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 = 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 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 The Spin-Boson Model Hamiltonian electronic nuclear coupling 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 cis trans hν e- Charge transport through single molecule junctions Electron transfer in mixed-valence compounds in solution hν e- V

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

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

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

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

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

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

Example: Coumarin 343 – TiO2 hν e-

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2

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

ET at dye-semiconductor interfaces: Coumarin 343 - TiO2 |k> |g> hν 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

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

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

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)

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)

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)

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)

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”