Multilayer Formulation of the Multi-Configuration Time-Dependent Hartree Theory

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

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

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 = 1015configurations • MCTDH with10single particle functions per group: 10×1000×5 + 105 = 1.5×105parameters • 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 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 = 10000 degrees of freedom

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

Model Scaling of the ML-MCTDH Theory

Simulating Time Correlation Functions • Examples • Imaginary Time Propagation and Monte Carlo Sampling

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

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

hν hν Intervalence Electron Transfer • Experiment: - Back ET in ≈ 100 – 200 fs • - Coherent structure in Pump-Probe signal

hν 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

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

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

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

hν e- Example: Coumarin 343 – TiO2

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

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

|k> |d> hν |g> ET at dye-semiconductor interfaces: Coumarin 343 - TiO2 population of the donor state 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 Kondov, Thoss, Wang, J. Phys. Chem. A 110 (2006) 1364

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

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

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

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

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

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”

Multilayer Formulation of the Multi-Configuration Time-Dependent Hartree Theory