Thomas-Fermi theory and beyond. Local-density ... H ¢. T. Vee. V. (2) is the Hamiltonian of a many-electron system in an external potential V. ¤. ¥ r. ¦ and with an ...
FOUNDATIONS OF DENSITY-FUNCTIONAL THEORY J. Hafner
¨ Materialphysik and Center for Computational Material Institut fur Science Universit¨at 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
3¯ h2 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
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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
h¯2 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
h¯2 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 ! J. H AFNER , A B - INITIO
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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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& &
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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
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& &
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 J. H AFNER , A B - INITIO
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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).
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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. J. H AFNER , A B - INITIO
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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)
