From Quantum Mechanics to Materials Design - uni …myweb.rz.uni-augsburg.de/~eyert/talks/Eyert_FocLect.pdf · Formalism Applications From Quantum Mechanics to Materials Design The

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

[email protected] From Quantum Mechanics to Materials Design


Outline

1 FormalismDefinitions and TheoremsApproximations

2 Applications



Outline


2 Applications



Calculated Electronic Properties

Moruzzi, Janak, Williams (IBM, 1978)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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

E/R

yd

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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

E/R

yd

Cohesive Energies=̂ Stability

2

2.5

3

3.5

4

4.5

5

5.5


WSR

/au

2

2.5

3

3.5

4

4.5

5

5.5


WSR

/au

Wigner-Seitz-Rad.=̂ Volume

0

500

1000

1500

2000

2500

3000

3500

4000


Bul

k m

odul

us/k

bar

0

500

1000

1500

2000

2500

3000

3500

4000


Bul

k m

odul

us/k

bar

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

=∑

i

[

−~

2

2m∇2

i

]

+12

e2

4πǫ0

∑

i,jj 6=i

1|ri − rj |

+∑

i

vext(ri)

where

∑

i

vext (ri) =12

e2

4πǫ0

∑

µν

µ6=ν

ZµZν

|Rµ − Rν |−

e2

4πǫ0

∑

µ

∑

i

Zµ

|Rµ − ri |

µ: ions with charge Zµ, i : electrons




Key Players

Electron Density Operator

ρ̂(r) =

N∑

i=1

δ(r − ri) =∑

αβ

χ∗α(r)χβ(r)a+

α aβ

χα: single particle state




Key Players

Electron Density Operator

ρ̂(r) =

N∑

i=1

δ(r − ri) =∑

αβ

χ∗α(r)χβ(r)a+

α aβ

χα: single particle state

Electron Density

ρ(r) = 〈Ψ|ρ̂(r)|Ψ〉 =∑

α

|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|∑

i

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!





Maps

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

vext (r)(1)=⇒ |Ψ0〉

(2)=⇒ ρ0(r)





Maps

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]︸︷︷︸

+12

e2

4πǫ0

∫

d3r∫

d3r′ρ(r)ρ(r′)|r − r′|

= G[ρ] +12

e2

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

e2

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 =

35

~2

2m

(

3π2) 2

3

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

α

∫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) =

[

−~

2

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]

δρ

„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

4π

e2

4πǫ0(3π2ρ)

13

vx (ρ) = −1π

e2

4πǫ0(3π2ρ)

13

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

E (

Ryd

)

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

E (

Ryd

)

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

E (

Ryd

)

xS



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

E (

Ryd

)

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

E (

Ryd

)

xSi

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

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30

ε

E (eV)

Im εxxIm εzz



Dielectric Function of Al2O3Real Part

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

FPASW

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30

ε

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

Si

W L G X W K

En

er

gy

(e

V)

− 1 0

0

Si bandgap

exp: 1.11 eV

GGA: 0.57 eV

Ge

W L G X W K

En

er

gy

(e

V)

− 1 0

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

PBE0

EPBE0xc =

14

EHFx +

34

EPBEx + EPBE

c

HSE03, HSE06

EHSExc =

14

EHF ,sr ,µx +

34

EPBE,sr ,µx + EPBE,lr ,µ

x + EPBEc

based on decomposition of Coulomb kernel

1r

= Sµ(r) + Lµ(r) =erfc(µr)

r+

erf(µr)r







GGA

W L G X W K

En

er

gy

(e

V)

− 1 0

0

Si bandgap

exp: 1.11 eV

GGA: 0.57 eV

HSE: 1.15 eV

HSE

W L G X W K

En

er

gy

(e

V)

− 1 0

0

1 0







GGA

W L G X W K

En

er

gy

(e

V)

− 1 0

0

Ge bandgap

exp: 0.67 eV

GGA: 0.09 eV

HSE: 0.66 eV

HSE

W L G X W K

En

er

gy

(e

V)

− 1 0

0

1 0




GGA

0

2

4

6

8

10

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

DO

S (

1/eV

)

(E - EV) (eV)

SrTiO3 Bandgap

GGA: ≈ 1.6 eV, exp.: 3.2 eV




GGA

0

2

4

6

8

10

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

DO

S (

1/eV

)

(E - EV) (eV)

HSE

0

2

4

6

8

10

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

DO

S (

1/eV

)

(E - EV) (eV)

SrTiO3 Bandgap

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



LaAlO3

GGA

0

5

10

15

20

25

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

DO

S (

1/eV

)

(E - EV) (eV)

LaAlO3 Bandgap

GGA: ≈ 3.5 eV, exp.: 5.6 eV



LaAlO3

GGA

0

5

10

15

20

25

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

DO

S (

1/eV

)

(E - EV) (eV)

HSE

0

5

10

15

20

25

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

DO

S (

1/eV

)

(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



Acknowledgments

Augsburg/München

Thank You for Your Attention!


Documents

From Quantum Mechanics to Materials Design - uni …myweb.rz.uni-augsburg.de/~eyert/talks/Eyert_FocLect.pdf · Formalism Applications From Quantum Mechanics to Materials Design The