^{1}and Graham R. Fleming

^{1,a)}

### Abstract

A new quantum dynamic equation for excitation energy transfer is developed which can describe quantum coherent wavelike motion and incoherent hopping in a unified manner. The developed equation reduces to the conventional Redfield theory and Förster theory in their respective limits of validity. In the regime of coherent wavelike motion, the equation predicts several times longer lifetime of electronic coherence between chromophores than does the conventional Redfield equation. Furthermore, we show quantum coherent motion can be observed even when reorganization energy is large in comparison to intersite electronic coupling (the Förster incoherent regime). In the region of small reorganization energy, slow fluctuation sustains longer-lived coherent oscillation, whereas the Markov approximation in the Redfield framework causes infinitely fast fluctuation and then collapses the quantum coherence. In the region of large reorganization energy, sluggish dissipation of reorganization energy increases the time electronic excitation stays above an energy barrier separating chromophores and thus prolongs delocalization over the chromophores.

We thank Dr. Yuan-Chung Cheng for valuable comments. This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231 and by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, U.S. Department of Energy under Contract No. DE-AC03-76SF000098. A.I. appreciates the support of the JSPS Postdoctoral Fellowship for Research Abroad.

I. INTRODUCTION

II. FORMULATION

III. DISCUSSION: NUMERICAL RESULTS

A. Weak electronic coupling

B. Strong electronic coupling

IV. CONCLUDING REMARKS

### Key Topics

- Phonons
- 43.0
- Excitation energies
- 13.0
- Integrodifferential equations
- 13.0
- Excited states
- 12.0
- Fluorescence spectra
- 12.0

## Figures

Time evolution of phonon modes in the present formalism. In panel (a), the lower parabola is the electronic ground state of site 1 while the upper one is the electronic excited state. The gray packets illustrate the phonon states. The wavy arrow stands for the reorganization process. In panel (b), the red line presents the time evolution of the population of site 1. The blue lines show the time evolution of the matrix elements of the auxiliary operators, ; from top to bottom, the values of are 1, 2, 3, 4, and 5. Panel (c) gives the emission spectrum calculated by the present theory, Eq. (2.23), as a function of a delay time after photoexcitation. In panels (b) and (c), the parameters are set to be , , and . The normalization of the spectra is such that the maximum value is unity.

Time evolution of phonon modes in the present formalism. In panel (a), the lower parabola is the electronic ground state of site 1 while the upper one is the electronic excited state. The gray packets illustrate the phonon states. The wavy arrow stands for the reorganization process. In panel (b), the red line presents the time evolution of the population of site 1. The blue lines show the time evolution of the matrix elements of the auxiliary operators, ; from top to bottom, the values of are 1, 2, 3, 4, and 5. Panel (c) gives the emission spectrum calculated by the present theory, Eq. (2.23), as a function of a delay time after photoexcitation. In panels (b) and (c), the parameters are set to be , , and . The normalization of the spectra is such that the maximum value is unity.

Intersite energy transfer rates from to , , as a function of reorganization energy, , predicted by the present theory, Eq. (2.23) (closed circles), the full-Redfield equation (open circles), and Förster theory (solid line). The other parameters are , , , and . For theses parameters, the intersite dynamics is dominantly incoherent for the entire region depicted.

Intersite energy transfer rates from to , , as a function of reorganization energy, , predicted by the present theory, Eq. (2.23) (closed circles), the full-Redfield equation (open circles), and Förster theory (solid line). The other parameters are , , , and . For theses parameters, the intersite dynamics is dominantly incoherent for the entire region depicted.

Emission spectrum from site 1 (a) and site 2 (b) calculated by the present theory, Eq. (2.23), as a function of a delay time after the photoexcitation of site 1. For the calculations, the parameters are chosen to be , , , , and . The normalization of the spectra is such that the maximum value of panel (a) is unity. Twenty equally spaced contour levels from 0.05 to 1 are drawn.

Emission spectrum from site 1 (a) and site 2 (b) calculated by the present theory, Eq. (2.23), as a function of a delay time after the photoexcitation of site 1. For the calculations, the parameters are chosen to be , , , , and . The normalization of the spectra is such that the maximum value of panel (a) is unity. Twenty equally spaced contour levels from 0.05 to 1 are drawn.

Time evolution of the population of site 1 calculated by the present theory, Eq. (2.23) (solid line) and the full-Redfield equation (dashed line) for various magnitudes of the reorganization energy . The other parameters are fixed to be , , , and .

Time evolution of the population of site 1 calculated by the present theory, Eq. (2.23) (solid line) and the full-Redfield equation (dashed line) for various magnitudes of the reorganization energy . The other parameters are fixed to be , , , and .

Adiabatic potential surface given by Eqs. (3.5). For the calculation, the parameters are chosen to be , , , , and . Six equally spaced contour levels from 0 to 500 are drawn. The local minimum located around corresponds to site 1, whereas that around is site 2. The point of origin corresponds to the Franck–Condon state.

Adiabatic potential surface given by Eqs. (3.5). For the calculation, the parameters are chosen to be , , , , and . Six equally spaced contour levels from 0 to 500 are drawn. The local minimum located around corresponds to site 1, whereas that around is site 2. The point of origin corresponds to the Franck–Condon state.

Time evolution of the population of site 1 calculated by the present theory, Eq. (2.23) (solid line) and the full-Redfield equation (dashed line) for various magnitudes of the reorganization energy . The other parameters are the same as those in Fig. 4, except .

Time evolution of the population of site 1 calculated by the present theory, Eq. (2.23) (solid line) and the full-Redfield equation (dashed line) for various magnitudes of the reorganization energy . The other parameters are the same as those in Fig. 4, except .

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