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ISSPI: Time-dependent DFT

http://dft.uci.edu

Kieron Burke and friends

UC Irvine Physics and Chemistry Departments

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Recent reviews of TDDFT

To appear in Reviews of Computational Chemistry

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Book: TDDFT from Springer

• Warning! By 2300, entire mass of universe will be TTDFT papers

Search ISI web of Science for topic ‘TDDFT’

TDDFT publications in recent years

0

50

100

150

200

250

300

1997 1999 2001 2003 2005

TDDFT pubs

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Road map

• TD quantum mechanics->TDDFT• Linear response• Overview of all TDDFT• Does TDDFT really work?• Complications for solids• Currents versus densities• Elastic scattering from TDDFT

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Basic points• TDDFT

– is an addition to DFT, using a different theorem– allows you to convert your KS orbitals into optical

excitations of the system– for excitations usually uses ground-state

approximations that usually work OK– has not been very useful for strong laser fields– is in its expansion phase: Being extended to whole

new areas, not much known about functionals– with present approximations has problems for solids– with currents is more powerful, but harder to follow– yields a new expensive way to get ground-state Exc.

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TD quantum mechanics

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Current and continuity

• Current operator:

• Acting on wavefunction:

• Continuity:

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Runge-Gross theorem (1984)

• Any given current density, j(r,t), and initial wavefunction, statistics, and interaction, there’s only one external potential, vext(r,t), that can produce it.

• Imposing a surface condition and using continuity, find also true for n(r,t).

• Action in RG paper is WRONG

• von Leeuwen gave a constructive proof (PRL98?)

TD Kohn-Sham equations

•Time-dependent KS equations:

• Density:

• XC potential:

dt

tdittv i

i

)()()(

2

1S

2 rrr

N

ii tt

1

2)()( rr

))](0(),0(;['

)'(')()( XC

3extS tnv

trdtvtv r

rr

rrr

Depends on entire history(MEMORY)

initial state(s) dependence(MEMORY)

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Road map

• TD quantum mechanics->TDDFT• Linear response• Overview of all TDDFT• Does TDDFT really work?• Complications for solids• Currents versus densities• Elastic scattering from TDDFT

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Optical response in box

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Excitations from DFT

• Many approaches to excitations in DFT

• There is no HK theorem from excited-state density (PRL with Rene Gaudoin)

• Would rather have variational approach (ensembles, constrained search, etc.)

• TDDFT yields a response approach, i.e, looks at TD perturbations around ground-state

For a given interaction and statistics:

HS:

KS: )]([S rv

)]([ext rv

-12

-8

-4

0

4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

potential

0

1

2

3

4

5

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

He atom density

)(ext rv)(S rv

In time-dependent external field

RG:

KS: ))](0(),0(),([S ttv r))](0(),([ext ttv r

)(ext tv r

)(S tv r

TDDFT linear response

where

)'()'](['

1')()( 0XC

3extS rrr

rrrr frdvv

)'()'](['

)'()']([')(

S0S3

ext03

rrr

rrrr

vrd

vrd

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

)(0 r)( r

)(ext rv)'(ext rv

)(S rv)(S rv

Density response

)''(

)()','(

tv

tntt

r

rrr

Key quantity is susceptibility

Dyson-like equation for a susceptibility:

)'()]([1

)()'(),'( 2210XC21

1S23

13

S rrrrrr

rrrrrr frdrd

Two inputs: KS susceptibility

jk kj

jjkjjk iff

0)(

)'()'()()()(),'(

**

S

rrrr

rr

and XC kernel

)','(),()',',( XC

XC ttvttf r

rrr

)()( ttn r )''( tv r

Dyson-like equation

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TDDFT linear response• Probe system with AC field of freq • Ask at what you find a self-sustaining response• That’s a transition frequency!• Need a new functional, the XC kernel, fxc[0](r,r’,)

• Almost always ignore -dependence (called adiabatic approximation)

• Can view as corrections to KS response

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Eigenvalue equations

'

2'' )(

~

qqqqq vv

')(4 HXC'2

'' qfqqqqqqqq

Casida’s matrix formulation (1996)

)r()r()r( *aiq

'rr)r();'r,r()r'()( HXC*

HXC ddfqfq qq

True transition frequencies

KS transition frequencies

Occupied KS orbital

Unoccupied KS orbital

),',('1 XC rrrr fIn this equation, fHXC is the Hartree-exchange-correlation kernel, , where fXC is the unknown XC kernel

q 'q

b

a

i

b

b

b

iaq

Transitions in TDDFT

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KS susceptibility

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How good the KS response is

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Extracting Exc

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Adiabatic approximation

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Road map

• TD quantum mechanics->TDDFT• Linear response• Overview of all TDDFT• Does TDDFT really work?• Complications for solids• Currents versus densities• Elastic scattering from TDDFT

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Overview of ALL TDDFT1. General Time-dependent Density Functional Theory

• Any e- system subjected to any

• Only unknown:

• Treat atoms and molecules in INTENSE laser fields

)(ext tv r

)]([XC tv r

2. TDDFT linear response to weak fields• Linear response:

• Only unknown: near ground state

• Treat electronic excitations in atoms + molecules + solids

)''()','('')( ext3 tvttdtrdt rrrr

)'()']([')]([)]([ 0XC3

0XC0XC rrrrr frdvv

)(XC tv r

3. Ground-state Energy from TDDFT

•Fluctuation–dissipation theorem: Exc from susceptibility

•Van der Waals; seamless dissociation

Basic approximation: ALDA ))((unifXC tv r

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Methodology for TDDFT

• In general: Propagate TDKS equations forward in time, and then transform the dipole moment, eg. Octopus code

• Linear response: Convert problem of finding transitions to eigenvalue problem (Casida, 1996).

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Green fluorescent ProteinTDDFT approach for Biological Chromophores,Marques et al, Phys Rev Lett 90, 258101 (2003)

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Success of TDDFT for excited states

• Energies to within about 0.4 eV• Bonds to within about 1%• Dipoles good to about 5%• Vibrational frequencies good to 5%• Cost scales as N2, vs N5 for CCSD• Available now in your favorite quantum

chemical code

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TDDFT results for vertical singlet excitations in Naphthalene

Elliot, Furche, KB, Reviews Comp Chem, sub. 07.

Naphthalene

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Road map

• TD quantum mechanics->TDDFT• Linear response• Overview of all TDDFT• Does TDDFT really work?• Complications for solids• Currents versus densities• Elastic scattering from TDDFT

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How good the KS response is

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Quantum defect of Rydberg series

• I=ionization potential, n=principal, l=angular quantum no.s

• Due to long-ranged Coulomb potential• Effective one-electron potential decays as -1/r.• Absurdly precise test of excitation theory, and

very difficult to get right.

2)(2

1

nlnnl I

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Be s quantum defect: expt

Top: triplet, bottom: singlet

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Be s quantum defect: KS

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Be s quantum defect: RPA

KS=triplet

RPA

fH

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Be s quantum defect: ALDAX

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Be s quantum defect: ALDA

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General notes

• Most papers are lin resp, looking at excitations: need gs potential, plus kernel

• Rydberg excitations can be bad due to poor potentials (then use OEP, or be clever!).

• Simple generalization to current TDDFT• Charge transfer fails, because little oscillator

strength in KS response.• Double excitations lost in adiabatic

approximation (but we can put them back in by hand)

• Typically not useful in strong fields• Exc schemes still under development

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Road map

• TD quantum mechanics->TDDFT• Linear response• Overview of all TDDFT• Does TDDFT really work?• Complications for solids• Currents versus densities• Elastic scattering from TDDFT

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Complications for solids and long-chain polymers

• Locality of XC approximations implies no corrections to (g=0,g’=0) RPA matrix element in thermodynamic limit!

• fH (r-r’) =1/|r-r’|, but fxcALDA = (3)(r-r’) fxc

unif(n(r))• As q->0, need q2 fxc -> constant to get effects.

• Consequences for solids with periodic boundary conditions:– Polarization problem in static limit– Optical response:

• Don’t get much correction to RPA, missing excitons• To get optical gap right, because we expect fxc to shift

all lowest excitations upwards, it must have a branch cut in w starting at EgKS

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Two ways to think of solids in fields

• A: Apply sin(qx), and take q->0– Keeps everything static– Needs great care to take

q->0 limit

• B: Turn on TD vector potential A(t)– Retains period of unit

cell– Need TD current DFT,

take w->0.

B

Au

Au

Au

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Relationship between q->0 and ->0

• Find terms of type: C/((q+ng)2-2)

• For n finite, no divergence; can interchange q->0 and ->0 limits

• For n=0:– if =0 (static), have to treat q->0 carefully to cancel

divergences– if doing q=0 calculation, have to do t-dependent, and

take ->0 at end

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Road map

• TD quantum mechanics->TDDFT• Linear response• Overview of all TDDFT• Does TDDFT really work?• Complications for solids• Currents versus densities• Elastic scattering from TDDFT

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TD current DFT

• RG theorem I actually proves functional of j(r,t).

• Easily generalized to magnetic fields• Naturally avoids Dobson’s dilemma: Gross-

Kohn approximation violates Kohn’s theorem.• Gradient expansion exists, called Vignale-Kohn

(VK).• TDDFT is a special case• Gives tensor fxc, simply related to scalar fxc

(but only for purely longitudinal case).

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Currents versus densities

• Origin of current formalism: Gross-Kohn approximation violates Kohn’s theorem.

• Equations much simpler with n(r,t).• But, j(r,t) more general, and can have B-fields.• No gradient expansion in n(rt).• n(r,t) has problems with periodic boundary

conditions – complications for solids, long-chain conjugated polymers

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Beyond explicit density functionals

• Current-density functionals– VK Vignale-Kohn (96): Gradient expansion in current– Various attempts to generalize to strong fields– But is just gradient expansion, so rarely

quantitatively accurate

• Orbital-dependent functionals– Build in exact exchange, good potentials, no self-

interaction error, improved gaps(?),…

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Basic problem for thermo limit

• Uniform gas:

• Uniform gas moving with velocity v:

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Polarization problem

• Polarization from current:

• Decompose current:where

• Continuity:

• First, longitudinal case:– Since j0(t) not determined by n(r,t), P is not!

• What can happen in 3d case (Vanderbilt picture frame)?– In TDDFT, jT (r,t) not correct in KS system

– So, Ps not same as P in general.

– Of course, TDCDFT gets right (Maitra, Souza, KB, PRB03).

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Improvements for solids: currents

• Current-dependence: Snijders, de Boeij, et al – improved optical response (excitons) via ‘adjusted’ VK

• Also yields improved polarizabilities of long chain conjugated polymers.

• But VK not good for finite systems

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Improvements for solids: orbital-dependence

• Reining, Rubio, etc.

• Find what terms needed in fxc to reproduce Bethe-Salpeter results.

• Reproduces optical response accurately, especially excitons, but not a general functional.

• In practice, folks use GW susceptibility as starting point, so don’t need effective fxc to have branch cut

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Our recent work

• Floquet theory• Double excitations • Understanding how it works

– Single- and Double-pole approximations

• X-ray spectra• Rydberg series from LDA potential• Quantum defects• Errors in DFT for transport• TDDFT for open systems• Elastic electron-atom scattering

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Road map

• TD quantum mechanics->TDDFT• Linear response• Overview of all TDDFT• Does TDDFT really work?• Complications for solids• Currents versus densities• Elastic scattering from TDDFT

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Elastic scattering from TDDFT

• Huge interest in low energy scattering from biomolecules, since resonances can lead to cleavage of DNA

• Traditional methods cannot go beyond 13 atoms

• Can we use TDDFT? Yes!

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Simple scheme for spherical case

• Eg e- scattering from H.• Put H- into spherical box, and consider E>0

states.• Old formula due to Fano (1935):

• Exact for any Rb beyond potential.

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Is KS a good starting place?

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Is the LDA potential good enough?

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TDDFT corrections

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Summary

• TDDFT is different from DFT• Linear response TDDFT turns KS orbital

differences into single optical excitations• Value is in semi-quantitative spectra

– Can help determine geometry– Identify significant excitations

• Troubles with strong fields• Troubles with solids• Current- or orbital-dependence are promising

alternatives for solids and long-chain polymers