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

ullrichc@missouri.edu

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:

protonmMMZZ

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

21)(

2

121)(

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

211

21

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||

121)(

2ˆ

11

2

rr

r

Even a two-electron problem (Helium) is very difficult:

),(),(||

11122 2121

2121

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||

121)(

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

xxxxxx

Nxx

The wave function is assumedto be a Slater determinant:

12 The Hartree-Fock method

)(||)()()](ˆ[

1

3 rrrrrr

*

j

N

k

kkj

nonlocX rdV

)()(||)()(

)(||

)()(2

1

3

32

rrrrrr

rrrrr

*

jjj

N

k

kk

j

rd

nrdV

This is the Hartree-Fock equation, alsoknown as the self-consistent field (SCF)equation.

||

)()( 3

rrrr 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

)()()()(21

)()(21

)()(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)()(

)()()()(21

xnxn

xnxnxV

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

rrrr

nnEnEnnEnE

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:

][||)()(

21)()(][

)()(][][][

333

30

nEnnrdrdVnrdnT

VnrdnWnTnE

xcs

rrrrrr

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

nnEnV xc

xc

Exchange-correlation (xc) potential:

][)()(||)()(

21][ 333

10 nEVnrdnnrdrdnE xcxc

N

jj

rrrrrr

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

)(],[

)](,[r

r

nnnE

nnV xcxc

23 Exact properties (I)

The Kohn-Sham Slater determinant is not meant to reproducethe full interacting many-body wave function:

),...,(det!

1),...,( 11 NjNKS xxN

xx

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

rrNnrdV

rNV H ||

)()(,)( 3

rrrrr 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

Vxc1)(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 )()()()(

21

|r-r|rrrr (evaluated with the

exact 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 iswhere 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 densityhas 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 dependson 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

VolEunif

HF

kinetic energydensity

exchange energydensity

29 The LDA exchange potential

)(][)]([

rr

nnEnV xc

xc

The LDA exchange potential is

313

3

3/13/42

3

3/423

)(314

)(334

4)(3

)()(

r

r

rr

r

n

n

nrdn

V LDAx

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)(

tAAtctAtncnerdE

snerdE

unifc

PBEc

unifx

PBEx

where/4,

)()(2|)(|)(,

)()(2|)(|)( Fs

sF

kkkn

ntkn

ns

rr

rrrr

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).