1. Multilayer Formulation of the Multi-Configuration Time-Dependent Hartree Theory
Haobin Wang
Department of Chemistry and Biochemistry
New Mexico State University
Las Cruces, New Mexico, USA
Collaborator: Michael Thoss
Support: NSF

2. Outline Conventional brute-force approach to wave packet propagation
Multi-configuration time-dependent Hartree (MCTDH) method
Multilayer formulation of MCTDH (ML-MCTDH)
Quantum simulation of time correlation functions
Application to ultrafast electron transfer reactions

3. Conventional Wave Packet Propagation
Dirac-Frenkel variational principle
Conventional Full CI Expansion (orthonormal basis)
Equations of Motion
Capability: <10 degrees of freedom (<~n10 configurations)
even for separable limit

4. Multi-Configuration Time-Dependent Hartree
Multi-configuration expansion of the wave function
Variations
Both expansion coefficients and configurations are time-dependent
Meyer, Manthe, Cederbaum, Chem. Phys. Lett. 165 (1990) 73

5. MCTDH Equations of Motion
Some notations

6. MCTDH Equations of Motion
Reduced density matrices and mean-field operators
The “single hole” function

7. Implementation of the MCTDH
Full CI expansion of the single particle functions (mode grouping and adiabatic basis contraction)
Only a few single particle functions are selected among the full CI space
Example: 5 single particle groups, each has 1000 basis functions
Conventional approach: = 1015 configurations
MCTDH with 10 single particle functions per group:
10×1000× = 1.5×105 parameters
Capability of the MCTDH theory: ~10×10 = 100 degrees of freedom

8. Multi-Layer Formulation of the MCTDH Theory
Multi-configurational expansion of the SP functions
More complex way of expressing the wave function
Two-layer MCTDH
Wang, Thoss, J. Chem. Phys. 119 (2003) 1289

9. The Multilayer MCTDH Theory
…….
Wang, Thoss, J. Chem. Phys. 119 (2003) 1289

10. The Multilayer MCTDH Theory
Wang, Thoss, J. Chem. Phys. 119 (2003) 1289

11. Exploring Dynamical Simplicity Using ML-MCTDH
Conventional
MCTDH
ML-MCTDH
Capability of the two-layer ML-MCTDH:
~10×10×10 = 1000 degrees of freedom
Capability of the three-layer ML-MCTDH:
~10×10×10×10 = degrees of freedom

12. The Scaling of the ML-MCTDH Theory
f: the number of degrees of freedom
L: the number of layers
N: the number of (contracted) basis functions
n: the number of single-particle functions

13. The Scaling of the ML-MCTDH Theory
The Spin-Boson Model
Hamiltonian
electronic
nuclear
coupling
Bath spectral density

14. Model Scaling of the ML-MCTDH Theory

15. Model Scaling of the ML-MCTDH Theory

16. Model Scaling of the ML-MCTDH Theory

17. Simulating Time Correlation Functions
Examples
Imaginary Time Propagation and Monte Carlo Sampling

18. Quantum Study of Transport Processes
Electron transfer at dye-semiconductor interfaces
Photochemical reactions hν
cis
trans hν e-
Charge transport through single molecule junctions
Electron transfer in mixed-valence compounds in solution hν e- V

19. Basic Models
pump
probe
|g>
|d>
|k> hν

20. Intervalence Electron Transferhν hν
Experiment: - Back ET in ≈ 100 – 200 fs
- Coherent structure in Pump-Probe signal

21. Photoinduced ET in Mixed-Valence Complexes
Experiment [Barbara et al., JPC A 104 (2000)
10637]: ET bimodal decay ≈ 100 fs / 2 ps
Wang, Thoss, J. Phys. Chem. A 107 (2003) 2126

22. Validity of Different Methods
Mean-field (Hartree)
Classical Ehrenfest
Self-consistent hybrid
Golden rule (NIBA)

23. Vibrational Dynamics in Intervalence ET
Ground state
Charge-Transfer State
Thoss, Wang, Domcke, Chem. Phys. 296 (2004) 217

24. Electron-transfer at dye-semiconductor interfaceshν e-
Zimmermann, Willig, et al., J. Chem. Phys. B 105 (2001) 9345

25. Example: Coumarin 343 – TiO2hν e-

26. ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

27. ET at dye-semiconductor interfaces: Coumarin 343 - TiO2
Absorption spectra
C343 in solution
C343 adsorbed on TiO2
experiment
simulation
Experiment: Huber et al., Chem. Phys. 285 (2002) 39

28. ET at dye-semiconductor interfaces: Coumarin 343 - TiO2
|k>
|g> hν
population of the donor state
Experiments: electron injection fs
Rehm, JCP 100 (1996) Murakoshi, Nanostr. Mat. 679 (1997) 221 Gosh, JPCB 102 (1998) Huber, Chem. Phys. 285 (2002) 39
Kondov, Thoss, Wang, J. Phys. Chem. A 110 (2006) 1364

29. ET at dye-semiconductor interfaces: Coumarin 343 - TiO2
|k>
|g> hν
vibrational dynamics
donor state
acceptor states
ω = 1612 cm-1

30. ET at dye-semiconductor interfaces: Coumarin 343 - TiO2
|k>
|g> hν
vibrational dynamics
donor state
acceptor states
ω = 133 cm-1
Vibrational motion induced by ultrafast ET

31. ET at dye-semiconductor interfaces
Electron injection dynamics - comparison of different methods hν
|d>
|k>
|g>
population of the donor state
ML-MCTDH
Ehrenfest
Mean-Field (Hartree)

32. ET at dye-semiconductor interfaces: Coumarin 343 - TiO2
Simulation of the dynamics including the coupling to the laser field
photoinduced electron injection dynamics
|d>
|k>
|g> hν
acceptor population
donor population
laser pulse (5 fs)

33. ET at dye-semiconductor interfaces: Coumarin 343 - TiO2
Simulation of the dynamics including the coupling to the laser field
photoinduced electron injection dynamics
|d>
|k>
|g> hν
acceptor population
donor population
laser pulse (20 fs)

34. ET at dye-semiconductor interfaces: Coumarin 343 - TiO2
Simulation of the dynamics including the coupling to the laser field
photoinduced electron injection dynamics
|d>
|k>
|g> hν
acceptor population
donor population
laser pulse
(40 fs)

35. ET at dye-semiconductor interfaces: Alizarin - TiO2
population of the donor state
Experiment: electron injection 6 fs
Huber, Moser, Grätzel, Wachtveitl, J. Phys. Chem. B 106 (2002) 6494

36. Summary of the ML-MCTDH Theory
Powerful tool to propagate wave packet in complex systems
Can reveal various dynamical information
population dynamics and rate constant
reduced wave packet motions
time-resolved nonlinear spectroscopy
dynamic/static properties: real and imaginary time
Current status
Has been implemented for certain potential energy functions: two-body, three-body, etc.
The (time-dependent) correlation DVR of Manthe
Challenges
Implementation: somewhat difficult
Long time dynamics: “chaos”