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Tutorial:Time-dependent density-functional theory
Carsten A. UllrichUniversity of Missouri
XXXVI National Meeting on Condensed Matter PhysicsAguas de Lindoia, SP, Brazil
May 13, 2013
2 Literature
Time-dependent Density-FunctionalTheory: Concepts and Applications(Oxford University Press 2012)
“A brief compendium of TDDFT” Carsten A. Ullrich and Zeng-hui YangarXiv:1305.1388(Brazilian Journal of Physics, Vol. 43)
C.A. Ullrich homepage:http://web.missouri.edu/~ullrichc
3 Outline
PART I:
● The many-body problem● Review of static DFT
PART II:
● Formal framework of TDDFT● Time-dependent Kohn-Sham formalism
PART III:
● TDDFT in the linear-response regime● Calculation of excitation energies
4 The electronic structure problem of matter
What atoms, molecules, and solids can exist, and with what properties?
What are the ground-state energies E and electron densities n(r)?
What are the bond lengths and angles?
What are the nuclear vibrations?
How much energy is needed to ionizethe system, or to break a bond?
R
E
R
5 The electronic-nuclear many-body problem
Consider a system with Ne electrons and Nn nuclei, with
nuclear mass and charge Mj and Zj , where j=1,..., Nn
n
e
N
N
RRR
rrr
,...,
,...,
1
1
electronic coordinates:
nuclear coordinates:
All electrons and all nuclei are quantum mechanical particles,forming an interacting (Ne + Nn ) -body system. For example,consider the hydrogen molecule H2:
protonmMM
ZZ
21
21 1
1R2R
1r
2r
We’ll use atomic units:
1 me
6 Nonrelativistic Schrödinger equation
RrRrRr ,,,ˆjjj EH
e n
en
ee
N
j
N
k kj
k
N
kj kj
kjN
jjjn
j
j
N
kj kj
N
jje
j
Z
ZZV
M
VH
1 1
1,
2
1
2
2
1)(
2
1
2
1)(
2,ˆ
Rr
RRR
rrrRr electronic
Hamiltonian
nuclearHamiltonian
electron-nuclearattraction
: external scalar potentials acting on the electrons/nuclei.)(),( Rr ne VV
7 Born-Oppenheimer approximation
e n
eee
N
j
N
k kj
k
N
kj kj
kjN
kj kj
N
j
jBO
Z
ZZH
1 1
1
2
2
11
2
1
2,ˆ
Rr
RRrrRr
► Decouple the electronic and nuclear dynamics. This is a good approximation since nuclei are much heavier than electrons.► Treat the nuclei as classical particles at fixed positions.► Write down electronic Hamiltonian for a given nuclear configuration (here we ignore any external fields):
This is just a constant
8 Potential-energy surfaces (diatomic molecule)
1
2
3
45
R
)( REl
DE
1: ground-state potential-energy surface2,3,4,5: excited-state potential-energy surfaces
molecular equilibrium
9 The electronic many-body problem
),...,,( 21 Nxxxantisymmetric N-electron wave function
),( jjjx r space and spincoordinate
),...,(),...,(ˆ11 NjjNj xxExxH
WVTVHN
kj kj
N
jj
N
j
j ˆˆˆ||
1
2
1)(
2ˆ
11
2
rr
r
Even a two-electron problem (Helium) is very difficult:
),(),(||
111
22 21212121
22
21 rrrr
rr jjj Err
This is a 6-dimensional partial differential equation!
10 The electronic many-body problem
Dirac (1929): “The fundamental laws necessary for the mathematical treatmentof a large part of physics and the whole of chemistry are thus completely known,and the difficulty lies only in the fact that applications of these laws leads toequations that are too complex to be solved.”
Expectation value of an observable: jjj OO ˆ
Energy spectrum:jjj HE ˆ
Probability density:
2
22 ),...,,,(...)( NjNj xxdxdxn
rr
11 The Hartree-Fock method
N
kj kj
N
jj
N
j
j VH||
1
2
1)(
2ˆ
11
2
rrr
Solving the full many-body Schrödinger equation is impossible. Instead, try a variational approach:
EH
0)()(ˆ)( 1
*3*
N
llll
j
rdH rrr
)()()(
)()()(
)()()(
!
1),...,(
21
22221
11211
1
NNNN
N
N
N
xxx
xxx
xxx
Nxx
The wave function is assumedto be a Slater determinant:
12 The Hartree-Fock method
)(||
)()()](ˆ[
1
3 rrr
rrr
*
j
N
k
kkj
nonlocX rdV
)()(||
)()(
)(||
)()(
2
1
3
32
rrrr
rr
rrr
rr
*
jjj
N
k
kk
j
rd
nrdV
This is the Hartree-Fock equation, alsoknown as the self-consistent field (SCF)equation.
||
)()( 3
rr
rr
nrdVH Hartree potential: local (multiplicative) operator
Nonlocal exchange potential,acting on the jth orbital. “Nonlocal” means that the orbital that is acted uponappears under the integral.
13 The Hartree-Fock method
The total Hartree-Fock ground-state energy is given by
N
ji
jiji
i
N
iiHF
rdrd
nnrdrd
VrdE
1,
**33
33
2
1
*3
)()()()(
2
1
)()(
2
1
)()(2
)(
|r-r|
rrrr
|r-r|
rr
rrr
“direct” energy
exchangeenergy
where the single-particle orbitals are those that come fromthe Hartree-Fock equation, solved self-consistently.
► Not accurate enough for most applications in chemistry► Very bad for solids: band gap too high, lattice constants too big, cohesive energy in metals much too small
14 Beyond Hartree-Fock: correlation
Exact ground-state energy:cHF
exact EEE 0correlation energy
Question: is the correlation energy positive, negative, or either?
Answer: the correlation energy is always negative!
This is because the HF energy comes from a variation under the constraint that the wave function is a single Slater determinant. An unconstrained minimization will give a lower energy, according to the Rayleigh-Ritz minimum principle.
How to get correlation energy? Wave-function based approaches(configuration interaction, coupled cluster) are accurate but expensive!
DFT: alternative theory, formally exact but more efficient!
15 One-electron example
)3(cos
)(cos)(2
20
x
xxn
20
20
0
0
)(8
)(
)(4
)()(
)()()()(2
1
xn
xn
xn
xnxV
xxxVx jjjj
)()( 00 xnx
}{ˆ0 jHVn
Is this always true??
16 The Hohenberg-Kohn Theorem
Recall: WVTH ˆˆˆˆ Hamiltonian for N-electron system
We can define the following map:
)(rV 0 )(0 rn000
ˆ EH2
00 ... n
Claim: this map from potentials to densities is uniquely invertible, i.e., it is a 1-1 map. In other words, it cannot happen that twodifferent potentials produce the same ground-state density:
)(rV
)(rV )(0 rn
where “different” means thatthe potentials differ by morethan just a constant:
cVV )()( rr
17 The Hohenberg-Kohn proof
Step 1: Show that)(rV
)(rV 0 cannot happen!
Proof by contradiction. Let but assume that ),()( rr VV ice 00 (trivial phase factor). Then we have
000
000
ˆˆˆ
ˆˆˆ
EWVT
EWVT
subtract: .ˆˆ
ˆˆ
00
0000
constEEVV
EEVV
ice 00but
Contradiction! This meansthat the assumption musthave been wrong. We have
.00
18 The Hohenberg-Kohn proof
Step 2: Show that0
0)(0 rn cannot happen when
ice 00
To prove this, we assume the contrary, namely that they bothgive the same density. Then we can use the Ritz variational principle,whereby The following inequality holds:.ˆ
000 HE
)()()(
ˆˆˆˆ
03
0
00000
rrr nVVrdE
VVHHE
)()()(
ˆˆˆˆ
03
0
00000
rrr nVVrdE
VVHHE
(simply inter-change primedand unprimed)
add: 0000 EEEE Contradiction!
Therefore )(rV0)(0 rn uniquely.
19 The Hohenberg-Kohn Theorem (1964)
)(rV )(0 rn1:1
Therefore, uniquely determines )(0 rn .ˆˆˆˆ WVTH The Hamiltonian formally becomes a functional of the density:
][][ˆ 00 nnH and all wave functions become densityfunctionals as well.
Every physical observable
is a functional of n0
][ˆ][][ 000 nOnnO
The energy is a density functional: ][ˆ][][ nHnnE
)()(for][
)()(for][
00
00
rr
rr
nnEnE
nnEnE
Minimum principle:
20 The Kohn-Sham formalism (1965)
The HK theorem can be proved for any type of particle-particleinteraction—in particular, it holds for noninteracting systems, too!Therefore, there exists a unique noninteracting system that reproducesa given ground-state density. This is the Kohn-Sham system.
We can write the total ground-state energy as follows:
][||
)()(
2
1)()(][
)()(][][][
333
30
nEnn
rdrdVnrdnT
VnrdnWnTnE
xcs
rr
rrrr
rr
kinetic energy functionalfor interacting systems
kinetic energy functionalfor noninteracting systems
][][][ nWnTnT s This defines the xc energy functional.
21 The Kohn-Sham equation
The Kohn-Sham many-body wave function is a single Slater determinant,whose single-particle orbitals follow from a self-consistent equation:
)()()]([)()(2
2
rrrrr jjjxcH nVVV
where is the exact ground-state density.
N
jj rn
1
20 |)(|)( r
)(
][)]([
rr
n
nEnV xc
xc
Exchange-correlation (xc) potential:
][)()(||
)()(
2
1][ 333
10 nEVnrd
nnrdrdnE xcxc
N
jj
rrrr
rr
The total ground-state energy can be written as
22 The Kohn-Sham equation: spin-DFT
In practice, almost all Kohn-Sham calculations are done withspin-dependent single-particle orbitals, even if the system isclosed-shell and nonmagnetic:
,),()()](,[)()(
2
2
rrrrr jjjxcH nnVVV
N
jj rnnn
1
20 |)(|)()()( rrr
)(
],[)](,[
rr
n
nnEnnV xc
xc
23 Exact properties (I)
The Kohn-Sham Slater determinant is not meant to reproducethe full interacting many-body wave function:
),...,(det!
1),...,( 11 NjNKS xx
Nxx
The Kohn-Sham energy eigenvalues do not have a rigorous physical meaning, except the highest occupied ones:
j
)()1()()( NINENENN ionization energy ofthe N-particle system
)()()1()1(1 NANENENN electron affinityof the N-particle system
,ia Eigenvalue differences between occupied and empty levels, cannot be interpreted as excitation energies of the many-body system.
24 Exact properties (II)
The asymptotic behavior of the KS potential for neutral systemsis very important. For an atom with nuclear charge +N, we have
rr
NnrdV
r
NV H ||
)()(,)( 3
rr
rrr for
If an electron is “far away” in the “outer regions” of the system,it should see the Coulomb potential of the remaining positive ion.Therefore,
rr
Vxc
1)(r for
25 Exact properties (III)
The exact Kohn-Sham formalism must be free of self-interaction. This implies that for a 1-electron system the Hartree and xc potential cancel out exactly.
We have, for
0]0,[][ jxcjH nEnE
,|)(|)( 2rr jjn
The self-Hartree energy is fully compensated by the exchange energy,
N
ji
jijiexactx rdrdE
1,
**33 )()()()(
2
1
|r-r|
rrrr (evaluated with theexact KS orbitals)
The self-correlation energy vanishes by itself.
0]0,[ jc nE
26 Exchange-correlation functionals
The exact xc energy functional is unknown and hasto be approximated in practice. There exist many approximations!
][nExc
K. Burke, J. Chem. Phys. 136, 150901 (2012)
27 The local-density approximation (LDA)
1r
2r
)( 1rnn
)( 2rnn
The xc energy of an inhomogeneous system is
where exc[n] is the xc energy density. )]([][ 3 rnerdnE xcxc
LDA: at each point r, replace the exact xc energy density with that of a uniform,homogeneous electron gas whose density
has the same value as n(r).
)(][ 3 rnerdnE unifxc
LDAxc
28 The homogeneous electron gas
nenene unifc
unifx
unifxc
The xc energy per unit volume of a uniform electron gas only depends
on the uniform density n. It can be separated into exchange and correlation.
The exchange energy can be calculated exactly from Hartree-Fock. The HF solutions are plane waves, and the total ground-state energy is
3
342
2
352
4
)3(
10
)3(
.
nn
Vol
EunifHF
kinetic energydensity
exchange energydensity
29 The LDA exchange potential
)(
][)]([
rr
n
nEnV xc
xc
The LDA exchange potential is
313
3
3/13/42
3
3/423
)(31
4
)(3
3
4
4
)(3
)()(
r
r
r
rr
n
n
nrd
nV LDA
x
The LDA correlation energy and correlation potential havemore complicated expressions (from Quantum Monte Carlo data).
30 Performance of the LDA
● Atomic and molecular ground-state energies within 1-5%
● Molecular equilibrium distances and geometries within ~3%
● Fermi surfaces of metals: within a few percent
● Vibrational frequencies and phonon energies within a few percent
● Lattice constants of solids within ~2%
Czonka et al., PRB 79,155107 (2009)
31 Shortcomings of the LDA
► The LDA is not self-interaction free. As a consequence, the xc potential goes to zero exponentially fast (not as -1/r):
and the KS energy eigenvalues are too low in magnitude.
► LDA does not produce any stable negative ions.
► LDA underestimates the band gap in solids ► Dissociation of heteronuclear molecules produces ions with fractional charges. Overestimates atomization energies.
► LDA in general not accurate enough for many chemical applications.
reV rLDAxc ,
)(NILDAN typically 30-50% too small
32 The Jacob’s Ladder of functionals
LDA
GGA
Meta-GGA
Hyper-GGAhybrids
RPA double hybrids
)(rn
)(rn
),(2 rn
exactxe
Unoccupied orbitals
1
2
3
4
5
Earth: the Hartree world
Heaven: chemical accuracy
33 Generalized Gradient Approximations (GGA)
)(),(),(),(],[ 3 rrrr nnnnerdnnE GGAxc
GGAxc
There exists hundreds of GGA functionals. The most famous arethe B88 exchange functional and the LYP correlation functional,
A.D. Becke, Phys. Rev. A 38, 3098 (1988)C. Lee, W. Yang, and R.G. Parr, Phys. Rev. B 37, 785 (1988)
and the PBE functional, J.P. Perdew, K. Burke, and M. Ernzerhof, PRL 79, 3865 (1996)
4220
223
03
223
1
/)1(1ln)(
3/11)(
tAAt
ctAtncnerdE
snerdE
unifc
PBEc
unifx
PBEx
where/4,
)()(2
|)(|)(,
)()(2
|)(|)( Fs
sF
kkkn
nt
kn
ns
rr
rr
rr
rr
34 Hybrid functionals
Hybrid functionals mix in a fraction of exact exchange:
GGAc
GGAx
exactx
hybridxc EEaaEE )1(
where a ~ 0.25. The most famous hybrid is B3LYP:
LYPc
LDAc
Bx
LDAx
exactx
LYPBxc EccEbEEaaEE )1()1( 883
where .81.0,72.0,20.0 cba
35 Mean absolute errors for large molecular test sets
V.N.Staroverov, G.E.Scuseria, J. Tao, and J.P. Perdew, JCP 119, 12129 (2003)
A good introduction to ground-state DFT:K. Capelle, Brazilian Journal of Physics 36, 1318 (2006).