This paper presents an evaluation of the performance of time-dependent density-functional response theory (TD-DFRT) for the calculation of high-lying bound electronic excitation energies of molecules. TD-DFRT excitation energies are reported for a large number of states for each of four molecules: N2, CO, CH2O, and C2H4. In contrast to the good results obtained for low-lying states within the time-dependent local density approximation (TDLDA), there is a marked deterioration of the results for high-lying bound states. This is manifested as a collapse of the states above the TDLDA ionization threshold, which is at ??HOMOLDA (the negative of the highest occupied molecular orbital energy in the LDA). The ??HOMOLDA is much lower than the true ionization potential because the LDA exchange-correlation potential has the wrong asymptotic behavior. For this reason, the excitation energies were also calculated using the asymptotically correct potential of van Leeuwen and Baerends (LB94) in the self-consistent field step. This was found to correct the collapse of the high-lying states that was observed with the LDA. Nevertheless, further improvement of the functional is desirable. For low-lying states the asymptotic behavior of the exchange-correlation potential is not critical and the LDA potential does remarkably well. We propose criteria delineating for which states the TDLDA can be expected to be used without serious impact from the incorrect asymptotic behavior of the LDA potential

Molecular excitation energies to high-lying bound states from time-dependent density-functional response theory: Characterization and correction of the time-dependent local density approximation ionization threshold

A scheme that reduces the calculations of ground-state properties of systems of interacting particles exactly to the solution of single-particle Hartree-type equations has obvious advantages. It is not surprising, then, that the density functional formalism, which provides a way of doing this, has received much attention in the past two decades. The quality of the energy surfaces calculated using a simple local-density approximation for exchange and correlation exceeds by far the original expectations. In this work, the authors survey the formalism and some of its applications (in particular to atoms and small molecules) and discuss the reasons for the successes and failures of the local-density approximation and some of its modifications.

The density functional formalism, its applications and prospects

Electronic excitations lie at the origin of most of the commonly measured spectra. However, the first-principles computation of excited states requires a larger effort than ground-state calculations, which can be very efficiently carried out within density-functional theory. On the other hand, two theoretical and computational tools have come to prominence for the description of electronic excitations. One of them, many-body perturbation theory, is based on a set of Green’s-function equations, starting with a one-electron propagator and considering the electron-hole Green’s function for the response. Key ingredients are the electron’s self-energy S and the electron-hole interaction. A good approximation for S is obtained with Hedin’s GW approach, using density-functional theory as a zero-order solution. First-principles GW calculations for real systems have been successfully carried out since the 1980s. Similarly, the electron-hole interaction is well described by the Bethe-Salpeter equation, via a functional derivative of S. An alternative approach to calculating electronic excitations is the time-dependent density-functional theory (TDDFT), which offers the important practical advantage of a dependence on density rather than on multivariable Green’s functions. This approach leads to a screening equation similar to the Bethe-Salpeter one, but with a two-point, rather than a four-point, interaction kernel. At present, the simple adiabatic local-density approximation has given promising results for finite systems, but has significant deficiencies in the description of absorption spectra in solids, leading to wrong excitation energies, the absence of bound excitonic states, and appreciable distortions of the spectral line shapes. The search for improved TDDFT potentials and kernels is hence a subject of increasing interest. It can be addressed within the framework of many-body perturbation theory: in fact, both the Green’s functions and the TDDFT approaches profit from mutual insight. This review compares the theoretical and practical aspects of the two approaches and their specific numerical implementations, and presents an overview of accomplishments and work in progress.

/pdf/electronic-excitations-density-functional-versus-many-body-2tgqd9dfpy.pdf

Electronic excitations: density-functional versus many-body Green's-function approaches

Preface Acknowledgements Notation Part I. Overview and Background Topics: 1. Introduction 2. Overview 3. Theoretical background 4. Periodic solids and electron bands 5. Uniform electron gas and simple metals Part II. Density Functional Theory: 6. Density functional theory: foundations 7. The Kohn-Sham ansatz 8. Functionals for exchange and correlation 9. Solving the Kohn-Sham equations Part III. Important Preliminaries on Atoms: 10. Electronic structure of atoms 11. Pseudopotentials Part IV. Determination of Electronic Structure, The Three Basic Methods: 12. Plane waves and grids: basics 13. Plane waves and grids: full calculations 14. Localized orbitals: tight binding 15. Localized orbitals: full calculations 16. Augmented functions: APW, KKR, MTO 17. Augmented functions: linear methods Part V. Predicting Properties of Matter from Electronic Structure - Recent Developments: 18. Quantum molecular dynamics (QMD) 19. Response functions: photons, magnons ... 20. Excitation spectra and optical properties 21. Wannier functions 22. Polarization, localization and Berry's phases 23. Locality and linear scaling O (N) methods 24. Where to find more Appendixes References Index.

Electronic Structure: Basic Theory and Practical Methods

During the past decade, computer simulations based on a quantum-mechanical description of the interactions between electrons and between electrons and atomic nuclei have developed an increasingly important impact on solid-state physics and chemistry and on materials science—promoting not only a deeper understanding, but also the possibility to contribute significantly to materials design for future technologies. This development is based on two important columns: (i) The improved description of electronic many-body effects within density-functional theory (DFT) and the upcoming post-DFT methods. (ii) The implementation of the new functionals and many-body techniques within highly efficient, stable, and versatile computer codes, which allow to exploit the potential of modern computer architectures. In this review, I discuss the implementation of various DFT functionals [local-density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, hybrid functional mixing DFT, and exact (Hartree-Fock) exchange] and post-DFT approaches [DFT + U for strong electronic correlations in narrow bands, many-body perturbation theory (GW) for quasiparticle spectra, dynamical correlation effects via the adiabatic-connection fluctuation-dissipation theorem (AC-FDT)] in the Vienna ab initio simulation package VASP. VASP is a plane-wave all-electron code using the projector-augmented wave method to describe the electron-core interaction. The code uses fast iterative techniques for the diagonalization of the DFT Hamiltonian and allows to perform total-energy calculations and structural optimizations for systems with thousands of atoms and ab initio molecular dynamics simulations for ensembles with a few hundred atoms extending over several tens of ps. Applications in many different areas (structure and phase stability, mechanical and dynamical properties, liquids, glasses and quasicrystals, magnetism and magnetic nanostructures, semiconductors and insulators, surfaces, interfaces and thin films, chemical reactions, and catalysis) are reviewed. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2008

Ab-initio simulations of materials using VASP: Density-functional theory and beyond.

We derive asymptotically exact results for the charge and spin densities far away from finite systems (atoms and molecules) and far outside solid surfaces. These results are then used to obtain the correct asymptotic form of the exchange-correlation potential of density-functional (DF) theory and to prove that, for all systems, the eigenvalue of the uppermost occupied DF orbital equals the exact ionization potential. For spin-polarized finite systems we show that the uppermost DF eigenvalue in each spin channel is also given by exact excitation energies.

Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues

We present fully self-consistent results for the self-energy of the electron gas within the GW approximation. This means that the self-consistent Green’s function G, as obtained from Dyson’s equation, is used not only for obtaining the self-energy but also for constructing the screened interaction W within the random-phase approximation. Such a theory is particle and energy conserving in the sense of Kadanoff and Baym. We find an increase in the weight of the quasiparticle as compared to ordinary non-self-consistent calculations but also to calculations with partial self-consistency using a fixed W. The quasiparticle bandwidth is larger than that of free electrons and the satellite structure is broad and featureless; both results clearly contradict the experimental evidence. The total energy, though, is as accurate as that from quantum Monte Carlo calculations, and its derivative with respect to particle number agrees with the Fermi energy as obtained directly from the pole of the Green’s function at the Fermi level. Our results indicate that, unless vertex corrections are included, non-self-consistent results are to be preferred for most properties except for the total energy. (Less)

https://www.itp.tugraz.at/LV/sormann/TFKP4/papers/Holm_98.pdf

Fully self-consistent GW self-energy of the electron gas

We report quasiparticle calculations of the newly observed wurtzite polymorph of InAs and GaAs. The calculations are performed in the GW approximation (based on a model dielectric function) using plane waves and pseudopotentials. For comparison we also report the study of the zinc-blende phase within the same approximations. In the InAs compound the In 4d electrons play a very important role: whether they are frozen in the core or not leads either to a correct or a wrong band ordering (negative gap) within the local-density appproximation (LDA). We have calculated the GW band structure in both cases. In the first approach, we have estimated the correction to the pd repulsion calculated within the LDA and included this effect in the calculation of the GW corrections to the LDA spectrum. In the second case, we circumvent the negative gap problem by first using the screened exchange approximation and then calculating the GW corrections starting from the so obtained eigenvalues and eigenfunctions. This approach, that can be thought of as a step towards self-consistency, leads to a more realistic band structure and was also used for GaAs. For both InAs and GaAs in the wurtzite phase we predict an increase of the quasiparticle gap with respect to the zinc-blende polytype.

Model GW band structure of InAs and GaAs in the wurtzite phase

The basic idea behind the present work is that an atom is not a linear perturbation of the electron gas. We have thus analyzed the exchange energy of the inhomogeneous electron gas to third order in the deviation from a constant density. We give the symmetry properties obeyed by the corresponding second-order response function Lx, and demonstrate how Lx gives rise to gradient corrections to the exchange energy. The expansion, which is taken up to sixth order in the density gradient, also includes the Laplacian of the density. In the case of a statically screened Coulomb interaction, we have calculated the coefficients of second- and fourth-order gradient terms both analytically and numerically. In analogy with the corresponding results from linear-response theory, the fourth-order coefficient is shown to diverge as the screening is made to vanish. For the bare Coulomb interaction we have not succeeded in obtaining analytical results, and, due to numerical problems at small-q vectors, our numerically obtained coefficients have an estimated uncertainty of 20%. (Less)

Gradient expansion of the exchange energy from second-order density response theory.

In these notes I have given a personally flavored exposeof static density- functional theory (DFT). I have started from standard many-body physics at a very elementary level and then gradually introduced the basic concepts of DFT. Successively more advanced topics are added and at the end I even discuss a few not yet published theories. The discussion represents many of the personal views of the author and there is no attempt at being comprehensive. I fully realize that I am often 'unfair' in treating the achievements of other researchers. Many topics of standard DFT are deliberately left out like, e.g., time- dependence, excitations, and magnetic or relativistic effects. These notes represent a compilation of a series of lectures given at at the EXC!TING Summer School DFT beyond the ground state at Riksgransen, Sweden in June of 2003. 1. Background Many reviews and articles on the topic of density- functional theory (DFT) start by the proclamation that, over the past so and so years, DFT has become, by far, the most prominent tool for the calculation of the ground-state properties of electronic systems. Although somewhat vacuous the statement is definitely true. In fact, DFT calculations of the electronic properties of real materials have nowadays turned into an extensive industrial endea- vor. The idea of using the density as the basic variable for the description of the energies of electronic systems goes back almost to the advent of quantum mechanics and the realization that the solution of the full equation of Schrodinger was beyond reach in most cases. The statistical atom of Gombas (1) and the approximations by Thomas (2) and Fermi (3) were early attempts in this direction. Then, of course, came the Hohenberg and Kohn (4) theorems in the mid sixties followed by the work by Kohn and Sham (5). They demonstrated that the electron density of a fully interacting system could actually, in a rigorous way, be obtained from simple one-electron theory. At that time, most researchers involved with the calculation of the electronic properties of atoms, molecules, and solids where strongly influenced by the school of J. C. Slater. Properties were calculated from one-electron theory using a statistical approximation but only for the effect of exchange and correlation. The latter was obtained in 1951 by Slater (6) as an average over the Fermi sea of the self-energy of the homogeneous electron gas treated within the Hartree-Fock approximation. As a matter of fact, from a numerical point of view, that theory was not very different from modern DFT within the local-density approximation (LDA).

/pdf/basic-density-functional-theory-an-overview-4dvv33yxu1.pdf

U. von Barth

Papers

Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues

Fully self-consistent GW self-energy of the electron gas

Model GW band structure of InAs and GaAs in the wurtzite phase

Gradient expansion of the exchange energy from second-order density response theory.

Basic density-functional theory - an overview