From Quantum Mechanics to Materials Design

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1 The Basics of Density Functional Theory Volker Eyert Center for Electronic Correlations and Magnetism Institute of Physics, University of Augsburg December 03, 2010

2 Outline Formalism 1 Formalism Definitions and Theorems Approximations 2

3 Outline Formalism 1 Formalism Definitions and Theorems Approximations 2

4 Calculated Electronic Properties Moruzzi, Janak, Williams (IBM, 1978) E/Ryd E/Ryd K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga 0 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Cohesive Energies ˆ= Stability WSR/au WSR/au K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga 2 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Wigner-Seitz-Rad. ˆ= Volume Bulk modulus/kbar Bulk modulus/kbar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga 0 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Compressibility ˆ= Hardness

5 Energy band structures from screened HF exchange Si, AlP, AlAs, GaP, and GaAs Experimental and theoretical bandgap properties Shimazaki, Asai, JCP 132, (2010)

6 Outline Formalism Definitions and Theorems Approximations 1 Formalism Definitions and Theorems Approximations 2

7 Key Players Formalism Definitions and Theorems Approximations Hamiltonian (within Born-Oppenheimer approximation) H = H el,kin + H el el + H ext ] [ 2 2m 2 i = i e 2 4πǫ 0 i,j j i 1 r i r j + i v ext (r i ) where v ext (r i ) = 1 2 i e 2 4πǫ 0 µν µ ν Z µ Z ν R µ R ν e2 4πǫ 0 µ i Z µ R µ r i µ: ions with charge Z µ, i: electrons

8 Key Players Formalism Definitions and Theorems Approximations Electron Density Operator ˆρ(r) = N δ(r r i ) = αβ i=1 χ α(r)χ β (r)a + α a β χ α : single particle state

9 Key Players Formalism Definitions and Theorems Approximations Electron Density Operator ˆρ(r) = N δ(r r i ) = αβ i=1 χ α(r)χ β (r)a + α a β χ α : single particle state Electron Density ρ(r) = Ψ ˆρ(r) Ψ = α χ α (r) 2 n α Ψ : many-body wave function, n α : occupation number Normalization: N = d 3 r ρ(r)

10 Key Players Formalism Definitions and Theorems Approximations Functionals Universal Functional (independent of ionic positions!) F = Ψ H el,kin + H el el Ψ Functional due to External Potential: Ψ H ext Ψ = Ψ v ext (r)δ(r r i ) Ψ i = d 3 r v ext (r)ρ(r)

11 Authors Formalism Definitions and Theorems Approximations Pierre C. Hohenberg Walter Kohn Lu Jeu Sham

12 Definitions and Theorems Approximations Hohenberg and Kohn, 1964: Theorems 1st Theorem The external potential v ext (r) is determined, apart from a trivial constant, by the electronic ground state density ρ(r). 2nd Theorem The total energy functional E[ρ] has a minimum equal to the ground state energy at the ground state density.

13 Definitions and Theorems Approximations Hohenberg and Kohn, 1964: Theorems 1st Theorem The external potential v ext (r) is determined, apart from a trivial constant, by the electronic ground state density ρ(r). 2nd Theorem The total energy functional E[ρ] has a minimum equal to the ground state energy at the ground state density. Nota bene Both theorems are formulated for the ground state! Zero temperature! No excitations!

14 Definitions and Theorems Approximations Hohenberg and Kohn, 1964: Theorems Maps Ground state Ψ 0 (from minimizing Ψ 0 H Ψ 0 ): v ext (r) (1) = Ψ 0 (2) = ρ 0 (r)

15 Definitions and Theorems Approximations Hohenberg and Kohn, 1964: Theorems Maps Ground state Ψ 0 (from minimizing Ψ 0 H Ψ 0 ): v ext (r) (1) = Ψ 0 (2) = ρ 0 (r) 1st Theorem v ext (r) (1) Ψ 0 (2) ρ 0 (r)

16 Formalism Definitions and Theorems Approximations Levy, Lieb, : Constrained Search Percus-Levy partition

17 Definitions and Theorems Approximations Levy, Lieb, : Constrained Search Variational principle E 0 = inf Ψ Ψ H Ψ = inf el,kin + H el el + H ext Ψ Ψ [ = inf inf el,kin + H el el Ψ + ρ(r) Ψ S(ρ) [ ] =: inf F LL [ρ] + d 3 r v ext (r)ρ(r) ρ(r) = inf ρ(r) E[ρ] ] d 3 r v ext (r)ρ(r) S(ρ): set of all wave functions leading to density ρ F LL [ρ]: Levy-Lieb functional, universal (independent of H ext )

18 Definitions and Theorems Approximations Levy, Lieb, : Constrained Search Levy-Lieb functional F LL [ρ] = inf Ψ H el,kin + H el el Ψ Ψ S(ρ) = T[ρ] + W xc [ρ] }{{} +1 e 2 d 3 r 2 4πǫ 0 = G[ρ] + 1 e 2 d 3 r 2 4πǫ 0 Functionals Kinetic energy funct.: T[ρ] Exchange-correlation energy funct.: W xc [ρ] Hartree energy funct.: 1 2 e 2 4πǫ 0 d 3 r d 3 r ρ(r)ρ(r ) r r d 3 r ρ(r)ρ(r ) r r d 3 r ρ(r)ρ(r ) r r not known! not known! known!

19 Definitions and Theorems Approximations Thomas, Fermi, 1927: Early Theory Approximations Failures ignore exchange-correlation energy functional: W xc [ρ]! = 0 approximate kinetic energy functional: T[ρ] = C F d 3 r (ρ(r)) 5 3, CF = atomic shell structure missing periodic table can not be described 2 no-binding theorem (Teller, 1962) 2 ( 3π 2) 2 3 2m

20 Definitions and Theorems Approximations Kohn and Sham, 1965: Single-Particle Equations Ansatz 1 use different splitting of the functional G[ρ] T[ρ] + W xc [ρ] = G[ρ]! = T 0 [ρ] + E xc [ρ] 2 reintroduce single-particle wave functions Imagine: non-interacting electrons with same density Density: ρ(r) = occ α χ α(r) 2 known! Kinetic energy funct.: T 0 [ρ] = occ α d 3 r χ α (r) [ 2 2m 2 ]χ α (r) known! Exchange-correlation energy funct.: E xc [ρ] not known!

21 Definitions and Theorems Approximations Kohn and Sham, 1965: Single-Particle Equations Euler-Lagrange Equations (Kohn-Sham Equations) ] δe[ρ] δχ α (r) ε αχ α (r) = [ 2 2m 2 + v eff (r) ε α χ α (r) =! 0 Effective potential: v eff (r) := v ext (r) + v H (r) + v xc (r) Exchange-correlation potential: v xc (r) := δe xc[ρ] δρ Single-particle energies : ε α (Lagrange-parameters, orthonormalization) not known!

22 Definitions and Theorems Approximations Kohn and Sham, 1965: Local Density Approximation Be Specific! Approximate exchange-correlation energy functional E xc [ρ] = ρ(r)ε xc (ρ(r))d 3 r Exchange-correlation energy density ε xc (ρ(r)) depends on local density only! is calculated from homogeneous, interacting electron gas Exchange-correlation potential [ ] v xc (ρ(r)) = ρ {ρε xc(ρ)} ρ=ρ(r)

23 Definitions and Theorems Approximations Kohn and Sham, 1965: Local Density Approximation Homogeneous, Interacting Electron Gas Split ε xc (ρ) = ε x (ρ) + ε c (ρ) Exchange energy density ε x (ρ) (exact for homogeneous electron gas) ε x (ρ) = 3 4π v x (ρ) = 1 π e 2 4πǫ 0 (3π 2 ρ) 1 3 e 2 4πǫ 0 (3π 2 ρ) 1 3 Correlation energy density ε c (ρ) Calculate and parametrize RPA (Hedin, Lundqvist; von Barth, Hedin) QMC (Ceperley, Alder; Vosko, Wilk, Nusair; Perdew, Wang)

24 Definitions and Theorems Approximations Kohn and Sham, 1965: Local Density Approximation Limitations and Beyond LDA exact for homogeneous electron gas (within QMC) Spatial variation of ρ ignored include ρ(r),... Generalized Gradient Approximation (GGA) Cancellation of self-interaction in v Hartree (ρ(r)) and v x (ρ(r)) violated for ρ = ρ(r) Self-Interaction Correction (SIC) Exact Exchange (EXX), Optimized Effective Potential (OEP) Screened Exchange (SX)

25 Outline Formalism 1 Formalism Definitions and Theorems Approximations 2

Iron Pyrite: FeS 2 Pyrite Pa 3 (T 6 h ) a = 5.

26 Iron Pyrite: FeS 2 Pyrite Pa 3 (T 6 h ) a = Å NaCl structure sublattices occupied by iron atoms sulfur pairs sulfur pairs 111 axes x S = rotated FeS 6 octahedra

27 FeS 2 : Equilibrium Volume and Bulk Modulus V 0 = a B 3 E 0 = Ryd B 0 = GPa E (Ryd) V (a B )

28 FeS 2 : From Atoms to the Solid E (Ryd) V (a B 3 )

29 FeS 2 : Structure Optimization E (Ryd) x S

30 Phase Stability in Silicon dia hex β-tin sc bcc fcc E (Ryd) V (a B ) diamond structure most stable pressure induced phase transition to β-tin structure

LTO(Γ)-Phonon in Silicon -1156.1000-1156.1010-1156.1020 E (Ryd) -1156.1030-1156.1040-1156.1050-1156.1060-1156.

31 LTO(Γ)-Phonon in Silicon E (Ryd) x Si phonon frequency: f calc = THz (f exp = THz)

32 Dielectric Function of Al 2 O 3 Imaginary Part FLAPW, Hosseini et al., 2005 FPLMTO, Ahuja et al., 2004 FPASW Im ε xx Im ε zz 0.4 ε E (ev)

33 Dielectric Function of Al 2 O 3 Real Part FLAPW, Hosseini et al., 2005 FPLMTO, Ahuja et al., 2004 FPASW Re ε xx Re ε zz 0.3 ε E (ev)

34 Hydrogen site energetics in LaNi 5 H n and LaCo 5 H n Enthalpy of hydride formation in LaNi 5 H n H min = 40kJ/molH 2 for H at 2b6c 1 6c 2 agrees with neutron data calorimetry: H min = (32/37)kJ/molH 2 Herbst, Hector, APL 85, 3465 (2004)

35 Hydrogen site energetics in LaNi 5 H n and LaCo 5 H n Enthalpy of hydride formation in LaCo 5 H n H min = 45.6kJ/molH 2 for H at 4e4h agrees with neutron data calorimetry: H min = 45.2kJ/molH 2 Herbst, Hector, APL 85, 3465 (2004)

36 Problems of the Past 0 10 W L G X W K 0 10 W L G X W K Si Ge Energy (ev) Energy (ev) Si bandgap exp: 1.11 ev GGA: 0.57 ev Ge bandgap exp: 0.67 ev GGA: 0.09 ev

37 Critical review of the Local Density Approximation Limitations and Beyond Self-interaction cancellation in v Hartree + v x violated Repair using exact Hartree-Fock exchange functional class of hybrid functionals PBE0 HSE03, HSE06 E PBE0 xc = 1 4 E HF x E PBE x + E PBE c E HSE xc = 1 4 E HF,sr,µ x E PBE,sr,µ x + E PBE,lr,µ x + E PBE c based on decomposition of Coulomb kernel 1 r = S µ(r) + L µ (r) = erfc(µr) + erf(µr) r r

38 Critical review of the Local Density Approximation 0 10 W L G X W K W L G X W K Limitations and Beyond Self-interaction cancellation in v Hartree + v x violated Repair using exact Hartree-Fock exchange functional class of hybrid functionals GGA Energy (ev) Si bandgap exp: 1.11 ev GGA: 0.57 ev HSE: 1.15 ev HSE Energy (ev)

39 Critical review of the Local Density Approximation 0 10 W L G X W K W L G X W K Limitations and Beyond Self-interaction cancellation in v Hartree + v x violated Repair using exact Hartree-Fock exchange functional class of hybrid functionals GGA Energy (ev) Ge bandgap exp: 0.67 ev GGA: 0.09 ev HSE: 0.66 ev HSE Energy (ev)

40 Critical review of the Local Density Approximation GGA 10 8 DOS (1/eV) (E - E V ) (ev) SrTiO 3 Bandgap GGA: 1.6 ev, exp.: 3.2 ev

41 Critical review of the Local Density Approximation GGA HSE DOS (1/eV) 6 4 DOS (1/eV) (E - E V ) (ev) (E - E V ) (ev) SrTiO 3 Bandgap GGA: 1.6 ev, HSE: 3.1 ev, exp.: 3.2 ev

42 LaAlO 3 GGA DOS (1/eV) (E - E V ) (ev) LaAlO 3 Bandgap GGA: 3.5 ev, exp.: 5.6 ev

43 LaAlO 3 GGA 25 HSE DOS (1/eV) DOS (1/eV) (E - E V ) (ev) (E - E V ) (ev) LaAlO 3 Bandgap GGA: 3.5 ev, HSE: 5.0 ev, exp.: 5.6 ev

44 Critical review of the Local Density Approximation Calculated vs. experimental bandgaps

45 Industrial Computational Materials Engineering Automotive Energy & Power Generation Aerospace Steel & Metal Alloys Glass & Ceramics Electronics Display & Lighting Chemical & Petrochemical Drilling & Mining

46 Summary Formalism Density Functional Theory exact (!) mapping of full many-body problem to an effective single-particle problem Local Density Approximation approximative treatment of exchange (!) and correlation considerable improvement: exact treatment of exchange very good agreement DFT/Exp. in numerous cases theory meets industry Further Reading V. Eyert and U. Eckern, PhiuZ 31, 276 (2000)

47 Summary Formalism Density Functional Theory exact (!) mapping of full many-body problem to an effective single-particle problem Local Density Approximation approximative treatment of exchange (!) and correlation considerable improvement: exact treatment of exchange very good agreement DFT/Exp. in numerous cases theory meets industry Further Reading V. Eyert and U. Eckern, PhiuZ 31, 276 (2000)

48 Summary Formalism Density Functional Theory exact (!) mapping of full many-body problem to an effective single-particle problem Local Density Approximation approximative treatment of exchange (!) and correlation considerable improvement: exact treatment of exchange very good agreement DFT/Exp. in numerous cases theory meets industry Further Reading V. Eyert and U. Eckern, PhiuZ 31, 276 (2000)

49 Summary Formalism Density Functional Theory exact (!) mapping of full many-body problem to an effective single-particle problem Local Density Approximation approximative treatment of exchange (!) and correlation considerable improvement: exact treatment of exchange very good agreement DFT/Exp. in numerous cases theory meets industry Further Reading V. Eyert and U. Eckern, PhiuZ 31, 276 (2000)

50 Acknowledgments Formalism

51 Acknowledgments Formalism Augsburg/München Thank You for Your Attention!

Metal-Insulator Transitions of the Vanadates. New Perspectives of an Old Mystery

Metal-Insulator Transitions of the Vanadates. New Perspectives of an Old Mystery : New Perspectives of an Old Mystery Volker Eyert Center for Electronic Correlations and Magnetism Institute of Physics, University of Augsburg November 15, 21 Outline Basic Theories 1 Basic Theories Density