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Xin Chen 1, Yinghua Wu 2 and Victor S. Batista Department of Chemistry, Yale University, New Haven, CT The MP/SOFT methodology for simulations of multidimensional quantum dynamics Funding: NSF CHE NSF ECCS NIH 2R01-GM DOE DE-FG02-07ER15909 US-Israel BSF Current Addresses: 1 Deptartment of Chemistry, Massachusetts Institute of Technology; 2 Department of Chemistry, Georgia Institute Technology. Funding: NSF, NIH, DOE, BSF

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Multiconfigurational time dependent Hartree (MCTDH) method Meyer, Burghardt, Cederbaum, Worth, Raab, Manthe, Makri, Miller, Thoss, Wang Time dependent Gauss-Hermit (TDGH) method Billing Multiple Spawning (MS) method Ben-Nun and Martinez Coupled Coherent States (CCS) technique Shalashilin and Child Matching-Pursuit Split-Operator Fourier Transform (MP/SOFT) method Wu, Chen, Batista Numerically Exact Methods for Multidimensional Wave-Packet Propagation Based on Expansions of Coherent-States/Configurations

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Time-Sliced Simulations of Quantum Processes

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Wu,Y.; Batista, V.S. J. Chem. Phys. (2003) 118, 6720 Wu,Y.; Batista, V.S. J. Chem. Phys. (2003) 119, 7606 Wu,Y.; Batista, V.S. J. Chem. Phys. (2004) 121, 1676 Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. (2005) 122, Wu,Y.; Herman, M.F.; Batista, V.S. J. Chem. Phys. (2005) 122, Wu,Y.; Batista, V.S. J. Chem. Phys. (2006) 124, Chen, X.; Batista, V.S. J. Chem. Phys. (2006) 125, Chen, X.; Batista, V.S. J. Photochem. Photobiol. (2007) 190, 274 Kim, J.; Wu, Y.; Batista, V.S. Israel J. Chem. (2009) 49, 187 MP/SOFT Method Trotter Expansion (Strang Splitting)

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Wu,Y.; Batista, V.S. J. Chem. Phys. 121, 1676 (2004)

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MP/SOFT Decomposition Dr. Rajdeep Saha

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Contraction Mapping

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Computation of Observables Time dependent reactant population: Absorption Spectrum:

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Electron Tunneling in Multidimensional Systems Model II Wu,Y.; Batista, V.S. J. Chem. Phys. 121, 1676 (2004)

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(2-dimensional Model II)

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2-dimensional (model II)

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2-dimensional (Model II)

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20-dimensional (model II) Benchmark calculation: Wu,Y.; Batista, V.S. J. Chem. Phys. 121, 1676 (2004)

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20-dimensional (Model II) Wu,Y.; Batista, V.S. J. Chem. Phys. 121, 1676 (2004)

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Thermal Correlation Functions Time-Dependent Boltzmann Ensemble Averages Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, (2005)

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Bloch Equation: MP/SOFT Integration Partition Function Boltzmann Matrix: Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, (2005)

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Initial classical density Initial quantum density Ground State, V 0 Excited State, V 1 Model System, cont’d

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Calculations of Thermal Correlation Functions Time-Dependent Position Ensemble Average Position-Position Correlation Function: Model System: Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, (2005)

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Time-Dependent Position Ensemble Average Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, (2005)

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Position-Position Correlation Function Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, (2005)

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Nonadiabatic Propagation Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. (2005) 122, Chen, X.; Batista, V.S. J. Chem. Phys. (2006) 125,

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MP/SOFT Nonadiabatic Propagation

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Embedded Strang Splitting Expansion Chen, X.; Batista, V.S. J. Chem. Phys. (2010) in prep.

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Wu,Y.; Herman, M.F.; Batista, V.S. J. Chem. Phys. 122, (2005) Single Avoided Crossing Time-Dependent Populations

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Single Avoided Crossing Transmission Probabilities Wu,Y.; Herman, M.F.; Batista, V.S. J. Chem. Phys. 122, (2005) P T (1) P T (2)

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Dual Avoided Crossings with quantum interferences between the crossings Stuckelberg oscillations in Transmission Probabilities Wu,Y.; Herman, M.F.; Batista, V.S. J. Chem. Phys. 122, (2005) P T (1) P T (2)

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Chen,X.; Batista, V.S. J. Chem. Phys. (2006) in prep. Nonadiabatic Dynamics of Pyrazine S 1 /S 2 Conical Intersection Chen,X.; Batista, V.S. J. Chem. Phys. (2006) 125, Benchmark Calcs.: 4-mode model

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Nonadiabatic Dynamics of Pyrazine S 1 /S 2 Conical Intersection Chen,X.; Batista, V.S. J. Chem. Phys. (2006) 125, Benchmark Calcs.: 24-mode model

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Benchmark Calcs.: 24-mode model

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Transient appearance of the C=O stretching mode of the keto*-state of HBT after excitation of the enol → enol* transition. Ultrafast Excited State Intramolecular Proton Transfer in HBT E. T. J. Nibbering, H. Fidder and E. Pines Annu. Rev. Phys. Chem. (2005) 56: nm IVR~ 750 fs, 15 ps ~50 fs Kim, J.; Wu, Y.; Batista, V.S. Israel J. Chem. (2009) 49, 187

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Energy (kcal/mol) OH bond length (a.u.) S0S0 S1S1 CCC bend. angle (degrees) Reaction Surface V 0 (r 1,r 2 ) KETO CCC bend. angle (degrees) ENOL

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t=0 fs t=50 fs t=100 fst=200 fst=300 fst=400 fst=500 fst=600 fst=700 fs t=800 fst=900 fs ESIPT or ESIHT ? Ultrafast Charge Redistribution in HBT Kim, J.; Wu, Y.; Batista, V.S. Israel J. Chem. (2009) 49, 187

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P R (t), (TDSCF) P R (t), (MP/SOFT) Tr [ρ 2 (t)], (MP/SOFT) ESIPT in HBT: 69-Dimensional MP/SOFT Wavepacket Propagation69-Dimensional MP/SOFT Wavepacket Propagation t = 50 fs

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UV-Vis Photoabsorption Spectrum of HBT MP/SOFT spectrum Experiments: Rini M,KummrowA, Dreyer J, Nibbering ETJ, Elsaesser T Faraday Discuss. 122:27–40 Rini M, Dreyer J, Nibbering ETJ, Elsaesser T Chem. Phys. Lett. 374:13–19

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t = 0 fs t = 500 fs Ultrafast Excited State Intramolecular Proton Transfer in HBT Wavenumber [cm -1 ] IR - S 1 (Calc.) Wavenumber [cm -1 ] IR - S 0 (Calc.)

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IR Intensity Wavenumber [cm -1 ] IR - S 0 (Exp.) IR - S 0 (Calc.) IR - S 1 (Calc.) Infrared spectra of HBT

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Wavenumber [cm -1 ] Theoretical Experimental Transient infrared spectra of HBT Experiments: Rini M,KummrowA, Dreyer J, Nibbering ETJ, Elsaesser T Faraday Discuss. 122:27–40 Rini M, Dreyer J, Nibbering ETJ, Elsaesser T Chem. Phys. Lett. 374:13–19

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Conclusions We have introduced the MP/SOFT method for time-sliced simulations of quantum processes in systems with many degrees of freedom. The MP/SOFT method generalizes the grid-based SOFT approach to non-orthogonal and dynamically adaptive coherent-state representations generated according to the matching-pursuit algorithm. The accuracy and efficiency of the resulting method were demonstrated in simulations of excited-state intramolecular proton transfer in 2-(2’- hydroxyphenyl)-oxazole (HPO), as modeled by an ab initio 35-dimensional reaction surface Hamiltonian, as well as in benchmark simulations of nonadiabatic quantum dynamics in pyrazine. Further, we have extended the MP/SOFT method for computations of thermal equilibrium density matrices (equilibrium properties of quantum systems), finite temperature time-dependent expectation values and time- correlation functions. The extension involves solving the Bloch equation via imaginary-time propagation of the density matrix in dynamically adaptive coherent-state representations, and the evaluation of the Heisenberg time- evolution operators through real-time propagation.

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