Lecture notes on Density functional Theory

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C.Fiolhais F.Nogueira M.Marques (Eds.) APrimerinDensity FunctionalTheory 131 Density Functionals for Non-relativistic Coulomb Systems in the New Century ∗ † John P. Perdew and Stefan Kurth ∗ Department of Physics and Quantum Theory Group, Tulane University, New Orleans LA 70118, USA perdewfrigg.phy.tulane.edu † Institut fur ¨ Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany kurthphysik.fu-berlin.de John Perdew 1.1 Introduction 1.1.1 Quantum Mechanical Many-Electron Problem The material world of everyday experience, as studied by chemistry and con- densed-matter physics, is built up from electrons and a few (or at most a few hundred) kinds of nuclei . The basic interaction is electrostatic or Coulom- bic: An electron at position r is attracted to a nucleus of charge Z at R by  the potential energy −Z/r−R, a pair of electrons at r and r repel one   another by the potential energy 1/r−r, and two nuclei at R and R repel   one another as Z Z/R−R. The electrons must be described by quantum mechanics, while the more massive nuclei can sometimes be regarded as clas- sical particles. All of the electrons in the lighter elements, and the chemically important valence electrons in most elements, move at speeds much less than the speed of light, and so are non-relativistic. In essence, that is the simple story of practically everything. But there is still a long path from these general principles to theoretical prediction of the structures and properties of atoms, molecules, and solids, and eventually to the design of new chemicals or materials. If we restrict our focus to the important class of ground-state properties, we can take a shortcut through density functional theory. These lectures present an introduction to density functionals for non- relativisticCoulombsystems.Thereaderisassumedtohaveaworkingknowl- edgeofquantummechanicsatthelevelofone-particlewavefunctionsψ(r)1. The many-electron wavefunction Ψ(r ,r ,...,r ) 2 is briefly introduced 1 2 N here, and then replaced as basic variable by the electron densityn(r). Various termsofthetotalenergyaredefinedasfunctionalsoftheelectrondensity,and some formal properties of these functionals are discussed. The most widely- used density functionals – the local spin density and generalized gradient C. Fiolhais, F. Nogueira, M. Marques (Eds.): LNP 620, pp. 1–55, 2003.  c Springer-Verlag Berlin Heidelberg 20032 John P. Perdew and Stefan Kurth approximations – are then introduced and discussed. At the end, the reader should be prepared to approach the broad literature of quantum chemistry and condensed-matter physics in which these density functionals are applied to predict diverse properties: the shapes and sizes of molecules, the crys- tal structures of solids, binding or atomization energies, ionization energies and electron affinities, the heights of energy barriers to various processes, static response functions, vibrational frequencies of nuclei, etc. Moreover, the reader’s approach will be an informed and discerning one, based upon an understanding of where these functionals come from, why they work, and how they work. These lectures are intended to teach at the introductory level, and not to serve as a comprehensive treatise. The reader who wants more can go to several excellent general sources 3,4,5 or to the original literature. Atomic units (in which all electromagnetic equations are written in cgs form, and 2 the fundamental constants , e , and m are set to unity) have been used throughout. 1.1.2 Summary of Kohn–Sham Spin-Density Functional Theory This introduction closes with a brief presentation of the Kohn-Sham 6 spin-density functional method, the most widely-used method of electronic- structurecalculationincondensed-matterphysicsandoneofthemostwidely- used methods in quantum chemistry. We seek the ground-state total energy E and spin densities n (r), n (r) for a collection of N electrons interacting ↑ ↓ with one another and with an external potential v(r) (due to the nuclei in most practical cases). These are found by the selfconsistent solution of an auxiliary (fictitious) one-electron Schr¨ odinger equation:   1 2 σ − ∇ +v(r)+u(n;r)+v (n ,n ;r) ψ (r)= ε ψ (r) , (1.1) ↑ ↓ ασ ασ ασ xc 2  2 n (r)= θ(µ −ε )ψ (r) . (1.2) σ ασ ασ α Here σ =↑ or ↓ is the z-component of spin, and α stands for the set of remaining one-electron quantum numbers. The effective potential includes a classical Hartree potential   n(r ) 3  u(n;r)= d r , (1.3)  r−r n(r)= n (r)+n (r) , (1.4) ↑ ↓ σ and v (n ,n ;r), a multiplicative spin-dependent exchange-correlation po- ↑ ↓ xc tential which is a functional of the spin densities. The step functionθ(µ −ε ) ασ in (1.2) ensures that all Kohn-Sham spin orbitals with ε µ are singly ασ1 Density Functionals for Non-relativistic Coulomb Systems 3 occupied, and those with ε µ are empty. The chemical potential µ is ασ chosen to satisfy  3 d rn(r)=N. (1.5) Because (1.1) and (1.2) are interlinked, they can only be solved by iteration to selfconsistency. The total energy is  3 E = T n ,n + d rn(r)v(r)+Un+E n ,n , (1.6) s ↑ ↓ xc ↑ ↓ where  1 2 T n ,n = θ(µ −ε )ψ − ∇ ψ  (1.7) s ↑ ↓ ασ ασ ασ 2 σ α isthenon-interactingkineticenergy,afunctionalofthespindensitiesbecause (asweshallsee)theexternalpotentialv(r)andhencetheKohn-Shamorbitals are functionals of the spin densities. In our notation,  3 ∗ ˆ ˆ ψ Oψ  = d rψ (r)Oψ (r) . (1.8) ασ ασ ασ ασ The second term of (1.6) is the interaction of the electrons with the external potential. The third term of (1.6) is the Hartree electrostatic self-repulsion of the electron density    1 n(r)n(r ) 3 3  Un= d r d r . (1.9)  2 r−r The last term of (1.6) is the exchange-correlation energy, whose functional derivative (as explained later) yields the exchange-correlation potential δE xc σ v (n ,n ;r)= . (1.10) ↑ ↓ xc δn (r) σ Not displayed in (1.6), but needed for a system of electrons and nuclei, is the electrostatic repulsion among the nuclei. E is defined to include everything xc else omitted from the first three terms of (1.6). If the exact dependence of E upon n and n were known, these equa- xc ↑ ↓ tions would predict the exact ground-state energy and spin-densities of a many-electron system. The forces on the nuclei, and their equilibrium posi- ∂E tions, could then be found from− . ∂R In practice, the exchange-correlation energy functional must be approxi- mated. The local spin density 6,7 (LSD) approximation has long been pop- ular in solid state physics:  LSD 3 E n ,n = d rn(r)e (n (r),n (r)) , (1.11) ↑ ↓ xc ↑ ↓ xc4 John P. Perdew and Stefan Kurth where e (n ,n ) is the known 8,9,10 exchange-correlation energy per par- xc ↑ ↓ ticle for an electron gas of uniform spin densities n , n . More recently, gen- ↑ ↓ eralized gradient approximations (GGA’s) 11,12,13,14,15,16,17,18,19,20,21 have become popular in quantum chemistry:  GGA 3 E n ,n = d rf(n ,n ,∇n ,∇n ) . (1.12) ↑ ↓ ↑ ↓ ↑ ↓ xc The input e (n ,n ) to LSD is in principle unique, since there is a pos- xc ↑ ↓ sible system in which n and n are constant and for which LSD is ex- ↑ ↓ act. At least in this sense, there is no unique input f(n ,n ,∇n ,∇n )to ↑ ↓ ↑ ↓ GGA. These lectures will stress a conservative “philosophy of approxima- tion” 20,21, in which we construct a nearly-unique GGA with all the known correct formal features of LSD, plus others. We will also discuss how to go beyond GGA. The equations presented here are really all that we need to do a practical calculation for a many-electron system. They allow us to draw upon the intuition and experience we have developed for one-particle systems. The many-body effects are in Un (trivially) and E n ,n (less trivially), but xc ↑ ↓ we shall also develop an intuitive appreciation for E . xc While E is often a relatively small fraction of the total energy of an xc atom, molecule, or solid (minus the work needed to break up the system into separated electrons and nuclei), the contribution from E is typically xc about 100% or more of the chemical bonding or atomization energy (the work needed to break up the system into separated neutral atoms). E is a kind of xc “glue”, without which atoms would bond weakly if at all. Thus, accurate ap- proximations toE are essential to the whole enterprise of density functional xc theory. Table 1.1 shows the typical relative errors we find from selfconsistent calculations within the LSD or GGA approximations of (1.11) and (1.12). Table 1.2 shows the mean absolute errors in the atomization energies of 20 molecules when calculated by LSD, by GGA, and in the Hartree-Fock ap- proximation. Hartree-Fock treats exchange exactly, but neglects correlation completely. While the Hartree-Fock total energy is an upper bound to the true ground-state total energy, the LSD and GGA energies are not. In most cases we are only interested in small total-energy changes asso- ciated with re-arrangements of the outer or valence electrons, to which the inner or core electrons of the atoms do not contribute. In these cases, we can replace each core by the pseudopotential 22 it presents to the valence electrons, and then expand the valence-electron orbitals in an economical and convenient basis of plane waves. Pseudopotentials are routinely com- bined with density functionals. Although the most realistic pseudopotentials are nonlocal operators and not simply local or multiplication operators, and although density functional theory in principle requires a local external po- tential, this inconsistency does not seem to cause any practical difficulties. There are empirical versions of LSD and GGA, but these lectures will only discuss non-empirical versions. If every electronic-structure calculation1 Density Functionals for Non-relativistic Coulomb Systems 5 Table1.1. Typicalerrorsforatoms,molecules,andsolidsfromselfconsistentKohn- Sham calculations within the LSD and GGA approximations of (1.11) and (1.12). Note that there is typically some cancellation of errors between the exchange (E ) x and correlation (E ) contributions to E . The “energy barrier” is the barrier to a c xc chemical reaction that arises at a highly-bonded intermediate state Property LSD GGA E 5% (not negative enough) 0.5% x E 100% (too negative) 5% c bond length 1% (too short) 1% (too long) structure overly favors close packing more correct energy barrier 100% (too low) 30% (too low) Table1.2. Mean absolute error of the atomization energies for 20 molecules, eval- uated by various approximations. (1hartree = 27.21eV) (From 20) Approximation Mean absolute error (eV) Unrestricted Hartree-Fock 3.1 (underbinding) LSD 1.3 (overbinding) GGA 0.3 (mostly overbinding) Desired “chemical accuracy” 0.05 were done at least twice, once with nonempirical LSD and once with nonem- pirical GGA, the results would be useful not only to those interested in the systems under consideration but also to those interested in the development and understanding of density functionals. 1.2 Wavefunction Theory 1.2.1 Wavefunctions and Their Interpretation We begin with a brief review of one-particle quantum mechanics 1. An 1 1 1 electron has spin s = and z-component of spin σ =+ (↑)or − (↓). 2 2 2 The Hamiltonian or energy operator for one electron in the presence of an external potential v(r)is 1 2 ˆ h =− ∇ +v(r) . (1.13) 2 The energy eigenstates ψ (r,σ) and eigenvalues ε are solutions of the time- α α independent Schr¨ odinger equation ˆ hψ (r,σ)= ε ψ (r,σ) , (1.14) α α α6 John P. Perdew and Stefan Kurth 2 3 andψ (r,σ) d r istheprobabilitytofindtheelectronwithspinσ involume α 3 element d r at r, given that it is in energy eigenstate ψ .Thus α   3 2 d rψ (r,σ) =ψψ=1 . (1.15) α σ ˆ Since h commutes with sˆ , we can choose the ψ to be eigenstates of sˆ , i.e., z α z we can choose σ =↑ or↓ as a one-electron quantum number. The Hamiltonian for N electrons in the presence of an external potential v(r)is2 N N    1 1 1 2 ˆ H =− ∇ + v(r)+ i i 2 2 r −r i j i=1 i=1 i j=  i ˆ ˆ ˆ = T +V +V . (1.16) ext ee ˆ The electron-electron repulsion V sums over distinct pairs of different elec- ee ˆ trons. The states of well-defined energy are the eigenstates of H: ˆ HΨ (r σ ,...,r σ )= E Ψ (r σ ,...,r σ ) , (1.17) k 1 1 N N k k 1 1 N N where k is a complete set of many-electron quantum numbers; we shall be interested mainly in the ground state or state of lowest energy, the zero- temperature equilibrium state for the electrons. Because electrons are fermions, the only physical solutions of (1.17) are those wavefunctions that are antisymmetric 2 under exchange of two elec- tron labels i and j: Ψ(r σ ,...,r σ,...,r σ ,...,r σ )= 1 1 i i j j N N − Ψ(r σ ,...,r σ ,...,r σ,...,r σ ) . (1.18) 1 1 j j i i N N There are N distinct permutations of the labels 1,2,...,N, which by (1.18) 2 2 3 3 all have the same Ψ.Thus NΨ(r σ ,...,r σ ) d r ...d r is the 1 1 N N 1 N 3 probability to find any electron with spin σ in volume element d r , etc., 1 1 and     1 3 3 2 2 d r ... d r NΨ(r σ ,...,r σ ) = Ψ =ΨΨ=1 . 1 N 1 1 N N N σ ...σ 1 N (1.19) 3 We define the electron spin densityn (r) so thatn (r)d r is the probabil- σ σ 3 ity to find an electron with spin σ in volume element d r at r.Wefind n (r) σ by integrating over the coordinates and spins of the (N−1) other electrons, i.e.,    1 3 3 2 n (r)= d r ... d r NΨ(rσ,r σ ,...,r σ ) σ 2 N 2 2 N N (N−1) σ ...σ 2 N    3 3 2 = N d r ... d r Ψ(rσ,r σ ,...,r σ ) . (1.20) 2 N 2 2 N N σ ...σ 2 N1 Density Functionals for Non-relativistic Coulomb Systems 7 Equations (1.19) and (1.20) yield   3 d rn (r)=N. (1.21) σ σ Based on the probability interpretation of n (r), we might have expected the σ right hand side of (1.21) to be 1, but that is wrong; the sum of probabilities of all mutually-exclusive events equals 1, but finding an electron atr doesnot  exclude the possibility of finding one at r , except in a one-electron system. 3 Equation (1.21) shows that n (r)d r is the average number of electrons of σ 3 spinσ in volume element d r. Moreover, the expectation value of the external potential is  N  3 ˆ V  =Ψ v(r )Ψ = d rn(r)v(r) , (1.22) ext i i=1 with the electron density n(r) given by (1.4). 1.2.2 WavefunctionsforNon-interactingElectrons As an important special case, consider the Hamiltonian forN non-interacting electrons:   N  1 2 ˆ H = − ∇ +v(r ) . (1.23) non i i 2 i=1 The eigenfunctions of the one-electron problem of (1.13) and (1.14) are spin orbitals which can be used to construct the antisymmetric eigenfunctions Φ ˆ of H : non ˆ H Φ = E Φ. (1.24) non non Letistandforr,σ andconstructtheSlaterdeterminantorantisymmetrized i i product 2  1 P Φ = √ (−1) ψ (P1)ψ (P2)...ψ (PN) , (1.25) α α α 1 2 N N P where the quantum label α now includes the spin quantum number σ. Here i P P is any permutation of the labels 1,2,...,N, and (−1) equals +1 for an even permutation and−1 for an odd permutation. The total energy is E = ε +ε +...+ε , (1.26) non α α α 1 2 N 2 and the density is given by the sum of ψ (r).Ifany α equals any α α i j i in (1.25), we find Φ = 0, which is not a normalizable wavefunction. This is the Pauli exclusion principle: two or more non-interacting electrons may not occupy the same spin orbital.8 John P. Perdew and Stefan Kurth As an example, consider the ground state for the non-interacting helium atom (N = 2). The occupied spin orbitals are ψ (r,σ)= ψ (r)δ , (1.27) 1 1s σ,↑ ψ (r,σ)= ψ (r)δ , (1.28) 2 1s σ,↓ and the 2-electron Slater determinant is     1 ψ (r ,σ ) ψ (r ,σ ) 1 1 1 2 1 1   Φ(1,2) = √   ψ (r ,σ ) ψ (r ,σ ) 1 2 2 2 2 2 2 1 √ = ψ (r )ψ (r ) (δ δ −δ δ ) , (1.29) 1s 1 1s 2 σ ,↑ σ ,↓ σ ,↑ σ ,↓ 1 2 2 1 2 which is symmetric in space but antisymmetric in spin (whence the total spin is S = 0). If several different Slater determinants yield the same non-interacting en- ergy E , then a linear combination of them will be another antisymmet- non ˆ ric eigenstate of H . More generally, the Slater-determinant eigenstates of non ˆ H define a complete orthonormal basis for expansion of the antisymmetric non ˆ eigenstates of H, the interacting Hamiltonian of (1.16). 1.2.3 Wavefunction Variational Principle The Schr¨ odinger equation (1.17) is equivalent to a wavefunction variational ˆ principle 2: ExtremizeΨHΨ subject to the constraintΨΨ = 1, i.e., set the following first variation to zero:  ˆ δ ΨHΨ/ΨΨ =0 . (1.30) The ground state energy and wavefunction are found by minimizing the ex- pression in curly brackets. The Rayleigh-Ritz method finds the extrema or the minimum in a re- strictedspaceofwavefunctions.Forexample,theHartree-Fockapproximation to the ground-state wavefunction is the single Slater determinant Φ that min- ˆ imizesΦHΦ/ΦΦ. The configuration-interaction ground-state wavefunc- tion 23 is an energy-minimizing linear combination of Slater determinants, restricted to certain kinds of excitations out of a reference determinant. The Quantum Monte Carlo method typically employs a trial wavefunction which is a single Slater determinant times a Jastrow pair-correlation factor 24. Those widely-used many-electron wavefunction methods are both approx- imate and computationally demanding, especially for large systems where density functional methods are distinctly more efficient. The unrestricted solution of (1.30) is equivalent by the method of La- grange multipliers to the unconstrained solution of  ˆ δ ΨHΨ−EΨΨ =0 , (1.31)1 Density Functionals for Non-relativistic Coulomb Systems 9 i.e., ˆ δΨ(H−E)Ψ=0 . (1.32) SinceδΨ isanarbitraryvariation,werecovertheSchr¨ odingerequation(1.17). ˆ ˆ Every eigenstate of H is an extremum ofΨHΨ/ΨΨ and vice versa. The wavefunction variational principle implies the Hellmann-Feynman and virial theorems below and also implies the Hohenberg-Kohn 25 density functional variational principle to be presented later. 1.2.4 Hellmann–Feynman Theorem ˆ Often the Hamiltonian H depends upon a parameter λ, and we want to λ know how the energy E depends upon this parameter. For any normalized λ ˆ variational solution Ψ (including in particular any eigenstate of H ), we λ λ define ˆ E =Ψ H Ψ  . (1.33) λ λ λ λ Then  ˆ  dE d ∂H λ λ ˆ    = Ψ H Ψ  +Ψ Ψ  . (1.34) λ λ λ λ λ   dλ dλ ∂λ  λ=λ The first term of (1.34) vanishes by the variational principle, and we find the Hellmann-Feynman theorem 26 ˆ dE ∂H λ λ =Ψ Ψ  . (1.35) λ λ dλ ∂λ Equation (1.35) will be useful later for our understanding ofE . For now, xc we shall use (1.35) to derive the electrostatic force theorem 26. Let r be i the position of the i-th electron, and R the position of the (static) nucleus I I with atomic number Z . The Hamiltonian I N     1 −Z 1 1 1 Z Z I I J 2 ˆ H = − ∇ + + + i 2 r −R 2 r −r 2 R −R i I i j I J i=1 i I i j=  i I J=I (1.36) depends parametrically upon the position R , so the force on nucleus I is I     ˆ ∂E ∂H   − = Ψ− Ψ   ∂R ∂R I I   Z (r−R ) Z Z (R −R ) I I I J I J 3 = d rn(r) + , (1.37) 3 3 r−R R −R I I J J=I just as classical electrostatics would predict. Equation (1.37) can be used to find the equilibrium geometries of a molecule or solid by varying all the R until the energy is a minimum and −∂E/∂R = 0. Equation (1.37) also I I forms the basis for a possible density functional molecular dynamics, in which10 John P. Perdew and Stefan Kurth the nuclei move under these forces by Newton’s second law. In principle, all we need for either application is an accurate electron density for each set of nuclear positions. 1.2.5 Virial Theorem The density scaling relations to be presented in Sect. 1.4, which constitute important constraints on the density functionals, are rooted in the same wavefunction scaling that will be used here to derive the virial theorem 26. ˆ LetΨ(r ,...,r )beanyextremumofΨHΨovernormalizedwavefunc- 1 N tions, i.e., any eigenstate or optimized restricted trial wavefunction (where ir- relevant spin variables have been suppressed). For any scale parameterγ 0, define the uniformly-scaled wavefunction 3N/2 Ψ (r ,...,r )= γ Ψ(γr ,...,γr ) (1.38) γ 1 N 1 N and observe that Ψ Ψ  =ΨΨ=1 . (1.39) γ γ The density corresponding to the scaled wavefunction is the scaled density 3 n (r)= γ n(γr) , (1.40) γ which clearly conserves the electron number:   3 3 d rn (r)= d rn(r)=N. (1.41) γ γ 1 leads to densities n (r) that are higher (on average) and more con- γ tracted than n(r), whileγ 1 produces densities that are lower and more expanded. ˆ ˆ ˆ Now consider what happens toH =T+V under scaling. By definition of Ψ,   d ˆ ˆ  Ψ T +VΨ  =0 . (1.42) γ γ  dγ γ=1 ˆ But T is homogeneous of degree -2 in r,so 2 ˆ ˆ Ψ TΨ  = γ ΨTΨ , (1.43) γ γ and (1.42) becomes   d ˆ ˆ  2ΨTΨ+ Ψ VΨ  =0 , (1.44) γ γ  dγ γ=1 or N  ˆ ∂V ˆ 2T− r · =0 . (1.45) i ∂r i i=11 Density Functionals for Non-relativistic Coulomb Systems 11 ˆ If the potential energy V is homogeneous of degree n, i.e., if n V(γr,...,γr )= γ V(r,...,r ) , (1.46) i N i N then −n ˆ ˆ Ψ VΨ  = γ ΨVΨ , (1.47) γ γ and (1.44) becomes simply ˆ ˆ 2ΨTΨ−nΨVΨ=0 . (1.48) For example, n = −1 for the Hamiltonian of (1.36) in the presence of a singlenucleus,ormoregenerallywhentheHellmann-Feynmanforcesof(1.37) vanish for the state Ψ. 1.3 Definitions of Density Functionals 1.3.1 Introduction to Density Functionals The many-electron wavefunction Ψ(r σ ,...,r σ ) contains a great deal of 1 1 N N information – all we could ever have, but more than we usually want. Because it is a function of many variables, it is not easy to calculate, store, apply or even think about. Often we want no more than the total energy E (and its changes), or perhaps also the spin densities n (r) and n (r), for the ground ↑ ↓ state. As we shall see, we can formally replace Ψ by the observables n and ↑ n as the basic variational objects. ↓ While a function is a rule which assigns a number f(x)toanumber x,a functional is a rule which assigns a number Ff to a function f.For ˆ example, hΨ= ΨHΨ is a functional of the trial wavefunction Ψ, given ˆ the Hamiltonian H. Un of (1.9) is a functional of the density n(r), as is the local density approximation for the exchange energy:  LDA 3 4/3 E n= A d rn(r) . (1.49) x x The functional derivative δF/δn(r) tells us how the functional Fn changes under a small variation δn(r):    δF 3 δF = d r δn(r) . (1.50) δn(r) For example,   LDA 3 4/3 4/3 δE = A d r n(r)+δn(r) −n(r) x x  4 3 1/3 = A d r n(r) δn(r) , x 312 John P. Perdew and Stefan Kurth so LDA δE 4 x 1/3 = A n(r) . (1.51) x δn(r) 3 Similarly, δUn = u(n;r) , (1.52) δn(r) where the right hand side is given by (1.3). Functional derivatives of various orders can be linked through the translational and rotational symmetries of empty space 27. 1.3.2 Density Variational Principle We seek a density functional analog of (1.30). Instead of the original deriva- tionofHohenberg,KohnandSham25,6,whichwasbasedupon“reductio ad absurdum”, we follow the “constrained search” approach of Levy 28, which is in some respects simpler and more constructive. Equation(1.30)tellsusthatthegroundstateenergycanbefoundbymini- ˆ mizingΨHΨoverallnormalized,antisymmetricN-particlewavefunctions: ˆ E = minΨHΨ . (1.53) Ψ We now separate the minimization of (1.53) into two steps. First we consider all wavefunctions Ψ which yield a given densityn(r), and minimize over those wavefunctions:  3 ˆ ˆ ˆ minΨHΨ = minΨT +V Ψ+ d rv(r)n(r) , (1.54) ee Ψ→n Ψ→n where we have exploited the fact that all wavefunctions that yield the same ˆ n(r) also yield the sameΨV Ψ. Then we define the universal functional ext min min ˆ ˆ ˆ ˆ Fn = minΨT +V Ψ =Ψ T +V Ψ  , (1.55) ee ee n n Ψ→n min where Ψ is that wavefunction which delivers the minimum for a given n. n Finally we minimize over all N-electron densities n(r): E = minE n v n  3 = min Fn+ d rv(r)n(r) , (1.56) n where of course v(r) is held fixed during the minimization. The minimizing density is then the ground-state density. The constraint of fixed N can be handled formally through introduction of a Lagrange multiplier µ :   3 3 δ Fn+ d rv(r)n(r)−µ d rn(r) =0 , (1.57)1 Density Functionals for Non-relativistic Coulomb Systems 13 which is equivalent to the Euler equation δF +v(r)=µ. (1.58) δn(r) µ is to be adjusted until (1.5) is satisfied. Equation (1.58) shows that the external potential v(r) is uniquely determined by the ground state density (or by any one of them, if the ground state is degenerate). The functional Fn is defined via (1.55) for all densities n(r) which are “N-representable”, i.e., come from an antisymmetric N-electron wave- function. We shall discuss the extension from wavefunctions to ensembles in Sect.1.4.5.ThefunctionalderivativeδF/δn(r)isdefinedvia(1.58)forallden- sities which are “v-representable”, i.e., come from antisymmetric N-electron ground-state wavefunctions for some choice of external potential v(r). This formal development requires only the total density of (1.4), and not the separate spin densities n (r) and n (r). However, it is clear how to get ↑ ↓ to a spin-density functional theory: just replace the constraint of fixed n in (1.54) and subsequent equations by that of fixed n and n . There are two ↑ ↓ practical reasons to do so: (1) This extension is required when the external potential is spin-dependent, i.e., v(r)→ v (r), as when an external magnetic σ field couples to the z-component of electron spin. (If this field also couples to the current density j(r), then we must resort to a current-density functional theory.) (2) Even when v(r) is spin-independent, we may be interested in the physical spin magnetization (e.g., in magnetic materials). (3) Even when neither(1)nor(2)applies,ourlocalandsemi-localapproximations(see(1.11) and (1.12)) typically work better when we use n and n instead of n. ↑ ↓ 1.3.3 Kohn–ShamNon-interactingSystem ˆ For a system of non-interacting electrons, V of (1.16) vanishes so Fn ee of (1.55) reduces to min min ˆ ˆ T n = minΨTΨ =Φ TΦ  . (1.59) s n n Ψ→n Although we can search over all antisymmetric N-electron wavefunctions min in (1.59), the minimizing wavefunction Φ for a given density will be a non- n interacting wavefunction (a single Slater determinant or a linear combination ˆ of a few) for some external potential V such that s δT s +v (r)=µ, (1.60) s δn(r) asin(1.58).In(1.60),theKohn-Shampotentialv (r)isafunctionalofn(r).If s there were any difference between µ and µ , the chemical potentials for inter- s acting and non-interacting systems of the same density, it could be absorbed14 John P. Perdew and Stefan Kurth intov (r). We have assumed thatn(r) is both interacting and non-interacting s v-representable. Now we define the exchange-correlation energy E nby xc Fn= T n+Un+E n , (1.61) s xc where Un is given by (1.9). The Euler equations (1.58) and (1.60) are con- sistent with one another if and only if δUn δE n xc v (r)= v(r)+ + . (1.62) s δn(r) δn(r) Thus we have derived the Kohn-Sham method 6 of Sect. 1.1.2. The Kohn-Sham method treats T n exactly, leaving only E ntobe s xc approximated. This makes good sense, for several reasons: (1) T n is typi- s cally a very large part of the energy, while E n is a smaller part. (2) T n xc s is largely responsible for density oscillations of the shell structure and Friedel types, which are accurately described by the Kohn-Sham method. (3) E n xc is somewhat better suited to the local and semi-local approximations than is T n, for reasons to be discussed later. The price to be paid for these benefits s is the appearance of orbitals. If we had a very accurate approximation for T directly in terms of n, we could dispense with the orbitals and solve the s Euler equation (1.60) directly for n(r). The total energy of (1.6) may also be written as   3 E = θ(µ −ε )ε −Un− d rn(r)v (n;r)+E n , (1.63) ασ ασ xc xc ασ where the second and third terms on the right hand side simply remove contributions to the first term which do not belong in the total energy. The first term on the right of (1.63), the non-interacting energy E , is the only non term that appears in the semi-empirical Huc ¨ kel theory 26. This first term includes most of the electronic shell structure effects which arise when T n s is treated exactly (but not when T n is treated in a continuum model like s the Thomas-Fermi approximation or the gradient expansion). 1.3.4 Exchange Energy and Correlation Energy E n is the sum of distinct exchange and correlation terms: xc E n= E n+E n , (1.64) xc x c where 29 min min ˆ E n=Φ V Φ −Un . (1.65) x ee n n min When Φ is a single Slater determinant, (1.65) is just the usual Fock inte- n gral applied to the Kohn-Sham orbitals, i.e., it differs from the Hartree-Fock1 Density Functionals for Non-relativistic Coulomb Systems 15 exchange energy only to the extent that the Kohn-Sham orbitals differ from the Hartree-Fock orbitals for a given system or density (in the same way that T n differs from the Hartree-Fock kinetic energy). We note that s min min ˆ ˆ Φ T +V Φ  = T n+Un+E n , (1.66) ee s x n n ˆ and that, in the one-electron (V = 0) limit 9, ee E n=−Un(N=1) . (1.67) x The correlation energy is E n= Fn−T n+Un+E n c s x min min min min ˆ ˆ ˆ ˆ =Ψ T +V Ψ −Φ T +V Φ  . (1.68) ee ee n n n n min ˆ Since Ψ is that wavefunction which yields density n and minimizes T + n ˆ V , (1.68) shows that ee E n≤ 0 . (1.69) c min ˆ Since Φ is that wavefunction which yields density n and minimizes T, n (1.68) shows that E n is the sum of a positive kinetic energy piece and a c negative potential energy piece. These pieces of E contribute respectively c to the first and second terms of the virial theorem, (1.45). Clearly for any one-electron system 9 E n=0 (N=1) . (1.70) c Equations (1.67) and (1.70) show that the exchange-correlation energy of a one-electron system simply cancels the spurious self-interaction Un. In the same way, the exchange-correlation potential cancels the spurious self- interaction in the Kohn-Sham potential 9 δE n x =−u(n;r)(N=1) , (1.71) δn(r) δE n c =0 (N=1) . (1.72) δn(r) Thus δE n 1 xc lim =− (N=1) . (1.73) r→∞ δn(r) r The extension of these one-electron results to spin-density functional theory is straightforward, since a one-electron system is fully spin-polarized.16 John P. Perdew and Stefan Kurth 1.3.5 Coupling-Constant Integration The definitions (1.65) and (1.68) are formal ones, and do not provide much intuitive or physical insight into the exchange and correlation energies, or much guidance for the approximation of their density functionals. These in- sights are provided by the coupling-constant integration 30,31,32,33 to be derived below. min,λ Let us define Ψ as that normalized, antisymmetric wavefunction n ˆ ˆ which yields density n(r) and minimizes the expectation value of T +λV , ee where we have introduced a non-negative coupling constant λ. When λ=1, min,λ min Ψ isΨ ,theinteractingground-statewavefunctionfordensityn.When n n min,λ min λ=0, Ψ is Φ , the non-interacting or Kohn-Sham wavefunction for n n density n. Varying λ at fixed n(r) amounts to varying the external potential v (r): At λ=1, v (r) is the true external potential, while at λ = 0 it is the λ λ Kohn-Sham effective potential v (r). We normally assume a smooth, “adia- s batic connection” between the interacting and non-interacting ground states as λ is reduced from 1 to 0. Now we write (1.64), (1.65) and (1.68) as E n xc     min,λ min,λ min,λ min,λ ˆ ˆ ˆ ˆ = Ψ T +λV Ψ  −Ψ T +λV Ψ  −Un ee ee n n n n λ=1 λ=0  1 d min,λ min,λ ˆ ˆ = dλ Ψ T +λV Ψ −Un . (1.74) ee n n dλ 0 The Hellmann-Feynman theorem of Sect. 1.2.4 allows us to simplify (1.74) to  1 min,λ min,λ ˆ E n= dλΨ V Ψ −Un . (1.75) xc ee n n 0 Equation (1.75) “looks like” a potential energy; the kinetic energy contri- bution to E has been subsumed by the coupling-constant integration. We xc should remember, of course, that only λ = 1 is real or physical. The Kohn- Sham system at λ = 0, and all the intermediate values of λ, are convenient mathematical fictions. Tomakefurtherprogress,weneedtoknowhowtoevaluatetheN-electron ˆ expectation value of a sum of one-body operators like T, or a sum of two- ˆ body operators like V . For this purpose, we introduce one-electron (ρ ) and ee 1 two-electron (ρ ) reduced density matrices 34 : 2     3 3 ρ (r σ,rσ)≡ N d r ... d r 1 2 N σ ...σ 2 N ∗  Ψ (r σ,r σ ,...,r σ )Ψ(rσ,r σ ,...,r σ ) , (1.76) 2 2 N N 2 2 N N     3 3 ρ (r ,r) ≡ N(N−1) d r ... d r 2 3 N σ ...σ 1 N  2 Ψ(r σ ,rσ ,...,r σ ) . (1.77) 1 2 N N1 Density Functionals for Non-relativistic Coulomb Systems 17 From (1.20), n (r)= ρ (rσ,rσ) . (1.78) σ 1 Clearly also     1 ∂ ∂  3  ˆ T =− d r · ρ (r σ,rσ) , (1.79) 1  2 ∂r ∂r σ  r=r    1 ρ (r ,r) 2 3 3  ˆ V  = d r d r . (1.80) ee  2 r−r  3  3 We interpret the positive number ρ (r ,r)d r d r as the joint probability of 2 3   3 finding an electron in volume element d r at r , and an electron in d r at r. By standard probability theory, this is the product of the probability of 3 3 finding an electron in d r (n(r)d r) and the conditional probability of finding 3   3  an electron in d r , given that there is one at r (n (r,r )d r ): 2   ρ (r ,r)= n(r)n (r,r ) . (1.81) 2 2  By arguments similar to those used in Sect. 1.2.1, we interpret n (r,r)as 2  the average density of electrons at r , given that there is an electron at r. Clearly then  3   d r n (r,r)= N−1 . (1.82) 2 min,λ For the wavefunction Ψ , we write n   λ  n (r,r)= n(r)+n (r,r ) , (1.83) 2 xc λ   an equation which defines n (r,r ), the density at r of the exchange- xc correlation hole 33 about an electron at r. Equations (1.5) and (1.83) imply that  3  λ  d r n (r,r)=−1 , (1.84) xc which says that, if an electron is definitely at r, it is missing from the rest of the system.  Because the Coulomb interaction 1/u is singular as u =r−r→ 0, the exchange-correlation hole density has a cusp 35,34 around u=0:      ∂ dΩ u λ λ  n (r,r+u) = λ n(r)+n (r,r) , (1.85) xc xc  ∂u 4π u=0  where dΩ /(4π) is an angular average. This cusp vanishes when λ=0, u and also in the fully-spin-polarized and low-density limits, in which all other λ electronsareexcludedfromthepositionofagivenelectron:n (r,r)=−n(r). xc We can now rewrite (1.75) as 33    1 n(r)¯ n (r,r ) xc 3 3  E n= d r d r , (1.86) xc  2 r−r18 John P. Perdew and Stefan Kurth where  1  λ  n ¯ (r,r)= dλn (r,r ) (1.87) xc xc 0 is the coupling-constant averaged hole density. The exchange-correlation en- ergy is just the electrostatic interaction between each electron and the coupling-constant-averaged exchange-correlation hole which surrounds it. The hole is created by three effects: (1) self-interaction correction, a classical effect which guarantees that an electron cannot interact with itself, (2) the Pauli exclusion principle, which tends to keep two electrons with parallel spins apart in space, and (3) the Coulomb repulsion, which tends to keep any two electrons apart in space. Effects (1) and (2) are responsible for the exchange energy, which is present even atλ = 0, while effect (3) is responsible for the correlation energy, and arises only for λ=0. min,λ=0 If Ψ is a single Slater determinant, as it typically is, then the one- n and two-electron density matrices at λ = 0 can be constructed explicitly from the Kohn-Sham spin orbitals ψ (r): ασ  λ=0  ∗  ρ (r σ,rσ)= θ(µ −ε )ψ (r )ψ (r) , (1.88) ασ ασ 1 ασ α λ=0    ρ (r ,r)= n(r)n(r)+n(r)n (r,r ) , (1.89) x 2 where λ=0  2  ρ (r σ,rσ)  λ=0  1 n (r,r)= n (r,r)=− (1.90) x xc n(r) σ is the exact exchange-hole density. Equation (1.90) shows that  n (r,r )≤ 0 , (1.91) x so the exact exchange energy    1 n(r)n (r,r ) x 3 3  E n= d r d r (1.92) x  2 r−r is also negative, and can be written as the sum of up-spin and down-spin contributions: ↑ ↓ E = E +E 0 . (1.93) x x x Equation (1.84) provides a sum rule for the exchange hole:  3   d r n (r,r)=−1 . (1.94) x Equations (1.90) and (1.78) show that the “on-top” exchange hole density is 36 2 2 n (r)+n (r) ↑ ↓ n (r,r)=− , (1.95) x n(r)1 Density Functionals for Non-relativistic Coulomb Systems 19 which is determined by just the local spin densities at positionr – suggesting areasonwhylocalspindensityapproximationsworkbetterthanlocaldensity approximations. The correlation hole density is defined by    n ¯ (r,r)= n (r,r)+n¯ (r,r ) , (1.96) xc x c and satisfies the sum rule  3   d r n ¯ (r,r)=0 , (1.97) c which says that Coulomb repulsion changes the shape of the hole but not its integral. In fact, this repulsion typically makes the hole deeper but more short-ranged, with a negative on-top correlation hole density: n ¯ (r,r)≤ 0 . (1.98) c Thepositivityof(1.77)isequivalentvia(1.81)and(1.83)totheinequality   n ¯ (r,r )≥−n(r ) , (1.99) xc which asserts that the hole cannot take away electrons that were not there  initially. By the sum rule (1.97), the correlation hole density n ¯ (r,r ) must c have positive as well as negative contributions. Moreover, unlike the exchange   hole density n (r,r ), the exchange-correlation hole density n ¯ (r,r ) can be x xc positive. To better understand E , we can simplify (1.86) to the “real-space ana- xc lysis” 37  ∞ N n ¯ (u) xc 2 E n= du4πu , (1.100) xc 2 u 0 where   1 dΩ u 3 n ¯ (u) = d rn(r) n ¯ (r,r+u) (1.101) xc xc N 4π is the system- and spherical-average of the coupling-constant-averaged hole density. The sum rule of (1.84) becomes  ∞ 2 du4πu n ¯ (u) =−1 . (1.102) xc 0 2 Asu increases from 0,n (u) rises analytically liken (0)+O(u ), while x x n ¯ (u) rises like n ¯ (0) +O(u) as a consequence of the cusp of (1.85). c c Because of the constraint of (1.102) and because of the factor 1/u in (1.100), E typically becomes more negative as the on-top hole densityn ¯ (u) gets xc xc more negative.

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