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

All of the results of the optical spectra match, generally, weak dozy chaos (γ << *E*). Strong dozy chaos (γ ≥ *E*) leads to the elucidation of important patterns in the reactions of proton transfers [12,51] and comparatively fresh temperature-dependent effects on electron transfers in Langmuir–Blodgett films [13,52]. In the case of strong dozy chaos, the dynamics of quantum-classical transitions become weakly dependent on dozy chaos, and the electronic component of the complete electron-nuclear amplitude of transitions can be fitted by the Gamow tunnel exponential, dependent on the transient phonon environment. This elementary method permit us to evade the consideration of the imaginary additive *i*γ in the spectral representation of the complete Green's function and to word the physical nature of the transient state, not in the concept of dozy chaos but in the concept of a large number of tunnel and over-barrier energy states providing the "quantum" transition of an elementary charged particle. This method was worked out [16] long before the development of quantum-classical (dozy-chaos) mechanics [2–9], and now we can say that the concept of a large number of tunnel and over-barrier states is a simplified version of the concept of dozy chaos.

The general result for the rate constant in the simplified version of dozy-chaos mechanics *K* is expressed in terms of the the Gamow tunnel exponential, dependent on the transient phonon-environment-energy ω<sup>1</sup> and one generating function [15,16]:

$$\mathcal{K} \propto \sum\_{\omega\_1 = -\infty}^{\infty} \sum\_{\omega\_1' = -\infty}^{\infty} \mathcal{G}\_0(\omega\_1, L) \mathcal{G}\_0 \* \left(\omega\_1', L\right) \times \frac{1}{\left(2\pi l\right)^3} \oint \frac{dx}{x^{\omega\_1 + 1}} \oint \frac{dy}{y^{\omega\_1' + 1}} \oint \frac{dz}{z^{\omega\_{12} + 1}} \mathcal{S}(\overline{n}\_1; x, y, z) \tag{34}$$

{

where the contours encircle the points *x* = 0, *y* = 0, and *z* = 0, correspondingly (cf. Equation (4)). The Gamow tunnel exponential is

$$G\_0 = G\_0(a, L) = \exp(-aL) \tag{35}$$

where the function α = α(ω1) is given by the following formula:

$$\alpha \equiv \alpha(\omega\_1) = \left[2m(I + \hbar\omega\_1)\right]^{1/2} / \hbar \tag{36}$$

(here, there, and everywhere, *J* ≡ *J*1). The generating function is as follows:

$$\begin{split} S(\overline{\boldsymbol{n}}\_{1};\boldsymbol{\chi},\boldsymbol{y},\boldsymbol{z}) &= \exp\Big\{-\sum\_{\mathbf{k}} \overline{q}\_{\mathbf{k}}^{2} (2\overline{\boldsymbol{n}}\_{\mathbf{k}1} + 1) + \frac{1}{2} \sum\_{\mathbf{k}} \overline{q}\_{\mathbf{k}}^{2} [(\overline{\boldsymbol{n}}\_{\mathbf{k}1} + 1)(\boldsymbol{x}^{\boldsymbol{\omega}\_{\mathbf{k}}} \boldsymbol{y}^{\boldsymbol{\omega}\_{\mathbf{k}}} + 1)\boldsymbol{z}^{\boldsymbol{\omega}\_{\mathbf{k}}} \\ &+ \overline{n}\_{\mathbf{k}1} (\boldsymbol{x}^{-\boldsymbol{\omega}\_{\mathbf{k}}} \boldsymbol{y}^{-\boldsymbol{\omega}\_{\mathbf{k}}} + 1)\boldsymbol{z}^{-\boldsymbol{\omega}\_{\mathbf{k}}}] \} \end{split} \tag{37}$$

The result (34) applies to both optical and nonradiative processes. In the case of optical processes, the heat energy ω<sup>12</sup> is determined from the law of conservation of energy (6) (ω<sup>12</sup> > 0—absorption and ω<sup>12</sup> < 0—luminescence), where the frequency Ω ≡ 0 in the cases of nonradiative endothermic and exothermic processes:

$$
\hbar \hbar \omega\_{12} = \hbar \mathfrak{z} - I\_1 < 0 \tag{38}
$$

and

$$
\hbar \omega\_{12} = f\_1 - f\_2 \equiv -\hbar \omega\_{21} > 0\tag{39}
$$

From the general result for the rate constant in the simplified version of dozy-chaos mechanics, Equations (34)–(39), in the framework of the Einstein model of nuclear vibrations (ωκ = constant ≡ ω), the simple expression for the rate constant *K* has been obtained [16]:

$$\begin{split} \mathcal{K} & \propto \exp\left\{ -\frac{2L}{a} - \frac{2E}{\hbar\omega} \left[ \coth\frac{\hbar\omega}{2k\_{\rm B}T} - \frac{\cosh t}{\sinh\left(\hbar\omega/2k\_{\rm B}T\right)} \right] \right. \\ & \left. + \left( \frac{\hbar\omega}{2k\_{\rm B}T} - t \right) \frac{\omega\_{12}}{\alpha\flat} - \frac{\hbar\omega \sinh\left(\hbar\omega/2k\_{\rm B}T\right)}{4E \cosh t} \left( \frac{\omega\_{12}}{\alpha\flat} \right)^{2} \right\} \end{split} \tag{40}$$

(cf. Equations (7) and (8)), where exp- <sup>−</sup>2*<sup>L</sup> a* is the Gamow exponential (cf. Equation (27)) and

$$t = \frac{\omega \text{ L}}{\sqrt{2 \text{J/m}}} \tag{41}$$

(cf. Equations (10) and (13)). If, as in the case of the complete theory for optical processes (Sections 2–4), we assume that the expression for the rate constant of the reverse process *K*rev is obtained by changing the sign in the heat energy ω<sup>12</sup> and in the donor–acceptor distance *L* (see Sections 10.1 and 10.3) in corresponding expression for the rate constant of the direct process *K* in the considered simplified version of dozy-chaos mechanics, then, applying this position to Equation (40), we obtain

$$\frac{K\_{\rm rev}}{K} = \exp\left(-\frac{\hbar\omega\_{12}}{k\_{\rm B}T}\right)\exp\left(\frac{4L}{a}\right) \equiv K\_{\rm eq}^{-1}\exp\left(\frac{4L}{a}\right) \tag{42}$$

where *<sup>K</sup>*eq exp- <sup>−</sup>4*<sup>L</sup> a* is the equilibrium constant of charged-particle-transfer reactions in the simplified version of dozy-chaos mechanics. In the limit *L* → 0, we obtain from Equation (42) the well-known detailed balance relationship in statistical physics and in the standard theory of many-phonon transitions [29,31].
