**8. Implementation of the Egorov Resonance in the Quasi-Symmetric Serious of Optical Band Shapes of a Representative Polymethine Dye**

Experimentally, the dynamic electron-nuclear-reorganization resonance (the Egorov resonance, see Section 7) manifests itself, for example, in polymethine dyes [3–9], namely, in the resonance nature

of the dependence of the shape of the optical absorption band on the length of the polymethine chain *L* (see Figure 2). The optical band with *n* = 3 corresponds to the Egorov resonance or is close to it.

**Figure 2.** Experimental [44,45] (**a**) and theoretical [6] (**b**) monomers' optical absorption spectra, dependent on the length of the polymethine chain *L* = 2(*n* + 2)d , where *d* are certain roughly equal bond lengths in the chain (thiapolymethinecyanine in methanol at room temperature; ε is the extinction coefficient) [4,9]. The optical absorption band with *n* = 3 corresponds to the dynamic electron-nuclear-reorganization resonance (the Egorov resonance, see Section 7) or is close to it. (Original citation)—Reproduced by permission of The Royal Society of Chemistry. For the short chains (*n* = 0, 1, 2, 3), the tunnel effects, associated with the quantity η in Equation (25), can be neglected (η = 1). For the long chains (*n* = 4, 5), the tunnel effects are small but they must be taken into account (η < 1). The absorption bands are computed by Equations (6)-(26) with η ≤ 1 instead of the Gamow tunnel factor (Equation (25)) when fitting them to the experimental data of Brooker and co-workers [44] (*a*) in terms of wavelength λmax, extinction εmax, and half-width *w*1/2 with a high degree of accuracy. The following parameters of the "dye + environment" system are used [6]: *m* = *m*e, where *m*<sup>e</sup> is the electron mass; *d* = 0.14 nm; <sup>ω</sup> = <sup>5</sup> <sup>×</sup> <sup>10</sup><sup>13</sup> <sup>s</sup><sup>−</sup>1; *<sup>n</sup>*refr = 1.33; for *<sup>n</sup>* = 0, 1, 2, 3, 4, 5, one has *<sup>J</sup>*<sup>1</sup> = (5.63, 5.40, 4.25, 3.90, 3.74, 3.40) eV, *J*<sup>1</sup> − *J*<sup>2</sup> = (1.71, 1.31, 1.11, 0.90, 0.74, 0.40) eV, *E* = (0.245, 0.248, 0.256, 0.275, 0.297, 0.496) eV, and γ = (0.402, 0.205, 0.139, 0.120, 0.129, 0.131) eV, respectively; for *n* = 0, 1, 2, 3, factor η = 1, and for *n* = 4, 5, factor η = 0.55, 0.1, respectively; *T* = 298 K.

To fit the theoretical result for the optical bands, which is given by Equations (6)–(26) to the corresponding experimental data (Figure 2a), we need estimated numerical values for the ground-state energies of the dye monomers, *J*1*M*, and also for the energy gaps between their ground and excited states, *J*1*<sup>M</sup>* − *J*2*M*. These estimates follow from literature data [5,46–49]: *J*1*<sup>M</sup>* 5 eV and *J*1*<sup>M</sup>* − *J*2*<sup>M</sup>* 1 eV. In addition, we need estimated numerical values for the reorganization energy of the nuclear environment of dye monomers, *EM*. The estimate of *EM* is found from the Egorov resonance (see Equations (29)–(31)) from the length of the optical chromophore [5,6]: <sup>√</sup>2*J*1*M*/*<sup>m</sup>* <sup>2</sup>*LM* <sup>=</sup> *EM* - , where *LM* is the length of the optical chromophore of the dye monomers (*LM* = 10*d*, *d* = 0.14 nm; see the caption to Figure 2).

Under resonance conditions (Equation (29)), the motion of the reorganization of the nuclei of the medium significantly contributes to the electronic transition in the optical π-electron chromophore—the polymethine chain with *n* = 3 as compared to the electronic transition in the optical π-electron chromophores of polymethine dyes with *n* - 3. As can be seen from the numerical data in the caption to Figure 2, the series of the dozy-chaos energies γ has a minimum at *n* = 3. Therefore, the appearance of the resonant band corresponding to *n* = 3 can also be interpreted as the transfer of chaos (dozy chaos) from the peak of a band into its wing(s) by a chaotic motion of the quantum-classical π-electronic state of the polymethine chain embedded in the medium as a result of the transition from "non-resonant" chains with *n* -3 to the "resonant" chain with *n* = 3.

Thus, the presence of symmetry in the shape of an optical band at high (room) temperatures is associated with a primitive, Franck–Condon picture of the dynamics of molecular "quantum" transitions. The loss of this symmetry and the appearance of a peak against the background of a wide band wing are related, as already noted in Section 6, to the effect of self-organization of transition dynamics in dozy-chaos mechanics, which is expressed, in particular, in the "pumping" of dozy chaos from one part of the optical band to another part (from the peak region to the wing). Therefore, the series for the shape of the optical bands of a representative polymethine dye, thiapolymethinecyanine, depending on the length of its polymethine chain, has a quasi-symmetric character with respect to the Egorov resonance (see Figure 2), which corresponds to the most organized quantum-classical transition.
