# Dynamic Symmetry in Dozy-Chaos Mechanics

^{†}

## Abstract

**:**

## 1. Introduction

## 2. On Dozy-Chaos Mechanics of Elementary Electron Transfers

## 3. General Formula for the Rate Constant of Electron Photo-Transfers

## 4. The Analytical Result for Optical Absorption Band Shapes and Its Invariance with Respect to the Change in the Sign of Dozy-Chaos Energy γ

## 5. Potential Box with a Movable Wall. Optical Absorption Band Shapes as Dependent on the Dozy-Chaos Energy γ: From Symmetry to Asymmetry

## 6. Passage to the Limit to the Standard Theory of Many-Phonon Transitions and the Symmetry of the Standard Result. The Reason for the Asymmetry of the Optical Absorption Band Shape in Dozy-Chaos Mechanics

## 7. The Egorov Resonance

## 8. Implementation of the Egorov Resonance in the Quasi-Symmetric Serious of Optical Band Shapes of a Representative Polymethine Dye

## 9. The Egorov Resonance and Its “Antisymmetric Twin”

## 10. Symmetry between Optical Absorption and Luminescence in the Standard Theory and Its Violation in Dozy-Chaos Mechanics as a Consequence of the Dynamic Organization of Quantum-Classical Transitions

#### 10.1. Luminescence and Absorption Spectra. Their Mirror Symmetry

#### 10.2. Optical Spectra, Nature of the Small Stokes Shift, and Dynamic Asymmetry of Luminescence and Absorption

#### 10.3. Luminescence and Absorption Spectra. Their Mirror Asymmetry

## 11. A Simplified Version of Dozy-Chaos Mechanics—Nonradiative Transitions

## 12. The Simplified Version of Dozy-Chaos Mechanics: Proton-Transfer Reactions. On Symmetry in the Brönsted Relationship

## 13. The Simplified Version of Dozy-Chaos Mechanics: Symmetrization of the Amplitude and Rate Constant of the Transition for the Case of Different Electron–Phonon Interactions on the Donor and Acceptor

## 14. Conclusions

## Funding

## Conflicts of Interest

## Data Availability Statement

## References

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**Figure 1.**Singularity in the rate of molecular quantum transitions: the optical absorption band shape dependent on the dozy chaos available to a given quantum transition; the band shape with the strongly pronounced peak (J-band) corresponds to the least dozy chaos [9]. The dozy-chaos-dependent optical absorption band is displaced to the red spectral region and narrowed. The position, the intensity, and the width of the optical absorption band are determined by the ratio between the dozy-chaos energy $\gamma $ and the reorganization energy $E$ (see Section 4). The smaller the value of $\gamma $ is, the higher the degree of organization of the molecular “quantum” transition and the higher the intensity and lower the width of the optical band. The position of the wing maximum is determined by the energy $E$, whereas the position of the peak is determined by the energy $\gamma $ [9].

**Figure 2.**Experimental [44,45] (

**a**) and theoretical [6] (

**b**) monomers’ optical absorption spectra, dependent on the length of the polymethine chain $L=2\left(n+2\right)\mathrm{d}$ , where $d$ are certain roughly equal bond lengths in the chain (thiapolymethinecyanine in methanol at room temperature; $\epsilon $ is the extinction coefficient) [4,9]. The optical absorption band with $n=3$ corresponds to the dynamic electron-nuclear-reorganization resonance (the Egorov resonance, see Section 7) or is close to it. (Original citation)—Reproduced by permission of The Royal Society of Chemistry. For the short chains ($n=0,1,2,3$), the tunnel effects, associated with the quantity $\eta $ in Equation (25), can be neglected ($\eta =1$). For the long chains ($n=4,5$), the tunnel effects are small but they must be taken into account ($\eta <1$). The absorption bands are computed by Equations (6)‒(26) with $\eta \le 1$ instead of the Gamow tunnel factor (Equation (25)) when fitting them to the experimental data of Brooker and co-workers [44] (

**a**) in terms of wavelength ${\lambda}_{\mathrm{max}}$, extinction ${\epsilon}_{\mathrm{max}}$, and half-width ${w}_{1/2}$ with a high degree of accuracy. The following parameters of the “dye + environment” system are used [6]: $m={m}_{\mathrm{e}}$, where ${m}_{\mathrm{e}}$ is the electron mass; $d=0.14$ nm; $\omega =5\times {10}^{13}\text{}{\mathrm{s}}^{-1}$; ${n}_{\mathrm{refr}}=1.33$; for $n=0,1,2,3,4,5$, one has ${J}_{1}=\left(5.63,5.40,4.25,3.90,3.74,3.40\right)$ eV, ${J}_{1}-{J}_{2}=\left(1.71,1.31,1.11,0.90,0.74,0.40\right)$ eV, $E=\left(0.245,0.248,0.256,0.275,0.297,0.496\right)$ eV, and $\gamma =\left(0.402,0.205,0.139,0.120,0.129,0.131\right)$ eV, respectively; for $n=0,1,2,3$, factor $\eta =1$, and for $n=4,5$, factor $\eta =0.55,\text{}0.1$, respectively; $T=298\text{}\mathrm{K}$.

**Figure 3.**Experimental [50] (

**a**) and theoretical [3] (

**b**) absorption and fluorescence spectra of J-aggregates. In the analytical result for the shape of the optical bands (Equations (6)–(26)), the transition from absorption spectrum to fluorescence spectrum is carried out by changing the sign before the heat energy $\hslash {\omega}_{12}$. See details in the Egorov, Vladimir (2018), Mendeley Data, V2, https://doi.org/10.17632/h4g2yctmvg.2.

**Figure 4.**(

**a**) The same as in Figure 3a. (

**b**) Theoretical absorption and fluorescence spectra [3], fitted to the experimental data [50] (see (

**a**)) in the J-aggregates. In the analytical result for the shape of the optical bands (Equations (6)–(26)), the transition from absorption spectrum to fluorescence spectrum is carried out by changing the sign before the heat energy $\hslash {\omega}_{12}$ and before the length of the optical chromophore (electron-charge-transfer distance) $L$ as well. See details in the Egorov, Vladimir (2018), Mendeley Data, V2, https://doi.org/10.17632/h4g2yctmvg.2.

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Egorov, V.V.
Dynamic Symmetry in Dozy-Chaos Mechanics. *Symmetry* **2020**, *12*, 1856.
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Egorov VV.
Dynamic Symmetry in Dozy-Chaos Mechanics. *Symmetry*. 2020; 12(11):1856.
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Egorov, Vladimir V.
2020. "Dynamic Symmetry in Dozy-Chaos Mechanics" *Symmetry* 12, no. 11: 1856.
https://doi.org/10.3390/sym12111856