**14. Conclusions**

In this final section, we will list all kinds of symmetries in dozy-chaos mechanics of elementary electron transfers considered in the article and discuss their physical meaning.

First of all, one should note the symmetry associated with the invariance of the expression for the rate constant of elementary electron transfers with respect to sign reversal in the dozy-chaos energy γ (Section 4). This invariance is consistent with the physical case that both the virtual acts of transformation of electron movements and energies into nuclear reorganization movements and energies and the reverse acts occur in the transient dozy-chaos state [4,7–9].

The result in the standard theory of many-phonon transitions [29], corresponding to high (that is, room) temperatures, is a symmetric Gaussian function for the shape of the optical absorption band. It completely neglects the dynamics of the transient molecular state. This result corresponds to the high values of the dozy-chaos energy γ in dozy-chaos mechanics (see Figure 1). Physically, the high values of γ in dozy-chaos mechanics correspond to the weak organization of the quantum-classical molecular transition (Section 1). With a decrease in the dozy-chaos energy γ, the organization of the quantum-classical transition increases, which is manifested in the appearance of a narrow optical absorption peak in the red region of the spectrum and strong asymmetry of the absorption band (Section 5, Figure 1). In other words, the presence of symmetry in the shape of an optical band at high (room) temperatures is associated with a primitive, Franck–Condon picture of molecular "quantum" transitions. The loss of this symmetry is associated with taking into account the effect of self-organization of the dynamics of transitions in dozy-chaos mechanics, which is expressed, in particular, in the "pumping" of dozy chaos from one part of the optical band (narrow peak) to another part (wide wing).

A series for the shape of optical absorption bands in polymethine dyes, depending on the length of the polymethine chain, has a quasi-symmetric and resonant character, where a certain "average" chain length corresponds to the resonance (Section 8, Figure 2). In theory, this resonance—the "center of symmetry" of the series—is the Egorov resonance (Section 7).

An important illustration of the dynamics of the transient state for the Egorov resonance (Equations (29)–(31)) is a qualitative picture of the dynamics based on the use of the Heisenberg uncertainty relation [6–8] (Section 9). In this picture, a quasiparticle called transferon corresponds to the Egorov resonance. This quasiparticle has an antisymmetric twin—an antiquasiparticle called dissipon (Equation (33)). The transferon is depicted by a narrow optical band and the dissipon by a broad one. Strictly speaking, the dissipon is a certain broad resonance rather than (narrow) resonance proper.

Dozy-chaos mechanics, where the transition from absorption spectra to luminescence spectra is carried out by changing only the sign in the heat energy ω12, as in the standard theory of many-phonon transitions [29], gives a mirror-symmetric picture of the shapes of absorption and luminescence bands (Section 10.1). However, the need to take into account the dynamics of the "quantum" transition in the theory leads to the need to change the sign in the donor–acceptor distance *L* as well. This, in turn, leads to the appearance of mirror asymmetry in the pattern of absorption and luminescence band shapes (Section 10.3): transitions with light emission give narrower bands in comparison with absorption bands. Physically, this means that, as a result of taking into account the chaotic dynamics of "quantum" transitions in dozy-chaos mechanics, transitions with emission of photons show themselves to be more organized in comparison with transitions with absorption of photons.

Nonradiative transitions are considered within the framework of a simplified version of dozy-chaos mechanics, in which the electronic component of the complete electron-nuclear amplitude of transitions is fitted by the Gamow tunnel exponential, dependent on the transient phonon environment (Section 11). As in dozy-chaos mechanics for optical processes in its full formulation, this simplified version of dozy-chaos mechanics is considered for the case of the same electron–phonon interactions on the donor and acceptor. Direct and reverse processes turn out to be related not by the standard detailed balance relationship known from statistical physics but by a new, more complex, detailed balance

relationship, which, in addition to the standard equilibrium constant, includes an exponential factor with the donor–acceptor distance in the exponent (Equation (42)).

Within the framework of the simplified version of dozy-chaos mechanics and the Einstein model of nuclear vibrations, the previously obtained [12] expressions for the Brönsted coefficients α and β for proton-transfer reactions (Section 12), which satisfy the well-known symmetric relation (Equation (47)), are given.

A simplified version of dozy-chaos mechanics is also considered for the case of electron–phonon interactions on the donor and acceptor of different magnitudes (Section 13), where a special procedure for the symmetrization of the total amplitude of the quantum-classical transition (Equation (51)) and the corresponding rate constant is performed. The analytical result obtained earlier [17,18] for the rate constant of nonradiative transitions (Equation (52)), which satisfies the new detailed balance relationship (Equation (42)), is presented.

In conclusion, we note that it is of interest to generalize dozy-chaos mechanics for optical processes in its full formulation (Sections 2–4) for the case of different electron–phonon interactions on the donor and acceptor, as well as to construct a theory of nonradiative dozy-chaos processes in its full version. An important point in the formulation of the problem in the theory of nonradiative dozy-chaos processes is the determination of the perturbation operator in the amplitude of the transition which causes the nonradiative transition. In the standard theory of many-phonon transitions [29], the well-known operator of nonadiabaticity [29,31] is taken as such an operator (see [2]). It is also of interest to generalize dozy-chaos mechanics to the case of nonlinear optics [1,10,19].

**Funding:** This work was supported by the Ministry of Science and Higher Education within the State assignment Federal Scientific Research Center "Crystallography and Photonics" Russian Academy of Sciences.

**Conflicts of Interest:** The author declares no conflict of interest.

**Data Availability Statement:** The data on which this article is based are available as an online resource with digital object identifier (doi) 10.5061/dryad.t0r3p and at the Egorov, Vladimir (2018), Mendeley Data, V2, https://doi.org/10.17632/h4g2yctmvg.2 [3].
