^{1}, Andreas Köhn

^{1,a)}, Michael E. Harding

^{1}, Gregor Diezemann

^{1}, Gerald Hinze

^{1}, Thomas Basché

^{1}and Jürgen Gauss

^{1}

### Abstract

Electronic excitation energy transfer (EET) rates in rylene diimide dyads are calculated using second-order approximate coupled-cluster theory and time-dependent density functional theory. We investigate the dependence of the EET rates on the interchromophoric distance and the relative orientation and show that Förster theory works quantitatively only for donor-acceptor separations larger than roughly . For smaller distances the EET rates are over- or underestimated by Förster theory depending on the respective orientation of the transition dipole moments of the chromophores. In addition to the direct transfer rates we consider bridge-mediated transfer originating from oligophenylene units placed between the chromophores. We find that the polarizability of the bridge significantly enhances the effective interaction. We compare our calculations to single molecule experiments on two types of dyads and find reasonable agreement between theory and experiment.

This work was supported by the Deutsche Forschungsgemeinschaft (SFB 625 “From Single Molecules to Nanoscopically Structured Materials”), the Carl-Zeiss-Stiftung, and the Fonds der Chemischen Industrie.

I. INTRODUCTION

II. DETERMINATION OF EET RATES FROM SINGLE MOLECULE EXPERIMENTS

III. THEORETICAL BACKGROUND

A. Förster theory and its limitations

B. Computational realization

IV. RESULTS

A. The isolated chromophores

B. The Coulomb interaction of PDI and TDI

C. Bridge-mediated transfer

D. Some notes on environmental effects

E. Comparison to the experiment

V. CONCLUSIONS

### Key Topics

- Electric dipole moments
- 25.0
- Density functional theory
- 20.0
- Absorption spectra
- 16.0
- Excited states
- 14.0
- Wave functions
- 10.0

## Figures

The dyads studied in Refs. 31–33 consist of the chromophores PDI and TDI linked via an oligophenylene bridge. For our quantum chemical calculations with the isolated chromophores we saturated the nitrogen sites with hydrogen atoms, whereas the experimental data for the single chromophores originate from measurements on Ph-PDI-Ph and Ph-TDI-Ph.

The dyads studied in Refs. 31–33 consist of the chromophores PDI and TDI linked via an oligophenylene bridge. For our quantum chemical calculations with the isolated chromophores we saturated the nitrogen sites with hydrogen atoms, whereas the experimental data for the single chromophores originate from measurements on Ph-PDI-Ph and Ph-TDI-Ph.

The results of B3LYP/SVP calculations for (a) the electronic coupling strength vs the donor-acceptor distance . The dashed lines denote the dipole-dipole approximated coupling strength and the straight line marks the full Coulomb coupling strength . (b) Ratio of the two calculated coupling strengths. Throughout the figure the (◻) denote the collinear alignment of the interacting chromophores, the (○) the coplanar case, and the (+) the cofacial alignment. The arrow indicates the geometry of dyad 1 (collinear, ).

The results of B3LYP/SVP calculations for (a) the electronic coupling strength vs the donor-acceptor distance . The dashed lines denote the dipole-dipole approximated coupling strength and the straight line marks the full Coulomb coupling strength . (b) Ratio of the two calculated coupling strengths. Throughout the figure the (◻) denote the collinear alignment of the interacting chromophores, the (○) the coplanar case, and the (+) the cofacial alignment. The arrow indicates the geometry of dyad 1 (collinear, ).

The results of B3LYP/SVP calculations for (a) the electronic coupling strength vs the donor-acceptor distance . The dashed lines denote the dipole-dipole approximated coupling and the straight line marks the full Coulomb coupling . (b) Ratio of the two calculated coupling strengths. Throughout the figure the (◻) denote , the (×) , the (▿) , the (▵) , the (◇) , and the (+) . For reasons of clarity not all alignments are displayed in panel (a).

The results of B3LYP/SVP calculations for (a) the electronic coupling strength vs the donor-acceptor distance . The dashed lines denote the dipole-dipole approximated coupling and the straight line marks the full Coulomb coupling . (b) Ratio of the two calculated coupling strengths. Throughout the figure the (◻) denote , the (×) , the (▿) , the (▵) , the (◇) , and the (+) . For reasons of clarity not all alignments are displayed in panel (a).

The results of B3LYP/SVP calculations for the electronic coupling of PDI and TDI with respect to the average geometry of dyad 2 (, , ). (a) The electronic coupling strength vs the donor-acceptor distance . The dashed line denotes the dipole-dipole approximated coupling and the straight line marks the full Coulomb coupling . (b) Ratio of the two calculated coupling strengths.

The results of B3LYP/SVP calculations for the electronic coupling of PDI and TDI with respect to the average geometry of dyad 2 (, , ). (a) The electronic coupling strength vs the donor-acceptor distance . The dashed line denotes the dipole-dipole approximated coupling and the straight line marks the full Coulomb coupling . (b) Ratio of the two calculated coupling strengths.

Influence of the -terphenyl bridge on the first excited singlet state of PDI: Shown are the isosurfaces of the (a) transition density and (b) difference density of PDI-3Ph as calculated with CC2/SVP.

Influence of the -terphenyl bridge on the first excited singlet state of PDI: Shown are the isosurfaces of the (a) transition density and (b) difference density of PDI-3Ph as calculated with CC2/SVP.

## Tables

The calculated electronic transitions of PDI in the gas phase. The values of the transition energies and the transition dipole moment magnitudes are listed. The absorption and emission transitions were calculated from the optimized ground and first excited state geometries, respectively. For comparison the absorption and emission maxima obtained by experiment and the absorption transition dipole moment magnitude extracted from the absorption spectrum of Ph-PDI-Ph in a toluene solution are given.

The calculated electronic transitions of PDI in the gas phase. The values of the transition energies and the transition dipole moment magnitudes are listed. The absorption and emission transitions were calculated from the optimized ground and first excited state geometries, respectively. For comparison the absorption and emission maxima obtained by experiment and the absorption transition dipole moment magnitude extracted from the absorption spectrum of Ph-PDI-Ph in a toluene solution are given.

The calculated absorption transitions of TDI in the gas phase. The values of the transition energies and the transition dipole moments are listed. For reasons of comparison the maximum and the transition dipole moment of the experimentally obtained spectrum of Ph-TDI-Ph in a toluene solution are given.

The calculated absorption transitions of TDI in the gas phase. The values of the transition energies and the transition dipole moments are listed. For reasons of comparison the maximum and the transition dipole moment of the experimentally obtained spectrum of Ph-TDI-Ph in a toluene solution are given.

Electronic couplings as obtained by quantum chemical calculations (CC2/SVP) considering separated chromophores (no bridge effects) for the dyads 1 and 2. The superscripts “full” and “” denote whether the respective electronic coupling was obtained using the full interaction operator [cf. Eq. (8)], or in the dipole-dipole approximation [cf. Eq. (9)].

Electronic couplings as obtained by quantum chemical calculations (CC2/SVP) considering separated chromophores (no bridge effects) for the dyads 1 and 2. The superscripts “full” and “” denote whether the respective electronic coupling was obtained using the full interaction operator [cf. Eq. (8)], or in the dipole-dipole approximation [cf. Eq. (9)].

The first five calculated absorption transitions of -terphenyl in the gas phase. The values of the transition energies and the transition dipole moments are listed.

The first five calculated absorption transitions of -terphenyl in the gas phase. The values of the transition energies and the transition dipole moments are listed.

Effect of the bridge on the electronic coupling of PDI and TDI, as calculated at the CC2/SVP level. is the direct coupling term given in Table III, is the additional contribution from the bridge. The superscripts “full” and “” denote the considered electronic coupling, i.e., the full [cf. Eq. (8)] and the dipole-dipole approximated [cf. Eq. (9)] coupling. The contribution of the bridge (index “D-B-A”) has been considered in different ways: For compound 1 (, ) we calculated the additional coupling resulting from the first-order perturbation approach of Eq. (11) and we furthermore considered the coupling between PDI-3Ph (cf. Fig. 5) and TDI, Eq. (13). For compound 2 (, ) we calculated the coupling between PDI-2Ph and 3Ph-TDI.

Effect of the bridge on the electronic coupling of PDI and TDI, as calculated at the CC2/SVP level. is the direct coupling term given in Table III, is the additional contribution from the bridge. The superscripts “full” and “” denote the considered electronic coupling, i.e., the full [cf. Eq. (8)] and the dipole-dipole approximated [cf. Eq. (9)] coupling. The contribution of the bridge (index “D-B-A”) has been considered in different ways: For compound 1 (, ) we calculated the additional coupling resulting from the first-order perturbation approach of Eq. (11) and we furthermore considered the coupling between PDI-3Ph (cf. Fig. 5) and TDI, Eq. (13). For compound 2 (, ) we calculated the coupling between PDI-2Ph and 3Ph-TDI.

The EET rates from both, experiments (Refs. 31–33) and quantum chemical calculations (CC2/SVP, this work) for the dyads 1 (, ) and 2 (, ). The superscripts “exp” and “Förster” denote the rates that have been directly measured and approximated with Eq. (1), respectively. The values given denote the rate average and the inverse time average determined from distributions as explained in Sec. II. The superscripts “full” and “” denote the approximations used for calculating the respective electronic couplings, i.e., the full [cf. Eq. (8)] and the dipole-dipole approximated [cf. Eq. (9)] coupling. To obtain the rates from the computed couplings the mean line shape overlaps of the experimental spectra were used [cf. Eq. (10)]. The value for dyad 1 is [from single molecule spectra at in PMMA [Refs. 32], for dyad 2 a value of [from single molecule spectra at room temperature in PMMA [Ref. 33] was used. The shielding factor was set to (with ) as in Refs. 31–33.

The EET rates from both, experiments (Refs. 31–33) and quantum chemical calculations (CC2/SVP, this work) for the dyads 1 (, ) and 2 (, ). The superscripts “exp” and “Förster” denote the rates that have been directly measured and approximated with Eq. (1), respectively. The values given denote the rate average and the inverse time average determined from distributions as explained in Sec. II. The superscripts “full” and “” denote the approximations used for calculating the respective electronic couplings, i.e., the full [cf. Eq. (8)] and the dipole-dipole approximated [cf. Eq. (9)] coupling. To obtain the rates from the computed couplings the mean line shape overlaps of the experimental spectra were used [cf. Eq. (10)]. The value for dyad 1 is [from single molecule spectra at in PMMA [Refs. 32], for dyad 2 a value of [from single molecule spectra at room temperature in PMMA [Ref. 33] was used. The shielding factor was set to (with ) as in Refs. 31–33.

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