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

The Hamiltonian for describing the elementary electron transfers in condensed media has the form [1–9]:

$$H = -\frac{\hbar^2}{2m}\Delta\_\mathbf{r} + V\_1(\mathbf{r}) + V\_2(\mathbf{r} - \mathbf{L}) + \sum\_\mathbf{k} V\_\mathbf{k}(\mathbf{r})q\_\mathbf{k} + \frac{1}{2}\sum\_\mathbf{k} \hbar \omega\_\mathbf{k} \left(q\_\mathbf{k}^2 - \frac{\partial^2}{\partial q\_\mathbf{k}^2}\right) \tag{1}$$

where 1 and 2 are the indices of the electron donor and acceptor, respectively; *m* is the effective mass of the electron; **r** is the electron's radius vector; *q*<sup>κ</sup> are the real normal phonon coordinates; ωκ are the eigenfrequencies of normal vibrations; κ is the phonon index; <sup>κ</sup> *V*κ(**r**)*q*<sup>κ</sup> is the electron–phonon coupling term. In comparison with the Hamiltonian in the standard theory of many-phonon transitions (see [29]), in the theory of elementary electron transfers, the Hamiltonian is complicated merely by an extra electron potential well *V*2(**r** − **L**) set apart from the original well *V*1(**r**) by the distance *L* ≡ |**L**| [5,6]. The nuclear reorganization energy *E* associated with the reorganization of the structure of the nuclear subsystem of the molecular system during electronic transitions in it (see Section 1), in this case, during elementary electron transfers in condensed matter, is defined as follows [2–4]

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

where *q*<sup>κ</sup> are the shifts of the normal phonon coordinates *<sup>q</sup>*κ, which correspond to the shifts in the equilibrium positions of the nuclei, caused by the presence of an electron in the medium on the donor 1 or on the acceptor 2.

The solution to the problem is sought by Green's function method:

$$G\_{H}(\mathbf{r}, \mathbf{r}'; q, q'; \; E\_{H}) = \sum\_{\mathbf{s}} \frac{\Psi\_{s}(\mathbf{r}, q) \, \Psi\_{\mathbf{s}} \* \left(\mathbf{r}', q'\right)}{E\_{H} - E\_{\mathbf{s}} - i\gamma} \tag{3}$$

where Ψ*s*(**r**, *q*) are the eigenfunctions of the total Hamiltonian *H* of the system—in our case, the Hamiltonian (1); (**r**, *q*) is the set of all electronic and nuclear (phonon) coordinates; *Es* are the eigenvalues of *H* and *EH* is the exact value of the total energy of the system; *i*γ (γ > 0) is the standard, infinitesimally small imaginary additive—the energy denominator vanishes when γ = 0; the aforementioned singularity in the rates of "quantum" transitions is eliminated by replacing γ in the energy denominator of Green's function (3) with a finite quantity [2–9]. The general formula for the rate constant of electron photo-transfers is obtained using the technique first described by Egorov [15,16], which generalizes the generating polynomial technique of Krivoglaz and Pekar [30,31] in the theory of many-phonon processes [29]; see the review article [2] for details.

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

The general result for the rate constant (optical absorption) *K* is expressed in terms of Green's function of the elementary electron-charge transfers and two generating functions (see [2,5]):

$$\begin{aligned} K &\propto \sum\_{\omega\_1=-\infty}^{\infty} \sum\_{\omega\_1'=-\infty}^{\infty} G^{\mathbb{E}}(\omega\_1, L) G^{\mathbb{E}} \ast \left(\omega\_1', L\right) \\ \times \frac{1}{(2\pi i)^3} \oint \frac{dx}{x^{\omega\_1+1}} \oint \frac{dy}{y^{\omega\_1'+1}} \oint \frac{dz}{z^{\omega\_1'2+1}} \mathcal{Q}(\overline{n}\_1; x, y, z) \mathcal{S}(\overline{n}\_1; x, y, z) \end{aligned} \tag{4}$$

where the contours encircle the points *x* = 0, *y* = 0, and *z* = 0, correspondingly. Green's function of the elementary electron-charge transfers *G*E(ω1, *L*) and the generating functions *Q*(*n*1; *x*, *y*, *z*) and *S*(*n*1; *x*, *y*, *z*) can be found in [2,5], where *n*<sup>1</sup> ≡ *n*κ1,*l*<sup>1</sup> (Planck's distribution function) is as follows

$$\overline{m}\_{\mathbf{x}1,l1} = \left[ \exp\left(\hbar\omega\_{\mathbf{x},l}/k\_{\mathbf{B}}T\right) - 1\right]^{-1} \tag{5}$$

The energy -Ω of the absorbed photon and the heat energy ω<sup>12</sup> > 0 are related by the law of conservation of energy:

$$
\hbar\hbar\Omega = J\_1 - J\_2 + \hbar\omega\_{12} \tag{6}
$$

*J*<sup>1</sup> is the electron binding energy on the donor 1 and *J*<sup>2</sup> is the electron-binding energy on the acceptor 2. The heat energy ω<sup>12</sup> < 0 corresponds to the inverse processes relative to optical absorption, i.e., to luminescence [3,29] (see Section 10). The wavelength λ, indicated on the *x*-axis in the figures below, corresponds to the frequency Ω in Equation (6) by the standard formula λ = 2π*c*/Ω*n*refr (*c* and *n*refr are the speed of light in vacuum and the refractive index, respectively). The conservation law (Equation (6)) corresponds to the entire shape of the optical band as a whole: by varying the heat energy ω12, we vary the frequency of light Ω and determine one or another part of the absorption band [2–9,29].
