Introduction to DFT and Density Functionals. by Michel Côté Université de Montréal Département de physique

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1 Introduction to DFT and Density Functionals by Michel Côté Université de Montréal Département de physique

ca/~michel_ cote/images_scientifiques/images.

2 Eamples Carbazole molecule Inside of diamant Réf: Jean-François Brière cote/images_scientifiques/images.shtml Réf: Mike Towler

3 3 Ψ = Ψ H t i < < + + = Ni Ne k j k j Ne j Ni j Ni j Ne j R R e Z Z r r e R r e Z M m H β α β α β α α α α α α α Condensed Matter Schrödinger's equation: Hamiltonian for electrons and atoms with coulomb interaction:

4 Electrons Using the Born-Oppenheimer approimation: HΨ = m N l= 1 l Ψ + N l= 1 U ion r Ψ + l 1 N e r r l, l = 1 l l Ψ = EΨ Kinetic energy Potential energy Interaction energy We are looking for a solution of the type of a wave function for many electrons: Ψ,,, 1 N The problem is easy to write down but the solution 4

5 Electrons system Ψ,,..., 1 N Storage required: =1000 data 10 electrons data bytes = Gb Impracticable!!! 5

6 Dirac s quote 199 «The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the eact application of these laws leads to equations much too complicated to be soluble» Réf: Quantum mechanics of many-electron systems, Proceedings of the Royal Society of London, pp

7 Dirac s quote 199 «It therefore becomes desirable that approimate practical methods of applying quantum mechanics should be developed, which can lead to an eplanation of the main features of comple atomic systems without too much computation.» Réf: Quantum mechanics of many-electron systems, Proceedings of the Royal Society of London, pp

8 8 Wavefunction approach: Hartree method = = = Ψ N l l l N N N ,,, φ φ φ φ = Ψ Ψ = N l l l l H E 1 φ φ λ 0 ion ion * V U m n d e U m E l l l H l l l l λ φ φ λ φ φ φ = + + = + + = = = N l l l n 1 * φ φ Lagrange multipliers to assure that the φ l remain orthogonal Same equation for all φ l

9 9! 1,,, 1! 1,,, N N N N N N N N l l l Pl P P N N N φ φ φ φ φ φ φ φ φ φ = = Φ = Ψ = Slater determinant Particules are not independant, change the position of one and all the others are affected. Pauli eclusion principle is respected. Hartree-Fock method

10 10 Hartree-Fock method * 1 ion d V U m l l l j N j j ss l H λ φ φ φ φ δ φ = + + = Echange energy Because of the echange term, the problem is much harder to resolve. Results are better than those of the Hartree method but still not very satisfying.

11 Configuration Interaction method Ψ,,, N = CiΦi 1,,, 1 N i Sum of Slater determinants configurations Must find the coefficients C i CI = configuration interaction CIS = CI with single ecitations only CISD = CI with single and double ecitations only Correlation energy: contribution over that of Hartree-Fock 11

12 Wavefunction methods Advantages: Control approimations Systematic approach H, HF, CIS, Upper bound variational principle Disadvantages: Very costly numerically up to 0-30 electrons, forget solids! 1

13 Progress in theoretical methods John A. Pople Nobel Prize 1998 in Chemistry: "for his development of computational methods in quantum chemistry "for his development of the density-functional theory Walter Kohn efficient fleible precise parameter free 13

14 Walter Kohn and Canada Toronto Nobel Prize in 1998 for Chemistry: Development of ab initio methods Density Functional Theory WK efficient, fleible, precise, parameters free efficientengland Sherbrooke fleible precise Harvard parameters free Walter Kohn Austria Walter Kohn died April 16,

15 Milestones : first-principles approach Precursor : Thomas-Fermi approimation 197 Inhomogeneous electron gas P. Hohenberg and W. Kohn, Phys. Rev. 136, B Self-consistent equations including echange and correlation effects W. Kohn and L. Sham, Phys. Rev. 140, A Ceperley, Alder 1980; Perdew, Zunger 1981 : computation and parametrization of the echange and correlation energy needed in the local density approimation 15

16 Most cited papers Papers published in APS journals PRL, PRA, PRB,.. RMP, most cited by papers published in APS journals S. Redner, Citation Statistics from 110 Years of Physical Review, Physics Today, June 005. Today, according to Google Scholar: K&S, 40k; H&K, 35k; PBE functional, 60k! 16

17 A basic reference on DFT and applications to solids Richard M. Martin Cambridge University Press, 004 Electronic Structure : Basic Theory and Practical Methods ISBN: For details, see http : // 17

18 Functionals of the density

19 19 What is a functional? It is a quantity that depends non on a variable but on a function. [ ] = = d v n n f A f function functionnal [ ] [ ] [ ] α αδ α α α α lim lim 0 0 n f n f n n f f f f + = + = Derivative function of

20 Density functional theory: First Hononberg-Kohn theorem Honenberg et Kohn, Physical Review, vol 136, B864, 1964 Assume: V r V r Ψ n r Ψ n r E = Ψ H Ψ < Ψ H Ψ = Ψ H Vr + V r Ψ E < E + [ V r Vr]nrdr Samething but starting from E: Combine: E < E + [Vr V r]nrdr E + E < E + E Contradiction!!! n r V r Ψ 0

21 First Hohenberg-Kohn theorem The ground state density nr of a many-electron system determines uniquely the eternal potential Vr, modulo one global constant. Consequence : formally, the density can be considered as the fundamental variable of the formalism, instead of the potential. No need for wavefunctions or Schrödinger equation! 1

22 The constrained-search approach to DFT M. Levy, Proc. Nat. Acad. Sci. USA, 76, Use the etremal principle of QM. E V = min{ Ψ Ĥ V Ψ } = min{ min{ Ψ Ĥ V Ψ }} Ψ n Ψ n = min{ min{ Ψ T ˆ + ˆV + ˆV Ψ }} int n Ψ n where = min n = min n F n { min{ Ψ T ˆ + ˆV Ψ + nrvrdr }} int Ψ n { F[ n] + nrvrdr} = min [ ] = min Ψ n { Ψ T ˆ + ˆV int Ψ } n { E V [ n] } is a universal functional of the density... Not known eplicitely!

23 The Kohn & Sham approach

24 The echange-correlation energy F[n] : large part of the total energy, hard to approimate Kohn & Sham Phys. Rev. 140, A : mapping of the interacting system on a non-interacting system [ ] = min F n Ψ n { Ψ T ˆ + ˆV int Ψ } If one considers a non-interacting electronic system : T s [ n] = min{ Ψ T ˆ Ψ } Ψ n Kinetic energy functional of the density Echange-correlation functional of the density : E c [ n] = F[ n] T s [ n] 1 nr 1 nr dr 1 dr r 1 - r Not known eplicitely! But let s suppose we know it 4

25 The Kohn-Sham potential So, we have to minimize: under constraint of total electron number E V [ n] = T s [ n] + Vrnr dr + 1 nr 1 nr dr 1 dr + E c n r 1 - r [ ] Introduction of Lagrange multipliers [ ] λ{ nrdr - N} = 0 = δ E V n % δt s ' & [ n] δn + Vr + [ ] nr 1 dr 1 + δe c n r 1 - r δnr λ* δnrdr If one considers the minimization for non-interacting electrons in a potential V KS r, with the same density nr, one gets 0 = $ % & δt s [ n] δn Identification : ' + V KS r λ δnrdr V KS r = Vr + nr 1 dr 1 + δe c n r 1 - r δnr The Kohn-Sham potential [ ] 5

26 The Kohn-Sham orbitals and eigenvalues Non-interacting electrons in the Kohn-Sham potential : # 1 & + V KS r $ % ' ψ i r = ε i ψ i r Density nr = V KS r = V et r + [ ] nr 1 dr 1 + δe c n r 1 - r δnr Hartree potential To be solved self-consistently! i ψ i * rψ i r Echange-correlation potential Note : by construction, at self-consistency, and supposing the echangecorrelation functional to be eact, the density will be the eact density, the total energy will be the eact one, but Kohn-Sham wavefunctions and eigenenergies correspond to a fictitious set of independent electrons, so they do not correspond to any eact quantity. 6

27 Minimum principle for the energy Using the variational principle for non-interacting electrons, one can show that the solution of the Kohn-Sham self-consistent system of equations is equivalent to the minimisation of E KS { ψ i } "# $ % = ψ i 1 ψ i + V et rnr dr + 1 i nr 1 nr dr 1 dr + E c n r 1 - r [ ] under constraints of orthonormalization for the occupied orbitals. ψ i ψ j = δ ij 7

28 Density Functional Theory : approimations

29 An eact result for the echange-correlation energy without demonstration The echange-correlation energy, functional of the density is the integral over the whole space of the density times the local echange-correlation energy per particle E c [ n] = nr 1 ε c r 1 ;n dr 1 while the local echange-correlation energy per particle is the electrostatic interaction energy of a particle with its DFT echange-correlation hole. ε c r 1 ;n= 1 n c r r 1 ;n dr r 1 r Sum rule : n c r r 1 ;n dr = 1 9

30 The local density approimation I Hypothesis : - the local XC energy per particle only depend on the local density - and is equal to the local XC energy per particle of an homogeneous electron gas of same density in a neutralizing background - «jellium» ε LDA c r 1 ; n = ε hom c nr 1 Gives ecellent numerical results! Why? 1 Sum rule is fulfilled Characteristic screening length indeed depend on local density 30

31 The local density approimation II Actual function : echange part + correlation part c 1/3 ε hom n = Cn 1/3 with C = 3 4π 3π for the correlation part, one resorts to accurate numerical simulations beyond DFT e.g. Quantum Monte Carlo Corresponding echange-correlation potential V appro c r = µ c nr µ c n = d nε c dn µ n = C 4 3 n1/3 = 4 3 ε hom n appro n [ n] V c r = δe c δnr 31

32 The local density approimation III To summarize : or and E LDA E LDA [ n] = T s [ n] + V et rnr dr + 1 { ψ i } nr 1 nr r 1 - r "# $ % = ψ i 1 ψ i + V et rnr dr + 1 V LDA KS r = V et r + i + nr 1 ε LDA c nr 1 dr 1 nr 1 r 1 - r dr 1 + µ hom c nr dr 1 dr + E LDA c [ n] nr 1 nr dr 1 dr r 1 - r 3

33 Beyond the local density approimation Generalized gradient approimations GGA E appro c [ n] = nr 1 ε appro c nr 1, nr 1, nr 1 dr 1 No model system like the homogeneous electron gas! Many different proposals, including one from Perdew, Burke and Ernzerhof, Phys. Rev. Lett. 77, , often abbreviated «PBE». Others : PW86, PW91, LYP... Also : «hybrid» functionals B3LYP, «eact echange» functional, «self-interaction corrected» functionals... 33

34 Eample Charge density of graphite 34

35 Accuracy, typical usage. If covalent bonds, metallic bonds, ionic bonds : -3% for the geometry bond lengths, cell parameters 0. ev for the bonding energies GGA problem with the band gap For weak bonding situations Hydrogen bonding, van derwaals, worse Treatment of a few thousand atoms is doable on powerful parallel computers Up to atoms is OK on a PC. 35

36 The band gap problem

37 The DFT bandgap problem I DFT is a ground state theory =>no direct interpretation of Kohn-Sham eigenenergies $ & 1 + V et r + % nr 1 r 1 - r ' dr 1 + V c r ψ r = ε i i ψ i r However { ε i } are similar to quasi-particle band structure : LDA / GGA results for valence bands are accurate... but NOT for the band gap E g KS = ε c - ε v The band gap can alternatively be obtained from total energy differences E g = EN+1 + EN-1 - EN = EN+1 - EN in the limit N where EN is the total energy of the N - electron system in [ correct epression! ] { } - { EN - EN-1 } ε i 37

38 The DFT bandgap problem II For LDA & GGA, the XC potential is a continuous functional of the number of electrons ε i = E f i E g KS = ε c - ε v [Janak's theorem] = E g = EN+1 + EN-1 - EN N In general, the XC potential might be discontinuous with the number of particle e.g. OEP E g KS E g N electrons N+1 electrons ε c - ε v = E g KS ΔV c ε c - ε v 38

39 The DFT bandgap problem III Comparison of LDA and GW band Structures with photoemission and Inverse photoemission eperiments for Silicon. From "Quasiparticle calculations in solids", by Aulbur WG, Jonsson L, Wilkins JW, in Solid State Physics 54,

40 Accuracy: doi: / strongly and 5 weakly bound solids

41 Pseudopotentials: Science 351, aad DOI: /science.aad3000

42 Pseudopotentials:

43 The band gap problem

44 The DFT bandgap problem I DFT is a ground state theory =>no direct interpretation of Kohn-Sham eigenenergies 1 + V et r + ε i nr 1 r 1 - r dr 1 + V c r ψ i r = ε i ψ i r However { } are similar to quasi-particle band structure : LDA / GGA results for valence bands are accurate... but NOT for the band gap E KS g = ε c - ε v The band gap can alternatively be obtained from total energy differences E g = EN+1 + EN-1 - EN = EN+1 - EN in the limit N where EN is the total energy of the N - electron system in [ correct epression! ] { } - { EN - EN-1 } ε i!40

45 The DFT bandgap problem II For LDA & GGA, the XC potential is a continuous functional of the number of electrons ε i = E f i E g KS = ε c - ε v [Janak's theorem] = E g = EN+1 + EN-1 - EN N In general, the XC potential might be discontinuous with the number of particle N electrons E g KS E g N+1 electrons ε c - ε v = E g KS ΔV c ε c - ε v!41

46 The DFT bandgap problem III Comparison of LDA and GW band Structures with photoemission and Inverse photoemission eperiments for Silicon. From "Quasiparticle calculations in solids", by Aulbur WG, Jonsson L, Wilkins JW, in Solid State Physics 54, !4

47 Relation to Many-body methods : the Sham-Schlüter equation Kohn-Sham hamiltonian: KS potential: choose such that it reproduces the interacting density KS Green function:!43

48 Relation to Many-body methods : the Sham-Schlüter equation Define interaction: Interacting hamiltonian: Dyson equation:!44

49 Relation to Many-body methods : the Sham-Schlüter equation!45

50 Relation to Many-body methods : the Sham-Schlüter equation!46

51 Relation to Many-body methods : the Sham-Schlüter equation!47

52 Thank you for your attention. Questions?!48

Introduction to DFT and Density Functionals. by Michel Côté Université de Montréal Département de physique

Introduction to DFT and Density Functionas by Miche Côté Université de Montréa Département de physique Eampes Carbazoe moecue Inside of diamant Réf: Jean-François Brière http://www.phys.umontrea.ca/ ~miche_cote/images_scientifiques/