FOUNDATIONS OF DENSITY-FUNCTIONAL
THEORY
J. Hafner
Institut für Materialphysik and Center for Computational Material
Science
Universität Wien, Sensengasse 8/12, A-1090 Wien, Austria
J. H AFNER , A B - INITIO
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Overview I
Hohenberg-Kohn theorem
Hellmann-Feynman theorem
Forces on atoms
Stresses on unit cell
Local-density approximation: density only
Thomas-Fermi theory and beyond
Local-density approximation: Kohn-Sham theory
Kohn-Sham equations
Variational conditions
Kohn-Sham eigenvalues
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Overview II
Exchange-correlation functionals
Local-density approximation (LDA)
Generalized gradient approximation (GGA), meta-GGA
Hybrid-functionals
Limitations of DFT
Band-gap problem
Overbinding
Neglect of strong correlations
Neglect of van-der-Waals interactions
Beyond LDA
LDA+U
GW, SIC,
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Density-functional theory - HKS theorem
Hohenberg-Kohn-Sham theorem:
(1) The ground-state energy of a many-body system is a unique functional
of the particle density, E0 E r .
(2) The functional E r has its minimum relative to variations δn r of the
particle density at the equilibrium density n0 r ,
nr
min E r
J. H AFNER , A B - INITIO
no r
0
(1)
δE n r
δn r
E n0 r
E
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Proof - HKS theorem
Reductio ad absurdum:
Vee
V
(2)
T
H
is the Hamiltonian of a many-electron system in an external potential V r
and with an electron-electron interaction Vee . In the ground-state this system
ψ0 H ψ0 and the particle density
has the energy E0 , with E0
n0 r
ψ0 r 2 . Let us assume that a different external potential V leads
to a different ground-state ψ0 , but to the same particle density:
ψ0 r 2 n0 r . According to the variational principle it follows
n0 r
that
E0
ψ0 H ψ0
ψ0 H V V ψ 0
(3)
E 0 ψ0 V V ψ 0
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Proof - HKS theorem
n0 r V r
V r d3r
(4)
E0
E0
Starting from
(5)
n0 r it follows
and using n0 r
ψ0 H ψ 0
E0
V r d3r
n0 r V r
d3r
(6)
V r
E0
n0 r V r
E0
E0
in direct contradiction to the results obtained above. Hence n0 r and n0 r
must be different and V r is a unique functional of n r .
The variational property of the Hohenberg-Kohn-Sham functional is a direct
consequence of the general variational principle of quantum mechanics.
J. H AFNER , A B - INITIO
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HKS theorem - Variational principle
With
Vee ψ
(7)
ψ T
d3r
Vee ψ
nrV r
ψ T
Fnr
Enr
Fnr
it follows
n0 r V r d 3 r
E n0 r
(8)
ψ0 V ψ 0
Vee ψ0
ψ0 T
F n0 r
n r V r d3r
Fn r
ψ V ψ
E n r
and hence
J. H AFNER , A B - INITIO
n0 r
0
(9)
nr
min E n r
δE n r
δn r
E n0 r
E0
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Hellmann-Feynman theorem: Forces and stresses
The external potential V is created by the ions located at the positions RI ,
V r
∑ v r RI . The ground-state energy and wavefunction depend on
I
RI as parameters. The force FI acting on an
the ionic coordinates, R
atom located at RI is given by
(10)
Ψ 0 H ∇I Ψ 0
Ψ 0 ∇I H Ψ 0
Ψ0 R
∇I H R
Ψ0 R
∇I Ψ 0 H Ψ 0
Ψ0 R
H R
Ψ0 R
∂
∂ RI
∇I E 0 R
FI
First and third terms in the derivative vanish due to variational property of
Forces acting on the ions are given by the expectation
the ground-state
value of the gradient of the electronic Hamiltonian in the ground-state. The
ground-state must be determined very accurately: errors in the total energy
are 2nd order, errors in the forces are 1st order !
J. H AFNER , A B - INITIO
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Hellmann-Feynman theorem: Forces and stresses
The stress tensor σi j describes the variation of the total energy under an
infinitesimal distortion of the basis vectors a k under a strain ti j :
σi j
∑ δi j
ti j a k
Ψ0
Ψ0
J. H AFNER , A B - INITIO
ak
∂
∂ti j H
σi j
(11)
j
j
i
ak
∂E ak
∂ti j
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DFT functional
Total-energy functional
V r n r d3r
E xc n
EH n
T n
En
(12)
T n
kinetic energy,
EH n
Hartree energy (electron-electron repulsion),
E xc n
exchange and correlation energies,
V r
external potential
- the exact form of T n and Exc is unknown !
Local density approximation - ”density only”:
- Approximate the functionals T n and Exc n by the corresponding
energies of a homogeneous electron gas of the same local density
Thomas-Fermi theory
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Thomas-Fermi theory
Kinetic energy:
t n r d3r
T n
(13)
3π2 2 3
5 3
nr
tn
3h̄2
10m
where t n is the kinetic energy of a noninteracting homogeneous electron
gas with the density n.
Electron-electron interaction: Coulomb repulsion only
e2
2
nr nr 3 3
d rd r
r r
(14)
E n
H
Add exchange-correlation term in modern versions. Variation of E n with
leads to the Thomas-Fermi equation
e
2
nr
d3r
r r
2 3
5
Cnr
3
J. H AFNER , A B - INITIO
V r
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0
(15)
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Kohn-Sham theory
d 3 rd 3 r
! " ! T nr
nr nr
r r
E xc n r
e2
2
n r V r d3r
Enr
(16)
(1) Parametrize the particle density in terms of a set of one-electron orbitals
representing a non-interacting reference system
2
(17)
r
∑ φi
nr
i
(2) Calculate non-interacting kinetic energy in terms of the φi r ’s, i.e.
T n T0 n ,
#$
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J. H AFNER , A B - INITIO
i
φi r
%
∑
T0 n
h̄2 2
∇ φi r d 3 r
2m
(18)
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Kohn-Sham theory II
(3) Local-density approximation for exchange-correlation energy
(19)
&
n r εxc n r d 3 r
E xc n r
where εxc n r is the exchange-correlation energy of a homogeneous
electron gas with the local density n r . For the exchange-part, a
Hartree-Fock calculation for a homogeneous electron gas with the density n
leads to
3e2
3π2 n r 1 3
(20)
εx n r
4π
(4) Determine the optimal one-electron orbitals using the variational
δi j
condition under the orthonormality constraint φi φ j
0
(21)
i j
δi j
φi φ j
∑ εi j
δ E nr
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Kohn-Sham theory III
Kohn-Sham equations (after diagonalizing εi j ):
(22)
δExc n r
δn r
with the exchange-correlation potential µxc n r
εi φ i r
φi r
µxc n r
nr
d3r
r r
e2
V r
h̄2 2
∇
2m
Total energy:
d3r
µxc n r
n r εxc n r
i
nr nr 3 3
d rd r
r r
1
2
∑ εi
E
*
'
() '* () 2
1
sum of ”one-electron energies”
”double-counting corrections”
(1)
(2)
(23)
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Kohn-Sham theory IV
Variational conditions
Total energy E n
nr
0
n0 r
(24)
δE n r
δn r
Kohn-Sham eigenvalues εi
εj
(25)
(26)
εi
+
0
0 with φi φ j
δ φi H KS φi
Norm of residual vector Ri
app
app
0
R φi
φi H KS φi
δ R φi
εi
φi
&
εi
H KS
R φi
No orthogonality constraint !
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Kohn-Sham theory V
Interpretation of the ”one-electron energies” εi
Hartree-Fock theory - Koopman’s theorem
0
E HF ni
1
E HF ni
εHF
i
(27)
= Ionisation energy if relaxation of the one-electron orbitals is
εHF
i
neglected.
Kohn-Sham theory:
Total energy is a nonlinear functional of the density
Koopmans theorem
not valid.
δE n r
εi ni r
φi r φ r
(28)
δni r
%
&
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Exchange-correlation functionals I
Definition of the exchange-correlation functional:
E xc n accounts for the difference between the exact ground-state energy
and the energy calculated in a Hartree approximation and using the
non-interacting kinetic energy T0 n ,
U xc n
(29)
T0 n
T n
-,
E xc n
&
exact and non-interacting kinetic energy functional
T n T0 n
U xc n
interaction of the electrons with their own exchange-correlation
hole nxc defined as (ρ2 is the two-particle density matrix)
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&
&
J. H AFNER , A B - INITIO
nxc r s; r s
ns r ns r
,
&
&
ρ2 r s; r s
(30)
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Exchange-correlation functionals II
Properties of the exchange-correlation hole
Locality
0
&
&
lim nxc r s; r s
(31)
!
" !
r r
Pauli principle for electrons with parallel spin
(32)
δs s
(33)
O
(34)
ns r
&
&
nxc r s; r s
Antisymmetric non-interacting wavefunction
&
&
nx r s; r s d 3 r
Normalization of two-particle density matrix
J. H AFNER , A B - INITIO
&
&
nc r s; r s d 3 r
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Exchange-correlation functionals III
Properties of the exchange-correlation functional
Adiabatic connection formula
d r
0
nxc λ r; r
dλ
r r
3
d rn r
n
3
1
2
E
xc
1
(35)
Lieb-Oxford bound
3
1 68
/
D
/
1 44
&
r d3r
n4
D
.
E xc n
(36)
plus scaling relations,.......
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Local (spin-)density approximation - L(S)DA
(37)
&
n r εxc n r d 3 r
E xc n r
&
&2
1
&0
Exchange-functional (for spin-polarized systems,
nr
nr
n n n )
4
3
4 3
6
4
7
6
3
1 21 3
(38)
6
εxp
εxp
nr 4 3
nr
n n4 3 n n4 3
1 1 21 3
nr
8 "6 6
: 7 "
6
9
5 f
εx
nr
3π2 1 3
&2
&
&0
εx n r
3e2
4π
n
n 2 for the paramagnetic (non-spinpolarized) and
with εxp εx n
f
nn
0 for the ferromagnetic (completely spin-polarized)
εx εx n
limits of the functional.
Correlation functional εc n r
nr
fitted to the ground-state energy of
a homogeneous electron gas calculated using quantum Monte Carlo
simulations and similar spin-interpolations.
;
4 3 &
4
3
&2
&
&0
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Semilocal functionals
Generalized gradient approximation - GGA
nr
∇n r
∇n r
d3r
&
f nr
&2
&
&0
&
2&
&
&0
nr
&2
&
&0
E xc n r
(39)
There are two different strategies for determining the function f :
(1) Adjust f such that it satisfies all (or most) known properties of the
exchange-correlation hole and energy.
(2) Fit f to a large data-set own exactly known binding energies of atoms
and molecules.
Strategy (1) is to be preferred, but many different variants: Perdew-Wang
(PW), Becke-Perdew (BP), Lee-Yang-Parr (LYP),
Perdew-Burke-Ernzernhof (PBE).
J. H AFNER , A B - INITIO
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Semilocal functionals
Meta-GGA
Include in addition a dependence on the kinetic energy density τ r of the
electrons,
∇φi r
2
∑
τr
nocc
i 1
(40)
In the meta-GGA’s, the exchange-correlation potential becomes
orbital-dependent !
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Hybrid functionals
General strategy: Mixing exact-exchange (i.e. Hartree-Fock) and
local-density energies, as suggested by the adiabatic connection formula
#
1 xc
U n
2 1
1 xc
U n
2 0
0
n dλ
n
Uλxc
(41)
nonlocal exchange energy of Kohn-Sham orbitals
potential energy for exchange and correlation
U0xc n
U1xc n
1
E
xc
Example: B3LYP functional
1
c
c EVW
N
c
cELY
P
x
bEB88
x
aEexact
x
a ELSDA
1
E xc n
(42)
&
&
x stand for the exchange part of the Becke88 GGA functional,
where EB88
c
ELY
P for the correlation part of the Lee-Yound-Parr local and GGA
c
functional, and EVW
N for the local Vosko-Wilk-Nusair correlation
functional. a b and c are adjustable parameters.
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Limitations of DFT I
Band-gap problem:
- HKS theorem not valid for excited states
band-gaps in
semiconductors and insulators are always underestimated
- Possible solutions: - Hybrid-functionals lead to better gaps
- LDA+U, GW, SIC increase correlation gaps
Overbinding:
- LSDA: too small lattice constants, too large cohesive energies, too high
bulk moduli
- Possible solutions: - GGA: overbinding largely corrected (tendency too
overshoot for the heaviest elements)
- The use of the GGA is mandatory for calculating adsorption energies, but
the choice of the ”correct” GGA is important.
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Limitations of DFT II
Neglect of strong correlations
- Exchange-splitting underestimated for narrow d- and f -bands
- Many transition-metal compounds are Mott-Hubbard or charge-transfer
insulators, but DFT predicts metallic state
- Possible solutions: - Use LDA+U, GW, SIC,
Neglect of van-der Waals interactions
- vdW forces arise from mutual dynamical polarization of the interacting
not included in any DFT functional
atoms
- Possible solution: - Approximate expression of dipole-dipole vdW forces
on the basis of local polarizabilities derived from DFT ??
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Beyond DFT
DFT+U
Describe on-site Coulomb-repulsion by Hubbard-Hamiltonian
J
U
∑
nm s nm
s
" m m s
(43)
<
2
s
mm s
nm s nm
∑
H
U
2
where nm s is the number operator for electrons with the magnetic quantum
number m and spin s, U E d n 1 E d n 1 2E d n and J a screened
exchange energy.
"
7
The DFT+U Hamiltonian includes contributions already accounted for in
subtract double-counting, adopt rotationally
the DFT functional
invariant formulation
U J
EDFT U EDFT
Tr ρs ρs ρs
(44)
∑
2
s
7
on-site density matrix ρsij of the d electrons
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Beyond DFT II
Calculate quasiparticle-excitation energies by low-order many-body
perturbation theory.
GW: Self-energy approximation approximated by
i
2π
&
&
G r r ; ω W r r ; ω dω
&
Σ r r ;ω
(45)
1
"
where G is the full interacting Green’s function and W the dynamically
screend Coulomb intreraction, described by the inverse dielectric matrix ε
and the bare Coulomb potential v,
r d3r
1
"
In practice, approximate forms of G and ε
&
r r ;ω v r
1
ε
"
&
W r r ;ω
(46)
have to be used.
SIC: Self-interaction corrections.
GW, SIC are not implemented in VASP. Results largely equivalent to
LDA+U.
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