^{1}, Frank Lee Emmert III

^{1}and Lyudmila V. Slipchenko

^{1}

### Abstract

The theory for modeling vibronic interactions in bichromophores was introduced in sixties by Witkowski and Moffitt [J. Chem. Phys.33, 872 (1960)10.1063/1.1731278] and extended by Fulton and Gouterman [J. Chem. Phys.35, 1059 (1961)10.1063/1.1701181]. The present work describes extension of this vibronic model to describe bichromophores with broken vibrational symmetry such as partly deuterated molecules. Additionally, the model is extended to include inter-chromophore vibrational modes. The model can treat multiple vibrational modes by employing Lanczos diagonalization procedure of sparse matrices. The developed vibronic model is applied to simulation of vibronic spectra of flexible bichromophore diphenylmethane and compared to high-resolution experimental spectra[J. A. Stearns, N. R. Pillsbury, K. O. Douglass, C. W. Müller, T. S. Zwier, and D. F. Plusquellic, J. Chem. Phys.129, 224305 (2008)10.1063/1.3028543].

This work was supported by National Science Foundation (NSF) Career Grant No. CHE-0955419 and Purdue Research Foundation. B.N. was partly supported by Andrews Graduate Fellowship. The authors thank Professor Tim Zwier, Dr. Christian Müller, and Dr. David Plusquellic for many fruitful discussions related to this work.

I. INTRODUCTION

II. THEORY

A. Intra-monomer modes

B. Inter-monomer modes

C. Intensities

III. MODEL SPECTRA

IV. MODELING VIBRONIC SPECTRUM OF DIPHENYLMETHANE

A. Computational details

B. DPM spectra

V. CONCLUSIONS

### Key Topics

- Polymers
- 32.0
- Wave functions
- 23.0
- Excited states
- 20.0
- Emission spectra
- 18.0
- Absorption spectra
- 11.0

## Figures

Potential energy surfaces for the ground (black) and excited (red) electronic state along vibrational mode *Q*. *E* _{ e } is the vertical excitation energy and is the displacement between the two minima.

Potential energy surfaces for the ground (black) and excited (red) electronic state along vibrational mode *Q*. *E* _{ e } is the vertical excitation energy and is the displacement between the two minima.

Model spectra of one intra-monomer vibrational mode in strong coupling regime with different frequencies on either monomer. ω_{ A } = 150 cm^{−1}, ω_{ B } = 150 + δ cm^{−1}, *b* _{ A } = *b* _{ B } = 1.0, *V* _{ AB } = 400 cm^{−1}. The first row is absorption, the second row is *S* _{1} emission, and the third row is *S* _{2} emission. δ = 0 in (a)–(c); δ = 30 cm^{−1} in (d)–(f); δ = 75 cm^{−1} in (g)–(i).

Model spectra of one intra-monomer vibrational mode in strong coupling regime with different frequencies on either monomer. ω_{ A } = 150 cm^{−1}, ω_{ B } = 150 + δ cm^{−1}, *b* _{ A } = *b* _{ B } = 1.0, *V* _{ AB } = 400 cm^{−1}. The first row is absorption, the second row is *S* _{1} emission, and the third row is *S* _{2} emission. δ = 0 in (a)–(c); δ = 30 cm^{−1} in (d)–(f); δ = 75 cm^{−1} in (g)–(i).

Model spectra of one intra-monomer vibrational mode in strong coupling regime with different displacements on either monomer. ω_{ A } = ω_{ B } = 150 cm^{−1}, *b* _{ A } = 1.0, *b* _{ B } = 1.0 + δ, *V* _{ AB } = 400 cm^{−1}. The first row is absorption, the second row is *S* _{1} emission, and the third row is *S* _{2} emission. δ = 0 in (a)–(c); δ = 0.3 in (d)–(f); δ = 0.6 in (g)–(i).

Model spectra of one intra-monomer vibrational mode in strong coupling regime with different displacements on either monomer. ω_{ A } = ω_{ B } = 150 cm^{−1}, *b* _{ A } = 1.0, *b* _{ B } = 1.0 + δ, *V* _{ AB } = 400 cm^{−1}. The first row is absorption, the second row is *S* _{1} emission, and the third row is *S* _{2} emission. δ = 0 in (a)–(c); δ = 0.3 in (d)–(f); δ = 0.6 in (g)–(i).

Model spectra of one intra-monomer vibrational mode in weak coupling regime with different displacements on either monomer. ω_{ A } = ω_{ B } = 300 cm^{−1}, *b* _{ A } = 0.6, *b* _{ B } = 0.6 − δ, *V* _{ AB } = 50 cm^{−1}. The first row is absorption, the second row is *S* _{1} emission, and the third row is *S* _{2} emission. δ = 0 in (a)–(c); δ = 0.2 in (d)–(f); δ = 0.4 in (g)–(i).

Model spectra of one intra-monomer vibrational mode in weak coupling regime with different displacements on either monomer. ω_{ A } = ω_{ B } = 300 cm^{−1}, *b* _{ A } = 0.6, *b* _{ B } = 0.6 − δ, *V* _{ AB } = 50 cm^{−1}. The first row is absorption, the second row is *S* _{1} emission, and the third row is *S* _{2} emission. δ = 0 in (a)–(c); δ = 0.2 in (d)–(f); δ = 0.4 in (g)–(i).

Model spectra of one inter-monomer vibrational mode with different frequencies in the ground and first and second excited states of the dimer. *V* _{ AB } = 300 cm^{−1}, *b* _{−} = *b* _{+} = 0.8 in all spectra. ω_{ g · s } = ω_{−} = ω_{+} = 100 cm^{−1} in (a)–(c); ω_{ g · s } = 100 cm^{−1}, ω_{−} = 150 cm^{−1}, ω_{+} = 100 cm^{−1} in (d)–(f); ω_{ g · s } = 100 cm^{−1}, ω_{−} = 150 cm^{−1}, ω_{+} = 80 cm^{−1} in (g)–(i). The first row is absorption, the second row is *S* _{1} (*S* _{−}) emission, and the third row is *S* _{2} (*S* _{+}) emission. Changing the frequency of one state does not change the spacing between frequency levels for the other state.

Model spectra of one inter-monomer vibrational mode with different frequencies in the ground and first and second excited states of the dimer. *V* _{ AB } = 300 cm^{−1}, *b* _{−} = *b* _{+} = 0.8 in all spectra. ω_{ g · s } = ω_{−} = ω_{+} = 100 cm^{−1} in (a)–(c); ω_{ g · s } = 100 cm^{−1}, ω_{−} = 150 cm^{−1}, ω_{+} = 100 cm^{−1} in (d)–(f); ω_{ g · s } = 100 cm^{−1}, ω_{−} = 150 cm^{−1}, ω_{+} = 80 cm^{−1} in (g)–(i). The first row is absorption, the second row is *S* _{1} (*S* _{−}) emission, and the third row is *S* _{2} (*S* _{+}) emission. Changing the frequency of one state does not change the spacing between frequency levels for the other state.

Model spectra of one inter-monomer vibrational mode with different displacement parameters for the *S* _{1} and *S* _{2} states of the dimer. *V* _{ AB } = 300 cm^{−1}, ω_{ g · s } = ω_{−} = ω_{+} = 100 cm^{−1} in all spectra. *b* _{−} = *b* _{+} = 0.8 in (a)–(c); *b* _{−} = 0.4, *b* _{+} = 0.8 in (d)–(f); *b* _{−} = 0.0, *b* _{+} = 0.8 in (g)–(i). The first row is absorption, the second row is *S* _{1} (*S* _{−}) emission, and the third row is *S* _{2} (*S* _{+}) emission. Changing the displacement for one state allows to suppress the Frank-Condon progression on this state while keeping it on the other.

Model spectra of one inter-monomer vibrational mode with different displacement parameters for the *S* _{1} and *S* _{2} states of the dimer. *V* _{ AB } = 300 cm^{−1}, ω_{ g · s } = ω_{−} = ω_{+} = 100 cm^{−1} in all spectra. *b* _{−} = *b* _{+} = 0.8 in (a)–(c); *b* _{−} = 0.4, *b* _{+} = 0.8 in (d)–(f); *b* _{−} = 0.0, *b* _{+} = 0.8 in (g)–(i). The first row is absorption, the second row is *S* _{1} (*S* _{−}) emission, and the third row is *S* _{2} (*S* _{+}) emission. Changing the displacement for one state allows to suppress the Frank-Condon progression on this state while keeping it on the other.

Potential energy surfaces of the symmetric torsion *T* mode in (a) the second excited state, (b) the first excited state, and (c) the ground state. The abscissa is the displacement from the optimized *S* _{1} geometry. Energy scales in frames (a)–(c) are different because near the *S* _{1} minimum, the PES of the *S* _{1} state is dominated by second order effects while PESs of the other two states are dominated by first order effects.

Potential energy surfaces of the symmetric torsion *T* mode in (a) the second excited state, (b) the first excited state, and (c) the ground state. The abscissa is the displacement from the optimized *S* _{1} geometry. Energy scales in frames (a)–(c) are different because near the *S* _{1} minimum, the PES of the *S* _{1} state is dominated by second order effects while PESs of the other two states are dominated by first order effects.

DPM spectra produced from parameters in Tables I and III with an electronic coupling constant of 155.8 cm^{−1}. Comparison of the calculated (red) and experimental (black) absorption spectra is shown in (a). Breakdown of the calculated spectrum by the electronic state, with the red trace representing the *S* _{1} (anti-symmetric) state and the blue trace representing the *S* _{2} (symmetric) state in (b). (c) and (d) Comparisons of the calculated (red) and experimental (black) emission spectra from the *S* _{1} and *S* _{2} origins, respectively.

DPM spectra produced from parameters in Tables I and III with an electronic coupling constant of 155.8 cm^{−1}. Comparison of the calculated (red) and experimental (black) absorption spectra is shown in (a). Breakdown of the calculated spectrum by the electronic state, with the red trace representing the *S* _{1} (anti-symmetric) state and the blue trace representing the *S* _{2} (symmetric) state in (b). (c) and (d) Comparisons of the calculated (red) and experimental (black) emission spectra from the *S* _{1} and *S* _{2} origins, respectively.

*S* _{2} “clump” emission spectra. The calculated spectrum (in red) is produced by adding *S* _{2} emission spectrum as in Fig. 8(d) with emissions from energetically close *S* _{1} vibrational states. Experiential spectrum is in black.

*S* _{2} “clump” emission spectra. The calculated spectrum (in red) is produced by adding *S* _{2} emission spectrum as in Fig. 8(d) with emissions from energetically close *S* _{1} vibrational states. Experiential spectrum is in black.

## Tables

Intra-monomer vibrational parameters for diphenylmethane as found from B3LYP/cc-pVTZ calculations on toluene.

Intra-monomer vibrational parameters for diphenylmethane as found from B3LYP/cc-pVTZ calculations on toluene.

Inter-monomer vibrational parameters as found from B3LYP/cc-pVTZ calculations on *S* _{0}, *S* _{1}, and *S* _{2} states of diphenylmethane.

Inter-monomer vibrational parameters as found from B3LYP/cc-pVTZ calculations on *S* _{0}, *S* _{1}, and *S* _{2} states of diphenylmethane.

Adjusted (fitted to experimental spectra) inter-monomer vibrational parameters. Calculated values were kept where appropriate.

Adjusted (fitted to experimental spectra) inter-monomer vibrational parameters. Calculated values were kept where appropriate.

Vertical *S* _{1} − *S* _{2} splittings computed at the ground state optimized geometry.^{a}

Vertical *S* _{1} − *S* _{2} splittings computed at the ground state optimized geometry.^{a}

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