An Introduction to Density Functional Theory

1 1An Introduction to Density FunctionalTheoryN. M. HarrisonDepartment of Chemistry, Imperial College of Science Technology andMedicine, SW7 2AY, London and CLRC, Daresbury Laboratory, Daresbury, Warrington, WA4 4 ADFor the past 30 years Density Functional Theory has been the dominant methodfor the quantum mechanical simulation of periodic systems. In recent years it has alsobeen adopted by quantum chemists and is now very widely used for the simulation ofenergy surfaces in molecules. In this lecture we introduce the basic concepts underlyingdensity Functional Theory and outline the features that have lead to its wide spreadadoption. Recent developments in exchange correlation functionals are introduced andthe performance of families of functionals lecture is intended for a researcher with little or no experience of quantummechanical simulations but with a basic (undergraduate) knowledge of quantummechanics.

2 We hope to provide sufficient background to enable informed judgements onthe applicability of a particular implementation of Density Functional Theory to a specificproblem in materials those who wish to go more deeply into the formalism of Density functionaltheory there are a number of reviews and books aimed at intermediate and advancedlevels available in the literature [1,2,3]. Where appropriate source articles are referred toin the The Solution of the Schr dinger EquationDuring the course of this lecture we will be primarily concerned with the calculationof the ground state energy of a collection of atoms. The energy may be computed bysolution of the Schr dinger equation which, in the time independent, non-relativistic, Born-Oppenheimer approximation is1;),..,,(),..,,(2121 NNEH rrrrrr = Equation 1 1 Atomic units are used Hamiltonian operator, H, consists of a sum of three terms; the kineticenergy, the interaction with the external potential (Vext) and the electron-electroninteraction (Vee).

3 That is; < ++ =NjijiextNiiVH||1212rrEquation 2In materials simulation the external potential of interest is simply theinteraction of the electrons with the atomic nuclei; = atNiextZV ||RrEquation 3 Here, ri is the coordinate of electron i and the charge on the nucleus at R isZ . Note that in order to simplify the notation and to focus the discussion on the mainfeatures of DFT the spin coordinate is omitted here and throughout this 1 is solved for a set of subject to the constraint that that the are anti-symmetric they change sign if the coordinates of any two electrons areinterchanged. The lowest energy eigenvalue, E0, is the ground state energy and theprobability Density of finding an electron with any particular set of coordinates {ri} is| 0| average total energy for a state specified by a particular , not necessarilyone of the eigenfunctions of Equation 1, is the expectation value of H, that is;[] = HdHEr*Equation 4 The notation [ ] emphasises the fact that the energy is a Functional of thewavefunction.

4 The energy is higher than that of the ground state unless correspondsto 0 which is the variational theorem;0][EE Equation 53 The ground state wavefunction and energy may be found by searching allpossible wavefunctions for the one that minimises the total energy. Hartree-Focktheory consists of an ansatz for the structure of - it is assumed to be anantisymmetric product of functions ( i) each of which depends in the coordinates of asingle electron, that is;[]NHFN ..det!1321= Equation 6where, det indicates a matrix determinant [4]. Substitution of this ansatz for into the Schr dinger equation results in an expression for the Hartree Fock energy; + + =NjijijijiNjijijjiiiextNiiiHFdddddVE,212 *211*,2122*11*2*)()()()(21)()()()(21)(21 )(rrrrrrrrrrrrrrrrrrr Equation 7 The second term is simply the classical Coulomb energy written in terms ofthe orbitals and the third term is the exchange energy.

5 The ground state orbitals aredetermined by applying the variation theorem to this energy expression under theconstraint that the orbitals are orthonormal. This leads to the Hartree-Fock (or SCF)equations;)(')'()',()('')'()(212rrrr rrrrrrriiiXiextdvdv =+ ++ Equation 8 Where the non-local exchange potential, vX, is such that:4 =NjijjiXddv')'(')'()(')'()',(*rrrrrrrrrr Equation 9 The Hartree-Fock equations describe non-interacting electrons under theinfluence of a mean field potential consisting of the classical Coulomb potential and anon-local exchange this starting point better approximations (correlated methods) for andE0 are readily obtained but the computational cost of such improvements is very highand scales prohibitively quickly with the number of electrons treated (for an excellentintroduction see ref.)

6 [4]). In addition, accurate solutions require a very flexibledescription of the wavefunction s spatial variation, a large and basis set is requiredwhich also adds to the expense for practical calculations. Many correlated methodshave been developed for molecular calculations [4]. The cost of the most commonlyused methods, MP2, MP3, MP4, CISD, CCSD, CCSD(T) formally scales with thenumber of electrons raised to the power of 5,6,7,6,6,7 respectively. In most casesCCSD(T) calculations are of sufficient accuracy to determine the chemical propertiesof systems to sufficient accuracy to predict chemical properties (stability, reactionrates ..) however, due to the computational expense the routine application of suchmethods to realistic models of systems of interest is not practical and not likely tobecome so despite rapid advances in computer technology.

7 In this context we mentionrecent advances in the solution of the Schr dinger equation using variational quantumMonte Carlo approach [5].The discussion above has established that direct solution of the Schr dingerequation is not currently feasible for systems of interest in condensed matter science this is a major motivation for the development and use of Density Functional question that arises is Is it necessary to solve the Schr dinger equationand determine the 3N dimensional wavefunction in order to compute the ground stateenergy ?2. Avoiding the Solution of the Schr dinger EquationThe Hamiltonian operator (Equation 2) consists of single electron and bi-electronic interactions operators that involve on the coordinates of one or twoelectrons only. In order to compute the total energy we do not need to know the 3 Ndimensional wavefunction.

8 Knowledge of the two-particle probability Density thatis, the probability of finding an electron at r1 and an electron at r2 is quantity of great use in analysing the energy expression is the second orderdensity matrix, which is defined as:5() ),..,,(),..,,(21),;,(4321''1'1*21'2'12 =Equation 10 The diagonal elements of P2, often referred to as the two-particle densitymatrix or pair Density , are;),;,(),(21212212rrrrrrPP=This is the required two electron probability function and completelydetermines all two particle operators. The first order Density matrix is defined in asimilar manner and may be written in terms of P2 as;2rrrrrrrdPNP),;,(12);(212'121'11 =Equation 11 Given P1 and P2 the total energy is determined exactly;212122111'11121),(||1),(||21'11r rrrrrrrrRrrrddPdPZPHtrEatN + = == Equation 12We conclude that the diagonal elements of the first and second order densitymatrices completely determine the total energy.

9 This appears to vastly simplify thetask in hand. The solution of the full Schr dinger equation for is not required it issufficient to determine P1 and P2 - and the problem in a space of 3N coordinates hasbeen reduced to a problem in a 6 dimensional based on the direct minimisation of E(P1,P2) suffer from thespecific problem of ensuring that the Density matrices are legal that is, they must beconstructible from an antisymmetric . Imposing this constraint is non trivial and iscurrently an unsolved problem [6,7]. In view of this we conclude that Equation 12does not lead immediately to a reliable method for computing the total energy withoutcalculating the many body observation which underpins Density Functional Theory is that we do noteven require P2 to find E the ground state energy is completely determined by thediagonal elements of the first order Density matrix the charge The Hohenburg-Kohn TheoremsIn 1964 Hohenburg and Kohn proved the two theorems [8].

10 The first theorem may bestated as follows;The electron Density determines the external potential (to within an additive constant).If this statement is true then it immediately follows that the electron densityuniquely determines the Hamiltonian operator (Equation 2). This follows as theHamiltonian is specified by the external potential and the total number of electrons,N, which can be computed from the Density simply by integration over all , in principle, given the charge Density , the Hamiltonian operator could beuniquely determined and this the wave functions (of all states) and all materialproperties and Kohn [8] gave a straightforward proof of this theorem, whichwas generalised to include systems with degenerate states in proof given by Levy in1979 [9]. It is said that the theoretical spectroscopist E.