^{1,a)}, Clara D. Christ

^{2,b)}and Irene Burghardt

^{3,a)}

### Abstract

The non-Markovian approach developed in the companion paper [Hughes *et al.*, J. Chem. Phys.131, 024109 (2009)], which employs a hierarchical series of approximate spectral densities, is extended to the treatment of nonadiabatic dynamics of coupled electronic states. We focus on a spin-boson-type Hamiltonian including a subset of system vibrational modes which are treated without any approximation, while a set of bath modes is transformed to a chain of effective modes and treated in a reduced-dimensional space. Only the first member of the chain is coupled to the electronic subsystem. The chain construction can be truncated at successive orders and is terminated by a Markovian closure acting on the end of the chain. From this Mori-type construction, a hierarchy of approximate spectral densities is obtained which approach the true bath spectral density with increasing accuracy. Applications are presented for the dynamics of a vibronic subsystem comprising a high-frequency mode and interacting with a low-frequency bath. The bath is shown to have a striking effect on the nonadiabatic dynamics, which can be rationalized in the effective-mode picture. A reduced two-dimensional subspace is constructed which accounts for the essential features of the nonadiabatic process induced by the effective environmental mode. Electronic coherence is found to be preserved on the shortest time scale determined by the effective mode, while decoherence sets in on a longer time scale. Numerical simulations are carried out using either an explicit wave function representation of the system and overall bath or else an explicit representation of the system and effective-mode part in conjunction with a Caldeira–Leggett master equation.

We thank Hiroyuki Tamura and Eric Bittner for useful discussions. This work was supported by the Agence Nationale de la Recherche (France) Project No. ANR-NT05-3-42315.

I. INTRODUCTION

II. THEORY

A. Generalized spin-boson Hamiltonian

B. System-bath Hamiltonian and effective-mode representation

C. Reduced-dimensional effective-mode chain including Markovian closure

D. Spectral densities

E. Electronic decoherence in the effective-mode picture

III. COMPUTATIONAL DETAILS

A. Density matrix calculations

B. Wavepacket calculations using MCTDH

IV. RESULTS AND DISCUSSION

A. System dynamics

B. System-environment dynamics: Lorentzian spectral density

1. System plus effective-mode dynamics

2. Influence of the residual bath: Friction and decoherence

C. Higher order hierarchy truncation

V. CONCLUSIONS

### Key Topics

- Coherence
- 25.0
- Non adiabatic reactions
- 18.0
- Friction
- 10.0
- Oscillators
- 9.0
- Population dynamics
- 9.0

## Figures

The diabatic potentials as defined in Eqs. (2) and (3) and parameters defined in Table I (state 1 depicted in black; state 2 in red). Also depicted are the initial coherent state (blue) on state 1 and snapshots of the densities for each electronic state at (state 1: full green line; state 2: dashed green line).

The diabatic potentials as defined in Eqs. (2) and (3) and parameters defined in Table I (state 1 depicted in black; state 2 in red). Also depicted are the initial coherent state (blue) on state 1 and snapshots of the densities for each electronic state at (state 1: full green line; state 2: dashed green line).

Pure system dynamics : (a) Lower state (state 2) population dynamics evaluated using the initial coherent state defined in Sec. IV. Also shown in red is the Rabi oscillation between the vibrational ground states of the two diabatic potentials; (b) electronic coherence —real and imaginary parts and absolute value; (c) position expectation values for the two states.

Pure system dynamics : (a) Lower state (state 2) population dynamics evaluated using the initial coherent state defined in Sec. IV. Also shown in red is the Rabi oscillation between the vibrational ground states of the two diabatic potentials; (b) electronic coherence —real and imaginary parts and absolute value; (c) position expectation values for the two states.

(a) Population dynamics for the upper state (state 1): The black curve depicts the populations computed using the system Hamiltonian only, the red curve depicts the populations computed using system and effective-mode Hamiltonian , the green curve depicts the populations computed from the Caldeira–Leggett equation as described in Eq. (16), and the blue curve depicts the populations computed from Eq. (1)–(6) by a 31 mode wavepacket calculation, where are depicted in part (b) below. For the dissipative examples, a value of was used for the friction coefficient. (b) Coupling coefficients defined in Eq. (6) that correspond to the spectral density of Eq. (31). The frequency spacing in (b) is .

(a) Population dynamics for the upper state (state 1): The black curve depicts the populations computed using the system Hamiltonian only, the red curve depicts the populations computed using system and effective-mode Hamiltonian , the green curve depicts the populations computed from the Caldeira–Leggett equation as described in Eq. (16), and the blue curve depicts the populations computed from Eq. (1)–(6) by a 31 mode wavepacket calculation, where are depicted in part (b) below. For the dissipative examples, a value of was used for the friction coefficient. (b) Coupling coefficients defined in Eq. (6) that correspond to the spectral density of Eq. (31). The frequency spacing in (b) is .

Trajectories in the 2D system vs effective-mode plane for (a) weak , (b) medium , and (c) strong coupling . The black curves correspond to the upper state (state 1) trajectories, the red curves correspond to the lower state trajectories, and the green curve indicates the crossing seam. The blue crosses indicate the initial condition , .

Trajectories in the 2D system vs effective-mode plane for (a) weak , (b) medium , and (c) strong coupling . The black curves correspond to the upper state (state 1) trajectories, the red curves correspond to the lower state trajectories, and the green curve indicates the crossing seam. The blue crosses indicate the initial condition , .

System-plus-effective-mode dynamics : Densities for the lower state (blue) and upper state (red) captured at (upper figure), (middle), and (lower figure). The extension of the grid is from in and from in .

System-plus-effective-mode dynamics : Densities for the lower state (blue) and upper state (red) captured at (upper figure), (middle), and (lower figure). The extension of the grid is from in and from in .

System-plus-effective-mode dynamics : (a) Upper state (state 1) population dynamics (blue) and electronic coherence—real and imaginary parts and absolute value—and (b) position expectation values for the two states.

System-plus-effective-mode dynamics : (a) Upper state (state 1) population dynamics (blue) and electronic coherence—real and imaginary parts and absolute value—and (b) position expectation values for the two states.

(a) Population and (b) coherence dynamics for the example of Sec. ???, computed using different values of .

(a) Population and (b) coherence dynamics for the example of Sec. ???, computed using different values of .

Time-dependent position expectation values for the two states computed for the nondissipative and, for different values, the dissipative case. The solid curves correspond to trajectories for the upper electronic state and the dashed curves correspond to trajectories for the lower electronic states. Due to the small amount of population transfer to the lower state for , the trajectory is not depicted for this state.

Time-dependent position expectation values for the two states computed for the nondissipative and, for different values, the dissipative case. The solid curves correspond to trajectories for the upper electronic state and the dashed curves correspond to trajectories for the lower electronic states. Due to the small amount of population transfer to the lower state for , the trajectory is not depicted for this state.

(a) Population, (b) coherence dynamics, and (c) position expectation values for the two states (dashed state), evaluated at different temperatures for the example of Sec. ???. The results were computed using a fixed value of for the Caldeira–Leggett equation as described in Eq. (16).

(a) Population, (b) coherence dynamics, and (c) position expectation values for the two states (dashed state), evaluated at different temperatures for the example of Sec. ???. The results were computed using a fixed value of for the Caldeira–Leggett equation as described in Eq. (16).

Population (a) and coherence (b) dynamics for the example of Sec. IV C using spectral densities approximated at the first, second, and third order levels of the effective-mode hierarchy , . For the bath a fixed value of was used. Part (c) depicts the couplings that are related to through Eq. (31).

Population (a) and coherence (b) dynamics for the example of Sec. IV C using spectral densities approximated at the first, second, and third order levels of the effective-mode hierarchy , . For the bath a fixed value of was used. Part (c) depicts the couplings that are related to through Eq. (31).

## Tables

System and effective-mode(s) parameters quoted in eVs.

System and effective-mode(s) parameters quoted in eVs.

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