We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.

Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions

"Chemists familiar with conventional quantum mechanics will applaud and benefit greatly from this particularly instructive, thorough and clearly written exposition of density functional theory: its basis, concepts, terms, implementation, and performance in diverse applications. Users of DFT for structure, energy, and molecular property computations, as well as reaction mechanism studies, are guided to the optimum choices of the most effective methods. Well done!" Paul von RaguE Schleyer "A conspicuous hole in the computational chemist's library is nicely filled by this book, which provides a wide-ranging and pragmatic view of the subject.[...It] should justifiably become the favorite text on the subject for practioneers who aim to use DFT to solve chemical problems." J. F. Stanton, J. Am. Chem. Soc. "The authors' aim is to guide the chemist through basic theoretical and related technical aspects of DFT at an easy-to-understand theoretical level. They succeed admirably." P. C. H. Mitchell, Appl. Organomet. Chem. "The authors have done an excellent service to the chemical community. [...] A Chemist's Guide to Density Functional Theory is exactly what the title suggests. It should be an invaluable source of insight and knowledge for many chemists using DFT approaches to solve chemical problems." M. Kaupp, Angew. Chem.

A Chemist's Guide to Density Functional Theory

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

Density functional theory (DFT) is a (in principle exact) theory of electronic structure, based on the electron density distribution n(r), instead of the many-electron wave function Ψ(r1,r2,r3,...). Having been widely used for over 30 years by physicists working on the electronic structure of solids, surfaces, defects, etc., it has more recently also become popular with theoretical and computational chemists. The present article is directed at the chemical community. It aims to convey the basic concepts and breadth of applications:  the current status and trends of approximation methods (local density and generalized gradient approximations, hybrid methods) and the new light which DFT has been shedding on important concepts like electronegativity, hardness, and chemical reactivity index.

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Density Functional Theory of Electronic Structure

This work presents theory, implementation, and validation of excited state properties obtained from time-dependent density functional theory (TDDFT). Based on a fully variational expression for the excited state energy, a compact derivation of first order properties is given. We report an implementation of analytic excited state gradients and charge moments for local, gradient corrected, and hybrid functionals, as well as for the configuration interaction singles (CIS) and time-dependent Hartree–Fock (TDHF) methods. By exploiting analogies to ground state energy and gradient calculations, efficient techniques can be transferred to excited state methods. Benchmark results demonstrate that, for low-lying excited states, geometry optimizations are not substantially more expensive than for the ground state, independent of the molecular size. We assess the quality of calculated adiabatic excitation energies, structures, dipole moments, and vibrational frequencies by comparison with accurate experimental data for a variety of excited states and molecules. Similar trends are observed for adiabatic excitation energies as for vertical ones. TDDFT is more robust than CIS and TDHF, in particular, for geometries differing significantly from the ground state minimum. The TDDFT excited state structures, dipole moments, and vibrational frequencies are of a remarkably high quality, which is comparable to that obtained in ground state density functional calculations. Thus, yielding considerably more accurate results at similar computational cost, TDDFT rivals CIS as a standard method for calculating excited state properties in larger molecules.

Adiabatic time-dependent density functional methods for excited state properties

A density-functional theory for ensembles of unequally weighted states is formulated on the basis of the generalized Rayleigh-Ritz principle of the preceding paper. From this formalism, two alternative approaches to the computation of excitation energies are derived, one equivalent to the equiensemble method proposed by Theophilou [J. Phys. C 12, 5419 (1979)], the other grounded on an expression relating the excitation energies to the Kohn-Sham single-particle eigenvalues.

Density-functional theory for ensembles of fractionally occupied states. I. Basic formalism.

A density-functional theory for superconductors at arbitrary temperature is described. It leads to equations of the Kohn-Sham type, which incorporate exchange and correlation effects into the Bogoliubov-de Gennes equations for an inhomogeneous superconductor. Further, this formalism yields exchange-correlation corrections to Eilenberger's expression for the thermodynamic potential of a superconductor, and the Ginzburg-Landau equation. Practical aspects of the application of the formalism are discussed.

Density-functional theory for superconductors.

The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues ${E}_{1}$\ensuremath{\le}${E}_{2}$\ensuremath{\le}...\ensuremath{\le}${E}_{M}$ of a Hamiltonian H, and given M real numbers ${w}_{1}$\ensuremath{\ge}${w}_{2}$\ensuremath{\ge}...\ensuremath{\ge}${w}_{M}$g0, an upper bound for the weighted sum ${w}_{1}$${E}_{1}$ +${w}_{2}$${E}_{2}$+...+${w}_{M}$${E}_{M}$ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles (${w}_{1}$=${w}_{2}$=...=${w}_{M}$). Applications of the generalized principle are discussed.

Rayleigh-Ritz variational principle for ensembles of fractionally occupied states

The two density-functional methods of calculating excitation energies proposed in the preceding paper, combined with the recently formulated quasi-local-density approximation for the equiensemble exchange-correlation energy functional [W. Kohn, Phys. Rev. A 34, 737 (1986)], are applied to the He atom. Although the splittings between nearly degenerate levels with different angular momenta are badly overestimated, in both approaches the averages over angular momentum and spin of the experimental excitation energies measured from the ionization threshold are reproduced within a few percent. The computed self-consistent ensemble-averaged densities and the Kohn-Sham potentials associated with them are discussed.

Density-functional theory for ensembles of fractionally occupied states. II. Application to the He atom.

By shifting the reference system for the local-density approximation (LDA) from the electron gas to other model systems, one obtains a new class of density functionals, which by design account for the correlations present in the chosen reference system. This strategy is illustrated by constructing an explicit LDA for the one-dimensional Hubbard model. While the traditional ab initio LDA is based on a Fermi liquid (the three-dimensional interacting electron gas), this one is based on a Luttinger liquid. First applications to inhomogeneous Hubbard models, including one containing a localized impurity, are reported.

Luiz N. Oliveira

Papers

Density-functional theory for ensembles of fractionally occupied states. I. Basic formalism.

Density-functional theory for superconductors.

Rayleigh-Ritz variational principle for ensembles of fractionally occupied states

Density-functional theory for ensembles of fractionally occupied states. II. Application to the He atom.

Density Functionals Not Based on the Electron Gas: Local-Density Approximation for a Luttinger Liquid