Multilayer Formulation of the Multi-Configuration

Time-Dependent Hartree Theory

Haobin WangDepartment of Chemistry and

BiochemistryNew Mexico State UniversityLas Cruces, New Mexico, USA

Collaborator: Michael ThossSupport: NSF

• 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

Outline

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

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

MCTDH Equations of Motion

• Some notations

MCTDH Equations of Motion

• Reduced density matrices and mean-field operators

The “single hole” function

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: 10005 = 1015 configurations MCTDH with 10 single particle functions per group: 10×1000×5 + 105 = 1.5×105 parameters

• Capability of the MCTDH theory: ~10×10 = 100 degrees of freedom

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

The Multilayer MCTDH Theory

Wang, Thoss, J. Chem. Phys. 119 (2003) 1289

…….

The Multilayer MCTDH Theory

Wang, Thoss, J. Chem. Phys. 119 (2003) 1289

Exploring Dynamical Simplicity Using 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 = 10000 degrees of freedom

Conventional

MCTDH

ML-MCTDH

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

• The Spin-Boson Model

The Scaling of the ML-MCTDH Theory

electronic

nuclear

coupling

• Hamiltonian

• Bath spectral density

Model Scaling of the ML-MCTDH Theory

Model Scaling of the ML-MCTDH Theory

Model Scaling of the ML-MCTDH Theory

Simulating Time Correlation Functions

• Examples

• Imaginary Time Propagation and Monte Carlo Sampling

Quantum Study of Transport Processes

Electron transfer at dye-semiconductor interfaces

Photochemical reactions

hν

e-

Electron transfer in mixed-valence compounds in solution

hν

e-

hν

cis

trans

V

Charge transport through single molecule junctions

pumpprobe

Basic Models

|g>

|d>|k>

hν

Intervalence Electron Transfer

• Experiment: - Back ET in ≈ 100 – 200 fs

- Coherent structure in Pump-Probe signal

hν

hν

Photoinduced ET in Mixed-Valence Complexes

Experiment [Barbara et al., JPC A 104 (2000)

10637]: ET bimodal decay ≈ 100 fs / 2 ps

hν

Wang, Thoss, J. Phys. Chem. A 107 (2003) 2126

Validity of Different Methods

Mean-field (Hartree)

Classical Ehrenfest

Self-consistent hybrid

Golden rule (NIBA)

Vibrational Dynamics in Intervalence ET

Thoss, Wang, Domcke, Chem. Phys. 296 (2004) 217

Charge-Transfer State Ground state

Electron-transfer at dye-semiconductor interfaces

Zimmermann, Willig, et al., J. Chem. Phys. B 105 (2001) 9345

hν

e-

Example: Coumarin 343 – TiO2

hν

e-

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

Absorption spectra

Experiment: Huber et al., Chem. Phys. 285 (2002) 39

C343 in solutionC343 adsorbed on TiO2

experiment

simulation

Experiments: electron injection 20 - 200 fs

Rehm, JCP 100 (1996) 9577 Murakoshi, Nanostr. Mat. 679 (1997) 221 Gosh, JPCB 102 (1998) 10208 Huber, Chem. Phys. 285 (2002) 39

ET at dye-semiconductor interfaces: Coumarin 343 -

TiO2

population of the donor state

|d>|k>

|g>

hν

Kondov, Thoss, Wang, J. Phys. Chem. A 110 (2006) 1364

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

vibrational dynamics

|d>|k>

|g>

hν

donor state

acceptor statesω = 1612 cm-1

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

vibrational dynamics

Vibrational motion induced by ultrafast ET

donor state

acceptor states

|d> |k>

|g>

hν

ω = 133 cm-1

ET at dye-semiconductor interfaces

ML-MCTDH

Ehrenfest

Mean-Field (Hartree)

hν

|d>|k>

|g>

Electron injection dynamics - comparison of different methods

population of the donor state

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

photoinduced electron injection dynamics

Simulation of the dynamics including the coupling to the laser field

|d>|k>

|g>

hν

laser pulse (5 fs)

donor population

acceptor population

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

photoinduced electron injection dynamics

Simulation of the dynamics including the coupling to the laser field

|d>|k>

|g>

hν

laser pulse (20 fs)

donor population

acceptor population

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

photoinduced electron injection dynamics

Simulation of the dynamics including the coupling to the laser field

|d>|k>

|g>

hν

laser pulse

donor population

acceptor population

(40 fs)

Experiment: electron injection 6 fsHuber, Moser, Grätzel, Wachtveitl, J. Phys. Chem. B 106 (2002) 6494

ET at dye-semiconductor interfaces: Alizarin - TiO2

population of the donor state

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”