TDDFTRealtime

Slide1 : Time-Dependent Density Functional Theory in Real Time Benjamin Levine, Chaehyuk Ko, Richard Martin, and Todd J. Martínez


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! Ab initio Quantum Dynamics


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: Semiclassical phase Cartesian degrees of freedom Position, momentum, width Classical evolution for R(t) and P(t). Variational principle for coefficients: Nuclear overlap matrix Hamiltonian matrix The Full Multiple Spawning (FMS) Method


Adaptive Basis Set : Prescription so far is insufficient… New Basis Functions must be added to augment classical mechanics R(t) time “Spawned” Basis Functions Nonadiabatic Coupling Regions Adaptive Basis Set


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 Multiple Spawning


Ab initio Multiple Spawning - Obstacles : 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)


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 Absorption and Resonance Raman Spectra


Absorption Spectra from TDDFT Dynamics : Absorption Spectra from TDDFT Dynamics AIMS-EOM-CCSD TDDFT is nearly as good as EOM-CCSD in Franck-Condon region…


Butadiene Bond Alternation – CASPT2 vs TDDFT : Butadiene Bond Alternation – CASPT2 vs TDDFT lines are caspt2, symbols are b3lyp 41Ag 31Ag 21Ag 21Ag 11Bu 21Ag and 41Ag have significant double excitation character in CAS – not represented in TDDFT


TDDFT Outside Franck-Condon Region : TDDFT Outside Franck-Condon Region Photoactive Yellow Protein Minimal energy path from CASSCF S1 is singly-excited; S2 is a double excitation Linear-response TDDFT does not describe doubly-excited states, in contrast to earlier reports


S1 PES for Ethylene : S1 PES for Ethylene 4.5 5.0 5.5 6.0 5.5 6.0 CASPT2 TDDFT CIS CIS and TDDFT similar and incorrect…


TDDFT and Conical Intersections : 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)


MECIs Determined from TDDFT : MECIs Determined from TDDFT


MECI Geometries from TDDFT : MECI Geometries from TDDFT TD-B3LYP CAS(4/4) CAS(4/4)*MSPT2


PES Behavior Near MECIs : PES Behavior Near MECIs CAS TD-B3LYP


Intersection Dimensionality in TDDFT : Intersection Dimensionality in TDDFT H2 O H1 y x TDDFT CAS TDDFT Intersections are N-1D!


Pseudospectral CIS/TDDFT : 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


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? Conclusions


Conical Intersection in Ethylene : Conical Intersection in Ethylene “Sudden Polarization” (Salem) Pyramidalized Minimum, but not CI


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 Benchmark Potential Energy Surface for Ethylene