^{1}, Tomáš Mančal

^{1}and Graham R. Fleming

^{1,a)}

### Abstract

Using the nonperturbative approach to the calculation of nonlinear optical spectra developed in a foregoing paper [Mančal *et al.*, J. Chem. Phys.124, 234504 (2006), preceding paper], calculations of two-dimensional electronic spectra of an excitonically coupled dimer model system are presented. The dissipative exciton transfer dynamics is treated within the Redfield theory and energetic disorder within the molecular ensemble is taken into account. The manner in which the two-dimensional spectra reveal electronic couplings in the aggregate system and the evolution of the spectra in time is studied in detail. Changes in the intensity and shape of the peaks in the two-dimensional relaxation spectra are related to the coherent and dissipative dynamics of the system. It is shown that coherent electronic motion, an electronic analog of a vibrational wave packet, can manifest itself in two-dimensional optical spectra of molecular aggregate systems as a periodic modulation of both the diagonal and off-diagonal peaks.

This work was supported by a grant from NSF.

I. INTRODUCTION

II. NONPERTURBATIVE CALCULATION OF NONLINEAR SIGNALS AND DEFINITION OF 2D OPTICAL SPECTRUM

III. DIMER MODEL SYSTEM

IV. EQUATIONS OF MOTION

A. Reduced density matrix description

B. Rotating wave approximation

C. Calculation scheme

V. 2D SPECTRA OF A DIMER MODEL SYSTEM: NUMERICAL RESULTS AND DISCUSSION

A. Dimer versus two uncoupled two-level systems

B. Dimer relaxation spectra

1. Coherent electronic motion

C. Population transfer

1. Shape of the peaks

2. Summary and outlook

VI. CONCLUSIONS

### Key Topics

- Coherence
- 23.0
- Dephasing
- 21.0
- Excitons
- 18.0
- Photon echoes
- 15.0
- Equations of motion
- 12.0

## Figures

The pulse scheme of a photon echo experiment. Three pulses with successive delays and are applied to the system. The time origin is conventionally set to the middle of the third pulse. The photon echo signal arises at times . In 2D spectroscopy we vary the first delay to record a two-dimensional signal (in and ) for a given delay .

The pulse scheme of a photon echo experiment. Three pulses with successive delays and are applied to the system. The time origin is conventionally set to the middle of the third pulse. The photon echo signal arises at times . In 2D spectroscopy we vary the first delay to record a two-dimensional signal (in and ) for a given delay .

The electronic-level scheme of the model dimer system. (a) Heterodimer in the molecular electronic states representation, with transition moments and and the excitonic coupling . (b) Heterodimer complex after diagonalization, i.e., in the eigenstate (exciton) representation.

The electronic-level scheme of the model dimer system. (a) Heterodimer in the molecular electronic states representation, with transition moments and and the excitonic coupling . (b) Heterodimer complex after diagonalization, i.e., in the eigenstate (exciton) representation.

Comparison of the 2D spectra (a) of two uncoupled monomers and (b) of the dimer . The parameters have been chosen to produce the same energy separation between two diagonal peaks. The electronic coupling between two monomers is revealed by the appearance of the cross peaks 21 and 12 in the 2D spectrum. Contour lines are drawn in 10% intervals at , for the absorptive real parts (left column) and refractive imaginary parts (right column) of . The level of 100% is determined from the highest peak value within the spectrum. Solid contour lines correspond to positive and dashed lines to negative amplitudes.

Comparison of the 2D spectra (a) of two uncoupled monomers and (b) of the dimer . The parameters have been chosen to produce the same energy separation between two diagonal peaks. The electronic coupling between two monomers is revealed by the appearance of the cross peaks 21 and 12 in the 2D spectrum. Contour lines are drawn in 10% intervals at , for the absorptive real parts (left column) and refractive imaginary parts (right column) of . The level of 100% is determined from the highest peak value within the spectrum. Solid contour lines correspond to positive and dashed lines to negative amplitudes.

2D relaxation spectra of the dimer calculated at population times (a) , (b) , (c) , (d) , (e) , (f) , (g) , and (h) . The exciton energy splitting corresponds to the modulation period of . Population times in (a), (c), and (e) and (b), (d), (f), and (g) correspond to the maxima and minima of periodic modulations (electronic coherence effect), respectively. At longer population times, (h), the intensity is transferred from the diagonal peak 22 to the cross peak 21 due to the population relaxation. The shape of the peaks also varies with the population time (see the discussion in the text).

2D relaxation spectra of the dimer calculated at population times (a) , (b) , (c) , (d) , (e) , (f) , (g) , and (h) . The exciton energy splitting corresponds to the modulation period of . Population times in (a), (c), and (e) and (b), (d), (f), and (g) correspond to the maxima and minima of periodic modulations (electronic coherence effect), respectively. At longer population times, (h), the intensity is transferred from the diagonal peak 22 to the cross peak 21 due to the population relaxation. The shape of the peaks also varies with the population time (see the discussion in the text).

The diagonal cuts of the real-part 2D relaxation spectra; absolute intensities (without scaling to the maximum value) are shown. Cuts correspond to the same population times (a) , (b) , (c) , (d) as in Fig. 4. The figure shows that diagonal peaks are also modulated by the motion of the excitonic wave packet.

The diagonal cuts of the real-part 2D relaxation spectra; absolute intensities (without scaling to the maximum value) are shown. Cuts correspond to the same population times (a) , (b) , (c) , (d) as in Fig. 4. The figure shows that diagonal peaks are also modulated by the motion of the excitonic wave packet.

Dimer homogeneously broadened 2D spectra which correspond to inhomogeneous case depicted on Figs. 4(d) and 4(e). Comparison shows that the form of 2D (inhomogeneous) spectrum can be obtained from elemental (homogeneous) spectral shapes. The presence of inhomogeneity can be understood in the way shown schematically on Fig. 7.

Dimer homogeneously broadened 2D spectra which correspond to inhomogeneous case depicted on Figs. 4(d) and 4(e). Comparison shows that the form of 2D (inhomogeneous) spectrum can be obtained from elemental (homogeneous) spectral shapes. The presence of inhomogeneity can be understood in the way shown schematically on Fig. 7.

The schematic explanation of the peak-shape formation for different correlation broadening cases. In the case where the energy fluctuations on both monomers are fully correlated (shown on the cross peak in the lower-right part of the figure), any change of the energy in one diagonal peak results in the same direction change in the other one. This shifts the position of the off-diagonal peak parallel with the diagonal. In the anticorrelated case (upper-left cross peak), the energetic changes in the diagonal peaks result in a shift of the cross peak that is orthogonal to the diagonal.

The schematic explanation of the peak-shape formation for different correlation broadening cases. In the case where the energy fluctuations on both monomers are fully correlated (shown on the cross peak in the lower-right part of the figure), any change of the energy in one diagonal peak results in the same direction change in the other one. This shifts the position of the off-diagonal peak parallel with the diagonal. In the anticorrelated case (upper-left cross peak), the energetic changes in the diagonal peaks result in a shift of the cross peak that is orthogonal to the diagonal.

The effect of correlated broadening on the dimer 2D spectra: real parts of the (a) homogeneous and inhomogeneous spectra calculated at time for the (b) fully correlated, (c) uncorrelated, and (d) fully anticorrelated fluctuations. Compare the shapes of the diagonal and cross peaks in different cases (see the discussion in the text).

The effect of correlated broadening on the dimer 2D spectra: real parts of the (a) homogeneous and inhomogeneous spectra calculated at time for the (b) fully correlated, (c) uncorrelated, and (d) fully anticorrelated fluctuations. Compare the shapes of the diagonal and cross peaks in different cases (see the discussion in the text).

## Tables

Difference frequencies /corresponding periods (fs) in the FMO complex.

Difference frequencies /corresponding periods (fs) in the FMO complex.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content