Tutorial:
Time-dependent density-functional theory
Carsten A. Ullrich
University of Missouri
XXXVI National Meeting on Condensed Matter Physics
Aguas de Lindoia, SP, Brazil
May 13, 2013
2
Literature
Time-dependent Density-Functional
Theory: Concepts and Applications
(Oxford University Press 2012)
“A brief compendium of TDDFT”
Carsten A. Ullrich and Zeng-hui Yang
arXiv:1305.1388
(Brazilian Journal of Physics, Vol. 43)
C.A. Ullrich homepage:
http://web.missouri.edu/~ullrichc
ullrichc@missouri.edu
Outline
3
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?
R
What are the nuclear vibrations?
How much energy is needed to ionize
the system, or to break a bond?
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
electronic coordinates:
r  r1 ,...,rN 
R  R1 ,...,R N
e
nuclear coordinates:
n

All electrons and all nuclei are quantum mechanical particles,
forming an interacting (Ne + Nn ) -body system. For example,
consider the hydrogen molecule H2:
Z1  Z 2  1
R1
r1
r2
M 1  M 2  m proton
R2
We’ll use atomic units:
e  m   1
6
Nonrelativistic Schrödinger equation
Hˆ r, R j r, R  E j  j r, R
 1 Ne
  2j
1
ˆ


 Ve (r j )  
H r, R   
 2 j k r  r

2
j 1 
k
j

Ne
 1 Ne Z j Z k
  2j
 Vn , j (R j )   
 
 2 jk R  R
 2M
j 1 
j
k
j

Nn
Ne Nn
 
j 1 k 1
Zk
rj  R k
electronic
Hamiltonian
nuclear
Hamiltonian
electron-nuclear
attraction
Ve (r), Vn (R): external scalar potentials acting on the electrons/nuclei.
7
Born-Oppenheimer approximation
► 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):
Ne
Hˆ BO r, R  
j 1
 2j
1 Ne
1
1 Ne Z j Z k
 
 
2 2 j  k r j  rk 2 j  k R j  R k
Ne Nn
 
j 1 k 1
Zk
rj  R k
This is just a constant
8
Potential-energy surfaces (diatomic molecule)
El (R)
4
5
3
2
ED
1
R
molecular equilibrium
1: ground-state potential-energy surface
2,3,4,5: excited-state potential-energy surfaces
The electronic many-body problem
9
( x1 , x2 ,...,xN )
x j  (r j , j )
space and spin
coordinate
antisymmetric N-electron wave function
Hˆ  j ( x1 ,...,xN )  E j j ( x1 ,...,xN )
N
Hˆ  
j 1
 2j
N
1 N
1
V (r j )  
 Tˆ  Vˆ  Wˆ
2 j 1
2 j  k | r j  rk |
Even a two-electron problem (Helium) is very difficult:
 12  22 1 1
1 

  

 j (r1 , r2 )  E j  j (r1 , r2 )
2 r1 r2 | r1  r2 | 
 2
This is a 6-dimensional partial differential equation!
10
The electronic many-body problem
Dirac (1929): “The fundamental laws necessary for the mathematical treatment
of 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 to
equations that are too complex to be solved.”
Expectation value of an observable:
O j   j Oˆ  j
Energy spectrum:
E j   j Hˆ  j
Probability density:
n j (r ) 
dx ... dx




2
N
 j (r,  , x2 ,..., x N )
2
11
The Hartree-Fock method
N
Hˆ  
j 1
 2j
N
1 N
1
  V (r j )  
2 j 1
2 j  k | r j  rk |
Solving the full many-body Schrödinger equation
is impossible. Instead, try a variational approach:
Hˆ   E
N
 

3
*
ˆ
  H     l  d r l (r )l (r )  0
*
 j (r ) 
l 1

The wave function is assumed
to be a Slater determinant:
 ( x1 ,..., xN ) 
1 ( x1 )
1 1 ( x2 )
N!

 2 ( x1 )   N ( x1 )
 2 ( x2 )   N ( x2 )



1 ( xN )  2 ( xN )   N ( xN )
The Hartree-Fock method
12
 2
n(r) 
3

 V (r )   d r
 j (r )


| r  r | 
 2
*
N


3
k (r ) k (r )
   d r
 j (r)   j j (r )
| r  r |
k 1
This is the HartreeFock equation, also
known as the selfconsistent field (SCF)
equation.
n(r)
VH (r )   d r 
Hartree potential: local (multiplicative) operator
| r  r |
3
nonloc
3  (r) k (r )
ˆ
[VX  j ](r)    d r 
 j (r)
| r  r |
k 1
N
*
k
Nonlocal exchange potential,
acting on the jth orbital.
“Nonlocal” means that the
orbital that is acted upon
appears under the integral.
The Hartree-Fock method
13
The total Hartree-Fock ground-state energy is given by
2



3
*
   d ri (r ) 
 V (r )i (r )
i 1
 2

1 3
n(r )n(r)
3
  d r  d r
“direct” energy
2
| r - r |
N
EHF
*
*



(
r
)

(
r
)

(
r
)

1 N
i
j
i
j (r )
3
3
   d r  d r
2 i , j 1
| r - r |
exchange
energy
where the single-particle orbitals are those that come from
the 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:
exact
0
E
 EHF  Ec
correlation 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
n0 ( x)  cos (x)
2
 cos 2 (3x)
1
 0 ( x)  n0 ( x)
  j( x)  V ( x) j ( x)   j j ( x)
2
2
ˆ  { }
n0( x) n0 ( x)
n

V

H
0
j
V ( x) 

4n0 ( x) 8n0 ( x)2
Is this always true??
The Hohenberg-Kohn Theorem
16
Recall:
Hˆ  Tˆ  Vˆ  Wˆ
Hamiltonian for N-electron system
We can define the following map:
V (r )
Hˆ 0  E00
0
n0   ... 0
2
n0 (r )
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 two
different potentials produce the same ground-state density:
V (r )
V (r )
n0 (r )
where “different” means that
the potentials differ by more
than just a constant:
V (r)  V (r)  c
The Hohenberg-Kohn proof
17
V (r )
Step 1: Show that
0
V (r )
Proof by contradiction. Let
cannot happen!
V (r)  V (r), but assume that
ic

0  0e (trivial phase factor). Then we have
Tˆ  Vˆ  Wˆ   E 
Tˆ  Vˆ   Wˆ   E 
0
0
subtract:
Vˆ  Vˆ 
0
0
0
0
0
 E0  E0 0
Vˆ  Vˆ   E0  E0  const.
but
ic

0  0e
Contradiction! This means
that the assumption must
have been wrong. We have
0  0 .
The Hohenberg-Kohn proof
18
0
Step 2: Show that
n0 (r )
0
cannot happen when
ic

0  0e
To prove this, we assume the contrary, namely that they both
give the same density. Then we can use the Ritz variational principle,
ˆ   . The following inequality holds:
whereby E    H
0
0
0
E0  0 Hˆ  0  0 Hˆ  Vˆ   Vˆ 0
 E0   d 3rV (r)  V (r)n0 (r)
E0  0 Hˆ 0  0 Hˆ   Vˆ  Vˆ  0
 E0   d 3rV (r)  V (r)n0 (r)
add:
E0  E0  E0  E0
Therefore
n0 (r )
(simply interchange primed
and unprimed)
Contradiction!
0
V (r )
uniquely.
19
The Hohenberg-Kohn Theorem (1964)
V (r )
1:1
n0 (r )
ˆ  Tˆ  Vˆ  Wˆ .
Therefore, n0 (r ) uniquely determines H
The Hamiltonian formally becomes a functional of the density:
Hˆ [n0 ]  [n0 ]
and all wave functions become density
functionals as well.
Every physical observable
is a functional of n0
The energy is a density functional:
Minimum principle:
O[n0 ]  [n0 ] Oˆ [n0 ]
E[n]  [n] Hˆ [n]
E[n]  E0
for n(r )  n0 (r )
E[n]  E0
for n(r )  n0 (r )
20
The Kohn-Sham formalism (1965)
The HK theorem can be proved for any type of particle-particle
interaction—in particular, it holds for noninteracting systems, too!
Therefore, there exists a unique noninteracting system that reproduces
a given ground-state density. This is the Kohn-Sham system.
We can write the total ground-state energy as follows:
kinetic energy functional
for interacting systems
E0 [n]  T [n]  W [n]   d 3r n(r )V (r )
1 3
n(r )n(r)
3
 Ts [n]   d r n(r )V (r )   d r  d r 
 E xc [n]
2
| r  r |
3
kinetic energy functional
for noninteracting systems
 T [n]  Ts [n]  W [n]
This defines the xc energy functional.
The Kohn-Sham equation
21
The Kohn-Sham many-body wave function is a single Slater determinant,
whose single-particle orbitals follow from a self-consistent equation:
 2

 V (r)  VH (r)  Vxc[n](r) j (r)   j j (r)

 2

N
2
where n0 (r)   |  j (r ) | is the exact ground-state density.
j 1
Exchange-correlation (xc) potential:
Exc[n]
Vxc [n](r ) 
n(r )
The total ground-state energy can be written as
1 3
n(r)n(r)
3
E0 [n]    j   d r  d r 
  d 3r n(r)Vxc (r)  Exc[n]
2
| r  r |
j 1
N
22
The Kohn-Sham equation: spin-DFT
In practice, almost all Kohn-Sham calculations are done with
spin-dependent single-particle orbitals, even if the system is
closed-shell and nonmagnetic:
 2

 V (r)  VH (r)  Vxc [n , n ](r) j (r)   j  j (r),

 2

N
n0 (r)  n (r)  n (r)  |  j (r ) |2

j 1
Exc[n , n ]
Vxc [n , n ](r) 
n (r)
 , 
23
Exact properties (I)
The Kohn-Sham Slater determinant is not meant to reproduce
the full interacting many-body wave function:
 KS ( x1 ,...,xN ) 
1
det j    ( x1 ,...,xN )
N!
The Kohn-Sham energy eigenvalues  j do not have a
rigorous physical meaning, except the highest occupied ones:
 N ( N )  E( N )  E( N 1)  I ( N )
ionization energy of
the N-particle system
 N 1 ( N  1)  E( N  1)  E( N )   A( N )
electron affinity
of the N-particle
system
  ,
Eigenvalue differences between occupied and empty levels, a
i
cannot be interpreted as excitation energies of the many-body system.
Exact properties (II)
24
The asymptotic behavior of the KS potential for neutral systems
is very important. For an atom with nuclear charge +N, we have
N
V (r )   ,
r
n(r)
N

VH (r )   d r

| r  r |
r
3
for
r 
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,
1
Vxc (r )  
r
for
r 
Exact properties (III)
25
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 n j (r) |  j (r) |2 ,
EH [n j ]  Exc[n j ,0]  0
The self-Hartree energy is fully compensated by the exchange energy,
*
*
N



(
r
)

(
r
)

(
r
)

1
i
j
i
j (r ) (evaluated with the
exact
3
3
Ex      d r  d r 
exact KS orbitals)
2   i , j 1
| r - r |
The self-correlation energy vanishes by itself.
Ec [n j ,0]  0
26
Exchange-correlation functionals
The exact xc energy functional Exc[n] is unknown and has
to be approximated in practice. There exist many approximations!
K. Burke, J. Chem. Phys. 136, 150901 (2012)
27
The local-density approximation (LDA)
r1
n  n(r1 )
r2
n  n(r2 )
The xc energy of an inhomogeneous system is
where exc[n] is the xc energy density.
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).
E xc [n]   d 3 r exc [n]( r )
unif
n(r ) 
E xcLDA [n]   d 3 r exc
28
The homogeneous electron gas
unif
n  exunif n  ecunif n
exc
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
unif
EHF
(3 2 n)5 3 (3 2 n) 4 3


2
3
Vol.
10
4
kinetic energy
density
exchange energy
density
29
The LDA exchange potential
Exc[n]
Vxc [n](r ) 
n(r )
The LDA exchange potential is

2


3

n(r)
LDA
3
Vx (r ) 
  d r 
n(r ) 
4 3
 

4/3



4 3
n(r )1/ 3

3
4 3
13
1
3
  3 n(r )
2 4/3



The LDA correlation energy and correlation potential have
more 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)
Shortcomings of the LDA
31
► The LDA is not self-interaction free. As a consequence,
the xc potential goes to zero exponentially fast (not as -1/r):
LDA
xc
V
 e
r
, r 
and the KS energy eigenvalues are too low in magnitude.

LDA
N
 I (N )
typically 30-50% too small
► 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.
32
The Jacob’s Ladder of functionals
Heaven: chemical accuracy
5
Earth: the Hartree world
RPA
double hybrids Unoccupied orbitals
4
Hyper-GGA
hybrids
exexact
3
Meta-GGA
2n(r), 
2
GGA
n(r)
1
LDA
n(r )
Generalized Gradient Approximations (GGA)
33
GGA
GGA
n (r), n (r), n (r), n (r)
E xc
[n , n ]   d 3r exc
There exists hundreds of GGA functionals. The most famous are
the B88 exchange functional and the LYP correlation functional,
and the
A.D. Becke, Phys. Rev. A 38, 3098 (1988)
C. Lee, W. Yang, and R.G. Parr, Phys. Rev. B 37, 785 (1988)
PBE functional, J.P. Perdew, K. Burke, and M. Ernzerhof, PRL 79, 3865 (1996)



ExPBE   d 3r exunif (n) 1   

2 2
1


s
/
3



PBE
c
E
where
 unif
 (1  At 2 ) t 2 / c0 
3
  d r ec (n)  nc0 ln 1 
2
2 4 
1

At

A
t 


3
| n(r) |
s(r) 
,
2n(r)k F (r)
| n(r) |
t (r) 
,
2n(r)k s (r)
k s  4k F / 
Hybrid functionals
34
Hybrid functionals mix in a fraction of exact exchange:
hybrid
xc
E
 aE
exact
x
 (1  a)E
GGA
x
E
GGA
c
where a ~ 0.25. The most famous hybrid is B3LYP:
ExcB3LYP  aExexact  (1  a)ExLDA  bExB88  cEcLDA  (1  c)EcLYP
where
a  0.20, b  0.72, c  0.81.
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).