Recent Advances in Density Functional Methods

The following sections are included:

• INTRODUCTION

• FOREWORD

• PREFACE

• CONTENTS

The differential form of the virial theorem for spherically symmetric systems is derived. As an application the kinetic energy density for atomic s and p shells in a bare Coulomb potential is obtained. The local virial theorem is presented. It is also formalized using the local temperature. The integral form of the virial theorem is derived. It can be related to the sum of orbital energies. It is shown that the orbital energy sum can be approximated by the integral of the pressure. Regional virial theorem is presented. The regional analogue of the Levy–Perdew relation can also be obtained. Spin virial theorem is derived. It provides a new way of checking the accuracy of spin orbitals.

Hierarchies of equations are derived for the kinetic, exchange–correlation and the exchange energy functionals. In the local density approximation the first equation of the hierarchies leads to the well-known Thomas–Fermi expression for the kinetic energy and the Slater–Gáspár–Kohn–Sham expression for the exchange energy. The truncation of hierarchies of the kinetic and exchange energies leads to lower and upper bounds to these energies.

Potential–phase relations for two–level and three-level spherically symmetric systems are derived. A numerical method is proposed to obtain the Kohn–Sham, exchange, exchange–correlation and Pauli potentials.

A thermodynamical description of the ground–state density functional theory is presented. The entropy derived contains a correction to the Sackur–Tetrode equation coming from the “nonideality”.

A relativistic density–functional theory for ensembles of unequally weighted states is formulated. This is a relativistic generalization of the formalism proposed by Gross, Oliveira and Kohn. Relativistic and nonrelativistic excitation energies obtained by local density approximation are presented.

Density Functional Theory is outlined and the problem of Self Interaction discussed. Correlation parameters for the electron pair interaction are introduced and general boundary conditions and those for the Generalized Local Spin Density, Hartree and Hartree-Fock theories derived. Correlation factors are derived for both the Fermi and Coulomb Holes. The connection with Quantum Field Theory is outlined and expressed in terms of the one- and two-density matrices, and the g-Hartree method introduced. The Correlation Factors in both DFT and QFT are shown to be related by g = ½ (1 + ƒ^c) where g is from the g-Hartree approach and ƒ^c from the Density Functional method.

In this article we consider the Kohn-Sham density-functional theory Hartree, Local Density and Gradient Expansion approximations to 0(∇²) from the perspective of the work formalism of electronic structure. In Kohn-Sham theory, the local potential representing electron correlations is obtained by functional differentiation of the electron-interaction energy functional of the density. Thus, in approximate Kohn-Sham theory, the electrons are considered correlated as assumed in the construction of the approximate energy functional. In the work formalism, the electron-interaction energy and local potential are both derived via Coulomb's law from a quantum-mechanical source charge distribution. This charge is the pair-correlation density. The electron-interaction potential is the work done to move an electron in the force field of the pair-correlation density, and the energy is the energy of interaction between the electronic and pair-correlation densities. The potentials of the three Kohn-Sham theory approximations considered can be rederived via a hierarchy within the work formalism by determining the appropriate source charge distribution in each case. These correspond to the zeroth-order and the sums to odd-order up to terms of 0(∇³) of the expansion of the pair-correlation density in gradients of the density about the uniform electron gas result. As a consequence of the work formalism derivation, electron correlations beyond those assumed within each approximation of Kohn-Sham theory then emerge. In the density-functional theory Hartree approximation, correlations due to the Pauli exclusion principle and Coulomb repulsion are observed to now exist. In the Local Density approximation, which within Kohn-Sham theory is assumed to be based on the uniform electron gas, the additional correlations explicitly account for the nonuniformity of the electronic density through a term proportional to its gradient. This then constitutes the fundamental reason for the accuracy of the approximation. The singular behavior of the Gradient Expansions for exchange and correlation in turn are a consequence of the corresponding additional correlations in these approximations. The structure of the correlations in each of the Kohn-Sham theory approximations considered as derived by the work formalism is then demonstrated by application to the nonuniform densities in atoms and at metal surfaces.

Several schemes which extend the Kohn-Sham construction to multideterminantal wavefunctions are possible. The following ones are presented:

• use of the eigenfunctions of Ŝ²

• selection of a reference orbital space

• splitting of the electron-electron interaction operator

With multideterminantal wavefunctions, it is necessary to replace the dependence of the functionals on the spin-density by that of a dependence on the on-top pair density.

Time-dependent density-functional response theory (TD-DFRT) is presented from the point of view of quantum chemistry. The extension of density-functional theory (DFT) into the time-domain is reviewed from the point of view of Runge, Gross, and Kohn. The basic working equations of TD-DFRT are then derived in a form analogous to the time-dependent Hartree–Fock (TDHF) equations used for molecular calculations. This is the first practical formulation of TD-DFRT for molecular applications, and the equations are presented in a more general form than has been the case for either atoms or solids. In particular, the present TD-DFRT equations anticipate applications to open-shell molecules based on spin-unrestricted DFT equations with fractional occupation numbers, and are general enough to accept time-dependent exchange-correlation functionals beyond the adiabatic approximation. The use of auxiliary function techniques to eliminate the four-center integrals that arise in TD-DFRT is discussed. The simple example of H₂ is used to illustrate the TD-DFRT method and its relationship to TDHF and to the usual ad hoc ΔSCF-based DFT treatment of excited states. The TD-DFRT method set forth here provides a powerful DFT technique for the calculation of such optical properties as dynamic polarizabilities and electronic excitation spectra, and can be readily extended to treat a number of other properties such as hyperpolarizabilities and intermolecular forces.

I review developments and applications of corrected effective medium (CEM) theories of chemical bonding that have occurred since the 1991 review¹ by T. J. Raeker and A. E. DePristo, “Theory of Chemical Bonding based upon the Atom-Homogeneous Electron Gas System.” The CEM theory is based upon replacement of an N-atom system by N individual reference systems each consisting of an atom in a suitably chosen ‘effective’ medium. While the original reference system was an atom embedded in jellium, a homogeneous electron gas with compensating positive background, new reference systems have been developed which yield substantially more accurate results and which allow critical evaluation of the fundamental limitations on the accuracy. Both topics are reviewed here. Most of the work since the last review has involved applications to systems containing large numbers of atoms with low symmetry. These topics form the bulk of the present review:

• monometallic and bimetallic clusters

• molecular chemisorption

• diffusion of metal atoms and clusters on metal surfaces

• bulk and surface mixing and alloy formation.

We present results of Linear Combination of Gaussian-Type Orbitals-Density Functional (LCGTO-DF) calculations on complexes of small molecules bonded to atoms and small clusters of transition metals. We first present our computational method and some test calculations illustrating the effect of different levels of approximation. The effects of gradient corrections to the exchange-correlation are particularly important and can be summarized as follows: binding energies decrease (and improve) substantially, bond lengths increase by 0.01-0.02Å for main-group elements and by up to 0.05Å for bonds involving a metal atom, and there is a corresponding decrease (and improvement) in bond stretch frequencies. We show extensive comparisons of calculated vibrational frequencies, infrared intensities, and metal-ligand binding energies with experimental data. The accuracy of theoretical predictions is not uniform: some complexes, e.g. those which are weakly bonded, are much more difficult to describe theoretically. The typical accuracy of calculations is: ±50 cm⁻¹ or 1–2% on vibrational frequencies associated with the ligand, ±50 cm⁻¹ or 5 –10% on those involving the metal atom(s), ±5−10% on isotopic shifts, ±5 kcal/mol on metal-ligand binding energies — with some notable failures where errors are ≈10–15 kcal/mol. We discuss the nature of the metal-ligand bond and implications for surface science along the lines of the cluster-surface analogy. We also give a personal perspective on recent and (possible) future applications of DFT to monoligand complexes and other models for the bonding of small molecules at metal centers of surfaces and catalysts.

This contribution is devoted to the impact of density functional theory in the most important building blocks of a comprehensive theoretical characterization and analysis of processes involving open-shell species. After a discussion of the theoretical background, I report the essentials of a number of case studies selected to show the potentialities and limits of local and gradient corrected density functionals in dealing with structural, spectroscopic, thermochemical and kinetic aspects. At the same time I introduce self-consistent hybrid methods obtained by adding some Hartree–Fock exchange to gradient-corrected functionals. The results obtained at this level for a number of properties are competitive with the most sophisticated post-Hartree–Fock approaches. The speed of the computations and the availability of analytical first and second derivatives allow the complete characterization of stationary points and the analysis of significant regions of potential energy surfaces for systems much larger than those which can be dealt with by conventional ab-initio methods. This paves the route toward the study of large amplitude motions, from conformational transitions to true chemical reactions.

The reliability of the Linear Combination of Gaussian-Type Orbitals-Density Functional (LCGTO-DFT) Method for the determination of physico-chemical properties is shown for a series of organic, inorganic, hydrogen bonded and Van der Waals systems. In particular, equilibrium structures, vibrational frequencies, binding energies, energetical parameters, valence and core level shifts, potential energy surfaces and solvent effects have been considered. The good agreement between our results and experimental ones and with high level correlated ab-initio methods further demonstrates that Gaussian DF method can be considered a practical, reliable and economical tool for predicting the electronic and spectroscopic properties of a quite large class of chemical systems.

Electron densities calculated for a variety of systems by means of density functional theory (DFT) are compared with the results given by several conventional ab initio methods, including Hartree-Fock (HF), second-order Møller-Plesset perturbation theory (MP2) and quadratic configuration interaction including single and double excitations (QCISD). The latter is used as a reference by which the accuracy of the DFT electron densities is measured. Among the properties discussed, particular attention is paid to certain topological properties, the Laplacian of the electron density and DFT spin densities. The chapter concludes with a suggestion for an approach for the development of new DFT functionals.

The following sections are included:

• INDEX