{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T01:52:14Z","timestamp":1775785934312,"version":"3.50.1"},"reference-count":49,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T00:00:00Z","timestamp":1725235200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Stavros Niarchos Foundation (SNF)","award":["011496"],"award-info":[{"award-number":["011496"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Ultrafast reaction processes take place when resonant features of nonlinear model systems are taken into account. In the targeted energy or electron transfer dimer model this is accomplished through the implementation of nonlinear oscillators with opposing types of nonlinearities, one attractive while the second repulsive. In the present work, we show that this resonant behavior survives if we take into account the vibrational degrees of freedom as well. After giving a summary of the basic formalism of chemical reactions we show that resonant electron transfer can be assisted by vibrations. We find the condition for this efficient transfer and show that in the case of additional interaction with noise, a distinct non-Arrhenius behavior develops that is markedly different from the usual Kramers-like activated transfer.<\/jats:p>","DOI":"10.3390\/e26090753","type":"journal-article","created":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T12:54:42Z","timestamp":1725281682000},"page":"753","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Targeted Energy Transfer Dynamics and Chemical Reactions"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0499-7691","authenticated-orcid":false,"given":"Natalya","family":"Almazova","sequence":"first","affiliation":[{"name":"Institute of Theoretical and Computational Physics, Department of Physics, University of Crete, 71003 Heraklion, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Serge","family":"Aubry","sequence":"additional","affiliation":[{"name":"Institute of Theoretical and Computational Physics, Department of Physics, University of Crete, 71003 Heraklion, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4793-0224","authenticated-orcid":false,"given":"Giorgos P.","family":"Tsironis","sequence":"additional","affiliation":[{"name":"Institute of Theoretical and Computational Physics, Department of Physics, University of Crete, 71003 Heraklion, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/S0031-8914(40)90098-2","article-title":"Brownian motion in a field of force and the diffusion model of chemical reactions","volume":"7","author":"Kramers","year":"1940","journal-title":"Physica"},{"key":"ref_2","unstructured":"Lindenberg, K., and West, B.J. (1996). The Nonequilibrium Statistical Mechanics of Open and Closed Systems, Wiley-VCH."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Malgaretti, P., Pagonabarraga, I., and Rubi, J.M. (2016). Rectification and Non-Gaussian Diffusion in Heterogeneous Media. Entropy, 18.","DOI":"10.3390\/e18110394"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1038\/s41560-024-01502-0","article-title":"Understanding entropic barriers","volume":"9","author":"Zeradjanin","year":"2024","journal-title":"Nat. Energy"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Schlawin, F., Gessner, M., Buchleitner, A., Sch\u00e4tz, T., and Skourtis, S.S. (2021). Continuously Parametrized Quantum Simulation of Molecular Electron-Transfer Reactions. PRX Quantum, 2.","DOI":"10.1103\/PRXQuantum.2.010314"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Chetverikov, A.P., Ebeling, W., and Velarde, M.G. (2016). Long-Range Electron Transport donor\u2013acceptor in Nonlinear Lattices. Entropy, 18.","DOI":"10.3390\/e18030092"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/S0005-2728(01)00202-X","article-title":"Electron transfer in photosystem I","volume":"1507","author":"Brettel","year":"2001","journal-title":"Biochim. Biophys. Acta (BBA) Bioenerg."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1038\/nature05678","article-title":"Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems","volume":"446","author":"Engel","year":"2007","journal-title":"Nature"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Lloyd, S. (2011). Quantum coherence in biological systems. J. Phys. Conf. Ser., 302.","DOI":"10.1088\/1742-6596\/302\/1\/012037"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mohseni, M., Rebentrost, P., Lloyd, S., and Aspuru-Guzik, A. (2008). Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys., 129.","DOI":"10.1063\/1.3002335"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Plenio, M.B., and Huelga, S.F. (2008). Dephasing-assisted transport: Quantum networks and biomolecules. New J. Phys., 10.","DOI":"10.1088\/1367-2630\/10\/11\/113019"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Kopidakis, G., Aubry, S., and Tsironis, G.P. (2001). Targeted Energy Transfer through Discrete Breathers in Nonlinear Systems. Phys. Rev. Lett., 87.","DOI":"10.1103\/PhysRevLett.87.165501"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/S0921-4526(00)00804-8","article-title":"Analytic conditions for targeted energy transfer between nonlinear oscillators or discrete breathers","volume":"296","author":"Aubry","year":"2001","journal-title":"Phys. B Condens. Matter"},{"key":"ref_14","unstructured":"Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., and Lee, Y.S. (2008). Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, Springer."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"C120","DOI":"10.1364\/JOSAB.430206","article-title":"Discovering nonlinear resonances through physics-informed machine learning","volume":"38","author":"Barmparis","year":"2021","journal-title":"J. Opt. Soc. Am. B"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Andronis, I., Arapantonis, G., Barmparis, G.D., and Tsironis, G.P. (2023). Quantum targeted energy transfer through machine learning tools. Phys. Rev. E, 107.","DOI":"10.1103\/PhysRevE.107.065301"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Xia, R., and Kais, S. (2020). Hybrid Quantum-Classical Neural Network for Calculating Ground State Energies of Molecules. Entropy, 22.","DOI":"10.3390\/e22080828"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1007\/s10867-005-1283-4","article-title":"A nonadiabatic theory for ultrafast catalytic electron transfer: A model for the Photosynthetic Reaction Center","volume":"31","author":"Aubry","year":"2005","journal-title":"J. Biol. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s10698-007-9033-2","article-title":"Structural formulas and explanation in organic chemistry","volume":"10","author":"Goodwin","year":"2008","journal-title":"Found. Chem."},{"key":"ref_20","unstructured":"Meyers, R.A. (2003). Organic Chemical Systems, Theory. Encyclopedia of Physical Science and Technology, Academic Press. [3rd ed.]."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Batsanov, S.S., and Batsanov, A.S. (2012). Introduction to Structural Chemistry, Springer.","DOI":"10.1007\/978-94-007-4771-5"},{"key":"ref_22","unstructured":"March, J. (1977). Advanced Organic Chemistry, New York McGraw-Hill."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Lagan\u00e0, A., and Parker, G.A. (2018). Chemical Reactions: Basic Theory and Computing, Springer.","DOI":"10.1007\/978-3-319-62356-6"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1021\/ja01355a027","article-title":"The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules","volume":"53","author":"Pauling","year":"1931","journal-title":"J. Am. Chem. Soc."},{"key":"ref_25","unstructured":"Aubry, S. (2014). A semiclassical non-adiabatic theory for elementary chemical reactions. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/S0009-2614(01)00132-4","article-title":"Mediation of ultrafast electron transfer in biological systems by conical intersections","volume":"338","author":"Worth","year":"2001","journal-title":"Chem. Phys. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"20265","DOI":"10.1039\/D0CP03464A","article-title":"A systematic model study quantifying how conical intersection topography modulates photochemical reactions","volume":"22","author":"Farfan","year":"2020","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"322","DOI":"10.3390\/e16010322","article-title":"Modeling Potential Energy Surfaces: From First-Principle Approaches to Empirical Force Fields","volume":"16","author":"Ballone","year":"2014","journal-title":"Entropy"},{"key":"ref_29","unstructured":"Marcus, R. (2024, July 30). Electron Transfer Reactions in Chemistry: Theory and Experiment. Nobel Lecture. Available online: https:\/\/www.nobelprize.org\/prizes\/chemistry\/1992\/marcus\/lecture\/."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1103\/RevModPhys.65.599","article-title":"Electron transfer reactions in chemistry. Theory and experiment","volume":"65","author":"Marcus","year":"1993","journal-title":"Rev. Mod. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/0304-4173(85)90014-X","article-title":"Electron transfers in chemistry and biology","volume":"811","author":"Marcus","year":"1985","journal-title":"Biochim. Biophys. Acta"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Baer, M. (2006). Beyond Born\u2013Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections, John Wiley & Sons.","DOI":"10.1002\/0471780081"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1103\/RevModPhys.62.251","article-title":"Reaction-rate theory: Fifty years after Kramers","volume":"62","author":"Talkner","year":"1990","journal-title":"Rev. Mod. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1039\/tf9262100536","article-title":"Elementary processes of photochemical reactions","volume":"21","author":"Franck","year":"1926","journal-title":"Trans. Faraday Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1182","DOI":"10.1103\/PhysRev.28.1182","article-title":"A Theory of Intensity Distribution in Band Systems","volume":"28","author":"Condon","year":"1926","journal-title":"Phys. Rev."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1103\/PhysRev.32.858","article-title":"Nuclear Motions Associated with Electron Transitions in Diatomic Molecules","volume":"32","author":"Condon","year":"1928","journal-title":"Phys. Rev."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0022-2852(62)90029-2","article-title":"The Franck-Condon principle and the structures of excited electronic states of molecules","volume":"8","author":"Coon","year":"1962","journal-title":"J. Mol. Spectrosc."},{"key":"ref_38","first-page":"467","article-title":"\u00dcber das Verhalten von Eigenwerten bei adiabatischen Prozessen","volume":"30","author":"Wigner","year":"1929","journal-title":"Phys. Z."},{"key":"ref_39","first-page":"64","article-title":"The Lagrangian in quantum mechanics","volume":"3","author":"Dirac","year":"1933","journal-title":"Phys. Zeitschirift Sowjetunion"},{"key":"ref_40","unstructured":"Feynman, R.P. (1942). The Principle of Least Action in Quantum Mechanics. [Ph.D. Thesis, Princeton University]."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Michaelian, K. (2023). The Non-Equilibrium Thermodynamics of Natural Selection: From Molecules to the Biosphere. Entropy, 25.","DOI":"10.20944\/preprints202305.1176.v1"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1514","DOI":"10.1002\/jcc.10231","article-title":"Exploring potential energy surfaces for chemical reactions: An overview of some practical methods","volume":"24","author":"Schlegel","year":"2003","journal-title":"J. Comput. Chem."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/j.physd.2005.02.003","article-title":"Targeted energy transfer by Fermi resonance","volume":"202","author":"Maniadis","year":"2005","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physd.2005.05.016","article-title":"Targeted energy transfer between a Rotor and a Morse oscillator: A model for selective chemical dissociation","volume":"207","author":"Memboeuf","year":"2005","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.physd.2003.08.001","article-title":"Classical and quantum targeted energy transfer between nonlinear oscillators","volume":"188","author":"Maniadis","year":"2004","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0771-050X(80)90013-3","article-title":"A family of embedded Runge-Kutta formulae","volume":"6","author":"Dormand","year":"1980","journal-title":"J. Comput. Appl. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1090\/S0025-5718-1986-0815836-3","article-title":"Some practical Runge-Kutta formulas","volume":"46","author":"Shampine","year":"1986","journal-title":"Math. Comput."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1488","DOI":"10.1137\/0913084","article-title":"Derivation of Efficient, Continuous, Explicit Runge\u2013Kutta Methods","volume":"13","author":"Owren","year":"1992","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"9251","DOI":"10.1021\/jacs.6b04777","article-title":"Marcus Bell-Shaped electron transfer kinetics observed in an Arrhenius plot","volume":"138","author":"Waskasi","year":"2016","journal-title":"J. Am. Chem. Soc."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/9\/753\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:47:25Z","timestamp":1760111245000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/9\/753"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,2]]},"references-count":49,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["e26090753"],"URL":"https:\/\/doi.org\/10.3390\/e26090753","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,2]]}}}