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

Size: px

Start display at page:

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

Dwayne Strickland
5 years ago
Views:

1 Introduction to DFT and Density Functionas by Miche Côté Université de Montréa Département de physique

ca/ ~miche_cote/images_scientifiques/ images.

2 Eampes Carbazoe moecue Inside of diamant Réf: Jean-François Brière ~miche_cote/images_scientifiques/ images.shtm Réf: Mike Tower

3 3 Ψ Ψ H t i < < + + i e k j k j e j i j i j e 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: Hamitonian for eectrons and atoms with couomb interaction:

4 Eectrons Using the Born-Oppenheimer approimation: HΨ m 1 Ψ + 1 U ion r Ψ + 1 e r r, 1 Ψ EΨ Kinetic energy Potentia energy Interaction energy We are ooking for a soution of the type of a wave function for many eectrons: Ψ,,, 1 The probem is easy to write down but the soution 4

5 Eectrons system Ψ,,..., 1 Storage required: data 10 eectrons data bytes Gb Impracticabe!!! 5

6 Dirac s quote 199 «The underying physica aws necessary for the mathematica theory of a arge part of physics and the whoe of chemistry are thus competey known, and the difficuty is ony that the eact appication of these aws eads to equations much too compicated to be soube» Réf: Quantum mechanics of many-eectron systems, Proceedings of the Roya Society of London, pp

7 7 Wavefunction approach: Hartree method Ψ ,,, Ψ Ψ H E 1 λ 0 ion ion * V U m n d e U m E H λ λ n 1 * Lagrange mutipiers to assure that the remain orthogona Same equation for a

8 8! 1,,, 1! 1,,, P P P Φ Ψ Sater determinant Particues are not independant, change the position of one and a the others are affected. Paui ecusion principe is respected. Hartree-Fock method

9 9 Hartree-Fock method * 1 ion d V U m j j j ss H λ δ + + Echange energy Because of the echange term, the probem is much harder to resove. Resuts are better than those of the Hartree method but sti not very satisfying.

10 Configuration Interaction method Ψ,,, CiΦi 1,,, 1 i Sum of Sater determinants configurations Must find the coefficients C i CI configuration interaction CIS CI with singe ecitations ony CISD CI with singe and doube ecitations ony Correation energy: contribution over that of Hartree-Fock 10

11 Wavefunction methods Advantages: Contro approimations Systematic approach H, HF, CIS, Upper bound variationa principe Disadvantages: Very costy numericay up to 0-30 eectrons, forget soids! 11

12 Progress in theoretica methods John A. Pope obe Prize 1998 in Chemistry: "for his deveopment of computationa methods in quantum chemistry "for his deveopment of the density-functiona theory Water Kohn efficient feibe precise parameter free 1

13 Water Kohn and Canada Toronto obe Prize in 1998 for Chemistry: Deveopment of ab initio methods Density Functiona Theory WK efficient, feibe, precise, parameters free efficient Engand Sherbrooke feibe precise Harvard parameters free Water Kohn Austria 13

14 Miestones : first-principes approach Precursor : Thomas-Fermi approimation 197 Inhomogeneous eectron gas P. Hohenberg and W. Kohn, Phys. Rev. 136, B Sef-consistent equations incuding echange and correation effects W. Kohn and L. Sham, Phys. Rev. 140, A Ceperey, Ader 1980; Perdew, Zunger 1981 : computation and parametrization of the echange and correation energy needed in the oca density approimation 14

15 Most cited papers Papers pubished in APS journas PRL, PRA, PRB,.. RMP, most cited by papers pubished in APS journas 15

16 A basic reference on DFT and appications to soids Richard M. Martin Cambridge University Press, 004 Eectronic Structure : Basic Theory and Practica Methods ISB: For detais, see http : // 16

17 Functionas of the density

18 18 What is a functiona? It is a quantity that depends non on a variabe but on a function. [ ] d v n n f A f function functionna [ ] [ ] [ ] α αδ α α α α im im 0 0 n f n f n n f f f f + + Derivative function of

19 Density functiona theory: First Hononberg-Kohn theorem Honenberg et Kohn, Physica Review, vo 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 19

20 First Hohenberg-Kohn theorem The ground state density nr of a many-eectron system determines uniquey the eterna potentia Vr, moduo one goba constant. Consequence : formay, the density can be considered as the fundamenta variabe of the formaism, instead of the potentia. o need for wavefunctions or Schrödinger equation! 0

21 The constrained-search approach to DFT M. Levy, Proc. at. Acad. Sci. USA, 76, Use the etrema principe of QM. E V min{ Ψ Ĥ V Ψ } min min{ Ψ Ĥ V Ψ } Ψ n Ψ n where min n min n min n F n { } { min{ Ψ T ˆ + ˆV + ˆV Ψ }} int Ψ n { min{ Ψ T ˆ + ˆV Ψ + nrvrdr }} int Ψ n { F[ n] + nrvrdr} min E V n n [ ] min Ψ n { Ψ T ˆ + ˆV int Ψ } { [ ]} is a universa functiona of the density... ot known epicitey! 1

22 The Kohn & Sham approach

23 The echange-correation energy F[n] : arge part of the tota 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 eectronic system : T s [ n] min{ Ψ T ˆ Ψ } Ψ n Kinetic energy functiona of the density Echange-correation functiona of the density : E c [ n] F[ n] T s [ n] 1 nr 1 nr dr 1 dr r 1 - r ot known epicitey! But et s suppose we know it 3

24 The Kohn-Sham potentia So, we have to minimize: E V [ n] T s [ n] + Vrnr dr + 1 under constraint of tota eectron number nr 1 nr dr 1 dr + E c n r 1 - r [ ] Introduction of Lagrange mutipiers 0 δ E V [ n] λ{ nrdr - } % 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 eectrons in a potentia 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 potentia [ ] 4

25 The Kohn-Sham orbitas and eigenvaues on-interacting eectrons in the Kohn-Sham potentia : # 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 potentia To be soved sef-consistenty! i ψ i * rψ i r Echange-correation potentia ote : by construction, at sef-consistency, and supposing the echangecorreation functiona to be eact, the density wi be the eact density, the tota energy wi be the eact one, but Kohn-Sham wavefunctions and eigenenergies correspond to a fictitious set of independent eectrons, so they do not correspond to any eact quantity. 5

26 Minimum principe for the energy Using the variationa principe for non-interacting eectrons, one can show that the soution of the Kohn-Sham sef-consistent system of equations is equivaent 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 orthonormaization for the occupied orbitas. ψ i ψ j δ ij 6

27 Density Functiona Theory : approimations

28 An eact resut for the echange-correation energy without demonstration The echange-correation energy, functiona of the density is the integra over the whoe space of the density times the oca echange-correation energy per partice E c [ n] nr 1 ε c r 1 ;n dr 1 whie the oca echange-correation energy per partice is the eectrostatic interaction energy of a partice with its DFT echange-correation hoe. ε c r 1 ;n 1 n c r r 1 ;n dr r 1 r Sum rue : n c r r 1 ;n dr 1 8

29 The oca density approimation I Hypothesis : - the oca XC energy per partice ony depend on the oca density - and is equa to the oca XC energy per partice of an homogeneous eectron gas of same density in a neutraizing background - «jeium» ε LDA c r 1 ; n ε hom c nr 1 Gives eceent numerica resuts! Why? 1 Sum rue is fufied Characteristic screening ength indeed depend on oca density 9

30 The oca density approimation II Actua function : echange part + correation part c 1/3 ε hom n Cn 1/3 with C 3 4π 3π for the correation part, one resorts to accurate numerica simuations beyond DFT e.g. Quantum Monte Caro Corresponding echange-correation potentia 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 30

31 The oca 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 31

32 Beyond the oca density approimation Generaized gradient approimations GGA E appro c [ n] nr 1 ε appro c nr 1, nr 1, nr 1 dr 1 o mode system ike the homogeneous eectron gas! Many different proposas, incuding one from Perdew, Burke and Ernzerhof, Phys. Rev. Lett. 77, , often abbreviated «PBE». Others : PW86, PW91, LYP... Aso : «hybrid» functionas B3LYP, «eact echange» functiona, «sef-interaction corrected» functionas... 3

33 Eampe Charge density of graphite 33

34 Accuracy, typica usage. If covaent bonds, metaic bonds, ionic bonds : -3% for the geometry bond engths, ce parameters 0. ev for the bonding energies GGA probem with the band gap For weak bonding situations Hydrogen bonding, van derwaas, worse Treatment of a few hundred atoms is OK on powerfu parae computers Up to atoms is OK on a PC. 34

35 The band gap probem

36 The DFT bandgap probem I DFT is a ground state theory > no direct interpretation of Kohn-Sham eigenenergies in $ & 1 + V et r + % ε i nr 1 r 1 - r ' dr 1 + V c r ψ r ε i i ψ i r However { } are simiar to quasi-partice band structure : LDA / GGA resuts for vaence bands are accurate... but OT for the band gap E g KS ε c - ε v The band gap can aternativey be obtained from tota energy differences E g E+1 + E-1 - E E+1 - E [ correct epression! ] { } - { E - E-1 } in the imit where E is the tota energy of the - eectron system ε i 36

37 The DFT bandgap probem II For LDA & GGA, the XC potentia is a continuous functiona of the number of eectrons ε i E f i E g KS ε c - ε v [Janak's theorem] E g E+1 + E-1 - E In genera, the XC potentia might be discontinuous with the number of partice e.g. OEP E g KS E g eectrons +1 eectrons ε c - ε v E g KS ΔV c ε c - ε v 37

38 The DFT bandgap probem III Comparison of LDA and GW band Structures with photoemission and Inverse photoemission eperiments for Siicon. From "Quasipartice cacuations in soids", by Aubur WG, Jonsson L, Wikins JW, in Soid State Physics 54,

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

Introduction to DFT and Density Functionals by Michel Côté Université de Montréal Département de physique Eamples Carbazole molecule Inside of diamant Réf: Jean-François Brière http://www.phys.umontreal.ca/~michel_