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The MP/SOFT methodology for simulations of multidimensional quantum dynamics. Xin Chen 1 , Yinghua Wu 2 and Victor S. Batista. Department of Chemistry, Yale University, New Haven, CT 06520-8107. Current Addresses: 1 Deptartment of Chemistry, Massachusetts Institute of Technology; - PowerPoint PPT Presentation
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Xin Chen1, Yinghua Wu2 and Victor S. BatistaDepartment of Chemistry, Yale University, New Haven,
CT 06520-8107
The MP/SOFT methodology for simulations of multidimensional quantum dynamics
Funding: • NSF CHE-0345984• NSF ECCS-0725118• NIH 2R01-GM043278-14• DOE DE-FG02-07ER15909• US-Israel BSF
tt Ht
i ˆ 0
/ˆ tHit e
where
Current Addresses: 1Deptartment of Chemistry, Massachusetts Institute of Technology;2Department of Chemistry, Georgia Institute Technology.
)ˆ(2
ˆˆ2
xVm
pH
Funding: NSF, NIH, DOE, BSF
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
Time-Sliced Simulations of Quantum Processes
Wu,Y.; Batista, V.S. J. Chem. Phys. (2003) 118, 6720Wu,Y.; Batista, V.S. J. Chem. Phys. (2003) 119, 7606Wu,Y.; Batista, V.S. J. Chem. Phys. (2004) 121, 1676Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. (2005) 122, 64102Wu,Y.; Herman, M.F.; Batista, V.S. J. Chem. Phys. (2005) 122, 114114Wu,Y.; Batista, V.S. J. Chem. Phys. (2006) 124, 224305Chen, X.; Batista, V.S. J. Chem. Phys. (2006) 125, 124313Chen, X.; Batista, V.S. J. Photochem. Photobiol. (2007) 190, 274Kim, J.; Wu, Y.; Batista, V.S. Israel J. Chem. (2009) 49, 187
MP/SOFT Method
Trotter Expansion (Strang Splitting)
0/ˆ tHi
t e
Wu,Y.; Batista, V.S. J. Chem. Phys. 121, 1676 (2004)
MP/SOFT Decomposition
Dr. Rajdeep Saha
Contraction Mapping
Min ε1⎡⎣ ⎤⎦⇔ ∂ ε1 / ∂ξ j =0,ξ j = xj , pj ,γ j }{
x j =Regj x$ψ
gj ψ
⎡
⎣⎢⎢
⎤
⎦⎥⎥
p j =Regj pµ ψ
gj ψ
⎡
⎣⎢⎢
⎤
⎦⎥⎥
1
2γ j
=Regj x$−xj( )
2ψ
gj ψ
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
Computation of Observables
Time dependent reactant population:
Absorption Spectrum:
Electron Tunneling in Multidimensional Systems
Model II
Wu,Y.; Batista, V.S. J. Chem. Phys. 121, 1676 (2004)
(2-dimensional Model II)
2-dimensional (model II)
2-dimensional (model II)
2-dimensional (Model II)
20-dimensional (model II)
Benchmark calculation:
Wu,Y.; Batista, V.S. J. Chem. Phys. 121, 1676 (2004)
20-dimensional (Model II)
Wu,Y.; Batista, V.S. J. Chem. Phys. 121, 1676 (2004)
Thermal Correlation Functions
Time-Dependent Boltzmann Ensemble Averages
Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, 64102 (2005)
Bloch Equation: MP/SOFT Integration
Partition Function Boltzmann Matrix:
Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, 64102 (2005)
Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, 64102 (2005)
Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, 64102 (2005) Initial classical density
Initial quantum density
Ground State, V0
Excited State, V1
Model System, cont’d
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, 64102 (2005)
Time-Dependent Position Ensemble Average
Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, 64102 (2005)
Position-Position Correlation Function
Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. 122, 64102 (2005)
Nonadiabatic Propagation
Chen, X., Wu,Y.; Batista, V.S. J. Chem. Phys. (2005) 122, 64102
Chen, X.; Batista, V.S. J. Chem. Phys. (2006) 125, 124313
MP/SOFT Nonadiabatic Propagation
Embedded Strang Splitting Expansion
Chen, X.; Batista, V.S. J. Chem. Phys. (2010) in prep.
Wu,Y.; Herman, M.F.; Batista, V.S. J. Chem. Phys. 122, 114114 (2005)
Single Avoided CrossingTime-Dependent Populations
Single Avoided CrossingTransmission Probabilities
Wu,Y.; Herman, M.F.; Batista, V.S. J. Chem. Phys. 122, 114114 (2005)
PT(1
)
PT(2
)
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, 114114 (2005)
PT(1
)
PT(2
)
Chen,X.; Batista, V.S. J. Chem. Phys. (2006) in prep.Nonadiabatic Dynamics of Pyrazine
S1/S2 Conical Intersection
Chen,X.; Batista, V.S. J. Chem. Phys. (2006) 125, 124313
Benchmark Calcs.: 4-mode model
Nonadiabatic Dynamics of PyrazineS1/S2 Conical Intersection
Chen,X.; Batista, V.S. J. Chem. Phys. (2006) 125, 124313
Benchmark Calcs.: 24-mode model
Benchmark Calcs.: 24-mode model
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) 56:337-67
310-350 nm
IVR~ 750 fs, 15 ps ~50 fs
Kim, J.; Wu, Y.; Batista, V.S. Israel J. Chem. (2009) 49, 187
Energy (kcal/mol)
Energy (kcal/mol)
OH bond length (a.u.)
S0
S1
OH bond length (a.u.)
CCC bend. angle
(degrees)
Reaction Surface V0(r1,r2)
KETO
CCC bend. angle
(degrees)
ENOL
t=0 fst=50 fst=100 fst=200 fst=300 fst=400 fst=500 fst=600 fst=700 fst=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
PR(t), (TDSCF)PR(t), (MP/SOFT)
Tr [ρ2(t)], (MP/SOFT)
ESIPT in HBT: 69-Dimensional MP/SOFT Wavepacket Propagation
t = 50 fs
UV-Vis Photoabsorption Spectrum of HBT
MP/SOFT spectrum
Experiments:Rini M,KummrowA, Dreyer J, Nibbering ETJ, Elsaesser T. 2003. Faraday Discuss. 122:27–40Rini M, Dreyer J, Nibbering ETJ, Elsaesser T. 2003. Chem. Phys. Lett. 374:13–19
t = 0 fs
t = 500 fs
Ultrafast Excited State Intramolecular Proton Transfer in HBT
Wavenumber [cm-1]1700 1600 1500 1400 1300 1200 1100 1000
IR - S1 (Calc.)
Wavenumber [cm-1]1700 1600 1500 1400 1300 1200 1100 1000
IR - S0 (Calc.)
IR I
nten
sity
IR I
nten
sity
IR I
nten
sity
Wavenumber [cm-1]
Wavenumber [cm-1]
Wavenumber [cm-1]
1700 1600 1500 1400 1300 1200 1100 1000
1700 1600 1500 1400 1300 1200 1100 1000
1700 1600 1500 1400 1300 1200 1100 1000
IR - S0 (Exp.)
IR - S0 (Calc.)
IR - S1 (Calc.)
Infrared spectra of HBT
Wavenumber [cm-1]
Wavenumber [cm-1]
TheoreticalExperimental
Transient infrared spectra of HBT
Experiments:Rini M,KummrowA, Dreyer J, Nibbering ETJ, Elsaesser T. 2003. Faraday Discuss. 122:27–40Rini M, Dreyer J, Nibbering ETJ, Elsaesser T. 2003. Chem. Phys. Lett. 374:13–19
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.
