#
Quantum–Classical Mechanics: Nano-Resonance in Polymethine Dyes^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Quantum–Classical Mechanics in Brief

## 3. Nano-Resonance

## 4. Nano-Resonance in Polymethine Dyes

## 5. Analytical Results in the Quantum–Classical Mechanics of Elementary Electron Transfers

## 6. Passage to the Standard Quantum–Mechanical Theory and the Formal Origin of Finite Gamma Values

## 7. Entanglement of Homogeneous and Inhomogeneous Effects in Quantum–Classical Mechanics

## 8. Introduction to History of Researching the Nature of the J-Band of J-Aggregates

## 9. J-Aggregation as a Structural Tuning into the Egorov Nano-Resonance

_{J}(Enr) = (6 + 2) 0.14 nm = 1.12 nm (see the captions to Figure 4 and Figure 5).

**bottom**) shows the results of fitting the theoretical shape of the optical absorption band (7)–(28) (with the replacement of the Gamow factor $\eta \equiv \mathrm{exp}\left(-\frac{4\theta}{1-{\xi}^{2}}\right)$ in formula (25) by the number 1, see rationale in Section 4), obtained on basis of quantum–classical mechanics [5,22,23,24,25,26], to Hertz’s experimental data on concentration $\mathrm{M}\leftrightarrow \mathrm{J}$-equilibrium [36,74]. It turns out that the number of molecules that form the J-chromophore is four, as in Hertz (cf. Figure 5). In addition, Figure 6B (

**bottom**) shows that the experimental data at high concentrations have a lower accuracy compared to Figure 6B (

**top**). The reason for this is the high probability of the formation of not only J-aggregates, but also colloidal particles, at high dye concentrations in an aqueous solution [22,23,24,25,26].

## 10. On Ad Hoc and By-Product Theoretical Approaches

## 11. Direct and Reverse Processes

## 12. Nano-Resonance as Antiadiabatic Invariant in the Dynamics of a Transient State and Its “Antisymmetric Twin”

## 13. Nano-Resonance in Two-Photon Absorption

## 14. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Egorov, V.V.; Thomas, S. Quantum-classical mechanics: On the problem of a two-photon resonance band shape in polymethine dyes. Nano-Struct. Nano-Objects
**2021**, 25, 100650. [Google Scholar] [CrossRef] - Egorov, V.V. Dynamic symmetry in dozy-chaos mechanics. Symmetry
**2020**, 12, 1856. [Google Scholar] [CrossRef] - Egorov, V.V. Dozy-chaos mechanics for a broad audience. Challenges
**2020**, 11, 16. [Google Scholar] [CrossRef] - Egorov, V.V. Quantum-classical electron as an organizing principle in nature. Int. J. Sci. Technol. Soc.
**2020**, 8, 93–103. [Google Scholar] [CrossRef] - Egorov, V.V. Quantum-classical mechanics as an alternative to quantum mechanics in molecular and chemical physics. Heliyon Phys.
**2019**, 5, e02579. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Egorov, V.V. Quantum-classical mechanics: Luminescence spectra in polymethine dyes and J-aggregates. Nature of the small Stokes shift. Results Phys.
**2019**, 13, 102252. [Google Scholar] [CrossRef] - Egorov, V.V. Nature of the optical band shapes in polymethine dyes and H-aggregates: Dozy chaos and excitons. Comparison with dimers, H*- and J-aggregates. R. Soc. Open Sci.
**2017**, 4, 160550. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Egorov, V.V. Optical lineshapes for dimers of polymethine dyes: Dozy-chaos theory of quantum transitions and Frenkel exciton effect. RSC Adv.
**2013**, 3, 4598–4609. [Google Scholar] [CrossRef] - Petrenko, A.; Stein, M. Toward a molecular reorganization energy-based analysis of third-order nonlinear optical properties of polymethine dyes and J-aggregates. J. Phys. Chem. A
**2019**, 123, 9321–9327. [Google Scholar] [CrossRef] - Born, M.; Oppenheimer, J.R. Quantum theory of the molecules. Ann. Phys.
**1927**, 84, 457–484. [Google Scholar] [CrossRef] - Perlin, Y.E. Modern methods in the theory of many-phonon processes. Sov. Phys. Uspekhi
**1964**, 6, 542–565. [Google Scholar] [CrossRef] - Frank-Kamenetskii, M.D.; Lukashin, A.V. Electron-vibrational interactions in polyatomic molecules. Sov. Phys. Uspekhi
**1975**, 18, 391–409. [Google Scholar] [CrossRef] - Bersuker, I.B.; Polinger, V.Z. Vibronic Interactions in Molecules and Crystals; Springer: New York, NY, USA, 1989. [Google Scholar]
- Stanke, M. Adiabatic, Born-Oppenheimer, and non-adiabatic approaches. In Handbook of Computational Chemistry; Leszczynski, J., Kaczmarek-Kedziera, A., Puzyn, T., Papadopoulos, M.G., Reis, H., Shukla, M.K., Eds.; Springer: Cham, Switzerland, 2017; pp. 173–223. [Google Scholar]
- Franck, J.; Dymond, E.G. Elementary processes of photochemical reactions. Trans. Faraday Soc.
**1925**, 21, 536–542. [Google Scholar] [CrossRef] - Condon, E.U. A theory of intensity distribution in band systems. Phys. Rev.
**1926**, 28, 1182–1201. [Google Scholar] [CrossRef] - Condon, E.U. Nuclear motions associated with electron transitions in diatomic molecules. Phys. Rev.
**1928**, 32, 858–872. [Google Scholar] [CrossRef] - Condon, E.U. The Franck-Condon principle and related topics. Am. J. Phys.
**1947**, 15, 365–374. [Google Scholar] [CrossRef] - Herzberg, G.; Spinks, J.W.T. Molecular Spectra and Molecular Structure. 1. Spectra of Diatomic Molecules; Prentice-Hall: New York, NY, USA, 1939. [Google Scholar]
- Herzberg, G. Molecular Spectra and Molecular Structure. 2. Infrared and Raman Spectra; D. Van Nostrand: Princeton, NJ, USA, 1945. [Google Scholar]
- Herzberg, G. Molecular Spectra and Molecular Structure. 3. Electronic Spectra and Electronic Structure of Polyatomic Molecules; Van Nostrand Reinhold: New York, NY, USA; London, UK, 1966. [Google Scholar]
- Egorov, V.V. Electron-transfer approach to the nature of the optical lineshape for molecular J-aggregates. Chem. Phys. Lett.
**2001**, 336, 284–291. [Google Scholar] [CrossRef] - Egorov, V.V. On electrodynamics of extended multiphonon transitions and nature of the J-band. Chem. Phys.
**2001**, 269, 251–283. [Google Scholar] [CrossRef] - Egorov, V.V. Nature of the optical transition in polymethine dyes and J-aggregates. J. Chem. Phys.
**2002**, 116, 3090–3103. [Google Scholar] [CrossRef] - Egorov, V.V.; Alfimov, M.V. Theory of the J-band: From the Frenkel exciton to charge transfer. Phys. Uspekhi
**2007**, 50, 985–1029. [Google Scholar] [CrossRef] - Egorov, V.V. Theory of the J-band: From the Frenkel exciton to charge transfer. Phys. Procedia
**2009**, 2, 223–326. [Google Scholar] [CrossRef] [Green Version] - Egorov, V.V. Discovery of Dozy Chaos and Discovery of Quanta: Analogy Being in Science and Perhaps in Human Progress. In Chaos and Complex Systems, Proceedings of the 4th International Interdisciplinary Chaos Symposium, Antalya, Turkey, 29 April–2 May 2012; Stavrinides, S.G., Banerjee, S., Caglar, H., Ozer, M., Eds.; Springer: Berlin, Germany, 2013; pp. 41–46. [Google Scholar] [CrossRef]
- Egorov, V.V. Dozy Chaos in Chemistry: Simplicity in Complexity. In Chaos and Complex Systems, Proceedings of the 4th International Interdisciplinary Chaos Symposium, Antalya, Turkey, 29 April–2 May 2012; Stavrinides, S.G., Banerjee, S., Caglar, H., Ozer, M., Eds.; Springer: Berlin, Germany, 2013; pp. 219–224. [Google Scholar] [CrossRef]
- Dirac, P.A.M. The quantum theory of the emission and absorption of radiation. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1927**, 114, 243–265. [Google Scholar] [CrossRef] [Green Version] - Fermi, E. Quantum theory of radiation. Rev. Mod. Phys.
**1932**, 4, 87–132. [Google Scholar] [CrossRef] - Berestetskii, V.B.; Lifshitz, E.M.; Pitaevskii, L.P. Quantum Electrodynamics, 2nd ed.; Elsevier: Amsterdam, The Netherlands, 1982. [Google Scholar]
- Davydov, A.S. Quantum Mechanics; Pergamon Press: Oxford, UK, 1976. [Google Scholar]
- Landau, L.D.; Lifshitz, E.M. Quantum Mechanics, Non-Relativistic Theory, 3rd ed.; Elsevier: Amsterdam, The Netherlands, 1977. [Google Scholar]
- Planck, M. On the law of distribution of energy in the normal spectrum. Ann. Phys.
**1901**, 309, 553–563. [Google Scholar] [CrossRef] - Brooker, L.G.S.; Sprague, R.H.; Smith, C.P.; Lewis, G.L. Color and constitution. I. Halochromism of anhydronium bases related to the cyanine dyes. J. Am. Chem. Soc.
**1940**, 62, 1116–1125. [Google Scholar] [CrossRef] - James, T.H. (Ed.) The Theory of the Photographic Process; Macmillan: New York, NY, USA, 1977. [Google Scholar]
- Egorov, V.V. Dryad Digital Repository. Data from: R. Soc. Open Sci.
**2017**, 4, 160550. [Google Scholar] [CrossRef] - Dähne, S. Color and constitution: One hundred years of research. Science
**1978**, 199, 1163–1167. [Google Scholar] [CrossRef] - Kachkovskii, A.D. The nature of electronic transitions in linear conjugated systems. Russ. Chem. Rev.
**1997**, 66, 647–664. [Google Scholar] [CrossRef] - Petrenko, A.; Stein, M. Molecular Reorganization energy as a key determinant of J-band formation in J-aggregates of polymethine dyes. J. Phys. Chem. A
**2015**, 119, 6773–6780. [Google Scholar] [CrossRef] [PubMed] - Egorov, V.V. Dynamic pumping of elementary charge transfer by environmental dissipative reorganization. Russ. J. Electrochem.
**2003**, 39, 86–96. [Google Scholar] [CrossRef] - Marcus, R.A. On the theory of oxidation-reduction reactions involving electron transfer. I. J. Chem. Phys.
**1956**, 24, 966–978. [Google Scholar] [CrossRef] [Green Version] - Marcus, R.A. Electrostatic free energy and other properties of states having nonequilibrium polarization. J. Chem. Phys.
**1956**, 24, 979–989. [Google Scholar] [CrossRef] [Green Version] - Marcus, R.A. On the theory of oxidation-reduction reactions involving electron transfer. II. Applications to data on the rates of isotopic exchange reactions. J. Chem. Phys.
**1957**, 26, 867–871. [Google Scholar] [CrossRef] - Marcus, R.A. On the theory of oxidation-reduction reactions involving electron transfer. III. Applications to data on the rates of organic redox reactions. J. Chem. Phys.
**1957**, 26, 872–877. [Google Scholar] [CrossRef] [Green Version] - Marcus, R.A.; Sutin, N. Electron transfers in chemistry and biology. Biochim. Biophys. Acta
**1985**, 811, 265–322. [Google Scholar] [CrossRef] - Marcus, R.A. Electron transfer reactions in chemistry. Theory and experiment. Rev. Mod. Phys.
**1993**, 65, 599–610. [Google Scholar] [CrossRef] [Green Version] - Huang, K.; Rhys, A. Theory of light absorption and non-radiative transitions in F-centres. Proc. R. Soc. A
**1950**, 204, 406–423. [Google Scholar] - Pekar, S.I. Theory of F-centers. Zh. Eksp. Teor. Fiz.
**1950**, 20, 510–522. (In Russian) [Google Scholar] - Pekar, S.I. To the theory of luminescence and light absorption by impurities in dielectrics. Zh. Eksp. Teor. Fiz.
**1952**, 22, 641–657. (In Russian) [Google Scholar] - Pekar, S.I. On the effect of lattice deformations by electrons on optical and electrical properties of crystals. Uspekhi Fiz. Nauk
**1953**, 50, 197–252. (In Russian) [Google Scholar] [CrossRef] - Lax, M. The Franck-Condon principle and its application to crystals. J. Chem. Phys.
**1952**, 20, 1752–1760. [Google Scholar] [CrossRef] - Krivoglaz, M.A.; Pekar, S.I. The shape of the spectra of the impurity light absorption and luminescence in dielectrics. Tr. Inst. Fiz. Akad. Nauk UKR SSR
**1953**, 4, 37–70. (In Russian) [Google Scholar] - Krivoglaz, M.A. The theory of thermal transitions. Zh. Eksp. Teor. Fiz.
**1953**, 25, 191–207. (In Russian) [Google Scholar] - Jelley, E.E. Spectral absorption and fluorescence of dyes in the molecular state. Nature
**1936**, 138, 1009–1010. [Google Scholar] [CrossRef] - Jelley, E.E. Molecular, nematic and crystal states of 1:1′-diethyl-ψ-cyanine chloride. Nature
**1937**, 139, 631–632. [Google Scholar] [CrossRef] - Scheibe, G. Variability of the absorption spectra of some sensitizing dyes and its cause. Angew. Chem.
**1936**, 49, 563. [Google Scholar] - Scheibe, G. On the variability of the absorption spectra in solutions and the secondary bonds as its cause. Angew. Chem.
**1937**, 50, 212–219. [Google Scholar] [CrossRef] - Davydov, A.S. Theory of Molecular Excitons; McGraw-Hill: New York, NY, USA, 1962. [Google Scholar]
- Franck, J.; Teller, E. Migration and photochemical action of excitation energy in crystals. J. Chem. Phys.
**1938**, 6, 861–872. [Google Scholar] [CrossRef] - Würthner, F.; Kaiser, T.E.; Saha-Möller, C.R. J-aggregates: From serendipitous discovery to supra-molecular engineering of functional dye materials. Angew. Chem. Int. Ed.
**2011**, 50, 3376–3410. [Google Scholar] [CrossRef] [PubMed] - Bricks, J.L.; Slominskii, Y.L.; Panas, I.D.; Demchenko, A.P. Fluorescent J-aggregates of cyanine dyes: Basic research and applications review. Methods Appl. Fluoresc.
**2018**, 6, 012001. [Google Scholar] [CrossRef] [Green Version] - Knapp, E.W. Lineshapes of molecular aggregates, exchange narrowing and intersite correlation. Chem. Phys.
**1984**, 85, 73–82. [Google Scholar] [CrossRef] - Makhov, D.V.; Egorov, V.V.; Bagatur’yants, A.A.; Alfimov, M.V. Numerical calculations of optical lineshapes for disordered molecular aggregates. Chem. Phys. Lett.
**1995**, 246, 371–380. [Google Scholar] [CrossRef] - Makhov, D.V.; Egorov, V.V.; Bagatur’yants, A.A.; Alfimov, M.V. Efficient approach to the numerical calculation of optical line shapes for molecular aggregates. J. Chem. Phys.
**1999**, 110, 3196–3199. [Google Scholar] [CrossRef] - Eisfeld, A.; Briggs, J.S. The J-band of organic dyes: Lineshape and coherence length. Chem. Phys.
**2002**, 281, 61–70. [Google Scholar] [CrossRef] - Spano, F.C. The spectral signatures of Frenkel polarons in H- and J-aggregates. Acc. Chem. Res.
**2010**, 43, 429–439. [Google Scholar] [CrossRef] [PubMed] - Frenkel, J. On the transformation of light into heat in solids. I. Phys. Rev.
**1931**, 37, 17–44. [Google Scholar] [CrossRef] - Frenkel, J. On the transformation of light into heat in solids. II. Phys. Rev.
**1931**, 37, 1276–1294. [Google Scholar] [CrossRef] - Kachkovski, O.; Tolmachov, O.; Slominskii, Y.; Kudinova, M.; Derevyanko, N.; Zhukova, O. Electronic properties of polymethine systems 7: Soliton symmetry breaking and spectral features of dyes with a long polymethine chain. Dyes Pigment.
**2005**, 64, 207–216. [Google Scholar] [CrossRef] - Guerrini, M.; Calzolari, A.; Varsano, D.; Corni, S. Quantifying the plasmonic character of optical excitations in a molecular J-aggregate. J. Chem. Theory Comput.
**2019**, 15, 3197–3203. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Egorov, V.V. Optical line shapes for polymethine dyes and their aggregates: Novel theory of quantum transitions and its correlation with experiment. J. Lumin.
**2011**, 131, 543–547. [Google Scholar] [CrossRef] - Egorov, V.V. Nature of the narrow optical band in H*-aggregates: Dozy-chaos-exciton coupling. AIP Adv.
**2014**, 4, 077111. [Google Scholar] [CrossRef] [Green Version] - Herz, A.H. Aggregation of sensitizing dyes in solution and their adsorption onto silver halides. Adv. Colloid Interface Sci.
**1977**, 8, 237–298. [Google Scholar] [CrossRef] - Kuhn, H.; Kuhn, C. Chromophore coupling effects. In J-Aggregates; Kobayashi, T., Ed.; World Scientific: Singapore, 1996; pp. 1–40. [Google Scholar]
- Zhao, Y.S.; Fu, H.; Peng, A.; Ma, Y.; Liao, Q.; Yao, J. Construction and optoelectronic properties of organic one-dimensional nanostructures. Acc. Chem. Res.
**2010**, 43, 409–418. [Google Scholar] [CrossRef] - Hunter, C.A.; Sanders, J.K.M. The nature of π-π interactions. J. Am. Chem. Soc.
**1990**, 112, 5525–5534. [Google Scholar] [CrossRef] - McGaughey, G.B.; Gagné, M.; Rappé, A.K. π-stacking interactions. J. Biol. Chem.
**1998**, 273, 15458–15463. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bloom, J.W.G.; Wheeler, S.E. Taking the aromaticity out of aromatic interactions. Angew. Chem.
**2011**, 123, 7993–7995. [Google Scholar] [CrossRef] - Martinez, C.R.; Iverson, B.L. Rethinking the term “π-stacking”. Chem. Sci.
**2012**, 3, 2191–2201. [Google Scholar] [CrossRef] [Green Version] - Wheeler, S.E.; Bloom, J.W.G. Toward a more complete understanding of noncovalent interactions involving aromatic rings. J. Phys. Chem. A
**2014**, 118, 6133–6147. [Google Scholar] [CrossRef] [Green Version] - Avakyan, V.G.; Shapiro, B.I.; Alfimov, M.V. Dimers, tetramers, and octamers of mono- and trimethyne thiacarbocyanine dyes. Structure, formation energy, and absorption band shifts. Dyes Pigment.
**2014**, 109, 21–33. [Google Scholar] [CrossRef] - Wheeler, S.E. Unraveling the origin of substituents effects in π-stacking interactions. In Noncovalent Forces, Challenges and Advances in Computational Chemistry and Physics 19; Scheiner, S., Ed.; Springer International Publishing: Cham, Switzerland, 2015; Chapter 14; pp. 421–442. [Google Scholar]
- Harrison, W.J.; Mateer, D.L.; Tiddy, G.J.T. Liquid-crystalline J-Aggregates formed by aqueous ionic cyanine dyes. J. Phys. Chem.
**1996**, 100, 2310–2321. [Google Scholar] [CrossRef] - Masunov, A.E.; Anderson, D.; Freidzon, A.Y.; Bagatur’yants, A.A. Symmetry-breaking in cationic polymethine dyes: Part 2. Shape of electronic absorption bands explained by the thermal fluctuations of the solvent reaction field. J. Phys. Chem. A
**2015**, 119, 6807–6815. [Google Scholar] [CrossRef] [PubMed] - Aviv, H.; Tischler, Y.R. Synthesis and characterization of a J-aggregating TDBC derivative in solution and in Langmuir-Blodgett films. J. Lumin.
**2015**, 158, 376–383. [Google Scholar] [CrossRef] - Hales, J.M.; Matichak, J.; Barlow, S.; Ohiro, S.; Yesudas, K.; Bredas, J.-L.; Perry, J.W.; Marder, S.R. Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit. Science
**2010**, 327, 1485–1488. [Google Scholar] [CrossRef]

**Figure 1.**The simplest demonstration of singularity in the rate of molecular “quantum” transitions [5,6,7,8]. (

**a**) The physical picture of quantum transitions in a potential box with a moving wall simulates the presence of a singularity in molecular “quantum” transitions when the wall moves without friction [5,6,7,8]. The case of a wall moving with friction corresponds to quantum–classical (dozy-chaos) mechanics [5,22,23,24], in which dozy chaos plays the role of friction. (

**b**) Shapes of optical absorption bands obtained from dozy-chaos mechanics, as a function of dozy-chaos intensity [5,6,7,8]. The band with the most pronounced peak (J-band) corresponds to the weakest dozy chaos. (Original citation)—Reproduced by permission of The Royal Society of Chemistry.

**Figure 2.**Shapes of optical absorption bands and Egorov nano-resonance (n = 3) in a series of polymethine dyes with varying lengths of polymethine chain L = 2 (n + 2) d, where d are certain roughly equal bond lengths (0.14 nm) in the chain (thiapolymethinecyanine in methanol at room temperature; $\epsilon $ is the extinction coefficient) [7,8]. (

**a**) Experimental data [35,36] and (

**b**) theoretical modeling [24,37] based on quantum–classical mechanics [5]. (Original citation)—Reproduced by permission of The Royal Society of Chemistry.

**Figure 3.**Alternation of positive (1) and negative (2) charges on carbon atoms along an ideal polymethine chain (IPS) in the ground state [8,38,39]. The distance between nitrogen atoms (N) is the length of the polymethine chain $L$. (Original citation)—Reproduced by permission of The Royal Society of Chemistry.

**Figure 4.**Optical absorption band shapes in thiapolymethinecyanines [36]. (

**a**) Experimental data on monomers (M), dimers (D), H-, H*- and J-aggregates [36]. (

**b**) Appropriate theoretical fit to these data [7,72]. The fitting parameters for the theoretical band shapes for the monomer, dimer, H-aggregate and H*-aggregate are given in [7,37]. The narrow, intense and red-shifted absorption J-band is computed from Equations (7)–(28), where the “J-aggregate + environment” system parameters are $q\equiv q\left(\mathrm{J}\right)=0.97e$, $m=1.7{m}_{\mathrm{e}}$, $\omega =5\times {10}^{13}{\mathrm{s}}^{-1}$, $d=0.14$ nm, L

_{J}(Enr) = (6 + 2) 0.14 nm = 1.12 nm, ${n}_{\mathrm{ref}}=1.33$, ${J}_{1}=5.4$ eV, ${J}_{1}-{J}_{2}=1.243$ eV, $E=0.22$ eV, $\gamma =0.088$ eV, and $T=298\mathrm{K}$. Regarding the narrow and blue-shifted H*-band, see in [7,37,73].

**Figure 5.**Scheme of polymethine dye chromophore elongation as a result of J-aggregation. The J-chromophore consists of four identical molecules forming a brickwork-like structure [7,36,74,75] in which the $\pi $-electron system of the polymethine chain joins with the $\pi $ -electron system of the benzene rings of two neighboring monomers ($d$ is a some unit bond length). The fourth molecule is “neutral” with respect to the formation of a common $\pi $ -electron system and acts as a stabilizer of the J-chromophore structure [6,7,22,23,24,25,26]. The J-aggregate is a thin and long “rod” composed of J-chromophores [6,7,22,23,24,25,26].

**Figure 6.**At the (

**top**): Hertz experimental data on the optical absorption of monomers (M) and J-aggregates (J) of benzimidazolocarbocyanine, which are in concentration equilibrium in an aqueous NaOH solution ($0.001{\mathrm{mol}\mathrm{L}}^{-1}$) at 25 °C [36,74]. (

**A**) Dye concentration is measured in micromole/liter: 0.5 (1), 1.0 (2), 5.0 (3), 10 (4), 100 (5), and 400 (6). (

**B**) Molar concentrations of the dye monomers (${C}_{\mathrm{M}}$) and J-aggregates (${C}_{\mathrm{J}}$ ) are taken from (

**A**). The number of molecules $N$ in the J-chromophore (Figure 5) is calculated from the law of mass action and is equal to 4. At the (

**bottom**): Theoretical band shapes fitted to Hertz’s experimental data (see at the (

**top**)) [22,23,24]. (

**A**) Absorption M-bands and J-bands are calculated from Equations (7)–(28) with $\eta =1$. The relative dye concentrations in the intermediate states are: 1% (1), 9% (2), 53% (3), 66% (4), 82% (5), and 99% (6). (

**B**) Molar concentrations of the dye monomers (${C}_{\mathrm{M}}$ ) and J-aggregates (${C}_{\mathrm{J}}$ ) are derived from the absolute concentrations reported by Herz (see the caption at the (

**top**)) and relative concentrations obtained by our theoretical fitting. The number of molecules $N$ in the J-chromophore (Figure 5) is calculated from the law of mass action and is equal to 4 (cf. (

**B**) at the (

**top**)). The fitting parameters for the “J-aggregate + environment” and “monomer + environment” systems are ${m}_{\mathrm{J}}=0.86{m}_{\mathrm{e}}$ and ${m}_{\mathrm{M}}=0.97{m}_{\mathrm{e}}$, $\omega =5\times {10}^{13}{\mathrm{s}}^{-1}$, $d=0.14$ nm, ${n}_{\mathrm{ref}}=1.33$; ${J}_{1\mathrm{J}}={J}_{1\mathrm{M}}=5$ eV, ${J}_{1\mathrm{J}}-{J}_{2\mathrm{J}}=1.11$ eV and ${J}_{1\mathrm{M}}-{J}_{2\mathrm{M}}=1.37$ eV, ${E}_{\mathrm{r},\mathrm{J}}=0.420$ eV and ${E}_{\mathrm{r},\mathrm{M}}=0.315$ eV, ${\gamma}_{\mathrm{J}}=0.067$ eV and ${\gamma}_{\mathrm{M}}=0.231$ eV; the total charges transferred along chromophores ${L}_{\mathrm{J}}=8d$ = 1.12 nm and ${L}_{\mathrm{M}}=6d$ = 0.84 nm of J-aggregate and monomer are ${q}_{\mathrm{J}}\cong \sqrt{2{\epsilon}_{\mathrm{d}}{L}_{\mathrm{J}}{E}_{\mathrm{r},\mathrm{J}}}\approx 1.28e$ and ${q}_{\mathrm{M}}\cong \sqrt{2{\epsilon}_{\mathrm{d}}{L}_{\mathrm{M}}{E}_{\mathrm{r},\mathrm{M}}}\approx 0.96e$, where permittivity ${\epsilon}_{\mathrm{d}}=2.5$ [75] (contribution from $\sigma $ -electrons and the solvent). Reprinted from [22], Copyright 2001, with permission from Elsevier.

**Figure 7.**Change in theoretical spectra of J-aggregate with deflection from the nano-resonance condition (Equations (4)–(6)) [40]. The black line describes a J-band, consisting of L-peak and D-wing, with the following parameters: $E=0.2\mathrm{eV}$, $\gamma =0.11\mathrm{eV}$, $L=1.68\mathrm{nm}$, ${J}_{1}=5.5\mathrm{eV}$, and ${J}_{2}=3.72\mathrm{eV}$; the red line describes a “J-band” with the following parameters: $E=0.4\mathrm{eV}$, $\gamma =0.15\mathrm{eV}$, $L=2.8\mathrm{nm}$, ${J}_{1}=5.5\mathrm{eV}$, and ${J}_{2}=4.0\mathrm{eV}$ [40]. Presented theoretical spectra generate changes in the sharp J-band of THIAMS dye (Table 3 in [40]) in aqueous solution at 0.80% w/w dye into a wide “J-band” at 3.53% w/w dye [84]. Reprinted (adapted) with permission from Petrenko A, Stein M. [40]. Copyright 2015, American Chemical Society.

**Figure 8.**Symmetric picture of absorption (see Figure 1b) [7,8] and luminescence spectra obtained by changing the sign in the heat energy $\Delta \to -\Delta $ (Equation (31)) in theoretical absorption spectra (Equations (7)–(27)) [6] according to standard quantum mechanics of condensed matter [11].

**Figure 9.**Comparison of theory with experiment for J-aggregates in Langmuir films. (

**a**) Absorption and fluorescence J-bands of a C18S4 monolayer obtained in a Langmuir film [86]. The characteristics of photoluminescence (PL) spectrum: a peak at 590 nm, a FWHM = 18 nm and a Stokes shift of 1 nm [86]. (

**b**) Theoretical absorption and fluorescence J-bands [6], fitted to the experimental data [86] (see (

**a**)). The absorption J-band is computed from Equations (7)–(27), where the “J-aggregate + environment” system parameters are $m={m}_{\mathrm{e}}$ (${m}_{\mathrm{e}}$ is the electron mass); ${n}_{\mathrm{ref}}=1.33$; $\omega =5\times {10}^{13}{\mathrm{s}}^{-1}$; ${J}_{1}=4\mathrm{eV}$; ${J}_{1}-{J}_{2}=1.19\mathrm{eV}$; $L=\left(6+2\right)d$ ($2d$ lengthening the optical chromophore shown in Figure 10 due to the J-aggregation [6,7,22,23,24,25,26,40] (see Figure 5), $d=0.14\mathrm{nm}$ [7,8,22,23,24,25,26,40,75], L

_{J}(Enr) = (6 + 2) 0.14 nm = 1.12 nm), then we obtain $E=0.349$ from the Egorov nano-resonance (4) (in Equation (12) parameter $\theta =1/2$); $\gamma =0.092\mathrm{eV}$; and $T=298\mathrm{K}$. The fluorescence J-band is symmetric to the absorption J-band and obtained from the latter by changing the sign in the heat energy $\Delta $ and increasing the energy gap ${J}_{1}-{J}_{2}=1.19\mathrm{eV}$ by 0.785 eV in order for the Stokes shift to be zero (cf. Figure 8).

**Figure 10.**Structure of the dye C18S4 [6] investigated experimentally in [86]. The length of the main optical chromophore, the polymethine chain [38], is defined as the distance $L$ between nitrogen atoms $\mathrm{N}$ and $\stackrel{\oplus}{\mathrm{N}}$ with $L=6d$, where $d$ designates roughly equal the C−C bond lengths of methine groups (cf. Figure 3), $d=0.14\mathrm{nm}$ [7,8,22,23,24,25,26,40,75].

**Figure 11.**Comparison of theory with experiment for J-aggregates in Langmuir films. (

**a**) Experimental absorption and fluorescence J-bands [86] shown in Figure 9a. (

**b**) Theoretical absorption and fluorescence J-bands [6], fitted to the experimental data [86] (see (

**a**)). The fluorescence J-band is asymmetric to the absorption J-band and obtained from the latter by changing the sign both in the heat energy Δ and in the J-chromophore length $L$, and increasing the energy gap ${J}_{1}-{J}_{2}=1.19\mathrm{eV}$ by 0.785 eV in order for the Stokes shift to be 1 nm (cf. Figure 9b). The rest of the fitting parameters are the same as those in Figure 9b.

**Figure 12.**Nondegenerate 2PA spectra for a solution of selenopyrylium-terminated polymethine dye Se-3C dissolved in chloroform. Different pump wavelengths were used to observe the full ND-2PA spectra. Solid circles are degenerate 2PA (D-2PA) cross sections derived from femtosecond-pulsed Z-scan measurements. 1 GM is defined as 1 × 10

^{–50}cm

^{4}s photon

^{–1}. Experimental uncertainties in the values were estimated to be ±10%. The linear absorption spectra are shown as reference. Reproduced from Hales et al. [87] with permission from the American Association for the Advancement of Science (AAAS).

**Figure 13.**Transformation of the theoretical spectra of a model polymethine dye with different deviations from the nano-resonance condition θ =

^{1}/

_{2}(see Equations (4)–(6), in Equation (12) parameter θ =

^{1}/

_{2}) [1,9]. The optical band that is near the nano-resonance band (black curve) corresponds to θ = 0.44, the optical band that is far beyond the nano-resonance band (red curve) corresponds to θ = 0.88. Both bands are computed from Equations (7)–(27), where the “dye + environment” system parameters are $m={m}_{\mathrm{e}}$ (${m}_{\mathrm{e}}$ is the electron mass); ${n}_{\mathrm{ref}}=1.33$; $\omega =5\times {10}^{13}{\mathrm{s}}^{-1}$; ${J}_{1}=5\mathrm{eV}$; ${J}_{1}-{J}_{2}=0.5\mathrm{eV}$; $E=1\mathrm{eV}$; ${L}_{\mathrm{black}}=0.384\mathrm{nm}$; ${L}_{\mathrm{red}}=2{L}_{\mathrm{black}}$; ${\gamma}_{\mathrm{black}}=0.45\mathrm{eV}$; ${\gamma}_{\mathrm{red}}=0.34\mathrm{eV}$; and $T=298\mathrm{K}$. A mathcad program file validating these results is available at http://dx.doi.org/10.17632/c4h9rm5xk6.1 (accessed on 19 June 2020). Reprinted from [1], Copyright 2021, with permission from Elsevier.

**Figure 14.**The theoretical spectra in Figure 13 are shown in photon-energy coordinates for convenience of discussion [1]. Additionally, for convenience, it is assumed that the energy gap ${J}_{1}-{J}_{2}\equiv 0$. A mathcad program file validating these results is available at http://dx.doi.org/10.17632/c4h9rm5xk6.1 (accessed on 19 June 2020). Reprinted from [1], Copyright 2021, with permission from Elsevier.

**Figure 15.**Transformation of the theoretical band shape for the two-photon absorption (Figure 14, red curve) with varying the reorganization energy $E$ [1]. (

**a**) $\gamma \equiv \mathrm{constant}=0.34\mathrm{eV}\equiv {\gamma}_{\mathrm{red}}$; $E=1.2\mathrm{eV}$ (blue), $1\mathrm{eV}$ (red), $0.7\mathrm{eV}$ (green), $0.45\mathrm{eV}$ (magic). (

**b**) ${\theta}_{0}\equiv {{E}_{\mathrm{red}}/\gamma}_{\mathrm{red}}\equiv \mathrm{constant}=2.94$; $E=1.2\mathrm{eV}$ (blue), $1\mathrm{eV}$ (red), $0.7\mathrm{eV}$ (green), $0.6\mathrm{eV}$ (magic). The magic band shape corresponds to the restored Egorov nano-resonance. Mathcad program files validating these results are available at http://dx.doi.org/10.17632/psdhby5hhg.1 (accessed on 3 December 2020). Reprinted from [1], Copyright 2021, with permission from Elsevier.

**Figure 16.**The same as in Figure 15, except for the wavelengths on the abscissa axis [1]. Mathcad program files validating these results are available at http://dx.doi.org/10.17632/psdhby5hhg.1 (accessed on 3 December 2020). Reprinted from [1], Copyright 2021, with permission from Elsevier.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Egorov, V.V.
Quantum–Classical Mechanics: Nano-Resonance in Polymethine Dyes. *Mathematics* **2022**, *10*, 1443.
https://doi.org/10.3390/math10091443

**AMA Style**

Egorov VV.
Quantum–Classical Mechanics: Nano-Resonance in Polymethine Dyes. *Mathematics*. 2022; 10(9):1443.
https://doi.org/10.3390/math10091443

**Chicago/Turabian Style**

Egorov, Vladimir V.
2022. "Quantum–Classical Mechanics: Nano-Resonance in Polymethine Dyes" *Mathematics* 10, no. 9: 1443.
https://doi.org/10.3390/math10091443