ISSPI: Time-dependent DFT

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ISSPI Time-dependent DFT
Kieron Burke and friends UC Irvine Physics and
Chemistry Departments
http//dft.uci.edu
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Recent reviews of TDDFT
To appear in Reviews of Computational Chemistry
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Book TDDFT from Springer
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TDDFT publications in recent years
Search ISI web of Science for topic TDDFT

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

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Road map

• TD quantum mechanics-gtTDDFT
• 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,
  theres 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?)

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TD Kohn-Sham equations

• Time-dependent KS equations
• Density
• XC potential

Depends on entire history(MEMORY)
initial state(s) dependence(MEMORY)
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Road map

• TD quantum mechanics-gtTDDFT
• 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

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TDDFT linear response
In time-dependent external field
For a given interaction and statistics HS KS
RG KS
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Density response
where
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Dyson-like equation
Key quantity is susceptibility
Dyson-like equation for a susceptibility
Two inputs KS susceptibility
and XC kernel
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TDDFT linear response

• Probe system with AC field of freq w
• Ask at what w you find a self-sustaining response
• Thats a transition frequency!
• Need a new functional, the XC kernel,
  fxcr0(r,r,w)
• Almost always ignore w-dependence (called
  adiabatic approximation)
• Can view as corrections to KS response

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Eigenvalue equations
Casidas matrix formulation (1996)
True transition frequencies
KS transition frequencies
Unoccupied KS orbital
Occupied KS orbital
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Transitions in TDDFT
In this equation, fHXC is the
Hartree-exchange-correlation kernel,
, where fXC is the unknown XC kernel
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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-gtTDDFT
• 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 TDDFT
1. General Time-dependent Density Functional
Theory
2. TDDFT linear response to weak fields
3. Ground-state Energy from TDDFT

• Fluctuationdissipation theorem Exc from
  susceptibility
• Van der Waals seamless dissociation

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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 Protein
TDDFT approach for Biological Chromophores, Marque
s 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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Naphthalene
TDDFT results for vertical singlet excitations in
Naphthalene Elliot, Furche, KB, Reviews Comp
Chem, sub. 07.
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Road map

• TD quantum mechanics-gtTDDFT
• 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

• Iionization potential, nprincipal, langular
  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.

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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
KStriplet
fH
RPA
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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-gtTDDFT
• 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 (g0,g0) RPA matrix element in
  thermodynamic limit!
• fH (r-r) 1/r-r, but fxcALDA d(3)(r-r)
  fxcunif(n(r))
• As q-gt0, need q2 fxc -gt constant to get effects.
• Consequences for solids with periodic boundary
  conditions
• Polarization problem in static limit
• Optical response
• Dont 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 E fields

• A Apply Esin(qx), and take q-gt0
• Keeps everything static
• Needs great care to take q-gt0 limit
• B Turn on TD vector potential A(t)
• Retains period of unit cell
• Need TD current DFT, take w-gt0.

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Relationship between q-gt0 and w-gt0

• Find terms of type C/((qng)2-w2)
• For n finite, no divergence can interchange
  q-gt0 and w-gt0 limits
• For n0
• if w0 (static), have to treat q-gt0 carefully to
  cancel divergences
• if doing q0 calculation, have to do t-dependent,
  and take w-gt0 at end

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Road map

• TD quantum mechanics-gtTDDFT
• 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 Dobsons dilemma Gross-Kohn
  approximation violates Kohns 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 Kohns 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 dont 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-gtTDDFT
• 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 Egt0
  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