**6. Passage to the Limit to the Standard Theory of Many-Phonon Transitions and the Symmetry of the Standard Result. The Reason for the Asymmetry of the Optical Absorption Band Shape in Dozy-Chaos Mechanics**

The limit passage from expressions (6)–(26) for the optical absorption *K* to the standard result in the theory of many-phonon transitions [29] can be realized by letting the dozy-chaos energy γ tend to infinity (θ<sup>0</sup> = *E*/γ → 0 according to Equation (12)) in Equation (10) for *t* (see Figure 3 in [2]) and to zero (θ<sup>0</sup> → ∞) in Equation (24) for *K<sup>e</sup>* <sup>0</sup> (see Equation (162) in [2]). An equation of the standard type for the optical absorption *K* (for *k*B*T* > ωκ/2) is thus obtained [2,5]:

$$K = \frac{a^2 \hbar}{\sqrt{4\pi \lambda\_\text{r} k\_\text{B} T}} \exp\left(-\frac{2L}{a}\right) \exp\left[-\frac{\left(\hbar \omega\_{12} - \lambda\_\text{r}\right)^2}{4\lambda\_\text{r} k\_\text{B} T}\right] \tag{27}$$

where *<sup>a</sup>* <sup>≡</sup> -/ √ 2*mJ*<sup>1</sup> and λ<sup>r</sup> ≡ 2*E*. A formula of this type was obtained by Markus in his electron-transfer model [32–37] and is often called the Marcus formula, and the energy λ<sup>r</sup> is called the reorganization energy of Marcus. Similar and more general formulas were previously obtained in the theory of many-phonon transitions (see [29,38]) for optical transitions by Huang and Rhys [39] and Pekar [40–42] (see also Lax [43] and Krivoglaz and Pekar [30]), and for nonradiative transitions, by Huang and Rhys [39] and Krivoglaz [31].

The result in the standard theory of many-phonon transitions, given by Equation (27) and corresponding to high (i.e., 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, large values of γ in dozy-chaos mechanics correspond to a pronounced chaos in the transient state and, hence, a weak organization of the quantum-classical molecular transition (see Section 1). With a decrease in the dozy-chaos energy γ, the transient state becomes less chaotic and the organization of the quantum-classical transition increases, which is manifested in the appearance of a

narrow optical absorption peak in the red spectral region and a strong asymmetry of the absorption band shape (see Figure 1).

We also note that the half-width of the Gaussian function for the shape of the optical absorption band (Equation (27))

$$w\_{1/2} = 2\sqrt{2\ln 2}\sqrt{2\lambda\_{\text{r}}k\_{\text{B}}T} \tag{28}$$

is determined both by the individual properties of the "donor-acceptor + medium" system, which are expressed in the reorganization energy λr, and by the properties of an ensemble of these systems, which are expressed in temperature *T*. In other words, even within the framework of the well-known standard theory of many-phonon transitions [29,38], the effects of homogeneous and inhomogeneous broadening in the optical band cannot be separated. The introduction into the theory of a new "homogeneous effect" in the form of the dozy-chaos energy γ in Green's function of the system (Equation (3)) further confuses homogeneous and inhomogeneous effects in the shape of an optical band, greatly complicating the analytical result for it (cf. Equation (27) and Equations (6)–(26)). A discussion of the physical meaning of each of the terms included in this complex analytical result (Equations (6)–(26)) can be found in [2,4]. Our complex result gives a greater variety of optical band shapes (see, e.g., Figure 1) compared to the two band shapes, Lorentzian and Gaussian, which are the result of homogeneous and inhomogeneous effects known from the standard quantum theory of spectral line broadening. These two differences can only be understood in an open quantum system framework where the quantum system is coupled to an external classical bath. In contrast to the standard quantum theory, where the dynamics of quantum transitions are not considered, in quantum-classical mechanics, this bath, which is already quantum here, enters the entire closed quantum "donor-acceptor + medium" system (see the last, phonon term in the Hamiltonian (1)) and becomes classical only in a dynamic (chaotic) transient state (see Green's function (3) with γ >> ω).
