Electronic Density Functional Theory

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(r 1 ,r 2 ,r 3 ,...). 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.

Over the past few decades, tremendous progress has been made in the development of computational methods for predicting the properties of materials. At the heart of this progress is density functional theory (DFT) [13, 17, 31, 39, 65], one of the most powerful and efficient computational modeling techniques for predicting electronic properties in chemistry, physics, and material science. Prior to the introduction of DFT in the 1960s [31, 39] the only obvious method for obtaining the electronic energies of materials required a direct solution of the many-body Schrödinger equation [62]. While the Schrödinger equation provides a rigorous path for predicting the electronic properties of any material system, analytical solutions for realistic systems having more than one interacting electron are out of reach. Moreover, since the Schrödinger equation is inherently a many-body formalism (3N spatial coordinates for N strongly interacting electrons), numerically accurate solutions of multi-electron systems are also impractical. Instead of the full 3N-dimensional Schrödinger equation, DFT recasts the electronic problem into a simpler yet mathematically equivalent 3-dimensional theory of non-interacting electrons (cf. Fig. 4.1). The exact form of this electron density, .D n.r//, hinges on the mathematical form of the exchange-correlation functional, E xc OEn.r/, which is crucial for providing accurate and efficient solutions to the many-body Schrödinger equation. Unfortunately, the exact form of the exchange-correlation functional is currently unknown, and all modern DFT functionals invoke various degrees of approximation.

We propose an in silico experiment to introduce the classical density functional theory (cDFT). Density functional theories, whether quantum or classical, rely on abstract concepts that are nonintuitive; however, they are at the heart of powerful tools and active fields of research in both physics and chemistry. They led to the 1998 Nobel Prize in chemistry. A DFT is illustrated here in its most simple and yet physically relevant form: the cDFT of an ideal fluid of classical particles. For illustration purposes, it is applied to the prediction of the molecular structure of liquid neon. The numerical experiment proposed therein is built around the writing of a cDFT code by students in Mathematica software. Students thus must deal with (i) the cDFT theory, (ii) some basic concepts of the statistical mechanics of simple fluids, (iii) functional minimization, and (iv) a useful functional programming language. This computational experiment is proposed during a molecular simulation class but may also be of interest in a quantum chemistry class to illustrate electronic DFT if the instruction highlights the analogies between quantum and classical DFTs.

For the past 30 years density functional theory has been the dominant method for the quantum mechanical simulation of periodic systems. In recent years it has also been adopted by quantum chemists and is now very widely used for the simulation of energy surfaces in molecules. In this lecture we introduce the basic concepts underlying density functional theory and outline the features that have lead to its wide spread adoption. Recent developments in exchange correlation functionals are introduced and the performance of families of functionals reviewed. The lecture is intended for a researcher with little or no experience of quantum mechanical simulations but with a basic (undergraduate) knowledge of quantum mechanics. We hope to provide sufficient background to enable informed judgements on the applicability of a particular implementation of density functional theory to a specific problem in materials simulation. For those who wish to go more deeply into the formalism of density functional theory there are a number of reviews and books aimed at intermediate and advanced levels available in the literature [1,2,3]. Where appropriate source articles are referred to in the text.

Density Functional Theory - Recent Advances, New Perspectives and Applications, 2021

Density Functional Theory (DFT) is a powerful and commonly employed quantum mechanical tool for investigating various aspects of matter. The research in this field ranges from the development of novel analytical approaches focused on the design of precise exchange-correlation functionals to the use of this technique to predict the molecular and electronic configuration of atoms, molecules, complexes, and solids in both gas and solution phases. The history to DFT’s success is the quest for the exchange-correlation functional, which utilizes density to represent advanced many-body phenomena inside one element formalism. If a precise exchange-correlation functional is applied, it may correctly describe the quantum nature of matter. The estimated character of the exchange-correlation functional is the basis for DFT implementation success or failure. Hohenberg-Kohn established that every characteristic of a system in ground state is a unique functional of its density, laying the foundati...

Progress in Theoretical Chemistry and Physics, 2003

2013

Se presentan los fundamentos matematicos de la teoria funcional de la densidad DFT en este trabajo. Empezamos con el inicio de la mecanica cuantica de esta teoria, es decir, el modelo de Thomas-Fermi (TF), que utiliza la densidad de electrones n (R), una funcion de solo 3 coordenadas, como la unica variable fisica. A continuacion mostramos la fundacion formal de DFT, los teoremas de Hohenberg y Kohn, expresados en una teoria bien establecida representada por pruebas twoexcited. Le mostramos al final del articulo como Kohn y Sham (KS) idearon una aplicacion practica y trajeron DFT en los calculos de la corriente principal de la estructura electronica.

2003

International Journal of Quantum Chemistry, 1992

Quantum chemistry is a very specialized subbranch of the natural sciences whose principal aim is to elucidate behaviors of real many-electron systems at the quantum-mechanical level of description. Quantum chemists wish to realize the belief that quantum mechanics is quite able to give everything of what already has happened, happens, and will happen with atoms, molecules, and solids in the real world: to, for instance, Why liquid water is too "schizophrenic"? or

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 densityfunctional approximation (DFA). In Kohn-Sham (KS) DFT, DFAs are applied to the unknown exchange-correlation (xc) energy. In orbitalfree 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-ofmagnitude lower computational cost. V V I I Preface This thesis is a summary of some of the work I was part of between 2012 and 2017 in the theoretical physics group at Linköping University. I am indebted to a large number of people, without whom this work would not have seen the light of day. First of all, I would like to thank my supervisor, Rickard Armiento, who successfully, and with great enthusiasm and perseverance, has guided me during these past years. Working with you has truly been a pleasure, and a privilege. I also wish to thank my co-supervisor, Igor Abrikosov, for his encouragement and support, and for making all of this happen in the first place. Ann Mattsson is greatly acknowledged for nice and insightful conversations. I am grateful for our current and future collaboration. In addition, I want to thank John Wills for his encouraging and nice ways. I have also been able to enjoy the warm hospitality of Stephan Kümmel at the University of Bayreuth and Olle Eriksson at Uppsala University, when visiting their groups. I want to acknowledge everyone in the theoretical physics group, from the weathered members of the old breed to the young and vigilant brothers in arms. Also, the support that I have received from my family and friends has been invaluable. Finally, I want to thank B, for her enduring support during the writing of this thesis, and for reminding me to continue to get after it.