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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
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FormalismApplications
Outline
1 FormalismDefinitions and TheoremsApproximations
2 Applications
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FormalismApplications
Outline
1 FormalismDefinitions and TheoremsApproximations
2 Applications
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FormalismApplications
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
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga
WSR
/au
2
2.5
3
3.5
4
4.5
5
5.5
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In
WSR
/au
Wigner-Seitz-Rad.=̂ Volume
0
500
1000
1500
2000
2500
3000
3500
4000
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga
Bul
k m
odul
us/k
bar
0
500
1000
1500
2000
2500
3000
3500
4000
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In
Bul
k m
odul
us/k
bar
Compressibility=̂ Hardness
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FormalismApplications
Energy band structures from screened HF exchange
Si, AlP, AlAs, GaP, and GaAs
Experimental andtheoretical bandgapproperties
Shimazaki, Asai,JCP 132, 224105 (2010)
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FormalismApplications
Definitions and TheoremsApproximations
Outline
1 FormalismDefinitions and TheoremsApproximations
2 Applications
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FormalismApplications
Definitions and TheoremsApproximations
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
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FormalismApplications
Definitions and TheoremsApproximations
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
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Definitions and TheoremsApproximations
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)
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Definitions and TheoremsApproximations
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)
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Definitions and TheoremsApproximations
Authors
Pierre C. Hohenberg Walter Kohn
Lu Jeu Sham
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FormalismApplications
Definitions and TheoremsApproximations
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.
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Definitions and TheoremsApproximations
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.
Nota bene
Both theorems are formulated for the ground state!
Zero temperature!
No excitations!
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Definitions and TheoremsApproximations
Hohenberg and Kohn, 1964: Theorems
Maps
Ground state |Ψ0〉 (from minimizing 〈Ψ0|H|Ψ0〉):
vext (r)(1)=⇒ |Ψ0〉
(2)=⇒ ρ0(r)
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Definitions and TheoremsApproximations
Hohenberg and Kohn, 1964: Theorems
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)
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Definitions and TheoremsApproximations
Levy, Lieb, 1979-1983: Constrained Search
Percus-Levy partition
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Definitions and TheoremsApproximations
Levy, Lieb, 1979-1983: Constrained Search
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 )
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Definitions and TheoremsApproximations
Levy, Lieb, 1979-1983: Constrained Search
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!
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Definitions and TheoremsApproximations
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)
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Definitions and TheoremsApproximations
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!
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Definitions and TheoremsApproximations
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)
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Definitions and TheoremsApproximations
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)
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Definitions and TheoremsApproximations
Kohn and Sham, 1965: Local Density Approximation
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)
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Definitions and TheoremsApproximations
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 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)
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FormalismApplications
Outline
1 FormalismDefinitions and TheoremsApproximations
2 Applications
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FormalismApplications
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
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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
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FeS2: From Atoms to the Solid
-16533
-16532
-16531
-16530
-16529
-16528
-16527
0 2000 4000 6000 8000 10000 12000
E (
Ryd
)
V (aB3)
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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
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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
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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)
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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
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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
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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)
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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)
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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
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FormalismApplications
Critical review of the Local Density Approximation
Limitations and Beyond
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
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FormalismApplications
Critical review of the Local Density Approximation
Limitations and Beyond
Self-interaction cancellation in vHartree + vx violated
Repair using exact Hartree-Fock exchange functional→ class of hybrid functionals
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
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FormalismApplications
Critical review of the Local Density Approximation
Limitations and Beyond
Self-interaction cancellation in vHartree + vx violated
Repair using exact Hartree-Fock exchange functional→ class of hybrid functionals
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
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FormalismApplications
Critical review of the Local Density Approximation
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
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FormalismApplications
Critical review of the Local Density Approximation
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
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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
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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
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Critical review of the Local Density Approximation
Calculated vs. experimental bandgaps
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Industrial Applications
Computational Materials Engineering
Automotive
Energy & PowerGeneration
Aerospace
Steel & Metal Alloys
Glass & Ceramics
Electronics
Display & Lighting
Chemical &Petrochemical
Drilling & Mining
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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)
[email protected] From Quantum Mechanics to Materials Design
FormalismApplications
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)
[email protected] From Quantum Mechanics to Materials Design
FormalismApplications
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)
[email protected] From Quantum Mechanics to Materials Design
FormalismApplications
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)
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FormalismApplications
Acknowledgments
Augsburg/München
Thank You for Your Attention!
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