ISSPI: Time-dependent DFT
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Transcript of ISSPI: Time-dependent DFT
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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
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50
100
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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