TimeDependent Density Functional Theory

1
Time-Dependent Density Functional Theory in Real
Time
Benjamin Levine, Chaehyuk Ko, Richard Martin,
and Todd J. Martínez
2
Ab initio Quantum Dynamics

• Ab Initio Quantum Dynamics ? On-the-fly
  solution of electronic and nuclear Schrödinger
  equations
• Multiple electronic states/Tunneling ? Need
  quantum nuclear dynamics
• Bond rearrangement ? Need to solve electronic
  Schrödinger equation
• Exact numerical solution of the nuclear
  Schrödinger equation is impractical for large
  systems.
• Ab initio quantum chemistry is local
• Nuclear Schrödinger equation is global
• Not a problem in Car-Parrinello b/c Newtonian
  mechanics is local
• Further considerations
• Need PESs for both ground and excited states.
• Matrix elements which couple electronic states
• Minimize the number of PESs evaluations b/c of
  extreme expense.
• Tailor the requirements of quantum dynamics to
  quantum chemistry
• and vice-versa!

3
The Full Multiple Spawning (FMS) Method

• Wavefunction ansatz

Nuclear wavefunction
Electronic state

• Nuclear wavefunction on each electronic state is
  a product of 3N frozen Gaussian basis functions

Position, momentum, width
Semiclassical phase
Cartesian degrees of freedom

• Classical evolution for R(t) and P(t).
• Variational principle for coefficients

Nuclear overlap matrix
Hamiltonian matrix
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Adaptive Basis Set

• Prescription so far is insufficient
• New Basis Functions must be added to augment
  classical mechanics

Spawned Basis Functions
R(t)
Nonadiabatic Coupling Regions
time
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Multiple Spawning
Time ?

• FMS is a hierarchy of methods
• Dynamics on a single electronic state?coupled
  frozen Gaussians (Heller,Metiu)
• Useful Approximations in Ab Initio Dynamics
• Independent First Generation Approximation
• Different initial Gaussian wavepackets are
  uncoupled
• Includes coherences between basis function and
  its children
• Neglects coherence between initial basis
  functions
• Saddle-Point Approximation for Integrals
• Use locality of Gaussians to evaluate integrals
  using local information

Centroid of ?i?j
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Ab initio Multiple Spawning - Obstacles

• Multireference ab initio methods are expensive
• Places constraints on
• Propagation time
• Number of trajectory basis functions
• Accuracy of electronic structure treatment (e.g.
  basis set)
• Is TDDFT a viable alternative?
• Vertical excitation energies (Electronic spectra)
• Excited State Gradients (Electronic spectra band
  shape)
• Global shape of excited state PES (Photodynamics)
• Conical intersection locations (Photodynamics)

7
Absorption and Resonance Raman Spectra
Time-domain Formulation of Spectroscopy (Heller)
Electronic Absorption Spectrum
Resonance Raman Excitation Profiles

• Anharmonicity and Duschinsky rotation included
• Coordinate dependence of transition dipole can
  be included

8
Absorption Spectra from TDDFT Dynamics
AIMS-EOM-CCSD
TDDFT is nearly as good as EOM-CCSD in
Franck-Condon region
9
Butadiene Bond Alternation CASPT2 vs TDDFT
41Ag
31Ag
21Ag and 41Ag have significant double
excitation character in CAS not
represented in TDDFT
21Ag
11Bu
21Ag
lines are caspt2, symbols are b3lyp
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TDDFT Outside Franck-Condon Region
Minimal energy path from CASSCF S1 is
singly-excited S2 is a double excitation
Photoactive Yellow Protein

• Linear-response TDDFT does not describe
• doubly-excited states, in contrast to earlier
  reports

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S1 PES for Ethylene
5.5
6.0
TDDFT
5.5
4.5
5.0
6.0
CIS
CASPT2
CIS and TDDFT similar and incorrect
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TDDFT and Conical Intersections

• Similar behavior in CIS and TDDFT
• Charge transfer (doubly-excited) state missing
• Expected?
• Does this imply failure in describing
  intersections?
• Searching for intersections
• Nonadiabatic coupling vector nontrivial in TDDFT
• Modified intersection search scheme
• Solve for Lagrange multiplier
• Optimize using conjugate gradient and numerical
  forces
• Ground state DFT (restricted) does not always
  succeed
• Packaged in Open Source MECI Optimization code
  (CIOpt)

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MECIs Determined from TDDFT
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MECI Geometries from TDDFT
TD-B3LYP CAS(4/4) CAS(4/4)MSPT2
15
PES Behavior Near MECIs
TD-B3LYP
CAS
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Intersection Dimensionality in TDDFT
TDDFT
y
x
H2 O H1
CAS
TDDFT Intersections are N-1D!
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Pseudospectral CIS/TDDFT

• Pseudospectral method
• Use grid and basis set
• Formal scaling advantage in
• J and K integrals
• Implemented for CIS and
• TDDFT in Jaguar code
• Accuracy ? 0.02eV
• Slight scaling advantage

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Conclusions

• Low-lying intersections often involve electronic
  states differing by a single excitation
• TDDFT predicts intersection locations and
  energetics quite well
• Insensitive to functionals and basis sets (as
  found for excitation energies)
• Ground state often found as deexcitation
• Should solve for ?, not ?2 2N x 2N vs N x N
  eigenvalue problem
• Unrestricted KS does not work well spin
  contamination problems
• Pseudospectral approximation promising in
  reducing computational cost
• Remaining difficulties
• Double excitations not present (butadiene,
  ethylene)
• Convergence difficulties in region of
  intersections
• Need to implement nonadiabatic coupling
• Dimensionality of intersections is incorrect
• Functionals accounting for Berry phase?
• Multi-reference DFT?

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Conical Intersection in Ethylene

• Sudden Polarization (Salem)
• Pyramidalized Minimum, but not CI

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Benchmark Potential Energy Surface for Ethylene

• CAS(2/6)SDCI
• aug-cc-pvdz basis set
• Same features as PES used in
• AIMS calculations
• Pyramidalized minimum on S1
• Conical Intersection near S1
• minimum
• V-State Vertical Excitation 7.8eV
• Experimental ?max 7.66eV

3s Rydberg State