Energy Density Functional Theory Of Many Electron Systems

This book is an outcome of the International Workshop on Electronic Density Functional Theory, held at Griffith University in Brisbane, Australia, in July 1996. Density functional theory, standing as it does at the boundary between the disciplines of physics, chemistry, and materials science, is a great mixer. Invited experts from North America, Europe, and Australia mingled with students from several disciplines, rapidly taking up the informal style for which Australia is famous. A list of participants is given at the end of the book. Density functional theory (DFT) is a subtle approach to the very difficult problem of predicting the behavior of many interacting particles. A major application is the study of many-electron systems. This was the workshop theme, embracing inter alia computational chemistry and condensed matter physics. DFT circumvents the more conceptually straightforward (but more computationally intensive) approach in which one solves the many-body Schrodinger equation. It relies instead on rather delicate considerations involving the electron number density. For many years the pioneering work of Kohn and Sham (the Local Density Ap proximation of 1965 and immediate extensions) represented the state of the art in DFT. This approach was widely used for its appealing simplicity and computability, but gave rather modest accuracy. In the last few years there has been a renaissance of interest, quite largely due to the remarkable success of the new generation of gradient functionals whose initiators include invitees to the workshop (Perdew, Parr, Yang).

This volume records the proceedings of a Forum on The Fundamentals of Electron Density, Density Matrix and Density Functional Theory in Atoms, Molecules and the Solid State held at the Coseners' House, Abingdon-on-Thames, Oxon. over the period 31st May - 2nd June, 2002. The forum consisted of 26 oral and poster presentations followed by a discussion structure around questions and comments submitted by the participants (and others who had expressed an interest) in advance of the meeting. Quantum mechanics provides a theoretical foundation for our under standing of the structure and properties of atoms, molecules and the solid state in terms their component particles, electrons and nuclei. (Rel ativistic quantum mechanics is required for molecular systems contain ing heavy atoms.) However, the solution of the equations of quantum mechanics yields a function, a wave function, which depends on the co ordinates, both space and spin, of all of the particles in the system. This functions contains much more information than is required to yield the energy or other property.

The prediction of ground state properties of atomistic systems is of vital importance in technological advances as well as in the physical sciences. Fundamentally, these predictions are based on a quantum-mechanical description of many-electron systems. One of the hitherto most prominent theories for the treatment of such systems is density functional theory (DFT). The main reason for its success is due to its balance of acceptable accuracy with computational efficiency. By now, DFT is applied routinely to compute the properties of atomic, molecular, and solid state systems. The general approach to solve the DFT equations is to use a density-functional approximation (DFA). In Kohn-Sham (KS) DFT, DFAs are applied to the unknown exchangecorrelation (xc) energy. In orbital-free DFT on the other hand, where the total energy is minimized directly with respect to the electron density, a DFA applied to the noninteracting kinetic energy is also required. Unfortunately, central DFAs in DFT fail to qualitatively capture many important aspects of electronic systems. Two prime examples are the description of localized electrons, and the description of systems where electronic edges are present. In this thesis, I use a model system approach to construct a DFA for the electron localization function (ELF). The very same approach is also taken to study the non-interacting kinetic energy density (KED) in the slowly varying limit of inhomogeneous electron densities, where the effect of electronic edges are effectively included. Apart from the work on model systems, extensions of an exchange energy functional with an improved KS orbital description are presented: a scheme for improving its description of energetics of solids, and a comparison of its description of an essential exact exchange feature known as the derivative discontinuity with numerical data for exact exchange. An emerging alternative route towards the prediction of the properties of atomistic systems is machine learning (ML). I present a number of ML methods for the prediction of solid formation energies, with an accuracy that is on par with KS DFT calculations, and with orders-of-magnitude lower computational cost. Att kunna förutsäga egenskaper hos atomistiska system utgör en viktigdel av vår teknologiska utveckling, samt spelar en betydande roll i defysikaliska vetenskaperna. Sådana förutsägelser bygger på en kvantmekaniskbeskrivning av mångelektronsystem. En av de mest framståendeteorierna för att behandla den här typen av system är täthetsfunktionalteorin(DFT). Den främsta orsaken till dess framgång är attden lyckas kombinera skaplig noggrannhet med en bra beräkningseffektivitet.DFT används numera rutinmässigt för att beräkna storheterhos atomer, molekyler, och fasta kroppar. Generellt sett löses ekvationerna inom DFT genom att man inför entäthetsfunktionalapproximation (DFA). I Kohn-Sham (KS) DFT, användsDFAer för att approximera utbytes-korrelationsenergin. Inom orbitalfriDFT, där målet är att direkt minimera den totala energin med avseendepå elektrontätheten, så approximerar man också den icke-interageranderörelseenergin hos elektronerna. Dessvärre så fallerar många centralaDFAer att kvalitativt beskriva många viktiga aspekter hos elektronsystem.Två viktiga exempel är beskrivningen av lokaliserade elektroner,samt beskrivningen av system där det förekommer elektronytor. I denna avhandling använder jag modellsystem för att konstruera enDFAför elektronlokaliseringsfunktionen (ELF). Samma tillvägagångssättappliceras sedan för att studera den kinetiska energitätheten i gränsen avlångsamt varierande elektrontätheter, där effekten av elektronytor effektivtinkluderas. Förutom arbetet som berör modellsystem, så presenterasen utökad variant av en utbytes-energifunktional med en förbättrad KSorbitalbeskrivning: ett schema för att förbättra dess energiegenskaperför solida material, samt en jämförelse av dess beskrivning av en viktigegenskap hos den exakta utbytesenergin, vilket utgörs av diskontinuiteteri dess derivata. Ett mera nyligen uppkommet samt alternativt sätt att kunna förutsägaegenskaper hos atomistiska system utgörs av maskinlärning (ML).Jag presenterar ett antal ML-modeller för att kunna förutsäga formeringsenergierhos fasta material med en noggrannhet som är i linje medresultat som uppnås av beräkningar med hjälp av KS DFT, och med enberäkningseffektivitet som är flera storleksordningar snabbare.

Density Functional Theory is a rapidly developing branch of many-particle physics that has found applications in atomic, molecular, solid-state and nuclear physics. This book describes the conceptual framework of density functional theory and discusses in detail the derivation of explicit functionals from first principles as well as their application to Coulomb systems. Both non-relativistic and relativistic systems are treated. The connection of density functional theory with other many-body methods is highlighted. The presentation is self-contained; the book is, thus, well suited for a graduate course on density functional theory.

In my original proposal to Springer for a book on Quantal Density Functional Theory, I had envisaged one that was as complete in its presentation as possible, describing the basic theory as well as the approximation methods and a host of applications. However,after workingon the bookforabout ?ve years, I realizedthat the goal was too ambitious, and that I would be writing for another ?ve years for it to be achieved. Fortunately,there was a natural breakin the material, and I proposed to my editor, Dr. Claus Ascheron, that we split the book into two components: the ?rst on the basic theoretical framework, and the second on approximation methods and applications. Dr. Ascheron consented, and I am thankful to him for agreeing to do so. Hence, we published Quantal Density Functional Theory in 2004, and are now publishing Quantal Density Functional Theory II: Approximation Methods and Applications. One signi?cant advantage of this, as it turns out, is that I have been able to incorporate in each volume the most recent understandings available. This volume, like the earlier one, is aimed at advanced undergraduates in physics and chemistry, graduate students and researchers in the ?eld. It is written in the same pedagogical style with details of all proofs and numerous ?gures provided to explain the physics. The book is independent of the ?rst volume and stands on its own. However, proofs given in the ?rst volume are not repeated here.

The first Nato Advanced Studies Institute entirely devoted to density functional theory was held in Portugal in September 1983. The proceedings of this School, publis hed in early 1985, is still used as a standard reference covering the basic development of the theory and applications in atomic, molecular, solid state and nuclear physics. Ho wever, astonishing progress has been achieved in the intervening years: The foundations of the theory have been extended to cover excited states and time dependent problems more fully, density functional theory of classical liquids and superconducting systems has been addressed and extensions to relativistic, that is, field theoretical systems, as well as a more thorough discussion of magnetic field problems have been presented. In addition, new functionals have been devised, for instance under the heading of ge neralised gradient expansions, and the number of applications in the traditional fields has steadily increased, in particular in chemistry. Applications in new fields, as for instance the structure of atomic clusters and the marriage of density functional theory with molecular dynamics and simulated annealing, have provided additional impetus to the field of density functional theory.

Density functional theory (DFT) is by now a well-established method for tackling the quantum mechanics of many-body systems. Originally applied to compute properties of atoms and simple molecules, DFT has quickly become a work horse for more complex applications in the chemical and materials sciences. The present set of lectures, spanning the whole range from basic principles to relativistic and time-dependent extensions of the theory, is the ideal introduction for graduate students or nonspecialist researchers wishing to familiarize themselves with both the basic and most advanced techniques in this field.