**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

From the general result of dozy-chaos mechanics of elementary electron transfers, Equation (4), the expression for the light absorption factor *K* (the optical extinction coefficient ε [2–9,29] is proportional to *K*), has been obtained. The obtained expression for *K* in the framework of the Einstein model of nuclear vibrations in the framework of the Einstein model of nuclear vibrations (ωκ = constant ≡ ω), although it is rather complex, is fully expressed in elementary functions and has the following form [2,5,6]:

$$K = K\_0 \exp \mathcal{W} \tag{7}$$

$$\begin{split} \mathcal{W} &= \frac{1}{2} \ln \left( \frac{\omega \tau \sinh \beta\_T}{4 \pi \cosh t} \right) - \frac{2}{\omega \tau} \Big( \coth \beta\_T - \frac{\cosh t}{\sinh \beta\_T} \Big) \\ &+ (\beta\_T - t) \frac{1}{\omega \tau \Theta} - \frac{\sinh \beta\_T}{4 \omega \tau \Theta^2 \cosh t} \end{split} \tag{8}$$

*Symmetry* **2020**, *12*, 1856

$$1 < \frac{1}{\omega \tau \Theta} \le \frac{2 \cosh t}{\omega \tau \sinh \beta\_T} \tag{9}$$

where <sup>β</sup>*<sup>T</sup>* <sup>≡</sup> ω/2*k*B*T*,

$$t = \frac{\omega \tau\_e}{\theta} \left[ \frac{A\mathbb{C} + BD}{A^2 + B^2} + \frac{2\Theta(\Theta - 1)}{\left(\Theta - 1\right)^2 + \left(\Theta / \theta\_0\right)^2} + \frac{\theta\_0^2}{\theta\_0^2 + 1} \right] \tag{10}$$

$$|\theta\_0| \gg \frac{E}{2f\_1} \tag{11}$$

$$\theta \equiv \frac{\tau\_{\rm c}}{\tau} = \frac{L \, E}{\hbar \sqrt{2f\_1/m}}, \ \Theta \equiv \frac{\tau'}{\tau} = \frac{E}{\hbar \omega\_{12}}, \ \theta\_0 \equiv \frac{\tau\_0}{\tau} = \frac{E}{\gamma} \tag{12}$$

$$
\tau\_{\rm e} = \frac{L}{\sqrt{2f\_1/m}}, \ \tau = \frac{\hbar}{E}, \ \tau' = \frac{1}{\alpha\_{12}}, \ \tau\_0 = \frac{\hbar}{\mathcal{V}} \tag{13}
$$

Here, we use the notation

$$A = \cos\left(\frac{\theta}{\theta\_0}\right) + \Lambda + \left(\frac{1}{\theta\_0}\right)^2 \text{N} \tag{14}$$

$$B = \sin\left(\frac{\theta}{\theta\_0}\right) + \frac{1}{\theta\_0} \mathbf{M} \tag{15}$$

$$\mathcal{L} = \theta \left[ \cos \left( \frac{\theta}{\theta\_0} \right) - \frac{1 - \xi^2}{2\theta\_0} \sin \left( \frac{\theta}{\theta\_0} \right) \right] + \mathcal{M} \tag{16}$$

$$D = \theta \left[ \sin \left( \frac{\theta}{\theta\_0} \right) + \frac{1 - \xi^2}{2\theta\_0} \cos \left( \frac{\theta}{\theta\_0} \right) \right] - \frac{2}{\theta\_0} \mathcal{N} \tag{17}$$

$$\text{and } \xi \equiv \left( 1 - \frac{E}{f\_1} \right)^{1/2} \left( f\_1 > E \text{ by definition} \right) \tag{18}$$

and where we finally have

$$\Lambda = -(\Theta - 1)^2 \mathbf{E} + \left[ \frac{(\Theta - 1)\theta}{\rho} + \Theta(\Theta - 2) \right] \mathbf{E}^{\frac{1-\rho}{1-\zeta}} \tag{19}$$

$$\mathbf{M} = 2\Theta(\Theta - 1)\mathbf{E} - \left[\frac{(2\Theta - 1)\theta}{\rho} + 2\Theta(\Theta - 1)\right] \mathbf{E}^{\frac{1-\rho}{1-\zeta}}\tag{20}$$

$$\mathbf{N} = \Theta \left[ \Theta \mathbf{E} - \left( \frac{\theta}{\rho} + \Theta \right) \mathbf{E}^{\frac{1-\rho}{1-\xi}} \right] \tag{21}$$

$$\mathcal{E} \equiv \exp\left(\frac{2\theta}{1+\underline{\xi}}\right), \rho \equiv \sqrt{\xi^2 + \frac{1-\underline{\xi}^\circ \mathcal{Z}}{\Theta}}\tag{22}$$

The factor *K*<sup>0</sup> becomes

$$K\_0 = K\_0^\epsilon K\_0^p \tag{23}$$

where

$$K\_0^\varepsilon = \frac{2\tau^3 I\_1}{m} \frac{\left(A^2 + B^2\right) \rho^3 \Theta^4 \xi}{\theta^2 \left[\left(\Theta - 1\right)^2 + \left(\frac{\Theta}{\vartheta\_0}\right)^2\right]^2 \left[1 + \left(\frac{1}{\vartheta\_0}\right)^2\right]} \cdot \eta \tag{24}$$

$$\text{rand } \eta \equiv \exp\left(-\frac{4\theta}{1-\xi^2}\right) \tag{25}$$

and

$$K\_0^p = \frac{1}{\omega \tau} \left[ 1 + \frac{\sinh(\beta\_T - 2t)}{\sinh \beta\_T} \right]^2 + \frac{\cosh(\beta\_T - 2t)}{\sinh \beta\_T} \tag{26}$$

Inequalities (9) and (11) are not any significant restrictions on the parameters of the system and associated with items of routine approximations made in the calculations (see [2]). The time scales τ*e*, τ, and τ0, given by Equation (13), control the dynamics of elementary electron-transfer processes. They are discussed in Section 9. The time scale τ (see Equation (13)) together with the law of conservation of energy (6) and the other parameters of a donor–acceptor system control the dynamics of producing the shape of optical bands [2,4].

Let us consider further the issues related to the change in the sign of the dozy-chaos energy γ. On the one hand, in standard quantum mechanics, where the value of γ is infinitesimal and where this value is introduced formally in order to avoid zero in the energy denominator of the spectral representation of Green's function (see Equation (3)), the sign of γ can be either positive or negative. On the other hand, in quantum-classical mechanics, although the value of γ becomes a finite value, the choice of its sign turns out to be insignificant here too. It is easy to show, for example, that our result for the light absorption factor *K*, given by Equations (6)–(26), is an even function of γ. For this, it is sufficient to consider those equations that include the dimensionless quantity θ<sup>0</sup> = *<sup>E</sup>* <sup>γ</sup> (see Equation (12)), in which the reorganization energy *E* is positive by definition (see Equation (2)). So, it is easy to see that the quantity *t* = *t*(Θ, θ0) (see Equation (10)) is an even function of θ0: in the nontrivial term *AC* + *BD*, the cofactors *A* and *C* are even functions of θ0, and the cofactors *B* and *D* are odd functions of θ0. Further, the factor *K<sup>e</sup>* <sup>0</sup> (see Equation (24)) is obviously an even function of θ0.

The invariance with respect to the change in the sign of the dozy-chaos energy γ 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]. For definiteness, we set γ > 0 here, there, and everywhere.
