TDDFT: applications and special topics

Carsten A. UllrichUniversity of Missouri

APS March Meeting, DenverMarch 2, 2014

2 Outline

PART I:

How to do a time-dependent Kohn-Sham calculation, and what to get out of it

PART II:

TDDFT in the linear-response regime:Double and charge-transfer excitations,spectroscopic properties in materials

PART III:

The future of TDDFT

3 The Runge-Gross theorem of TDDFT

N

kj kj

N

jj

j tVtHrr

r1

2

1,

2)(ˆ

1

2Consider an N-electronsystem with Hamiltonian

tV ,r )(t tn ,r Hti ˆ

0fixed

)(ˆ)( tnt

t,rjuniquelydetermined

uniquelydetermined

RG step 1 RG step 2

Therefore: tV ,r tn ,r1:1

4

ttVtVtVtt

i jxcHextj ,,,,2

,2

rrrrr

Consider an N-electron system, starting from a stationary state.

Solve a set of static KS equations to get a set of N ground-state orbitals:

Njtjj ,...,1,, 0)0( rr

rrrrr )0()0(0

2

2 jjjxcHext VV,tV

The N static KS orbitals are taken as initial orbitals and will be propagated in time:

N

jj ttn

1

2,, rr Time-dependent density:

Time-dependent Kohn-Sham scheme

5

0t T time

start withselfconsistent

KS ground state

propagateuntil here

I. Propagate TtttVi jold

KSj ,, 02

2

II. With the density 2j

j ttn calculate the new KS potential

tnVtnVtVtV xcHextnew

KS

III. Selfconsistency is reached if TtttVtV newKS

oldKS ,, 0

for all Ttt ,0

Time-dependent self-consistency (I)

6

tett jtHi

j ,,ˆ

rr Propagate a time step :t

Crank-Nicolson algorithm:

2ˆ1

2ˆ1ˆ

tHi

tHie tHi

tHtttHt ji

ji ,ˆ1,ˆ1 22 rr

Problem: H must be evaluated at the mid point 2tt But we know the density only for times t

Numerical time propagation

7

Predictor Step:

tj ttHttj )1()1( ˆ

nth Corrector Step:

tj ttHtt nnj )1()1( ˆ

ttHtH

ttHn

)(

21 ˆˆ

2ˆ

Selfconsistency is reached if remains unchanged for)(tn Ttt ,0upon addition of another corrector step in the time propagation.

Time-dependent self-consistency (II)

8

1

2

3

Prepare the initial state, usually the ground state, by a static DFT calculation. This gives the initial orbitals: )0,()0( rj

Solve TDKS equations selfconsistently, using an approximatetime-dependent xc potential which matches the static one usedin step 1. This gives the TDKS orbitals: ),(),( tntj r r

Calculate the relevant observable(s) as a functional of ),( tn r

Summary of TDKS scheme: 3 steps

9 Observables: the time-dependent density

The simplest observable is the time-dependent density itself: ),( tn r

Electron density map of the myoglobinmolecules, obtained using time-resolvedX-ray scattering. A short-lived CO groupappears during the photolysis process[Schotte et al., Science 300, 1944 (2003)]

10 Observables: the particle number

Unitary time propagation: spaceall Ntnrd ),(3 r

During an ionizationprocess, chargemoves away fromthe system.

Numerically, we candescribe this on a finite grid with anabsorbing boundary.

The number of bound/escaped particles at time t is approximately given by

boundescvolumeanalyzingbound NNNtnrdN ,),(3 r

11 Observables: moments of the density

dipole moment: zyxtnrrdtd ,,,),()( 3 r

sometimes one wants higher moments, e.g. quadrupole moment:

),()3()( 23 tnrrrrdtq r

One can calculate the Fourier transform of the dipole moment:

f

i

t

t

ti

fi

dtetdtt

d )(

1)(

Dipole power spectrum:

3

1

2)()(

tdD

12 Example: Na9+ cluster in a strong laser pulse

off resonance on resonance

not muchionization

a lot ofionization

Intensity: I=1011 W/cm2

13 Example: dipole power spectrum of Na9+ cluster

14 Implicit density functionals

We have learned that in TDDFT all quantum mechanical observables become density functionals:

)]([ˆ)]([)]([ tntntn

Some observables (e.g., the dipole moment), can easily beexpressed as density functionals. But there are also difficult cases!

► Probability to find the system in a k-fold ionized state

)()()( 00 tttPl

kl

klk

Projector on eigenstateswith k electrons in thecontinuum

15 Implicit density functionals

►Photoelectron kinetic-energy spectrum

dEtdEEPN

k

kE

t

2

1

)(lim)(

►State-to-state transition amplitude (S-matrix)

)(lim, tS ft

fi

All of the above observables are easy to express in terms of thewave function, but very difficult to write down as explicit density functionals.

Not knowing any better, people often calculate them approximately usingthe KS Slater determinant instead of the exact wave function. This isan uncontrolled approximation, and should only be done with great care.

16 Ionization of a Na9+ cluster in a strong laser pulse

25 fs laser pulses0.87 eV photon energyI=4x1013 W/cm2

For implicit observablessuch as ion probabilitiesone needs to make twoapproximations:

(1) for the xc potential in the TDKS calculation

(2) for calculating the observable from the TDKS orbitals.

17 Outline

PART I:

How to do a time-dependent Kohn-Sham calculation, and what to get out of it

PART II:

TDDFT in the linear response regime:double and charge transfer excitations,spectroscopic properties in materials

PART III:

The future of TDDFT

18 The Casida formalism for excitation energies

Y

X

10

01

Y

X

AK

KA**

Excitation energies follow from eigenvalue problem(Casida 1995):

rrrrrr

rr

aixcaiaiia

aiiaiaaaiiaiia

frdrdK

KA

,,1

,*33

,

,,

For real orbitals we can rewrite this as

ai

aiaiaiiaiaaiaiaaii ZZK 2,

2 2

121212122 2121221 HxcHxc ff 2-level approx.:

19 The Casida formalism for excitation energies

The Casida formalism gives, in principle, the exact excitation energiesand oscillator strengths. In practice, three approximations are required:

► KS ground state with approximate xc potential

► The inifinite-dimensional matrix needs to be truncated

► Approximate xc kernel (usually adiabatic):

r

rrr

n

Vf

statxcadia

xc

,

advantage: can use any xc functional from static DFT (“plug and play”)disadvantage: no frequency dependence, no memory

→ missing physics

20 Excited states with TDDFT: general trends

● Even with the simplest functionals, TDDFT gives significant improvement over KS excitation energies

● Try to do your ground-state calculation with a good xc functional. Then, the KS excitations will already be close to the exact excitations, and the TDDFT corrections can be small.

● The degree of accuracy of TDDFT for excited-state energies, forces and geometries is basically the same as that which can be achieved with the same functionals for the ground state.

● However, there are challenging cases, which we will discuss next.

21 Single versus double excitations

22 Single versus double excitations

j

jnjn C

The nth excited many-body state can be expanded in a series ofSlater determinants:

► “Single Excitation”: the expansion of is dominated by some singly excited Slater determinant.

► “Double Excitation”: the expansion of is dominated by some doubly excited Slater determinant.

► There are many-body states that have both singly and doubly excited character, so that a clear assignment cannot be made.

n

n

23

),(),,(||

1,,),( 1

331 rrr

rrrrr

nfrdrdn xcs

Single versus double excitations

Has poles at KS single excitations. The exact response function hasmore poles (single, doubleand multiple excitations).

Gives dynamical corrections tothe KS excitation spectrum. Shifts the single KS poles to thecorrect positions, and createsnew poles at double andmultiple excitations.

► Adiabatic approximation (fxc does not depend on ω): only single excitations!► ω-dependence of fxc will generate additional solutions of the Casida equations, which corresponds to double/multiple excitations.

24 Double excitations in linear polyenes

butadiene

● Lowest excitation, is optically dark and has significant double-excitation character● The necessary configuration mixing requires ω-dependent xc kernel● One can construct such an xc kernel explicitly for cases where there are close lying single and double excitations that are well isolated from the rest of the spectrum (“Dressed” TDDFT), see Maitra et al., JCP 120, 5932 (2004)

gg AA 11 21

25 Charge-transfer excitations

Zincbacteriochlorin-Bacteriochlorin complex(light-harvesting in plantsand purple bacteria)

TDDFT error: 1.4 eV

TDDFT predicts CT states energetically well below local fluorescing states. Predicts CT quenching of the fluorescence. Not observed!Dreuw and Head-Gordon, JACS 126, 4007 (2004)

26 Charge-transfer excitations: large separation

RAI ad

exactct

1

(ionization potential of donor minuselectron affinity of acceptor plus Coulomb energy of the charged fragments)

What do we get in TDDFT? Let’s look at the two-level approximation:

)()(),,()()(2 32 rrrrrr

dH

aLHxc

dH

aL

dH

aL

SPAct

frdrd

The highest occupied orbital of the donor and the lowest unoccupiedorbital of the acceptor have exponentially vanishing overlap!

dH

aL

TDDFTct

For all (semi)local xc approximations,TDDFT significantly underestimatescharge-transfer energies!

27 Charge-transfer excitations: exchange

R

rdrd

HFdH

HFaL

dH

dH

aL

aLHFd

HHFa

LTDHFct

1

||

)()()()(

,,

32,,

rr

rrrr

and use Koopmans theorem!

TDHF reproduces -1/R behavior of charge-transfer energies correctly. Therefore, hybrid functionals (such as B3LYP) will give some improvement over LDA and GGA.

Even better are the so-called range-separated hybrids: Let

||

|)|(1

||

|)|(

||

1

rr

rr

rr

rr

rr

ff )erfc()(,)( xxfexf x

GGAc

HFLRx

GGASRxxc EEEE

28 Charge-transfer excitations: long-range correction

Long-range corrected (LC) functionals (i.e., range-sparated hybrids)perform very well for charge-transfer excitations.

Tawada et al., JCP 120,8425 (2004)

29

What are the properties of the unknown exact xc kernel that must be well-modelled to get long-range CT energies correct ?

Exponential dependence on the fragment separation R,

fxc ~ exp(aR)

For transfer between open-shell species, need strong frequency-dependence.

Gritsenko & Baerends (PRA, 2004), Maitra (JCP, 2005), Tozer (JCP, 2003) Tawada et al. (JCP, 2004)

Step in Vxc re-aligns the 2 atomic HOMOs near-degeneracy of molecular HOMO & LUMO static correlation, crucial double excitations frequency-dependence!

(It’s a rather ugly kernel…)

“LiH”

step

Long-range charge-transfer excitations

30 Metals: particle-hole continuum and plasmons

In ideal metals, all single-particle states inside the Fermi sphereare filled. A particle-hole excitation connects an occupied single-particle state inside the sphere with an empty state outside.

From linear response theory, one can show that the plasmon dispersiongoes as

...)( 2 qq pl m

nepl

24

pl

31 Plasmon excitations in bulk metals

Quong and Eguiluz, PRL 70, 3955 (1993)

● In general, excitations in (simple) metals very well described by ALDA. ●Time-dependent Hartree (=RPA) already gives the dominant contribution

● fxc typically gives some (minor) corrections (damping!)

●This is also the case for 2DEGs in doped semiconductor heterostructures

Al

Gurtubay et al., PRB 72, 125114 (2005)

Sc

32 Plasmon excitations in metal clusters

Yabana and Bertsch, PRB 54,4484 (1996)

Surface plasmons (“Mie plasmon”) in metal clusters are very well reproducedwithin ALDA. Plasmonics: mainly using classical electrodynamics, not quantum response

Calvayrac et al., Phys. Rep.337, 493 (2000)

33 Insulators: three different gaps

Band gap: xcKSgg EE ,

Optical gap:exciton

gopticalg EEE 0

The Kohn-Sham gap approximates the opticalgap (neutral excitation),not the band gap!

34 Elementary view of Excitons

Mott-Wannier exciton:weakly bound, delocalizedover many lattice constants

An exciton is a collective interband excitation:single-particle excitations arecoupled by Coulomb interaction

Real space: Reciprocal space:

35 Excitonic features in the absorption spectrum

● Sharp peaks below the onset of the single-particle optical gap

● Redistribution of oscillator strength: enhanced absorption close to the onset of the continuum

36

G. Onida, L. Reining, A. Rubio, RMP 74, 601 (2002)S. Botti, A. Schindlmayr, R. Del Sole, L. Reining, Rep. Prog. Phys. 70, 357 (2007)

RPA and ALDA both bad!

►absorption edge red shifted (electron self-interaction)

►first excitonic peak missing (electron-hole interaction)

Silicon

Why does ALDA fail?

Optical absorption of insulators

37 Linear response in periodic systems

),(),()(

),(),(),(

21211

21

kkk

kkk

GGGGGGG

GGGGGGGG

2

1

xc

ss

fV

Optical properties are determined by the macroscopic dielectric function:

2~)( nmac (Complex index of refraction)

For cubic symmetry, one can prove that

1

00

1

0),(lim)(

GGGG k

kmac

),()(),(1 kkk GGGGGGG VTherefore, one needs the

inverse dielectric matrix:

38 The xc kernel for periodic systems

rGq

qGG

GG,

rGq qrr )(,

)( ),(),,(

i

FBZHxc

iHxc efef

TDDFT requires the following matrix elements as input:

00

00

,,

)(,

)(, ),(

GqkkGqkk

rGqGG

q GG

rGqGG kkqkk

iaia

ii

aHxcFBZ

ai

iaiia ieafaeiK

Most important: long-range limit of “head” :)0( GG)0( q

qkk0q

qr ai

i aei20,

1),(

qf exact

xc q00 q

but .const),(0, q00 q ALDA

xcf Therefore, no excitonsin ALDA!

39

● LRC (long-range corrected) kernel (with fitting parameter α):

GGGG Gqq

2, ||

LRCxcf

Long-range xc kernels for solids

● “bootstrap” kernel (S. Sharma et al., PRL 107, 186401 (2011)

)0,(

)0,(),(

00

1

, q

qq GG

GGs

bootxcf

● Functionals from many-body theory: (requires matrix inversion)

exact exchangeexcitonic xc kernel from Bethe-Salpeter equation

40 Excitons with TDDFT: “bootstrap” xc kernel

S. Sharma et al., PRL 107, 186401 (2011)

41

► TDDFT works well for metallic and quasi-metallic systems already at the level of the ALDA. Successful applications for plasmon modes in bulk metals and low-dimensional semiconductor heterostructures.

► TDDFT for insulators is a much more complicated story:

● ALDA works well for EELS (electron energy loss spectra), but not for optical absorption spectra

● Excitonic binding due to attractive electron-hole interactions, which require long-range contribution to fxc

● At present, the full (but expensive) Bethe-Salpeter equation gives most accurate optical spectra in inorganic and organic materials (extended or nanoscale), but TDDFT is catching up.

● Several long-range XC kernels have become available (bootstrap, meta-GGA), with promising results. Stay tuned!

Extended systems - summary

42 Outline

PART I:

How to do a time-dependent Kohn-Sham calculation, and what to get out of it

PART II:

TDDFT in the linear response regime:Double and charge transfer excitations,spectroscopic properties in materials

PART III:

The future of TDDFT

43 The future of TDDFT: biological applications

C.A. Rozzi, S.M. Falke, N. Spallanzani, A. Rubio, E. Molinari, D. Brida, M. Maiuri, G.Cerullo, H. Schramm, J. Christoffers, and C. Lienau, Nature Commun. 4, 1602 (2013)

(TD)DFT can handle big systems (103—106 atoms).

Many applications to large organic systems (DNA, light-harvestingcomplexes, organic solar cells) will become possible.

Charge-transfer excitations and van der Waals interactions canbe treated from first principles.

Carotene-porphyrin-fullerenelight-harvesting triad:time-dependent simulationof charge-transfer processincluding ionic motion(Ehrenfest dynamics)

44 The future of TDDFT: materials science

K. Yabana, S. Sugiyama, Y. Shinohara, T. Otobe, and G.F. Bertsch, PRB 85, 045134 (2012)

● Combined solution of TDKS and Maxwell’s equations● Strong fields acting on crystalline solids: dielectric breakdown, coherent phonons, hot carrier generation● Coupling of electron and nuclear dynamics allows description of relaxation and dissipation (TDDFT + Molecular Dynamics)

Si

Vacuum Si

45 The future of TDDFT: open formal problems

► Development of nonadiabatic xc functionals (needed for double excitations, dissipation, etc.)

► TDDFT for open systems: nanoscale transport in dissipative environments. Some theory exists, but applications so far restricted to simple model systems

► Strongly correlated systems. Mott-Hubbard insulators, Kondo effect, Coulomb blockade. Requires subtle xc effects (discontinuity upon change of particle number)

► Formal extensions: finite temperature, relativistic effects…

TDDFT will remain an exciting field of research for many years to come!

46 TDDFT computer codes

Commercial: ADF, Gaussian, Jaguar, Q-Chem, Turbomole, VASP, WIEN2k,…

Open-source: ABINIT, DP, EXC, Exciting, GAMESS, NWChem, Octopus, Parsec, Quantum Espresso, SIESTA, CPMD, Smeagol, YAMBO, ELK,…

47 Literature

Time-dependent Density-FunctionalTheory: Concepts and Applications(Oxford University Press 2012)

“A brief compendium of TDDFT” Carsten A. Ullrich and Zeng-hui YangarXiv:1305.1388Brazilian Journal of Physics 44, 154 (2014)

C.A. Ullrich homepage:http://web.missouri.edu/~ullrichc

ullrichc@missouri.edu