# Dozy-Chaos Mechanics for a Broad Audience

## Abstract

**:**

**2019**, 5, e02579), is introduced to a wide general readership.

## 1. Introduction

## 2. Dozy-Chaos Mechanics on a Qualitative Level of Consideration

#### 2.1. Full-Fledged Electron-Nuclear Motion in the Transient State of Molecular Quantum Transitions, Singularity in Their Rates, and the Franck–Condon Principle as a Primitive Singularity Damper

#### 2.2. Potential Box with a Movable Wall and Dozy Chaos as a Damper of the Singularity

## 3. Dozy-Chaos Mechanics: Hamiltonian, Green’s Function, and Dozy Chaos

## 4. Dozy-Chaos Mechanics, Quantum Mechanics, and Classical Mechanics: Some Analogy

## 5. Dynamic Electron–Nuclear–Reorganization Resonance

## 6. Absorption and Luminescence Spectra

## 7. Optical Electron-Transfer-Polymethine-Chain Chromophore

## 8. Implementation of the Dynamic Electron–Nuclear–Reorganization Resonance in the Optical Band Shape

## 9. Dozy-Chaos Mechanics and the Franck–Condon Principle

## 10. Conclusions

## Funding

## Conflicts of Interest

## References

- 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] - 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] - 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] - 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] [Green Version] - Struganova, I.A.; Lim, H.; Morgan, S.A. The Influence of Inorganic Salts and Bases on the Formation of the J-band in the Absorption and Fluorescence Spectra of the Diluted Aqueous Solutions of TDBC. J. Phys. Chem. B
**2002**, 106, 11047–11050. [Google Scholar] [CrossRef] - Struganova, I.A.; Hazell, M.; Gaitor, J.; McNally-Carr, D.; Živanović, S. Influence of Inorganic Salts and Bases on the J-Band in the Absorption Spectra of Water Solutions of 1,1‘-Diethyl-2,2‘-cyanine Iodide. J. Phys. Chem. A
**2003**, 107, 2650–2656. [Google Scholar] [CrossRef] - Zhang, Z.; Achilefu, S. Synthesis and Evaluation of Polyhydroxylated Near-Infrared Carbocyanine Molecular Probes. Org. Lett.
**2004**, 6, 2067–2070. [Google Scholar] [CrossRef] - Abdel-Halim, S.T.; Awad, M.K. Absorption, fluorescence, and semiempirical ASED-MO studies on a typical Brooker’s merocyanine dye. J. Mol. Struct.
**2005**, 754, 16–24. [Google Scholar] [CrossRef] - Li, C.; Greenwood, T.R.; Bhujwalla, Z.M.; Glunde, K. Synthesis and Characterization of Glucosamine-Bound Near-Infrared Probes for Optical Imaging. Org. Lett.
**2006**, 8, 3623–3626. [Google Scholar] [CrossRef] [PubMed] - Eisfeld, A.; Briggs, J. The shape of the J-band of pseudoisocyanine. Chem. Phys. Lett.
**2007**, 446, 354–358. [Google Scholar] [CrossRef] - Roden, J.; Eisfeld, A.; Briggs, J. The J- and H-bands of dye aggregate spectra: Analysis of the coherent exciton scattering (CES) approximation. Chem. Phys.
**2008**, 352, 258–266. [Google Scholar] [CrossRef] [Green Version] - Kaiser, T.E.; Stepanenko, V.; Würthner, F. Fluorescent J-Aggregates of Core-Substituted Perylene Bisimides: Studies on Structure−Property Relationship, Nucleation−Elongation Mechanism, and Sergeants-and-Soldiers Principle. J. Am. Chem. Soc.
**2009**, 131, 6719–6732. [Google Scholar] [CrossRef] [PubMed] - Kaiser, T.E.; Scheblykin, I.G.; Thomsson, D.; Würthner, F. Temperature-dependent exciton dynamics in J-Aggregates—When disorder plays a role. J. Phys. Chem. B
**2009**, 113, 15836–15842. [Google Scholar] [CrossRef] - Kalimuthu, P.; John, S.A. Nanostructured Aggregates ofmeso-Tetramesitylporphyrin on Solid Substrate. Langmuir
**2009**, 25, 12414–12418. [Google Scholar] [CrossRef] - Bouit, P.-A.; Aronica, C.; Toupet, L.; Le Guennic, B.; Andraud, C.; Maury, O. Continuous Symmetry Breaking Induced by Ion Pairing Effect in Heptamethine Cyanine Dyes: Beyond the Cyanine Limit. J. Am. Chem. Soc.
**2010**, 132, 4328–4335. [Google Scholar] [CrossRef] - Würthner, F.; Kaiser, T.E.; Saha-Möller, C.R. J-Aggregates: From serendipitous discovery to supramolecular engineering of functional dye materials. Angew. Chem. Int. Ed.
**2011**, 50, 3376–3410. [Google Scholar] [CrossRef] - Somaschi, N.; Mouchliadis, L.; Coles, D.; Perakis, I.E.; Lidzey, D.G.; Lagoudakis, P.G.; Savvidis, P.G. Ultrafast polariton population build-up mediated by molecular phonons in organic microcavities. Appl. Phys. Lett.
**2011**, 99, 143303-1–143303-3. [Google Scholar] [CrossRef] [Green Version] - Matsumoto, S.; Horiguchi-Babamoto, E.; Eto, R.; Sato, S.; Kobayashi, T.; Naito, H.; Shirod, M.; Takahashie, H. J-aggregate structure in a chloroform solvate of a 2,3-dicyanopyrazine dye—Separation of two-dimensional stacking dye layers by solvate formation. Dyes Pigm.
**2012**, 95, 431–435. [Google Scholar] [CrossRef] [Green Version] - Dubinina, T.V.; Tomilova, L.G.; Zefirov, N.S. Synthesis of phthalocyanines with an extended system of pi-electron conjugation. Russ. Chem. Rev.
**2013**, 82, 865–895. [Google Scholar] [CrossRef] - Suponitsky, K.Y.; Masunov, A.E. Supramolecular step in design of nonlinear optical materials: Effect of π…π stacking aggregation on hyperpolarizability. J. Chem. Phys.
**2013**, 139, 094310. [Google Scholar] [CrossRef] - Frost, J.E.; Jones, G.A. A quantum dynamical comparison of the electronic couplings derived from quantum electrodynamics and Förster theory: Application to 2D molecular aggregates. New J. Phys.
**2014**, 16, 113067. [Google Scholar] [CrossRef] [Green Version] - Bergendahl, L.T.; Paterson, M.J. Excited states of porphyrin and porphycene aggregates: Computational insights. Comput. Theor. Chem.
**2014**, 1040–1041, 274–286. [Google Scholar] [CrossRef] - Rubia-Payá, C.; Giner-Casares, J.J.; Miguel, G.; Martín-Romero, M.T.; Möbius, D.; Camacho, L. Aggregation and structural study of the monolayers formed by an amphiphilic thiapentacarbocyanine. RSC Adv.
**2015**, 5, 32227–32238. [Google Scholar] [CrossRef] - Masunov, A.E.; Anderson, D.; Freidzon, A.Y.; Bagaturyants, 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] - Bricks, J.L.; Slominskii, Y.L.; Panas, I.; Demchenko, A.P.; Slominsky, Y.L.; Demchenko, A. Fluorescent J-aggregates of cyanine dyes: Basic research and applications review. Methods Appl. Fluoresc.
**2018**, 6. [Google Scholar] [CrossRef] [Green Version] - Hestand, N.J.; Spano, F.C. Expanded Theory of H- and J-Molecular Aggregates: The Effects of Vibronic Coupling and Intermolecular Charge Transfer. Chem. Rev.
**2018**, 118, 7069–7163. [Google Scholar] [CrossRef] - Guerrini, M.; Cocchi, C.; Calzolari, A.; Varsano, D.; Corni, S. Interplay between Intra- and Intermolecular Charge Transfer in the Optical Excitations of J-Aggregates. J. Phys. Chem. C
**2019**, 123, 6831–6838. [Google Scholar] [CrossRef] [Green Version] - 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] [Green Version] - Von Weber, A.; Stanley, P.; Jakob, M.; Kartouzian, A.; Heiz, U. Tunable Induced Circular Dichroism in Thin Organic Films. J. Phys. Chem. C
**2019**, 123, 9255–9261. [Google Scholar] [CrossRef] - Egorov, V.V.; Alfimov, M.V. Theory of the J-band: From the Frenkel exciton to charge transfer. Phys. Usp.
**2007**, 50, 985–1029. [Google Scholar] [CrossRef] - Egorov, V.V. Theory of the J-band: From the Frenkel exciton to charge transfer. Phys. Proc.
**2009**, 2, 223–326, [Proc. 15th Int. Conf. Lumin. Opt. Spectr. Cond. Matter—ICL ’2008, Lyon, France, 7–11 July 2008]. [Google Scholar] [CrossRef] [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, [Proc. 17th Int. Conf. on Dynamical Processes in Excited States of Solids (DPC’10), Argonne Nat. Lab., Argonne, Illinois, USA, 20–25 June 2010]. [Google Scholar] [CrossRef] - Egorov, V.V. Discovery of Dozy Chaos and Discovery of Quanta: Analogy Being in Science and Perhaps in Human Progress; Stavrinides, S.G., Banerjee, S., Caglar, H., Ozer, M., Eds.; In Proceedings of the Chaos and Complex Systems: Proceedings of the 4th International Interdisciplinary Chaos Symp., Antalya, Turkey, 29 April–2 May, 2012; Springer: Berlin, Germany, 2013; pp. 41–46. [Google Scholar] [CrossRef]
- Egorov, V.V. Dozy Chaos in Chemistry: Simplicity in Complexity; Stavrinides, S.G., Banerjee, S., Caglar, H., Ozer, M., Eds.; In Proceedings of the Chaos and Complex Systems: Proceedings of the 4th International Interdisciplinary Chaos Symp., Antalya, Turkey, 29 April–2 May, 2012; Springer: Berlin, Germany, 2013; pp. 219–224. [Google Scholar] [CrossRef]
- 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] - 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] - 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. [Google Scholar] [CrossRef] [Green Version] - Egorov, V.V. Where and why quantum mechanics ceases to work in molecular and chemical physics. In Proceedings of the European XFEL Theory Seminar, Schenefeld, Hamburg, Germany, 6 March 2018; Available online: https://indico.desy.de/indico/event/20069/ (accessed on 21 February 2018).
- Egorov, V.V. Quantum-classical mechanics: Luminescence spectra in polymethine dyes and J-aggregates. Nature of the small Stokes shift. Res. Phys.
**2019**, 13. [Google Scholar] [CrossRef] - Mustroph, H. Potential-Energy Surfaces, the Born-Oppenheimer Approximations, and the Franck-Condon Principle: Back to the Roots. Chem. Phys. Chem.
**2016**, 17, 2616–2629. [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] - 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. (Leipzig)
**1901**, 309, 553–563. [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] - Brooker, L.G.S.; Sprague, R.H.; Smyth, C.P.; Lewis, G.L. Color and Constitution. I. Halochromism of Anhydronium Bases Related to the Cyanine Dyes1. 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. [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 Pigm.
**2005**, 64, 207–216. [Google Scholar] [CrossRef] - James, N.S.; Chen, Y.; Joshi, P.; Ohulchanskyy, T.Y.; Ethirajan, M.; Henary, M.; Strekowsk, L.; Pandey, R.K. Evaluation of Polymethine Dyes as Potential Probes for Near Infrared Fluorescence Imaging of Tumors: Part–1. Theranostics
**2013**, 3, 692–702. [Google Scholar] [CrossRef] - König, S.G.; Krämer, R. Accessing Structurally Diverse Near-Infrared Cyanine Dyes for Folate Receptor-Targeted Cancer Cell Staining. Chem. A Eur. J.
**2017**, 23, 9306–9312. [Google Scholar] [CrossRef] - Usama, S.M.; Thavornpradit, S.; Burgess, K. Optimized Heptamethine Cyanines for Photodynamic Therapy. ACS Appl. Bio Mater.
**2018**, 1, 1195–1205. [Google Scholar] [CrossRef] - Atchison, J.; Kamila, S.; Nesbitt, H.; Logan, K.A.; Nicholas, D.M.; Fowley, C.; Davis, J.; Callan, J.F.; McHale, A.P.P.; Callan, J.F. Iodinated cyanine dyes: A new class of sensitisers for use in NIR activated photodynamic therapy (PDT). Chem. Commun.
**2017**, 53, 2009–2012. [Google Scholar] [CrossRef] - Bender, C.M. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys.
**2007**, 70, 947–1118. [Google Scholar] [CrossRef] [Green Version] - Moiseyev, N. Non-Hermitian Quantum Mechanics; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Sergi, A. Matrix Algebras in Non-Hermitian Quantum Mechanics. Commun. Theor. Phys.
**2011**, 56, 96–98. [Google Scholar] [CrossRef] [Green Version] - Sergi, A.; Zloshchastiev, K.G. Non-hermitian quantum dynamics of a two-level system and models of dissipative environments. Int. J. Mod. Phys. B
**2013**, 27. [Google Scholar] [CrossRef] - Sergi, A. Embedding quantum systems with a non-conserved probability in classical environments. Theor. Chem. Acc.
**2015**, 134. [Google Scholar] [CrossRef] [Green Version] - Zloshchastiev, K. Non-Hermitian Hamiltonians and stability of pure states. Eur. Phys. J. D
**2015**, 69. [Google Scholar] [CrossRef] [Green Version] - Sergi, A.; Giaquinta, P.V. Linear Quantum Entropy and Non-Hermitian Hamiltonians. Entropy
**2016**, 18. [Google Scholar] [CrossRef] [Green Version] - Znojil, M. Non-Hermitian interaction representation and its use in relativistic quantum mechanics. Ann. Phys. (NY)
**2017**, 385, 162–179. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Singularity in the rate of molecular quantum transitions: a potential box with a movable wall (

**a**) and the optical absorption band shape dependent on the dozy chaos available to a given quantum transition (

**b**); the band shape with the strongly pronounced peak (J-band) corresponds to the least dozy chaos [37]. The wall is fastened to the abscissa axis by a freely movable hinge and can move with a certain friction or without friction against the axis. Such a wall simulates the environmental nuclear reorganization in the molecular “quantum” transitions, where dozy chaos plays the role of friction. In the theory [6], this results in the dozy-chaos dependent optical absorption band being displaced to the red spectral region and narrowed (

**b**). The position, the intensity and the width of the optical absorption band are determined by the ratio between the dozy-chaos energy $\gamma $ and the reorganization energy $E$ (see Section 3). The smaller the value of $\gamma $ is, the higher the degree of organization of the molecular “quantum” transition, and the more the intensity and less the width of the optical band (

**b**). The position of the wing maximum is determined by the energy $E$, whereas the position of the peak is determined by the energy $\gamma $ [37]. (Original citation)—reproduced by permission of The Royal Society of Chemistry.

**Figure 2.**Distribution function of black light $\varphi \left(\lambda ,T\right)$ (Equation (4)) and singularity in this function in the framework of classical mechanics (on the right). The wavelength $\lambda $, indicated on the x-axis, corresponds to the frequency $\mathsf{\Omega}$ by the standard formula $\lambda =2\pi c/\mathsf{\Omega}$ ($c$ is the speed of light in vacuum).

**Figure 3.**Ideal polymethine state (IPS) [50,51]. Charges reside on carbon atoms of the polymethine chain in the ground state; charges: 1, positive; 2, negative [37]. The polymethine chain length $L$ is defined as the distance between nitrogen atoms N. (Original citation)—Reproduced by permission of The Royal Society of Chemistry.

**Figure 4.**Experimental [52,53] (

**a**) and theoretical [4] (

**b**) monomer’s optical absorption spectra dependent on the length of the polymethine chain $L=2\left(n+2\right)d$, where $d$ are certain roughly equal bond lengths in the chain (thiapolymethinecyanine in methanol at room temperature; $\epsilon $ is the extinction coefficient) [37,39,54]. The optical absorption band with $n=3$ corresponds to the dynamic electron–nuclear–reorganization resonance (the Egorov resonance, see Section 5) or is close to it. (Original citation)—reproduced by permission of The Royal Society of Chemistry.

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Egorov, V.V.
Dozy-Chaos Mechanics for a Broad Audience. *Challenges* **2020**, *11*, 16.
https://doi.org/10.3390/challe11020016

**AMA Style**

Egorov VV.
Dozy-Chaos Mechanics for a Broad Audience. *Challenges*. 2020; 11(2):16.
https://doi.org/10.3390/challe11020016

**Chicago/Turabian Style**

Egorov, Vladimir V.
2020. "Dozy-Chaos Mechanics for a Broad Audience" *Challenges* 11, no. 2: 16.
https://doi.org/10.3390/challe11020016