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

Until now, both in the case of the complete theory for optical processes (Sections 2–4) and in the case of its simplified version for nonradiative processes (Section 11), we have considered the case of the same electron–phonon interaction when a light charged particle, in particular an electron, is localized on the donor or on the acceptor. In other words, it was assumed that the reorganization energy *E* ≡ *E*<sup>1</sup> = *E*<sup>2</sup> (Equation (2)). For example, "quantum" transitions and the corresponding shapes of optical bands in polymethine dyes are well described by the case of the same value of the electron–phonon interaction on the donor and on the acceptor, because charge alternation occurs in the polymethine chain upon optical excitation [1,3–6]. In this section, we will briefly consider the case of different electron–phonon interactions when an electron is localized on a donor or acceptor. This corresponds to different magnitude shifts of the normal phonon coordinates *q*κ<sup>1</sup> and *q*κ<sup>2</sup> (in the case of the same interaction, *q*<sup>κ</sup> <sup>≡</sup> *q*κ<sup>1</sup> <sup>=</sup> <sup>−</sup>*q*κ<sup>2</sup> [2]) and an obvious redefinition of the reorganization energy *<sup>E</sup>* (Equation (2)):

$$E\_{1,2} = \frac{1}{2} \sum\_{\kappa} \hbar \omega\_{\kappa} \overline{q}\_{\text{x1},2}^2 \tag{48}$$

where *E*<sup>1</sup> - *E*2. For example, in the case of nonradiative processes, the change in sign in the heat energy ω<sup>12</sup> (Equations (38) and (39)) and in the donor–acceptor distance *L* ≡ |**L** ≡ **L**12| (Equation (1)) is associated with the permutation of indices 1 and 2 in the reverse order. The assumption *E*<sup>1</sup> - *E*<sup>2</sup> leads to asymmetry with respect to the permutation of indices 1 and 2 in the expression for the rate constant of transitions and the loss of connection between forward and reverse processes, expressed in Equation (42). To restore this connection, it is necessary to symmetrize the expression for the amplitude and rate constant of electron transfers with respect to different values of the electron–phonon interaction at the donor and at the acceptor, which leads to the case of reorganization energies *E*<sup>1</sup> -*E*2.

The symmetrization method proposed in [14,17,18] consists of the fact that, in addition to the transition amplitude [2–4]

$$A\_{12} = \langle \Psi\_2(\mathbf{r} - \mathbf{L}, q) \Big| \mathbf{V} \Big| \Psi\_1(\mathbf{r}, q) \rangle \tag{49}$$

which, in view of taking the wave function Ψ<sup>2</sup> in the Born–Oppenheimer adiabatic approximation Ψ<sup>2</sup> = ΨBO <sup>2</sup> and taking into account the entire dynamics of the transition only in the wave function <sup>Ψ</sup><sup>1</sup> = *<sup>G</sup> <sup>V</sup>* <sup>Ψ</sup>BO <sup>1</sup> (*<sup>G</sup>* is Green's function of the Hamiltonian *<sup>H</sup>* <sup>−</sup> *<sup>V</sup>*, *<sup>V</sup>* <sup>≡</sup> κ *<sup>V</sup>*κ(**r**)(*q*<sup>κ</sup> <sup>−</sup>*q*κ) [2–4]), can be called the amplitude of the transition on the acceptor *A<sup>a</sup>* <sup>12</sup>, we introduce into the theory also the amplitude

$$A\_{12}^d = A\_{21}^a \tag{50}$$

in which, on the contrary, the wave function Ψ<sup>1</sup> is taken in the adiabatic approximation Ψ<sup>1</sup> = ΨBO <sup>1</sup> , and the entire dynamics of the transition are taken into account only in the wave function <sup>Ψ</sup><sup>2</sup> = *GV* <sup>Ψ</sup>BO <sup>2</sup> . This new amplitude *A<sup>d</sup>* <sup>12</sup> can be called the amplitude of the transition on the donor. Then, the half-sum of these two amplitudes is taken as the total transition amplitude:

$$A\_{12} = \frac{A\_{12}^d + A\_{12}^a}{2} = A\_{21} \tag{51}$$

Since the symmetrization is carried out only with respect to the electron–phonon interaction, in Equation (51), the permutation of indices 1 and 2 in the quantity **L**<sup>12</sup> is not performed and the sign of *L* ≡ |**L**| ≡ |**L**12| does not change.

Using Equation (51), for the case of different electron–phonon interactions on the donor and acceptor in the framework of the Einstein model of nuclear vibrations (ωκ = constant ≡ ω), the simple analytical expression for the rate constant has been obtained [17,18]:

$$K \propto \frac{1}{2} \left[ \left( \frac{E\_1 e^{-t} + E\_2 e^t}{E\_1 t^t + E\_2 e^{-t}} \right)^{\frac{\nu\_{12}}{2\omega}} + \left( \frac{E\_1 e^{-t} + E\_2 e^t}{E\_1 t^t + E\_2 e^{-t}} \right)^{-\frac{\nu\_{12}}{2\omega}} \right]$$

$$\times \exp\left\{ -\frac{2L}{a} - \frac{E\_1 + E\_2}{\hbar \omega} \coth \frac{\hbar \omega}{2k\_\mathrm{B} T} + \frac{\sqrt{(E\_1 e^{-t} + E\_2 e^t)(E\_1 t^t + E\_2 e^{-t})}}{\hbar \omega \sinh(\hbar \omega / 2k\_\mathrm{B} T)} \right. \tag{52}$$

$$+ \left( \frac{\hbar \omega}{2k\_\mathrm{B} T} - t \right) \frac{\omega \nu\_{12}}{\omega} - \frac{\hbar \omega \sinh(\hbar \omega / 2k\_\mathrm{B} T)}{2 \sqrt{(E\_1 e^{-t} + E\_2 e^t)(E\_1 t^t + E\_2 e^{-t})}} \left( \frac{\omega \nu\_{12}}{a^t} \right)^2 \right\}$$

where *e*±*<sup>t</sup>* ≡ exp(±*t*). Substituting *E*<sup>2</sup> = *E*<sup>1</sup> ≡ *E* into Equation (52), we obtain Equation (40) for the rate constant in the case of the same electron–phonon interaction on the donor and acceptor. It is easy to see that Equation (52) satisfies the relationship of detailed balance in the simplified version of dozy-chaos mechanics (Equation (42)).
