time-dependent density-functional theory

Time-dependent density-functional theory

Carsten A. UllrichUniversity of Missouri-Columbia

APS March Meeting 2010, New Orleans

Neepa T. MaitraHunter College & Graduate Center

CUNY

Outline

1. A survey of time-dependent phenomena

2. Fundamental theorems in TDDFT

3. Time-dependent Kohn-Sham equation

4. Memory dependence

5. Linear response and excitation energies

6. Optical processes in Materials

7. Multiple and charge-transfer excitations

8. Current-TDDFT

9. Nanoscale transport

10. Strong-field processes and control

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1. Survey Time-dependent Schrödinger equation

),,...,(ˆ)(ˆˆ),,...,( 11 tWtVTtt

i NN rr rr

kinetic energy operator:

electron interaction:

N

j

j

mT

1

22

2ˆ

N

kjkj kj

eW,

2

21ˆ

rr

The TDSE describes the time evolution of a many-body state

starting from an initial state under the influence of an

external time-dependent potential

,t ,0t

.,ˆ1

N

jj tVtV r

From now on, we’ll (mostly) use atomic units (e = m = h = 1).

Start from nonequilibrium initial state, evolve in static potential:

t=0 t>0

1. Survey Real-time electron dynamics: first scenario

Charge-density oscillations in metallic clusters or nanoparticles (plasmonics)

New J. Chem. 30, 1121 (2006)Nature Mat. Vol. 2 No. 4 (2003)

Start from ground state, evolve in time-dependent driving field:

t=0 t>0

Nonlinear response and ionization of atoms and molecules in strong laser fields

1. Survey Real-time electron dynamics: second scenario

1. Survey Coupled electron-nuclear dynamics

High-energy proton hitting ethene

T. Burnus, M.A.L. Marques, E.K.U. Gross,Phys. Rev. A 71, 010501(R) (2005)

● Dissociation of molecules (laser or collision induced)● Coulomb explosion of clusters● Chemical reactions

Nuclear dynamicstreated classically

For a quantum treatment of nuclear dynamics within TDDFT (beyond the scope of this tutorial), see O. Butriy et al., Phys. Rev. A 76, 052514 (2007).

1. Survey Linear response

),( tr

),( tr

tickle the systemobserve how thesystem respondsat a later time

tVtttddtn ,,,,),( 11 rrrrr density

responseperturbationdensity-density

response function

1. Survey Optical spectroscopy

● Uses weak CW laser as Probe

● System Response has peaks at electronic excitation energies

Marques et al., PRL 90, 258101 (2003)

Greenfluorescentprotein

Vasiliev et al., PRB 65, 115416 (2002)

Theory

Energy (eV)

Phot

oabs

orpt

ion

cros

s sec

tion

Na2 Na4

Outline








8. Current-TDDFT



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2. Fundamentals Runge-Gross Theorem

Runge & Gross (1984) proved the 1-1 mapping for fixed T and W:

n(r t) vext(r t)

For a given initial-state y0, the time-evolving one-body density n(r t) tells you everything about the time-evolving interacting electronic system, exactly.

0

This follows from :

0, n(r,t) unique vext(r,t) H(t) (t) all observables

For any system with Hamiltonian of form H = T + W + Vext , e-e interaction

kinetic external potential

Consider two systems of N interacting electrons, both starting in the same Y 0 , but evolving under different potentials vext(r,t) and vext’(r,t) respectively:

RG prove that the resulting densities n(r,t) and n’(r,t) eventually must differ, i.e.

vext’(t), ’(t)o

vext(t), (t)

2. Fundamentals Proof of the Runge-Gross Theorem (1/4)

same

Assume time-analytic potentials:

The first part of the proof shows that the current-densities must differ. Consider Heisenberg e.o.m for the current-density in each system,

;t )

the part of H that differs in the two systems

initial density

if initially the 2 potentials differ, then j and j’ differ infinitesimally later ☺

At the initial time:


*

Take divergence of both sides of * and use the eqn of continuity,

…

As vext(r,t) – v’ext(r,t) = c(t), and assuming potentials are time-analytic at t=0,

there must be some k for which RHS = 0

proves j(r,t) vext(r,t)1-1

Yo

The second part of RG proves 1-1 between densities and potentials:


If vext(r,0) = v’ext(r,0), then look at later times by repeatedly using Heisenberg e.o.m :

1st part of RG ☺

…

1-1 mapping between time-dependent densities and potentials, for a given initial state

≡ u(r) is nonzero for some k, but must taking the div here be nonzero?

Yes!By reductio ad absurdum: assume

Then assume fall-off of n0 rapid enough that surface-integral 0

integrand 0, so if integral 0, then 0u contradiction


…

samei.e.

n v for given 0, implies any observable is a functional of n and 0

-- So map interacting system to a non-interacting (Kohn-Sham) one, that reproduces the same n(r,t). All properties of the true system can be extracted from TDKS “bigger-faster-cheaper” calculations of spectra and dynamicsKS “electrons” evolve in the 1-body KS potential:

functional of the history of the density

and the initial states

-- memory-dependence (see more shortly!)

If begin in ground-state, then no initial-state dependence, since by HK,

0 = 0[n(0)] (eg. in linear response). Then

2. Fundamentals The TDKS system

The KS potential is not the density-functional derivative of any action !

If it were, causality would be violated:

Vxc[n,0,F0](r,t) must be causal – i.e. cannot depend on n(r t’>t)

But if

2. Fundamentals Clarifications and Extensions

But how do we know a non-interacting system exists that reproduces a given interacting evolution n(r,t) ?

van Leeuwen (PRL, 1999) for time-analytic potentials and densities

(& under mild restrictions of the choice of the KS initial state F0)

But RHS must be symmetric in (t,t’) symmetry-causality paradox.

van Leeuwen (PRL 1998): an action, and variational principle, may be defined, using Keldysh contours in complex-time.

Vignale (PRA 2008): usual real-time action is just fine IF include boundary terms

then

2. Fundamentals Clarifications and Extensions

Restriction to time-analytic potentials means RG is technically not valid for many potentials, eg adiabatic turn-on, although RG is assumed in practise.

van Leeuwen (Int. J. Mod. Phys. B. 2001) extended the RG proof in the linear response regime to the wider class of Laplace-transformable potentials.

The first step of the RG proof showed a 1-1 mapping between currents and potentials TD current-density FT

In principle, must use TDCDFT (not TDDFT) for

-- response of periodic systems (solids) in uniform E-fields (see later…)

-- in presence of external magnetic fields (Ghosh & Dhara, PRA 1988)

In practice, approximate functionals of current are simpler where spatial non-local dependence is important

(Vignale & Kohn, 1996; Vignale, Ullrich & Conti 1997) … Stay tuned!








8. Current-TDDFT



Outline

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3. TDKS Time-dependent Kohn-Sham scheme (1)

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:

3. TDKS Time-dependent Kohn-Sham scheme (2)

Only the N initially occupied orbitals are propagated. How can this be sufficient to describe all possible excitation processes?? Here’s a simple argument:

Expand TDKS orbitals in complete basis of static KS orbitals,

1

)0(,k

kjkj tat rr

A time-dependent potential causes the TDKS orbitals to acquire admixtures ofinitially unoccupied orbitals.

finite for Nk

3. TDKS Adiabatic approximation

r-rrr tnrdtVH

,, 3

tnVxc ,r

depends on density at time t(instantaneous, no memory)

is a functional of tttn ,,rThe time-dependent xc potential has a memory!

Adiabatic approximation: rr )(, tnVtnV gsxc

adiaxc

(Take xc functional from static DFT and evaluate with time-dependent density)

ALDA: ),(

2

hom2 )(,),(tnn

xcLDAxc

ALDAxc nd

nedtnVtVr

rr

3. TDKS Time-dependent selfconsistency (1)

0t T time

start withselfconsistent

KS ground state

propagateuntil here

I. Propagate TtttVi jold

KSj ,, 022

II. With the density 2

j

j ttn calculate the new KS potential

tnVtnVtVtV xcHextnew

KS

III. Selfconsistency is reached if TtttVtV newKS

oldKS ,, 0

for all Ttt ,0

3. TDKS Numerical time Propagation

tett jtHi

j ,, ˆ rr Propagate a time step :t

Crank-Nicholson algorithm:2ˆ12ˆ1ˆ

tHitHie 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

3. TDKS Time-dependent selfconsistency (2)

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.

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

3. TDKS Summary of TDKS scheme: 3 Steps

3. TDKS Example: two electrons on a 2D quantum strip

periodicboundaries(travelling

waves)

hard walls

(standing waves)z

x

C.A. Ullrich, J. Chem. Phys. 125, 234108 (2006)

L

Charge-density oscillations

Δ

initial-state density

exactLDA

● Initial state: constant electric field, which is suddenly switched off

● After switch-off, free propagation of the charge-density oscillations

Step 1: solve full 2-electron Schrödinger equation

0,,1,,22 21

2121

22

21

tt

itzVtzV rrrr

Step 2: calculate the exact time-dependent density

tzntrdss

,,,21 ,

222

3 rr

Step 3: find that TDKS system which reproduces the density

021

2

2

t,zt

it,zVt,zVt,zVdzd

xcH

22 t,z

3. TDKS Construction of the exact xc potential

3. TDKS Construction of the exact xc potential

Ansatz: titnt ,exp2,, rrr

2

22

,21,

,ln81,ln

41

,,,

tt

tntn

tVtVtV Hxc

rr

r r

rr r

AxcV

dynxcV

density

adiabatic Vxc

exact Vxc

3. TDKS 2D quantum strip: charge-density oscillations

● The TD xc potential can be constructed from a TD density● Adiabatic approximations get most of the qualitative behavior right, but there are clear indications of nonadiabatic (memory) effects● Nonadiabatic xc effects can become important (see later)








8. Current-TDDFT



Outline

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Almost all calculations ignore memory, and use an “adiabatic approximation” :

vxc

functional dependence on history, n(r t’<t), and on initial states

Just take xc functional from static DFT and evaluate on instantaneous density

But what about the exact functional?

4. Memory Memory dependence

Maitra, Burke, Woodward (PRL 2002): Exact condition relating initial-state dependence and history-dependence.

parametrizesdensity

Hessler, Maitra, Burke, (J. Chem. Phys, 2002); see also other examples in the Literature handout

Any adiabatic (or even semi-local-in-time) approximation would incorrectly predict the same vc at both times.

Eg. Time-dependent Hooke’s atom –exactly solvable

2 electrons in parabolic well, time-varying force constant

k(t) =0.25 – 0.1*cos(0.75 t)

• Development of History-Dependent Functionals: Dobson, Bunner & Gross (1997), Vignale, Ullrich, & Conti (1997), Kurzweil & Baer (2004), Tokatly (2005,2007)

4. Memory Example of history dependence

But is there ISD? That is, if we start in different 0’s, can we get the same n(r t), for all t, by evolving in different potentials? i.e.

0n(r t) vext(r t)1-1RG:

n (r t)Evolve 0 in v(t)

? Evolve 0 in v (t) same n ?~ ~

If no, then ISD redundant, i.e. the functional dependence on the density is enough.

The answer is: No! for one electron, but, Yes! for 2 or more electrons

t

4. Memory Initial-state dependence

Maitra & Burke, (PRA 2001)(2001, E); Chem. Phys. Lett. (2002).

( Consequence for Floquet DFT: No 1-1 mapping between densities and time-periodic potentials. )

If we start in different 0’s, can we get the same n(r t) by evolving in different potentials? Yes!

• Say this is the density of an interacting system. Both top and middle are possible KS systems.

vxc different for each. Cannot be captured by any adiabatic approximation

A non-interacting example: Periodically driven HO

Re and Im parts of 1st and 2nd Floquet orbitals

Doubly-occupied Floquet orbital with same n

4. Memory Example of initial-state dependence

4. Memory Time-dependent optimized effective potential

C.A.Ullrich, U.J. Gossmann, E.K.U. Gross, PRL 74, 872 (1995)H.O. Wijewardane and C.A. Ullrich, PRL 100, 056404 (2008)

rrrr

rr

..),(),(),(),(

),(),(0

1

**

1

3

cctttt

tutVrdtdi

kjjkk

N

j

t

xcjxc

),(),(

1),( * tA

ttu

j

ixc

jxcj rr

r

exact exchange:

N

k

kkj

jxj

tttrd

ttu

1

**3

*

,,,,

1,rr

rrrr

r

where








8. Current-TDDFT



Outline

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5. Linear Response TDDFT in linear response

Poles at KS excitations

Poles at true excitations

adiabatic approx: no w-dep

Need (1) ground-state vS,0[n0](r), and its bare excitations

(2) XC kernel

Yields exact spectra in principle; in practice, approxs needed in (1) and (2).

Petersilka, Gossmann, Gross, (PRL, 1996)

Well-separated single excitations: SMA

When shift from bare KS small: SPA

Useful tools for analysis: “single-pole” and “small-matrix” approximations (SPA,SMA)

Zoom in on a single KS excitation, q = i a

Quantum chemistry codes cast eqns into a matrix of coupled KS single excitations (Casida 1996) : Diagonalize

5. Linear Response Matrix equations (a.k.a. Casida’s equations)

q = (i a)

Excitation energies and oscillator strengths

From Burke & Gross, (1998); Burke, Petersilka & Gross (2000)

5. Linear Response How it works: atomic excitation energies

Compare different functional approxs (ALDA, EXX), and also with SPA. All quite similar for He.

TDDFT linear response fromexact helium KS ground state:

Exp. SPASMA

LDA + ALDA lowest excitations

Vasiliev, Ogut, Chelikowsky, PRL 82, 1919 (1999)

full matrix

5. Linear Response Atomic excitations: Rydberg states

Generally, KS excitations themselves are good zero-order approximations to the exact energies – except when they are missing !LDA/GGA KS potentials asymptotically decay exponentially (ground-state lectures)

No -1/r tail no Rydberg excitations.

Either paste a tail on (eg LB94, or some kind of hybrid…)

OR, use a clever trick to obtain their energies:

A. Wasserman & K. Burke, Phys. Rev. Lett. (2005);

Quantum defect theory:

determined by short-range part of v

5. Linear Response A comparison of functionals

Study of various functionals over a set of ~ 500 organic compounds, 700 excited singlet states

From: D. Jacquemin, V. Wathelet, E. A. Perpete, C. Adamo, J. Chem. Theory Comput. (2009).

5. Linear response General trends

Energies typically to within about “0.4 eV”

Bonds to within about 1%

Dipoles good to about 5%

Vibrational frequencies good to 5%

Cost scales as N3, vs N5 for wavefunction methods of comparable accuracy (eg CCSD, CASSCF)

Available now in many electronic structure codes

TDDFT Sales Tag

Unprecedented balance between accuracy and efficiency

Can study big molecules with TDDFT !

5. Linear response Examples

Photo-excitation of a light-harvesting supra-molecular triad: a TDDFT study, N. Spallanzani, C. A. Rozzi, D. Varsano, T. Baruah, M. R. Pederson, F. Manghi, and A. Rubio, J. Phys. Chem. (2009)

carotenoid-diaryl-porphyrin-C60 -- Can study candidates for solar cells, eg.(632 valence electrons! )

Circular dichroism spectra of chiral fullerenes: D2C84

F. Furche and R. Ahlrichs, JACS 124, 3804 (2002).

5. Linear response Examples








8. Current-TDDFT



Outline

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6. TDDFT in solids Excitations in finite and extended systems

j j

jj cciEE

nnww

ww

..

ˆˆlim,,

0

00

0

rrrr

jThe full many-body response function has poles at the exact excitation energies

x xxx x

wIm

wRe

wIm

wRe

finite extended

► Discrete single-particle excitations merge into a continuum (branch cut in frequency plane)► New types of collective excitations appear off the real axis (finite lifetimes)

6. TDDFT in solids Metals vs. insulators

Excitation spectrum of simple metals:

● single particle-hole continuum (incoherent)● collective plasmon mode

plasmon

Optical excitationsof insulators:

● interband transitions● excitons (bound electron-hole pairs)

6. TDDFT in solids Excitations in bulk metals

Quong and Eguiluz, PRL 70, 3955 (1993)

Plasmon dispersion of Al

►RPA (i.e., Hartree) gives already reasonably good agreement►ALDA agrees very well with exp.

In general, (optical) excitation processes in (simple) metals are very welldescribed by TDDFT within ALDA.

Time-dependent Hartree already gives the dominant contribution, and fxc typically gives some (minor) corrections.

This is also the case for 2DEGs in doped semiconductor heterostructures(quantum wells, quantum dots).

Excitons are bound electron-hole pairs created in optical excitationsof insulators.

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

Frenkel exciton: tightly bound, localized ona single (or a few) atoms

6. TDDFT in solids Elementary view of excitons

)()(2

222

rr

Er

emr

r

● is exciton wave function● derived from TDHF linearized Semiconductor Bloch equation● includes dielectric screening

)(r

Cu2O GaAs

R.G. Ulbrich, Adv. Solid State Phys. 25, 299 (1985)R.J. Uihlein, D. Frohlich, and R. Kenklies,

PRB 23, 2731 (1981)

6. TDDFT in solids Wannier equation and excitonic Rydberg series

)()()( wwwww cvcvcv F kqq

kqkqq

)()(),,()()(2)( **332 rrrrrr qqkkkq cvxcvc frdrdF ww

● Finite atomic/molecular system: single-pole approximation involves two discrete levels

● “Single-pole approximation” for excitons involves two entire bands ● Excitons are a collective phenomenon!

● TDDFT Wannier equation: nonlocal e-h interaction (in real space)

from linearized TDDFT semiconductor Bloch equations (Tamm-Dancoff approx.):

V. Turkowski, A. Leonardo, C.A.Ullrich, PRB 79, 233201 (2009)

6. TDDFT in solids Simplified calculation of exciton binding energies

6. TDDFT in solids Optical absorption of insulators

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 the ALDA fail??

6. TDDFT in solids Optical absorption of insulators: failure of ALDA

Optical absorption requires imaginary part of macroscopic dielectric function:

GGGqq ImlimIm

0Vmac

where

0,00,

,GGG

G

VVfV xcKSKS

2~ q Long-range excluded, so RPA is ineffective

Needs component tocorrect

21 q

KS0q limit:

But ALDA is constant for :0q

0,lim hom

0

wqff xcq

ALDAxc

6. TDDFT in solids Long-range XC kernels for solids

● LRC (long-range correlation) kernel (with fitting parameter α): 2q

f LRCxc

q

● TDOEP kernel (X-only):

rrrr

rrrr

nn

ff k

kkkOEP

x 2,

2*

Simple real-space form: Petersilka, Gossmann, Gross, PRL 76, 1212 (1996)TDOEP for extended systems: Kim and Görling, PRL 89, 096402 (2002)

● “Nanoquanta” kernel (L. Reining et al, PRL 88, 066404 (2002)

)2,6(5,66,45,43,54,33,12,1 10

10

PGGWGGPf xc

quasiparticleGreen’s function

screened Coulomb

interaction

independentquasiparticlepolarizability

6. TDDFT in solids Optical absorption of insulators, again

F. Sottile et al., PRB 76, 161103 (2007)

Silicon

Kim & Görling

Reining et al.

6. TDDFT in solids Optical absorption of molecular chains

D. Varsano, A. Marini, and A. Rubio, PRL 101, 133002 (2008)

Undistorted H-chain: no gap,delocalized exciton.ALDA works well

Peierls-distorted H-chainhas optical gap and localizedexcitons. ALDA fails.

long-range works well in all cases.

xcf

6. TDDFT in solids Extended systems - summary

► 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

● some long-range XC kernels have become available, but the best ones are quite complicated.

● 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.








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Outline

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meaning, semi-local in space and local in time

Rydberg states

Polarizabilities of long-chain molecules

Optical response/gap of solids

Molecular Dissociation

Long-range charge transfer

Conical intersections

Double excitations

Local/semilocal approx inadequate. Need Im fxc to open gap and 1/q2

Cure with orbital- dependent fnals (exact-exchange/sic), or Nanoquanta kernel or TD current-DFT

Adiabatic approx for fxc fails.

Frequency-dependent kernel derived for some of these cases

7. Where the usual approxs. fail Ailments – and some Cures (I)

Haunted by static correlation in the ground-state.

• Some strong-field dynamics calculations

• Certain electronic quantum control problems

• Momentum distributions (eg in ion-recoil experiments)

• Non-sequential double ionization

• Coupled correlated electron-ion dynamics

• Electronic transport

or, are questionable…

Adiabatic approx fails -- memory-dependence crucial

TD Static correlation

Cannot extract observable simply from KS system

-- Need essential derivative discontinuity lacking in approx

7. Where the usual approxs. fail Ailments – and some Cures (II)

Interacting systems: generally involve mixtures of (KS) SSD’s that may have 1,2,3…electrons in excited orbitals.

single-, double-, triple- excitations

Non-interacting systems eg. 4-electron atom

Eg. single excitations

near-degenerate

Eg. double excitations

Types of Excitations7. Where the usual approxs. fail Double Excitations

c – poles at true states that are mixtures of singles, doubles, and higher excitations

c S -- poles only at single KS excitations, since one-body operator can’t connect Slater determinants differing by more than one orbital.

has more poles than s

? How does fxc generate more poles to get states of multiple excitation character?

7. Where the usual approxs. fail Double Excitations

Now consider:

How do these different types of excitations appear in the TDDFT response functions?

Exactly Solve a Simple Model: one KS single (q) mixing with a nearby double (D)

Strong non-adiabaticity!

Invert and insert into Dyson-like eqn for kernel dressed SPA (i.e. w-dependent):


General case: Diagonalize many-body H in KS subspace near the double ex of interest, and require reduction to adiabatic TDDFT in the limit of weak coupling of the single to the double

Maitra, Zhang, Cave,& Burke JCP (2004); Alternate derivations: Casida JCP (2005); Romaniello et al (JCP 2009); Gritsenko & Baerends PCCP (2009)


usual adiabatic matrix element

dynamical (non-adiabatic) correction

So: (i) scan KS orbital energies to see if a double lies near a single,

(ii) apply this kernel just to that pair(iii) apply usual ATDDFT to all other

excitations

Example: Short-chain polyenes

Lowest-lying excitations notoriously difficult to calculate due to significant double-excitation character.Cave, Zhang, Maitra, Burke, CPL (2004)

• Note importance of accurate double-excitation description in coupled electron-ion dynamics – propensity for curve-crossing

Levine, Ko, Quenneville, Martinez, Mol. Phys. (2006)


Example: Dual Fluorescence in DMABN in Polar Solvents

“anomalous” Intramolecular Charge Transfer (ICT)

“normal”“Local” Excitation (LE)

TDDFT resolved the long debate on ICT structure (neither “PICT” nor “TICT”), and elucidated the mechanism of LE -- ICT reaction

Rappoport & Furche, JACS 126, 1277 (2004).

Success in predicting ICT structure – How about CT energies ??

7. Where the usual approxs. fail Long-Range Charge-Transfer Excitations

Eg. Zincbacteriochlorin-Bacteriochlorin complex(light-harvesting in plants and purple bacteria)

Dreuw & Head-Gordon, JACS 126 4007, (2004).

TDDFT predicts CT states energetically well below local fluorescing states. Predicts CT quenching of the fluorescence.

! Not observed !

TDDFT error ~ 1.4eV

TDDFT typically severely underestimates long-range CT energies

Important process in

biomolecules, large enough

that TDDFT may be only

feasible approach !


We know what the exact energy for charge transfer at long range should be:

Why TDDFT typically severely underestimates this energy can be seen in SPA

-As,2-I1

(Also, usual g.s. approxs underestimate I)

Why do the usual approximations in TDDFT fail for these excitations?

exact

i.e. get just the bare KS orbital energy difference: missing xc contribution to acceptor’s electron affinity, Axc,2, and -1/R


~0 overlap

Dreuw, Weisman, Head-Gordon, JCP (2003)

Tozer, JCP (2003)

Eg. Tawada, Tsuneda, S. Yanagisawa, T. Yanai, & K. Hirao, J. Chem. Phys. (2004): “Range-separated” interaction in TDDFT matrix, with parameter m

Short-ranged, use GGA

Long-ranged, use HF, gives -1/R

Eg. Vydrov, Heyd, Krukau, & Scuseria (2006), 3 parameter range-separated, SR/LR decomposition…

Eg. Stein, Kronik, and Baer, JACS 131, 2818 (2009) – range-separated hybrid, but with non-empirically determined m

Eg. Dreuw, Weisman, & Head-Gordon, JCP (2003) – use CIS curve but shifted vertically to match SCF-DFT to account for correlation

Eg. Zhao & Truhlar (2006) M06-HF – empirical functional with 35 parameters!!! Ensures -1/R.

Eg. Heßelmann, Ipatov, Görling, PRA 80, 012507 (2009) – exact-exchange kernel


Many approaches to try to fix TDDFT for CT:

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








8. Current-TDDFT



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1

2

TDDFT does not apply for time-dependent magnetic fields or forelectromagnetic waves. These require vector potentials.

The original RG proof is for finite systems with potentials thatvanish at infinity (step 2). Extended/periodic systems can be tricky:

● TDDFT works for periodic systems if the time-dependent potential is also periodic in space.

● The RG theorem does not apply when a homogeneous electric field (a linear potential) acts on a periodic system.

N.T. Maitra, I. Souza, and K. Burke, PRB 68, 045109 (2003)

tB

▲ ▼

A(t), E(t)

ring geometry:

8. TDCDFT Situations not covered by the RG theorem

Continuity equation only involves longitudinal part of the current density:

tt

tnL ,, rjr

In general, time-dependent currents are not V-representable.This makes sense, since j is vector (3 components), and V is scalar (1 component).

R. D’Agosta and G. Vignale, PRB 71, 245103 (2005)

ttt TL ,,, rjrjrj

ttt TL ,,, rjrjrj

If comes from a potential tV ,r

then cannot come from ., tV r[both have the same , and this would violate the RG theorem] tn ,r

8. TDCDFT V-representability of current densities

generalization of RG theorem: Ghosh and Dhara, PRA 38, 1149 (1988) G. Vignale, PRB 70, 201102 (2004)

N

jjKSjKScjKS tVttH

1

21 ,,21ˆ rrAp

ttt TL ,,, rjrjrj The full current is uniquely determined by the pair ofscalar and vector potentials

N

kj kj

N

jjjcj tVttH

rrrrAp 1

21,,

21)(ˆ

1

21

AV,

8. TDCDFT TDCDFT

uniquely determined up to gauge transformation

wwwww ,,,,,', 1,1,1,3

1 rArArArr rj xcHextKSrd�

KS current-current response tensor: diamagnetic + paramagnetic part

rrrrrrr

jkkj

kj jk

jk PPi

ffn mmm w

w,

0 21,,

rrrr **kjjk

kjP mmm where

8. TDCDFT TDCDFT in the linear response regime

:,1, wrAext external perturbation. Can be atrue vector potential, or a gaugetransformed scalar perturbation:

1,1,1

extext Vi

w

A

rrrjrA w

ww ,, 13

21, rdiH gauge transformed

Hartree potential

ww

w ,,, 13

21, rj rrrA ALDA

xcALDAxc frd

i

www ,,,, 13

1, rjrrrA xcxc frd�

ALDA:

the xc kernel isnow a tensor!

8. TDCDFT Effective vector potential

Visualize electron dynamics as the motion (and deformation)of infinitesimal fluid elements:

t,r t,r

Nonlocality in time (memory) implies nonlocality in space!

Dobson, Bünner, and Gross, PRL 79, 1905 (1997)I.V. Tokatly, PRB 71, 165104 and 165105 (2005), PRB 75, 125105 (2007)

8. TDCDFT Nonlocality in space and time

● txn ,0

xx0

An xc functional that depends only on the local density (or its gradients) cannot see the motion of the entire slab.

A density functional needs to have a long range to seethe motion through the changes at the edges.

8. TDCDFT Ultranonlocality and the density

● automatically satisfies zero-force theorem/Newton’s 3rd law ● automatically satisfies the Harmonic Potential theorem ● is local in the current, but nonlocal in the density ● introduces dissipation/retardation effects

8. TDCDFT TDCDFT beyond the ALDA: the VK functional

G. Vignale and W. Kohn, PRL 77, 2037 (1996)G. Vignale, C.A. Ullrich, and S. Conti, PRL 79, 4878 (1997)

ww

ww ,,,0

1,1, r r

rArA xcALDAxcxc ni

c �

jkxcjkjkkjxcjkxc 11,1,1,~

32vv~ vv

xc viscoelastic stress tensor:

rrjrv 0/,, nww velocity field

2

22

2

,34,,~

,,~

dnednfnf

inn

nfinn

unifxcT

xcL

xcxc

Txcxc

www

w

ww

w

In contrast with the classical case, the xc viscosities have both realand imaginary parts, describing dissipative and elastic behavior:

wwww

wwww

iB

iS

dynxc

xc

~

~ shear modulus

dynamicalbulk modulus

reflect thestiffness of Fermi surfaceagainst defor-mations

8. TDCDFT XC viscosity coefficients

GK: E.K.U. Gross and W. Kohn, PRL 55, 2850 (1985)NCT: R. Nifosi, S. Conti, and M.P. Tosi, PRB 58, 12758 (1998)QV: X. Qian and G. Vignale, PRB 65, 235121 (2002)

LxcfIm L

xcfRe

TxcfIm T

xcfRe

8. TDCDFT xc kernels of the homogeneous electron gas

2

22

2

)0()0(

)0(34)()0(

nSf

nS

dnnedf

xcTxc

xcunifxcL

xc

The shear modulus of the electron liquid does not disappear for(as long as the limit q0 is taken first). Physical reason:

.0w

● Even very small frequencies <<EF are large compared to relaxation rates from electron-electron collisions.● The zero-frequency limit is taken such that local equilibrium is not reached.● The Fermi surface remains stiff against deformations.

8. TDCDFT Static limits of the xc kernels

(A) In the (quasi)-static ω→0 limit:● Polarizabilities of π-conjugated polymers● Nanoscale transport● Stopping power of slow ions in metals

These applications profit from the fact that VK does notreduce to the ALDA in the static limit.

(B) To describe excitations at finite frequencies:

● atomic and molecular excitation energies● plasmon excitations in doped semiconductor structures● optical properties of bulk metals and insulators

Here the picture is less clear, but it seems that VK worksfor metallic systems but can fail for systems with a gap.

8. TDCDFT Applications of the VK functional

ALDA overestimatespolarizabilities of longmolecular chains.The long-range VKfunctional producesa counteracting field,due to the finite shearmodulus at .0w

M. van Faassen et al., PRL 88, 186401 (2002) and JCP 118, 1044 (2003)

8. TDCDFT TDCDFT for conjugated polymers

Nazarov, Pitarke, Takada, Vignale, and Chang, PRB 76, 205103 (2007)

► Stopping power measures friction experienced by a slow ion moving in a metal due to interaction with conduction electrons► ALDA underestimates friction (only single-particle excitations)► TDCDFT gives better agreement with experiment: additional contribution due to viscosity

rrddf

nnQ

xc

xc

33

0

00

,,Im

ˆˆ

wwwrr

vrvr

friction coefficient:

xcparticle

glesin QQQ (Winter et al.)

(VK)

(ALDA)

8. TDCDFT Stopping power of electron liquids

► TDCDFT overcomes several formal limitations of TDDFT: ● allows treatment of electromagnetic waves, vector potentials, uniform applied electric fields. ● works for all extended systems. One does not need the condition that the current density vanishes at infinity.

► But TDCDFT is also practically useful in situations that could, in principle, be fully described with TDDFT: ● Upgrading to the current density can be a more “natural” way to describe dynamical systems. ● Helps to deal with the ultranonlocality problem of TDDFT ● Provides ways to construct nonadiabatic approximations

► The VK functional is a local xc vector potential beyond the ALDA. ● Works well for many metallic and quasi-metallic systems, but has problems for systems with a gap. ● More work is needed to construct current-dependent xc functionals.

8. TDCDFT TDCDFT: discussion








8. Current-TDDFT



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Koentopp, Chang, Burke, and Car (2008)

EfEfETdEI RL

2

two-terminal Landauer formula

Transmission coefficient, usually obtained fromDFT-nonequilibrium Green’s function

9. Transport DFT and nanoscale transport

Problems: ● standard xc functionals (LDA,GGA) inaccurate ● unoccupied levels not well reproduced in DFT transmission peaks can come out wrong conductances often much overestimated need need better functionals (SIC, orbital-dep.) and/or TDDFT

Current response: www ,'rE,'r,r'rd,rj eff�

03

wwww ,'rE,'rEE'rdTI xcHext

F 300

XC piece of voltage drop: Current-TDDFT

Sai, Zwolak, Vignale, Di Ventra, PRL 94, 186810 (2005)

dznn

AeR z

c

dyn

4

2

234

dynamical resistance: ~10% correction

9. Transport TDDFT and nanoscale transport: weak bias

(A) Current-TDDFT and Master equation Burke, Car & Gebauer, PRL 94, 146803 (2005)

(B) TDDFT and Non-equilibriumGreen’s functions

Stefanucci & Almbladh, PRB 69, 195318 (2004)

9. Transport TDDFT and nanoscale transport: finite bias

● periodic boundary conditions (ring geometry), electric field induced by vector potential A(t)● current as basic variable● requires coupling to phonon bath for steady current

● localized system● density as basic variable● steady current via electronis dephasing with continuum of the leads

► (A) and (B) agree for weak bias and small dissipation► some preliminary results are available – stay tuned!








8. Current-TDDFT



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In addition to an approximation for vxc[n;0,F0](r,t), also need an

approximation for the observables of interest, also with functional dependence A[n;0,F0]

Certainly measurements involving only density (eg dipole moment) can be extracted directly from KS – no functional approximation needed for the observable. But generally not the case.

We’ll take a look at: High-harmonic generation (HHG) Above-threshold ionization (ATI) Non-sequential double ionization (NSDI) Attosecond Quantum Control Correlated electron-ion dynamics

Is the relevant KS quantity physical ?

10. Strong-field processes TDDFT for strong fields

Erhard & Gross, (1996)

Eg. He

correlation reduces peak heights by ~ 2 or 3

TDHF

10. Strong-field processes High Harmonic Generation

HHG: get peaks at odd multiples of laser frequency

Measures dipole moment,

|d(w)|2 = ∫ n(r,t) r d3rso directly available from TD KS system

L’Huillier (2002)

Nguyen, Bandrauk, and Ullrich, PRA 69, 063415 (2004).

Eg. Na-clusters

30 Up

l= 1064 nm I = 6 x 1012 W/cm2 pulse length 25 fs

• TDDFT is the only computationally feasible method that could compute ATI for something as big as this!

• ATI measures kinetic energy of electrons – not directly accessible from KS. Here, approximate T by KS kinetic energy.

• TDDFT yields plateaus much longer than the 10 Up predicted by quasi-classical one-electron models

10. Strong-field processes Above-threshold ionization

ATI: Measure kinetic energy of ejected electrons

L’Huillier (2002)

Knee forms due to a switchover from a sequential to a non-sequential (correlated) process of double ionization.Knee missed by all single-orbital theories eg TDHF

TDDFT can get it, but it’s difficult :• Knee requires a derivative discontinuity, lacking in most approxs• Need to express pair-density as purely a density functional – uncorrelated

expression gives wrong knee-height. (Wilken & Bauer (2006))

10. Strong-field processes Non-sequential double ionization

TDDFT1

2

TDDFT c.f. TDHF

Exact c.f. TDHF

21

Lappas & van Leeuwen (1998),Lein & Kummel (2005)

10. Strong-field processes Non-sequential double ionization: Momentum

• Generally time-dependent KS momentum distributions don’t have anything to do with the true p-distribution

( in principle the true p-dist is a functional of the KS system…but what functional?! – “observable problem”)

Ion-recoil p-distributions computed from exact KS orbitals are poor, eg.

Wilken and Bauer, PRA 76, 023409 (2007)

A. Rajam, P. Hessler, C. Gaun, N. T. Maitra, J. Mol. Struct. (Theochem), TDDFT Special Issue 914, 30 (2009);

TDKS

Is difficult: Consider pumping He from (1s2) (1s2p)

Problem!! The KS state remains doubly-occupied throughout – cannot evolve into a singly-excited KS state. Simple model: evolve two electrons in a harmonic potential from ground-state (KS doubly-occupied 0) to the first excited state (0,1) :

• KS system achieves the exact target excited-state density, but with a doubly-occupied ground-state orbital !! The exact vxc(t) and observables are unnatural and difficult to approximate.

• If instead, optimized final-state overlap – max possible with KS is ½ while true is 0.98

, 10. Strong-field processes Electronic quantum control

Maitra, Woodward, & Burke (2002), Werschnik, Gross & Burke (2007)

Eg. Collisions of O atoms/ions with graphite clusters

Isborn, Li. Tully, JCP 126, 134307 (2007)

10. Strong-field processes Coupled electron-ion dynamics

Classical nuclei coupled to quantum electrons, via Ehrenfest coupling, i.e.

=

http://www.tddft.orgCastro, Appel, Rubio, Lorenzen, Marques, Oliveira, Rozzi, Andrade, Yabana, Bertsch

Freely-available TDDFT code for strong and weak fields:

10. Strong-field processes Coupled electron-ion dynamics

!! essential for photochemistry, relaxation,

electron transfer, branching ratios,

reactions near surfaces...

How about Surface-Hopping a la Tully with TDDFT ?

Simplest: nuclei move on KS PES between hops. But, KS PES ≠ true PES, and generally, may give wrong forces on the nuclei.

Should use TDDFT-corrected PES (eg calculate in linear response).

Many recent interesting applications.

Trajectory hopping probabilities cannot always be simply extracted – e.g. they depend on the coefficients of the true (not accessible in TDDFT), and on non-adiabatic couplings (only ground-excited accessible in TDDFT)

Craig, Duncan, & Prezhdo PRL 2005, Tapavicza, Tavernelli, Rothlisberger, PRL 2007, Maitra, JCP 2006, Tavernelli, Curchod, Rothlishberger, JCP 2009

Classical Ehrenfest method misses electron-nuclear correlation(“branching” of trajectories)

To learn more…

Time-Dependent Density Functional Theory, edited by M.A.L. Marques, C.A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E.K.U. Gross, Springer Lecture Notes in Physics, Vol. 706 (2006)

(see handouts for TDDFT literature list)

NWChem octopusGaussian YamboQChem ABINITGAMESS ParsecTurbomole SIESTAADF

Codes with TDDFT capabilities

Acknowledgments

• Peter Elliott• Harshani Wijewardane• Volodymyr Turkowski• Aritz Leonardo• Fedir Kyrychenko• Ednilsom Orestes• Daniel Vieira• Yonghui Li• David Tempel • Arun Rajam• Christian Gaun• August Krueger• Gabriella Mullady• Allen Kamal• Sharma Goldson• Chris Canahui• Izabela Raczckowsa

• Giovanni Vignale (Missouri)• Kieron Burke (UC Irvine)• Ilya Tokatly (San Sebastian)• Irene D’Amico (York/UK)• Klaus Capelle (Sao Paulo/Brazil)• Robert van Leeuwen (Jyväskylä/Finland)• Meta van Faassen (Groningen)• Adam Wasserman (Purdue)• Hardy Gross (MPI Halle)• Tchavdar Todorov (Queen’s, Belfast)• Ali Abedi (MPI Halle)

Collaborators: Students/Postdocs:

time-dependent density-functional theory

Documents

survey of time

nr t vextr t

fixed t

time evolution

timedependent driving

t unique vextr

body density nr t

survey linear response