intro-to-dft

intro-to-dft - An Introduction to Density Functional Theory...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
1 An Introduction to Density Functional Theory N. M. Harrison Department of Chemistry, Imperial College of Science Technology and Medicine, SW7 2AY, London and CLRC, Daresbury Laboratory, Daresbury, Warrington, WA4 4AD 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. 1. The Solution of the Schrödinger Equation During the course of this lecture we will be primarily concerned with the calculation of the ground state energy of a collection of atoms. The energy may be computed by solution of the Schrödinger equation – which, in the time independent, non- relativistic, Born-Oppenheimer approximation is 1 ; ) ,..., , ( ) ,..., , ( 2 1 2 1 N N E H r r r r r r Ψ = Ψ Equation 1 1 Atomic units are used throughout.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 The Hamiltonian operator, H, consists of a sum of three terms; the kinetic energy, the interaction with the external potential ( V ext ) and the electron-electron interaction ( V ee ). That is; < - + + - = N j i j i ext N i i V H | | 1 2 1 2 r r Equation 2 In materials simulation the external potential of interest is simply the interaction of the electrons with the atomic nuclei; - - = at N i ext Z V a a a | | R r Equation 3 Here, r i is the coordinate of electron i and the charge on the nucleus at R a is Z a . Note that in order to simplify the notation and to focus the discussion on the main features of DFT the spin coordinate is omitted here and throughout this article. Equation 1 is solved for a set of Ψ subject to the constraint that that the Ψ are anti- symmetric – they change sign if the coordinates of any two electrons are interchanged. The lowest energy eigenvalue, E 0 , is the ground state energy and the probability density of finding an electron with any particular set of coordinates { r i } is | Ψ 0 | 2 . The average total energy for a state specified by a particular
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 26

intro-to-dft - An Introduction to Density Functional Theory...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online