviates the need for extensive optimization of the virtual orbitals for the purpose of interpreting excitations. To highlight the importance of the realistic nature of the KS virtuals, including the Rydberg orbitals, we discuss and reassess the nature of the 21 Ag state of s-trans-1,3-butadiene, which has been widely considered as a prototype excitation with large double excitation character. Charge-transfer and true double excitations cannot be handled by the simple ALDA kernel used here.