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FormalismApplications

From Quantum Mechanics to Materials DesignThe Basics of Density Functional Theory

Volker Eyert

Center for Electronic Correlations and MagnetismInstitute of Physics, University of Augsburg

December 03, 2010

Volker@Eyert.de From Quantum Mechanics to Materials Design

Outline

1 FormalismDefinitions and TheoremsApproximations

2 Applications

Outline

2 Applications

Calculated Electronic Properties

Moruzzi, Janak, Williams (IBM, 1978)

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In

Cohesive Energies=̂ Stability

Wigner-Seitz-Rad.=̂ Volume

Compressibility=̂ Hardness

Energy band structures from screened HF exchange

Si, AlP, AlAs, GaP, and GaAs

Experimental andtheoretical bandgapproperties

Shimazaki, Asai,JCP 132, 224105 (2010)

Definitions and TheoremsApproximations

Outline

2 Applications

Key Players

Hamiltonian (within Born-Oppenheimer approximation)

H = Hel ,kin + Hel−el + Hext

2m∇2

4πǫ0

i,jj 6=i

1|ri − rj |

vext(ri)

vext (ri) =12

4πǫ0

µ6=ν

ZµZν

|Rµ − Rν |−

4πǫ0

|Rµ − ri |

µ: ions with charge Zµ, i : electrons

Key Players

Electron Density Operator

ρ̂(r) =

δ(r − ri) =∑

χ∗α(r)χβ(r)a+

α aβ

χα: single particle state

Key Players

Electron Density Operator

ρ̂(r) =

δ(r − ri) =∑

χ∗α(r)χβ(r)a+

α aβ

χα: single particle state

Electron Density

ρ(r) = 〈Ψ|ρ̂(r)|Ψ〉 =∑

|χα(r)|2nα

|Ψ〉: many-body wave function, nα: occupation number

Normalization: N =∫

d3r ρ(r)

Key Players

Functionals

Universal Functional (independent of ionic positions!)

F = 〈Ψ|Hel ,kin + Hel−el |Ψ〉

Functional due to External Potential:

〈Ψ|Hext |Ψ〉 = 〈Ψ|∑

vext(r)δ(r − ri)|Ψ〉

d3r vext(r)ρ(r)

Authors

Pierre C. Hohenberg Walter Kohn

Lu Jeu Sham

Hohenberg and Kohn, 1964: Theorems

1st Theorem

The external potential vext(r) is determined, apart from a trivialconstant, by the electronic ground state density ρ(r).

2nd Theorem

The total energy functional E [ρ] has a minimum equal to theground state energy at the ground state density.

1st Theorem

The external potential vext(r) is determined, apart from a trivialconstant, by the electronic ground state density ρ(r).

2nd Theorem

The total energy functional E [ρ] has a minimum equal to theground state energy at the ground state density.

Nota bene

Both theorems are formulated for the ground state!

Zero temperature!

No excitations!

Ground state |Ψ0〉 (from minimizing 〈Ψ0|H|Ψ0〉):

vext (r)(1)=⇒ |Ψ0〉

(2)=⇒ ρ0(r)

Ground state |Ψ0〉 (from minimizing 〈Ψ0|H|Ψ0〉):

vext (r)(1)=⇒ |Ψ0〉

(2)=⇒ ρ0(r)

1st Theorem

vext(r)(1)⇐⇒|Ψ0〉

(2)⇐⇒ρ0(r)

Levy, Lieb, 1979-1983: Constrained Search

Percus-Levy partition

Variational principle

E0 = inf|Ψ〉

〈Ψ|H|Ψ〉

= inf|Ψ〉

〈Ψ|Hel ,kin + Hel−el + Hext |Ψ〉

= infρ(r)

inf|Ψ〉∈S(ρ)

〈Ψ|Hel ,kin + Hel−el |Ψ〉 +

d3r vext(r)ρ(r)]

=: infρ(r)

FLL[ρ] +

d3r vext(r)ρ(r)]

= infρ(r)

E [ρ]

S(ρ): set of all wave functions leading to density ρFLL[ρ]: Levy-Lieb functional, universal (independent of Hext )

Levy-Lieb functional

FLL[ρ] = inf|Ψ〉∈S(ρ)

〈Ψ|Hel ,kin + Hel−el |Ψ〉

= T [ρ] + Wxc[ρ]︸︷︷︸

4πǫ0

d3r∫

d3r′ρ(r)ρ(r′)|r − r′|

= G[ρ] +12

4πǫ0

d3r∫

d3r′ρ(r)ρ(r′)|r − r′|

Functionals

Kinetic energy funct.: T [ρ] not known!

Exchange-correlation energy funct.: Wxc[ρ] not known!

Hartree energy funct.: 12

4πǫ0

∫d3r

∫d3r′ ρ(r)ρ(r′)

|r−r′| known!

Thomas, Fermi, 1927: Early Theory

Approximations

ignore exchange-correlation energy functional:

Wxc[ρ]!= 0

approximate kinetic energy functional:

T [ρ] = CF

d3r (ρ(r))53 , CF =

3π2) 2

Failures1 atomic shell structure missing

→ periodic table can not be described2 no-binding theorem (Teller, 1962)

Kohn and Sham, 1965: Single-Particle Equations

Ansatz1 use different splitting of the functional G[ρ]

T [ρ] + Wxc[ρ] = G[ρ]!= T0[ρ] + Exc[ρ]

2 reintroduce single-particle wave functions

Imagine: non-interacting electrons with same density

Density: ρ(r) =∑occ

α |χα(r)|2 known!

Kinetic energy funct.:

T0[ρ] =∑occ

∫d3r χ∗

α(r)[

− ~2

2m∇2]

χα(r) known!

Exchange-correlation energy funct.: Exc[ρ] not known!

Kohn and Sham, 1965: Single-Particle Equations

Euler-Lagrange Equations (Kohn-Sham Equations)

δE [ρ]

δχ∗α(r)

− εαχα(r) =

2m∇2 + veff (r) − εα

χα(r) != 0

Effective potential: veff (r) := vext(r) + vH(r) + vxc(r)

Exchange-correlation potential: not known!

vxc(r) :=δExc[ρ]

„Single-particle energies“:εα (Lagrange-parameters, orthonormalization)

Kohn and Sham, 1965: Local Density Approximation

Be Specific!

Approximate exchange-correlation energy functional

Exc[ρ] =

ρ(r)εxc(ρ(r))d3r

Exchange-correlation energy density εxc(ρ(r))depends on local density only!is calculated from homogeneous, interacting electron gas

Exchange-correlation potential

vxc(ρ(r)) =

∂ρ{ρεxc(ρ)}

ρ=ρ(r)

Homogeneous, Interacting Electron Gas

Splitεxc(ρ) = εx (ρ) + εc(ρ)

Exchange energy density εx (ρ)(exact for homogeneous electron gas)

εx (ρ) = −3

4πǫ0(3π2ρ)

vx (ρ) = −1π

4πǫ0(3π2ρ)

Correlation energy density εc(ρ)Calculate and parametrize

RPA (Hedin, Lundqvist; von Barth, Hedin)QMC (Ceperley, Alder; Vosko, Wilk, Nusair; Perdew, Wang)

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 vHartree(ρ(r)) and vx (ρ(r))violated for ρ = ρ(r)→ Self-Interaction Correction (SIC)→ Exact Exchange (EXX),

Optimized Effective Potential (OEP)→ Screened Exchange (SX)

Outline

2 Applications

Iron Pyrite: FeS2

Pyrite

Pa3̄ (T 6h )

a = 5.4160 Å

“NaCl structure”sublattices occupied by

iron atomssulfur pairs

sulfur pairs ‖ 〈111〉 axes

xS = 0.38484

rotated FeS6 octahedra

FeS2: Equilibrium Volume and Bulk Modulus

-16532.950

-16532.900

-16532.850

-16532.800

-16532.750

-16532.700

-16532.650

-16532.600

850 900 950 1000 1050 1100 1150 1200

V (aB3)

V0 = 1056.15 aB3

E0 = -16532.904060 Ryd

B0 = 170.94 GPa

FeS2: From Atoms to the Solid

-16533

-16532

-16531

-16530

-16529

-16528

-16527

0 2000 4000 6000 8000 10000 12000

V (aB3)

FeS2: Structure Optimization

-16532.910

-16532.900

-16532.890

-16532.880

-16532.870

-16532.860

-16532.850

0.374 0.378 0.382 0.386 0.390 0.394

Phase Stability in Silicon

-578.060

-578.055

-578.050

-578.045

-578.040

-578.035

-578.030

-578.025

-578.020

-578.015

-578.010

80 90 100 110 120 130 140 150 160 170

V (aB3)

diahex

β-tinsc

bccfcc

diamond structure most stable

pressure induced phase transition to β-tin structure

LTO(Γ)-Phonon in Silicon

-1156.1070

-1156.1060

-1156.1050

-1156.1040

-1156.1030

-1156.1020

-1156.1010

-1156.1000

0.118 0.12 0.122 0.124 0.126 0.128 0.13 0.132

phonon frequency: fcalc = 15.34 THz (fexp = 15.53 THz)

Dielectric Function of Al2O3Imaginary Part

FLAPW, Hosseini et al., 2005FPLMTO, Ahuja et al., 2004 FPASW

0 5 10 15 20 25 30

E (eV)

Im εxxIm εzz

Dielectric Function of Al2O3Real Part

FLAPW, Hosseini et al., 2005FPLMTO, Ahuja et al., 2004

0 5 10 15 20 25 30

E (eV)

Re εxxRe εzz

Hydrogen site energetics in LaNi5Hn and LaCo5Hn

Enthalpy of hydride formation in LaNi5Hn

∆Hmin = −40kJ/molH2

for H at 2b6c16c2

agrees with

neutron data

calorimetry:∆Hmin =−(32/37)kJ/molH2

Herbst, Hector,APL 85, 3465 (2004)

Hydrogen site energetics in LaNi5Hn and LaCo5Hn

Enthalpy of hydride formation in LaCo5Hn

∆Hmin = −45.6kJ/molH2

for H at 4e4h

agrees with

neutron data

calorimetry:∆Hmin =−45.2kJ/molH2

Herbst, Hector,APL 85, 3465 (2004)

Problems of the Past

W L G X W K

− 1 0

Si bandgap

exp: 1.11 eV

GGA: 0.57 eV

W L G X W K

− 1 0

Ge bandgap

exp: 0.67 eV

GGA: 0.09 eV

Critical review of the Local Density Approximation

Self-interaction cancellation in vHartree + vx violated

Repair using exact Hartree-Fock exchange functional→ class of hybrid functionals

EPBE0xc =

EHFx +

EPBEx + EPBE

HSE03, HSE06

EHSExc =

EHF ,sr ,µx +

EPBE,sr ,µx + EPBE,lr ,µ

x + EPBEc

based on decomposition of Coulomb kernel

= Sµ(r) + Lµ(r) =erfc(µr)

erf(µr)r

W L G X W K

− 1 0

Si bandgap

exp: 1.11 eV

GGA: 0.57 eV

HSE: 1.15 eV

W L G X W K

− 1 0

W L G X W K

− 1 0

Ge bandgap

exp: 0.67 eV

GGA: 0.09 eV

HSE: 0.66 eV

W L G X W K

− 1 0

-8 -6 -4 -2 0 2 4 6 8 10

(E - EV) (eV)

SrTiO3 Bandgap

GGA: ≈ 1.6 eV, exp.: 3.2 eV

-8 -6 -4 -2 0 2 4 6 8 10

(E - EV) (eV)

-8 -6 -4 -2 0 2 4 6 8 10

(E - EV) (eV)

SrTiO3 Bandgap

GGA: ≈ 1.6 eV, HSE: ≈ 3.1 eV, exp.: 3.2 eV

LaAlO3

-8 -6 -4 -2 0 2 4 6 8 10

(E - EV) (eV)

LaAlO3 Bandgap

GGA: ≈ 3.5 eV, exp.: 5.6 eV

LaAlO3

-8 -6 -4 -2 0 2 4 6 8 10

(E - EV) (eV)

-8 -6 -4 -2 0 2 4 6 8 10

(E - EV) (eV)

LaAlO3 Bandgap

GGA: ≈ 3.5 eV, HSE: ≈ 5.0 eV, exp.: 5.6 eV

Calculated vs. experimental bandgaps

Industrial Applications

Computational Materials Engineering

Automotive

Energy & PowerGeneration

Aerospace

Steel & Metal Alloys

Glass & Ceramics

Electronics

Display & Lighting

Chemical &Petrochemical

Drilling & Mining

Summary

Density Functional Theory

exact (!) mapping of full many-body problem to an effectivesingle-particle problem

Local Density Approximation

approximative treatment of exchange (!) and correlation

considerable improvement: exact treatment of exchange

Applications

very good agreement DFT/Exp. in numerous cases

theory meets industry

Further Reading

V. Eyert and U. Eckern, PhiuZ 31, 276 (2000)

Summary

Applications

Further Reading

Summary

Applications

Further Reading

Summary

Applications

Further Reading

Acknowledgments

Augsburg/München

Thank You for Your Attention!

From Quantum Mechanics to Materials Design - uni...

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