Electronic Structure Calculations and Density Functional Theory

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Ecole normale supérieure UPMC Formation ModPhyChem Lyon, 09/10/2017

1 Electronic Structure Calculations and Density Functional Theory Rodolphe Vuilleumier Pôle de chimie théorique Département de chimie de l ENS CNRS Ecole normale supérieure UPMC Formation ModPhyChem Lyon, 09/10/2017 Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 1 / 56

2 Many applications of electronic structure calculations Geometries and energies, equilibrium constants Reaction mechanisms, reaction rates... Vibrational spectroscopies Properties, NMR spectra,... Excited states Force fields From material science to biology through astrochemistry, organic chemistry Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 2 / 56

3 Length /,("-%0,11,("2,"3,4)(",3"2,"1'56+,+&" and time scales noyaux et électrons 0,1 nm 10 nm 1µm 1ps 1ns 1µs 1s Variables atomes et molécules macroscopiques Modèles Gros-Grains! (champs) #$%&'(%')$*+," #-('(%')$*+," #.%&'(%')$*+,"!" Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 3 / 56

4 Potential energy surface for the nuclei c The LibreTexts libraries Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 4 / 56

5 Born-Oppenheimer approximation Total Hamiltonian of the system atoms+electrons: Ĥ T = ˆT N + ˆV NN + Ĥ M I m e approximate system wavefunction: Ψ S ( R I, r 1,..., r N ) = Ψ( R I )Ψ 0 ( r 1,..., r N ; R I ) Ψ 0 ( r 1,..., r N ; R I ): the ground state electronic wavefunction { } of the electronic Hamiltonian Ĥ at fixed ionic configuration RI Energy of the electrons at this ionic configuration: E 0 ( R I ) = Ψ 0 Ĥ Ψ 0 ionic Hamiltonian for the evolution of the nuclei wavefunction: Ĥ N = ˆT N + ˆV NN + E 0 (ˆ RI ) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 5 / 56

6 Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 6 / 56

7 Interacting electrons Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 7 / 56

8 Interacting electrons Interacting electrons Hamiltonian (in atomic units) for an N electrons system Ĥ = ˆT + ˆV ee + ˆV ext kinetic energy operator N ˆT = 1 2 i=1 ˆ 2 i electron-electron repulsion ˆV ee = ˆ r i ˆ r j i j where ˆ r i is the position operator for electron i Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 8 / 56

9 Interacting electrons Ground state wavefunction satisfies (fermions) We have Ψ 0 ( r 1,..., r N ) function of 3N variables Ψ 0 ( r 1,..., r i,..., r j,..., r N ) = Ψ 0 ( r 1,..., r j,..., r i,..., r N ) ĤΨ 0 ( r 1,..., r N ) = E 0 Ψ 0 ( r 1,..., r N ) or, alternatively, Ψ 0 can be obtained from a variational principle: E 0 = min Ψ Ĥ Ψ. Ψ( r 1,..., r N ) The minimization of Ψ Ĥ Ψ is realized for Ψ = Ψ 0 (non-degenerate ground state). Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 9 / 56

10 Interacting electrons Early calculations Linear Combination of Atomic Orbitals (LCAO) for one electron atomic orbitals to construct many-electron wavefunctions Two fundamental papers Boys, S. F., G. B. Cook, C. M. Reeves, et I. Shavitt Automatic Fundamental Calculations of Molecular Structure. Nature 178 (4544): doi: / a0. Boys, S. F., et G. B. Cook Mathematical Problems in the Complete Quantum Predictions of Chemical Phenomena. Reviews of Modern Physics 32 (2): doi: /revmodphys Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 10 / 56

11 Interacting electrons Gaussian basis set for one electron orbitals (Pople) Primitive cartesian gaussian functions: g( r, η, l, A) = N c (x A x ) lx (y A y ) ly (z A z ) lz e η( r A) 2 (1) r: Electron coordinate A: Atomic coordinate l: Angular momentum Contracted cartesian gaussian functions: ϕ µ ( r) = i d µi g µi ( r) (2) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 11 / 56

12 The Hartree-Fock method Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 12 / 56

13 The Hartree-Fock method Hartree-Fock approximation Wavefunction ansatz Ψ HF ( r 1,..., r N ) = 1 N! φ 1 ( r 1 ) φ 2 ( r 1 )... φ N ( r 1 ).. φ 1 ( r N ) φ N ( r N ) single Slater determinant (φ i single electron orbitals) Ψ HF 0 obtained by minimization of Ψ HF Ĥ Ψ HF However Ψ 0 is not in general a single determinant except for non-interacting particles (correlation effects) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 13 / 56

14 The Hartree-Fock method Solution of the Hartree-Fock method (restricted case) For an even number of electrons being paired in each orbital, the orbitals φ i ( r) are solutions of N/2 1 2 ˆ 2 + ˆV en + j=1 ( 2 ˆ ) J j ˆK j φ i ( r) = ˆF φ i ( r) = ɛ i φ i ( r) with Jˆ j φ i ( r) = φ j ( r 1 ) r r φ j( r )φ i ( r) d 3 r (Coulomb operator) and ˆK j φ i ( r) = φ j ( r 1 ) r r φ i( r )φ j ( r) d 3 r (Exchange operator) Self-Consistent Field method Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 14 / 56

15 Post Hartree-Fock methods Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 15 / 56

16 Post Hartree-Fock methods Full Configuration Interaction (FCI) Construct all possible Slater determinants for a given basis set, D I Construct the electronic Hamiltonian H IJ on this basis set of wavefunctions and diagonilize Computer cost grows very quickly with system size and basis-set size Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 16 / 56

17 Post Hartree-Fock methods Perturbation theory Møller-Plesset method The Hartree-Fock Slater determinant D HF is an eigenvector of the Fock operator ˆF We are looking for the eigenvector of Ĥ Rewrite Ĥ as ) Ĥ = ˆF + (Ĥ ˆF ) and treat (Ĥ ˆF as a perturbation Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 17 / 56

18 Post Hartree-Fock methods Other methods recipes to define a basis of Slater determinant Active space methods: CASSCF, CASPT2 Specified by m active electrons in n active orbitals Coupled-cluster methods: CCSD, CCSD(T) Ψ = e ˆT D HF with ˆT = 1 + ˆT 1 + ˆT Multi-reference conference interaction (MRCI), Valence Bond Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 18 / 56

19 Post Hartree-Fock methods Stochastic methods: Quantum Monte-Carlo Here, one flavor: Variational Monte-Carlo Rewritting of the ground-state energy: E 0 = = Πi d 3 r i Ψ 0 ({ r})ĥ Ψ 0 ({ r}) Πi d 3 r i Ψ 0 ({ r}) 2 Πi d 3 r i Ψ 0 ({ r}) 2 Ĥ Ψ 0({ r}) Ψ 0 ({ r}) Πi d 3 r i Ψ 0 ({ r}) 2 Sample the local energy Ĥ Ψ 0({ r}) Ψ 0 ({ r}) with the probability density Ψ 0 ({ r}) 2 with a Monte-Carlo method Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 19 / 56

20 Post Hartree-Fock methods Stochastic methods: Quantum Monte-Carlo Here, one flavor: Variational Monte-Carlo Allows complicated Ansatz for the wavefunction Ψ 0 : Ψ = eĵ I c I D I ˆ J = J({R K r i }, {r i r j }): Jastrow factor VMC: optimize variationally all parameters of the Ansatz Note that for the exact ground state the local energy is constant: Ĥ Ψ 0 ({ r}) Ψ 0 ({ r}) = E 0 Other falvors: diffusive Monte-Carlo... Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 20 / 56

21 Post Hartree-Fock methods Stochastic methods: Full-CI Quantum Monte-Carlo Diagonalization of a large matrix H IJ in the basis of all possible Slater determinants D I : Ψ 0 = I C I D I H IJ C J = C I Stockastic diagonalisation using a set of walkers such that the average population n I for determinant D I is the eingvector coefficient: n I = C I Evolution equation: n I (β + τ) = τ H IJ n J (β) τ(h II S)n I (β) J I Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 21 / 56

22 Post Hartree-Fock methods Outline of Density Functional Theory Instead of using the full electronic wavefunction Ψ 0 ( r 1,..., r N ), the electronic ground state of a system can be entirely described by its electron density n( r ) (Hohenberg-Kohn theorem) or, alternatively, by N one-electron orbitals φ 1 ( r ),..., φ N ( r ) giving rise to the same density n( r ): N d 3 r 2... d 3 r N Ψ ( r,..., r N ) 2 = n( r ) = N φ i ( r ) 2 i=1 (Kohn-Sham theorem) n( r ) or φ 1 ( r ),..., φ N ( r ) are obtained by minimization of a density functional Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 22 / 56

23 Table of contents 1 Interacting electrons 2 The Hartree-Fock method 3 Post Hartree-Fock methods 4 Density Functional Theory Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 23 / 56

24 The Hohenberg-Kohn theorem The Hohenberg-Kohn theorem Existence of a map n( r ) v( r ) v( r ) Ψ 0 n( r ) HK theorem Knowledge of n( r ) gives knowledge of v( r ) and of Ψ 0, thus all properties of the electronic system and in particular any expectation value of an observable Ô = Ψ Ô Ψ, are functionals of n( r ) (O[n] for the observable Ô). Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 24 / 56

25 The Hohenberg-Kohn theorem Energy functional The total energy of the system is now a functional E[n] of the density. We can write it as E[n] = d 3 r n( r )v( r ) + F HK [n] with F HK [n] = T [n] + V ee [n] T [n] = Ψ[n] ˆT Ψ[n] V ee [n] = Ψ[n] ˆV ee Ψ[n] F HK is universal (defined only for v-representable densities) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 25 / 56

26 The Hohenberg-Kohn theorem Variational principle For a given external potential v 0 ( r ), we define an energy functional E v0 [n] as E v0 [n] = d 3 r n( r )v 0 ( r ) + F HK [n] then E v0 [n] is minimal for n = n 0, the ground state density of v 0 : E 0 = E[n 0 ] < F HK [n] + d 3 r n( r )v 0 ( r ) for n( r ) n 0 ( r ) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 26 / 56

27 The Hohenberg-Kohn theorem Some remarks about the F HK functional Unfortunately F HK is not known and one has to make approximations for it. It can be shown that F HK [n] = min Ψ n Ψ ˆT + ˆV ee Ψ, realised for Ψ = Ψ[n] Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 27 / 56

28 Thomas-Fermi functional The Thomas-Fermi Functional F HK [n] = d 3 r C F n( r ) d 3 rd 3 r n( r )n( r ) r r C F n( r ) 5 3 : Kinetic energy of a uniform electron gas of independent electrons with density n = n( r ) (C F = 3 ( 3π 2 ) ) 10 J[n] = 1 d 3 rd 3 r n( r )n( r ) 2 r r : Hartree Mean-field term Minimization with the constraint F HK [n] n( r ) d 3 rn( r ) = N leads to + v 0 ( r) = µ Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 28 / 56

29 A simple exactly soluble problems Spinless fermions in a flat box Particle in a box ɛ j = π2 2 j 2 ; φ j (x) = 2 sin πjx; j = 1, 2, 3,... E(N) = LDA (TF) approximation N j=1 T LDA = ɛ j = π2 6 N(N + 1 )(N + 1) = T 2 dx t(n(x)); t(n) = π2 6 n3 n LDA (x) = cte = N T LDA = π2 6 N3 Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 29 / 56

30 A simple exactly soluble problems Spinless fermions in a flat box n(x) ,2 0,4 0,6 0,8 1 x Exact and LDA densities for 3 fermions in a box Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 30 / 56

31 A simple exactly soluble problems TF theory for neutral atoms Let s consider the case of an atom, then v 0 ( r) = Z r For a neutral atom, we have µ = 0, then if we call φ the electrostatic potential: 5 3 C F n(r) 2 3 = φ(r) 2 φ(r) = n(r) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 31 / 56

32 A simple exactly soluble problems 1 n(r) α 1 r 6 as r 2 E 0 = c 1 Z 7/3 3 n 0 (r) = Z 2 n 1 (Z 1/3 r) c 1 and n 1 are the exact limit behaviour as Z E. H. Lieb and B. Simon, Phys. Rev. Lett. 31, 681 (1973). E. H. Lieb, Rev. Mod. Phys. 53, 603 (1981). Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 32 / 56

33 A simple exactly soluble problems Regions of electronic cloud 1 Inner core region of size Z 1/3 with density Z 2 and Z electrons 2 Mantle core with n α 1 r 6 (independant of Z) and size also Z 1/3 3 A complicated intermediate region 4 Outer shell of size 1 and Z 2/3 electrons. Chemistry takes place here! 5 Outside region where density decays exponentially. Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 33 / 56

34 A simple exactly soluble problems TF functional leads to densities with wrong asymptotic behavior no bonding however gives reasonable energies for true electronic densities recover correct densities as Z Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 34 / 56

35 Local Density Approximation to the kinetic energy Local Density Approximation to the kinetic energy Approximation for many electron systems Works best for large N Local approximation are crudely correct but miss crucial details for chemistry Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 35 / 56

36 The Kohn-Sham theorem The Kohn-Sham method Postulate an auxiliary set of one electron orbitals {φ i } solution of a system of N non-interacting electrons, in an external potential v s, Ĥ s φ i = ɛ i φ i φ i φ j = δ ij such as Ĥ s = v s ( r ) Ψ s = 1 N! det[φ 1φ 2... φ N ] n s ( r ) = N φ i ( r ) 2 n( r ) i=1 If this is possible n( r ) is said to be non-interacting v-representable Then from a Hohenberg-Kohn theorem for non-interacting electrons, the map n( r ) v s ( r ) {φ i ( r )} is unique Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 36 / 56

37 The Kohn-Sham theorem Total energy decomposition I The total energy of the system can be decomposed into E[n] = T s [n] + d 3 r n( r )v( r ) + J[n] + E xc [n] with recall that φ i φ i [n] and T s [n] = N φ i φ i i=1 T s [n] = min {φ i } n φ i φ j =δ ij N φ i φ i i=1 (cures the major deficiencies of TF functional) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 37 / 56

38 The Kohn-Sham theorem Total energy decomposition II J[n] = 1 2 d 3 rd 3 r n( r )n( r ) r r : Hartree term and finally E xc [n] is the exchange correlation energy: E xc [n] = (T [n] T s [n]) + (V ee [n] J[n]) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 38 / 56

39 The Kohn-Sham theorem The exchange-correlation potential Define the exchange-correlation potential as n( r ) v xc [n]( r ) = v s [n]( r ) r r d3 r v ext ( r ) To calculate the density n 0 ( r ) corresponding to fixed v 0 ( r ), we have to solve ( 1 n0 ( r ) ) v 0 ( r ) + r r d3 r + v xc [n 0 ]( r ) φ i ( r ) = ɛ i φ i ( r ) with and n 0 ( r ) = N φ i ( r ) 2 i=1 v xc [n]( r ) = E xc[n] n( r ) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 39 / 56

40 The Kohn-Sham theorem Kohn-Sham energies ɛ i Act as the Lagrange multipliers for d 3 r φ i ( r ) 2 = 1 Chemical potential: µ = ɛ HOMO + ɛ LUMO 2 Mermin: generalization to finite temperature T and grand-canonical ensemble Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 40 / 56

41 The Kohn-Sham theorem Local density approximation approximate the xc energy by Exc LDA = d 3 r ɛ xc (n( r )) n( r ) ɛ xc (n) is the exchange and correlation energy per electron of the homogeneous electron gas with density n Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 41 / 56

42 The Kohn-Sham theorem ɛ xc (n) = ɛ x (n) + ɛ c (n) ɛ x (n): Dirac exchange energy ɛ x (n) = C x n 1 3 ; Cx = 3 4 ( ) π (when added to the TF functional, give the Thomas-Fermi-Dirac functional) ɛ c (n): fit to a quantum Monte-Carlo calculation (Ceperley-Alder) two parametrisations using Padé approximants (Volsko, Wilk, Nussair and Perdew-Zunger) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 42 / 56

43 The Kohn-Sham theorem Generalized gradient approximation From 1965 till 1986: dark ages of DFT. Determinaton of exact results. GGA: E xc [n] = d 3 r e xc (n( r )) F (s( r )) s = n 2k F n ; k F = (3π 2 n) 1/3 Most used for molecular systems: BLYP (fit to closed shell atoms) separates exchange (B88) and correlation (LYP) Also common are GGA s due to Perdew (ab initio): P86, PW91, PBE Largely correct for LDA overbinding LDA PBE BLYP 1 ev 1/3 ev 1/6 ev Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 43 / 56

44 The Kohn-Sham theorem Functional zoology LDA GGA PBE, revpbe, PW91,BP86, BLYP, HCTCH etc. Meta-GGA TPSS Hybrid B3LYP, PBE0... Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 44 / 56

The Kohn-Sham theorem Water molecule: electronic density ρ sum of atomic densities Optimized electronic density 0.1 0.

45 The Kohn-Sham theorem Water molecule: electronic density ρ sum of atomic densities Optimized electronic density Formation energy E H2 O = 0.5 A.U. = 1300 kj.mol 1 Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 45 / 56

46 The Kohn-Sham theorem Relaxation about half the formation energy Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 46 / 56

47 The Kohn-Sham theorem Water dimer geometry LDA BP BLYP expt. d OO (Å) OHO (deg) E 2 (kj.mol 1 ) M. Sprik et al., J. Chem. Phys. 105, 1142 (1996) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 47 / 56

electronic densities of the separate species Rodolphe

48 The Kohn-Sham theorem Water dimer electronic density Total electronic density Difference with the sum of the electronic densities of the separate species Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 48 / 56

49 The Kohn-Sham theorem The exchange-correlation hole P( r, r ): probability to find one electron in r and one electron in r P( r, r ) = N(N 1) Ψ ( r, r, r 3..., r N ) 2 d 3 r 3... d 3 r N P( r, r ) = n( r ) ( n( r ) + n xc ( r, r ) ) n xc ( r, r ) exchange-correlation hole U xc [n] = V ee [n] J[n] = 1 2 Universal property: d 3 rd 3 r n( r )n xc( r, r ) r r n xc ( r, r ) d 3 r = 1 exactly Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 49 / 56

50 The Kohn-Sham theorem Example: exchange hole in Helium 1s 1 2s 1 nhrl nhrl r 0 = 0.8 r 0 = 2.5 Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 50 / 56

51 The Kohn-Sham theorem exchange-correlation potential for Be atom 0-0,5 v xc (r) -1 exact LDA -1/r -1, r courtesy of A. Savin Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 51 / 56

52 be Density Functional Theory The Kohn-Sham theorem n(x) = exp( 2 x ) + 2 exp( 4 x L ) ( exchange-correlation potential for H+He= discontinuities where the H-atom is at 0 and the He ion is at L. For this two electron system, w the molecular orbital as φ(x) = n(x)/2 and the Kohn-Sham potential by inversion. 3 v S (x) n(x) 2 1d H-He+ density d H-He+ potential x Figure 14.2: Density and Kohn-Sham potential for 1d H-He +. is shown in Fig. 14.2, for L = 4. The apparent step in the potential between the a Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 52 / 56

53 The Kohn-Sham theorem Some defects of common functionals lack of long distance dispersion (vdw term 1 r 6 ) self-interaction error stretch H 2 Derivative discontinuity and Fe 2+ -Fe 3+ problem Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 53 / 56

54 The Kohn-Sham theorem Underlying approximations The development of the theory given here depends on two explicit approximations: first, the validity of Schrödinger s many-particle equation and the antisymmetry condition and, second, the sufficiency of the Born-Oppenheimer approximation in which nuclear and electronic motions are separated Boys, Cook, Reeves, Shavitt (1956) Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 54 / 56

55 References For further reading I A. Szabo and N. S. Ostlung. Modern Quantum Chemistry. Dover, R. Martins. Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, R. G. Parr and W. Yang. Density Functional Theory of Atoms and Molecules. Oxford University Press, R. M. Dreizler and E. K. U. Gross. Density Functional Theory. Springer-Verlag, Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 55 / 56

56 References For further reading II K. Burke. The abc of dft. Rodolphe Vuilleumier (ENS UPMC) Electronic Structure and DFT ModPhyChem 56 / 56

Density Functional Theory for Electrons in Materials

Density Functional Theory for Electrons in Materials Richard M. Martin Department of Physics and Materials Research Laboratory University of Illinois at Urbana-Champaign 1 Density Functional Theory for