{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T21:02:09Z","timestamp":1792789329042,"version":"4.5.0"},"reference-count":189,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,29]],"date-time":"2022-12-29T00:00:00Z","timestamp":1672272000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"JSPS KAKENHI","award":["JP20H00373"],"award-info":[{"award-number":["JP20H00373"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Quantum chaos is reviewed from the viewpoint of \u201cwhat is molecule?\u201d, particularly placing emphasis on their dynamics. Molecules are composed of heavy nuclei and light electrons, and thereby the very basic molecular theory due to Born and Oppenheimer gives a view that quantum electronic states provide potential functions working on nuclei, which in turn are often treated classically or semiclassically. Therefore, the classic study of chaos in molecular science began with those nuclear dynamics particularly about the vibrational energy randomization within a molecule. Statistical laws in probabilities and rates of chemical reactions even for small molecules of several atoms are among the chemical phenomena requiring the notion of chaos. Particularly the dynamics behind unimolecular decomposition are referred to as Intra-molecular Vibrational energy Redistribution (IVR). Semiclassical mechanics is also one of the main research fields of quantum chaos. We herein demonstrate chaos that appears only in semiclassical and full quantum dynamics. A fundamental phenomenon possibly giving birth to quantum chaos is \u201cbifurcation and merging\u201d of quantum wavepackets, rather than \u201cstretching and folding\u201d of the baker\u2019s transformation and the horseshoe map as a geometrical foundation of classical chaos. Such wavepacket bifurcation and merging are indeed experimentally measurable as we showed before in the series of studies on real-time probing of nonadiabatic chemical reactions. After tracking these aspects of molecular chaos, we will explore quantum chaos found in nonadiabatic electron wavepacket dynamics, which emerges in the realm far beyond the Born-Oppenheimer paradigm. In this class of chaos, we propose a notion of Intra-molecular Nonadiabatic Electronic Energy Redistribution (INEER), which is a consequence of the chaotic fluxes of electrons and energy within a molecule.<\/jats:p>","DOI":"10.3390\/e25010063","type":"journal-article","created":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T03:31:17Z","timestamp":1672371077000},"page":"63","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Quantum Chaos in the Dynamics of Molecules"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5624-1810","authenticated-orcid":false,"given":"Kazuo","family":"Takatsuka","sequence":"first","affiliation":[{"name":"Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lichtenberg, A.J., and Lieberman, M.A. (1992). Regular and Chaotic Dynamics, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4757-2184-3"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ozorio de Almeida, A.M. (1988). Hamiltonian Systems, Chaos and Quantization, Cambridge University Press.","DOI":"10.1017\/CBO9780511564161"},{"key":"ref_3","unstructured":"Tabor, M. (1989). Chaos and Integrability in Nonlinear Dynamics, John Wiley."},{"key":"ref_4","unstructured":"Giannoni, M.-J., Voros, A., and Zinn-Justin, J. (1991). Chaos and Quantum Physics, North Holland Publisher."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Reichl, L.E. (1992). The Transition to Chaos, Springer.","DOI":"10.1007\/978-1-4757-4352-4"},{"key":"ref_6","unstructured":"Nakamura, K. (1993). Quantum Chaos, Cambridge University Press."},{"key":"ref_7","first-page":"105","article-title":"New Way of Understanding Semiclassical Quantization","volume":"XC","author":"Gaspard","year":"1995","journal-title":"Adv. Chem. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Casati, G., and Chirikov, B. (1995). (Eds.) Quantum Chaos: Between Order and Disorder, Cambridge University Press.","DOI":"10.1017\/CBO9780511599989"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Haake, F. (2010). Quantum Signatures of Chaos, Springer.","DOI":"10.1007\/978-3-642-05428-0"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Berggren, K.-F., and \u00c5berg, S. (2001). (Eds.) Quantum Chaos Y2K, The Royal Swedish Academy of Sciences. Physica Scripta.","DOI":"10.1142\/4775"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1140\/epjst\/e2016-02649-0","article-title":"Fundamental concepts of quantum chaos","volume":"225","author":"Robnik","year":"2016","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"L030201","DOI":"10.1103\/PhysRevE.103.L030201","article-title":"Quantum Chaos and the Correspondence Principle","volume":"103","author":"Wang","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1088\/0031-8949\/40\/3\/013","article-title":"Quantum chaology, not quantum chaos","volume":"40","author":"Berry","year":"1989","journal-title":"Phys. Scr."},{"key":"ref_14","unstructured":"Child, M.S. (1974). Molecular Collision Theory, Academic Press."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Child, M.S. (1991). Semiclassical Mechanics with Molecular Applications, Clarendon Press.","DOI":"10.1093\/oso\/9780198556541.001.0001"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/S0370-1573(01)00052-7","article-title":"Introduction to the theory of electronic non-adiabatic coupling terms in molecular systems","volume":"358","author":"Baer","year":"2002","journal-title":"Phys. Rep."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Yang, X., and Liu, K. (2004). Modern Trends in Chemical Reaction Dynamics Part I, Chapter 8, World Scientific.","DOI":"10.1142\/9789812565808"},{"key":"ref_18","unstructured":"Yarkony, W.D.D.R., and K\u00f6ppel, H. (2004). (Eds.), Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, World Scientific."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Nakamura, H. (2012). Nonadiabatic Transition: Concepts, Basic Theories and Applications, World Scientific. [2nd ed.].","DOI":"10.1142\/8009"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Baer, M. (2006). Beyond Born\u2013Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections, Wiley.","DOI":"10.1002\/0471780081"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"10196","DOI":"10.1021\/jp072233j","article-title":"Generalization of classical mechanics for nuclear motions nonadiabatically coupled with electron wavepacket dynamics and in quantum-classical mixed representation","volume":"111","author":"Takatsuka","year":"2007","journal-title":"J. Phys. Chem. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"134109","DOI":"10.1063\/1.2987302","article-title":"Phase-space averaging and natural branching of nuclear paths for nonadiabatic electron wavepacket dynamics","volume":"129","author":"Yonehara","year":"2008","journal-title":"J. Chem. Phys."},{"key":"ref_23","first-page":"93","article-title":"Nonadiabatic chemical dynamics in intermediate and intense laser fields","volume":"144","author":"Takatsuka","year":"2010","journal-title":"Adv. Chem. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4987","DOI":"10.1039\/c0cp00937g","article-title":"Exploring dynamical electron theory beyond the Born-Oppenheimer framework: From chemical reactivity to non-adiabatically coupled electronic and nuclear wavepackets on-the-fly under laser field","volume":"13","author":"Takatsuka","year":"2011","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1021\/cr200096s","article-title":"Fundamental approaches to nonadiabaticity: Towards a chemical theory beyond the Born-Oppenheimer paradigm","volume":"112","author":"Yonehara","year":"2012","journal-title":"Chem. Rev."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Takatsuka, K., Yonehara, T., Hanasaki, K., and Arasaki, Y. (2015). Chemical Theory beyond the Born\u2014Oppenheimer Paradigm, World Scientific.","DOI":"10.1142\/9291"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1246\/bcsj.20200388","article-title":"Electron dynamics in molecular elementary processes and chemical reactions","volume":"94","author":"Takatsuka","year":"2021","journal-title":"Bull. Chem. Soc. Jpn."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1086\/109234","article-title":"The applicability of the third integral of motion","volume":"69","author":"Henon","year":"1964","journal-title":"Astron. J."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"012108","DOI":"10.1103\/PhysRevA.89.012108","article-title":"Towards many-dimensional real-time quantum theory for heavy-particle dynamics. I. Semiclassics in the Lagrange picture of classical phase flow","volume":"89","author":"Takahashi","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"012109","DOI":"10.1103\/PhysRevA.89.012109","article-title":"Towards many-dimensional real-time quantum theory for heavy particle dynamics. II. Beyond semiclassics by quantum smoothing of singularity in quantum-classical correspondence","volume":"89","author":"Takatsuka","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_31","unstructured":"Helgaker, T., J\u00f8rgensen, P., and Olsen, J. (2013). Molecular Electronic-Structure Theory, John Wiley."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1404","DOI":"10.1103\/PhysRevLett.78.1404","article-title":"Global Representation of Maslov-type Semiclassical Wavefunction and Its Spectrum in a Small Number of Classical Trajectories","volume":"78","author":"Takatsuka","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3256","DOI":"10.1103\/PhysRevA.59.3256","article-title":"Semiclassical Theory for Maslov-type Wave Packet. Hierarchy below the Semiclassical Feynman Kernel","volume":"59","author":"Takatsuka","year":"1999","journal-title":"Phys. Rev. A"},{"key":"ref_34","first-page":"69","article-title":"Classical-Limit Quantum Mechanics and the Theory of Molecular Collisions","volume":"25","author":"Miller","year":"1974","journal-title":"Adv. Chem. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1791","DOI":"10.1063\/1.1665328","article-title":"Energy Spectrum According to Classical Mechanics","volume":"11","author":"Gutzwiller","year":"1970","journal-title":"J. Math. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1063\/1.1665596","article-title":"Periodic Orbits and Classical Quantization Conditions","volume":"12","author":"Gutzwiller","year":"1971","journal-title":"J. Math. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Gutzwiller, M.C. (1990). Chaos in Classical and Quantum Mechanics, Springer.","DOI":"10.1007\/978-1-4612-0983-6"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"144101","DOI":"10.1063\/1.2173997","article-title":"On the validity range of the Born-Oppenheimer approximation: A semiclassical study for all-particle quantization of three-body Coulomb systems","volume":"124","author":"Takahashi","year":"2006","journal-title":"J. Chem. Phys."},{"key":"ref_39","unstructured":"Born, M., and Huang, K. (1954). Dynamical Theory of Crystal Lattices, Oxford University Press."},{"key":"ref_40","unstructured":"Laidler, K.J. (1987). Chemical Kinetics, 3rd ed, Harpeer & ROW."},{"key":"ref_41","unstructured":"Steinfeld, J.I., Francisco, J.S., and Hase, W.L. (1989). Chemical Kinetics and Dynamics, Prentice Hall."},{"key":"ref_42","unstructured":"Fueno, T. (1999). (Ed.) The Transition State: A Theoretical Approach, Kodansha."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1823","DOI":"10.1063\/1.1682181","article-title":"Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants","volume":"61","author":"Miller","year":"1974","journal-title":"J. Chem. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1021\/ar00028a007","article-title":"Beyond Transition-State Theory: A Rigorous Quantum Theory of Chemical Reaction Rates","volume":"26","author":"Miller","year":"1993","journal-title":"Acc. Chem. Res."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"8512","DOI":"10.1021\/jp400102r","article-title":"Adiabatic\/Nonadiabatic State-to-State Reactive Scattering Dynamics Implemented on Graphics Processing Units","volume":"117","author":"Zhang","year":"2013","journal-title":"J. Phys. Chem. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"3311","DOI":"10.1021\/acs.jpca.2c01209","article-title":"Accurate Calculation of Rate Constant and Isotope Effect for the F + H2 Reaction by the Coupled 3D Time-Dependent Wave Packet Method on the Newly Constructed Ab Initio Ground Potential Energy Surface","volume":"126","author":"Naskar","year":"2022","journal-title":"J. Phys. Chem. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1002\/anie.200461459","article-title":"Methyl group rotation driven by proton transfer through a long-range chemical interaction","volume":"44","author":"Ushiyama","year":"2005","journal-title":"Angew. Chem. Int. Ed."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1063\/1.1700424","article-title":"Unimolecular Dissociations and Free Radical Recombination Reactions","volume":"20","author":"Marcus","year":"1952","journal-title":"J. Chem. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/0003-4916(62)90221-X","article-title":"A unified theory of nuclear reactions. II","volume":"19","author":"Feshbach","year":"1962","journal-title":"Ann. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Newton, R.G. (1982). Scattering Theory of Waves and Particles, Springer.","DOI":"10.1007\/978-3-642-88128-2"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1063\/1.449560","article-title":"Fluorescence and stimulated emission S1\u2192S0 spectra of acetylene: Regular and ergodic regions","volume":"83","author":"Abramson","year":"1985","journal-title":"J. Chem. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"6330","DOI":"10.1063\/1.461554","article-title":"Vibrationally highly excited acetylene as studied by dispersed fluorescence and stimulated emission pumping spectroscopy: Vibrational assignment of the feature states","volume":"95","author":"Yamanouchi","year":"1991","journal-title":"J. Chem. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2813","DOI":"10.1063\/1.463021","article-title":"Intramolecular vibrational relaxation and forbidden transitions in the SEP spectrum of acetylene","volume":"97","author":"Jonas","year":"1992","journal-title":"J. Chem. Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"7350","DOI":"10.1063\/1.465716","article-title":"Intramolecular vibrational redistribution of energy in the stimulated emission pumping spectrum of acetylene","volume":"99","author":"Jonas","year":"1993","journal-title":"J. Chem. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"11226","DOI":"10.1063\/1.472920","article-title":"Vibrational relaxation and energy localization in polyatomics: Effects of high order resonances on flow rates and the quantum ergodicity transition","volume":"105","author":"Leitner","year":"1996","journal-title":"J. Chem. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s42004-019-0252-y","article-title":"Stable chaos and delayed onset of statisticality in unimolecular dissociation reactions","volume":"3","author":"Karmakar","year":"2020","journal-title":"Comm. Chem."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1080\/00018732.2015.1109817","article-title":"Quantum ergodicity and energy flow in molecules","volume":"64","author":"Leitner","year":"2015","journal-title":"Adv. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"11139","DOI":"10.1039\/D0CP01413C","article-title":"Intramolecular vibrational energy redistribution and the quantum ergodicity transition: A phase space perspective","volume":"22","author":"Karmakar","year":"2020","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1002\/(SICI)1097-461X(1999)75:4\/5<523::AID-QUA17>3.0.CO;2-U","article-title":"Influence of Quantum Energy Flow and Localization on Molecular Isomerization in Gas and Condensed Phases","volume":"75","author":"Leitner","year":"1999","journal-title":"Intern. J. Quant. Chem."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"6726","DOI":"10.1073\/pnas.0409083102","article-title":"Vibrational energy relaxation in proteins","volume":"102","author":"Fujisaki","year":"2005","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"144910","DOI":"10.1063\/1.2191038","article-title":"Time-dependent perturbation theory for vibrational energy relaxation and dephasing in peptides and proteins","volume":"124","author":"Fujisaki","year":"2006","journal-title":"J. Chem. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1681","DOI":"10.1063\/1.455114","article-title":"Leitner, D.M.; Berry, R.S. Melting and phase space transitions in small clusters: Spectral characteristics, dimensions, and K entropy","volume":"89","author":"Beck","year":"1988","journal-title":"J. Chem. Phys."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1103\/PhysRevA.30.919","article-title":"Melting of clusters and melting","volume":"30","author":"Berry","year":"1984","journal-title":"Phys. Rev. A."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"3261","DOI":"10.1021\/j100322a038","article-title":"Stochastic Molecular Dynamics Study of Cyclohexane Isomerization","volume":"92","author":"Kuharski","year":"1988","journal-title":"J. Phys. Chem."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1389","DOI":"10.1021\/jp067110y","article-title":"Nonempirical statistical theory for atomic evaporation from nonrigid clusters: Applications to the absolute rate constant and kinetic energy release","volume":"111","author":"Fujii","year":"2007","journal-title":"J. Phys. Chem. A"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"114318","DOI":"10.1063\/1.2894848","article-title":"Nonempirical statistical theory for molecular evaporation from nonrigid clusters","volume":"128","author":"Fujii","year":"2008","journal-title":"J. Chem. Phys."},{"key":"ref_67","first-page":"25","article-title":"Temperature, geometry, and variational statistical structure in microcanonical ensemble for structural isomerization dynamics of clusters: A multichannel chemical reaction beyond the transition-state concept","volume":"130","author":"Takatsuka","year":"2005","journal-title":"Adv. Chem. Phys."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"8613","DOI":"10.1063\/1.471550","article-title":"Non-Ergodicity and Two Sub-Phases in the Coexistence Region in Isomerization Dynamics of Ar7-like Molecules","volume":"104","author":"Seko","year":"1996","journal-title":"J. Chem. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"10356","DOI":"10.1063\/1.472965","article-title":"A Scrutiny of the Premise of the Rice-Ramsberger-Kassel-Marcus Theory in Isomerization Reaction of an Ar7-type Molecule","volume":"105","author":"Seko","year":"1996","journal-title":"J. Chem. Phys."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"3263","DOI":"10.1063\/1.478191","article-title":"Concept of Inter-basin Mixing and Extension of the Liapunov Exponen in Multiple Potential-basin Dynamics as Structural Isomerization of Clusters","volume":"110","author":"Takatsuka","year":"1999","journal-title":"J. Chem. Phys."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"2552","DOI":"10.1063\/1.1305866","article-title":"Microcanonical Temperature and Its Arrhenius Relation to Lifetimes in Isomerization Dynamics of Clusters","volume":"113","author":"Takatsuka","year":"2000","journal-title":"J. Chem. Phys."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"4768","DOI":"10.1063\/1.477088","article-title":"Origin of the complex dynamics in structural isomerization of small clusters: The effects of potential topography","volume":"109","author":"Seko","year":"1998","journal-title":"J. Chem. Phys."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"8310","DOI":"10.1063\/1.460116","article-title":"Topper, R.Q. Cylindrical manifolds in phase space as mediators of chemical reaction dynamics and kinetics. I. Theory","volume":"94","author":"Mehta","year":"1991","journal-title":"J. Chem. Phys."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"8329","DOI":"10.1063\/1.460065","article-title":"Topper, R.Q. Cylindrical manifolds in phase space as mediators of chemical reaction dynamics and kinetics. II. Numerical considerations and applications to models with two degrees of freedom","volume":"94","author":"Mehta","year":"1991","journal-title":"J. Chem. Phys."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0009-2614(80)80596-3","article-title":"Quantum dynamics in classically integrable and non-integrable regions","volume":"76","author":"Davis","year":"1980","journal-title":"Chem. Phys. Lett."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"1567","DOI":"10.1103\/PhysRevA.23.1567","article-title":"Quantum ergodicity for time-dependent wave-packet dynamics","volume":"23","author":"Hutchinson","year":"1980","journal-title":"Phys. Rev. A"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"4175","DOI":"10.1063\/1.444328","article-title":"Quantum dynamics of the H\u00e9non\u2013Heiles system","volume":"77","author":"Bixon","year":"1982","journal-title":"J. Chem. Phys."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"2578","DOI":"10.1063\/1.447051","article-title":"Wave packet dynamics and chaos in the H\u00e9on\u2013Heiles system","volume":"80","author":"Feit","year":"1984","journal-title":"J. Chem. Phys."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"6914","DOI":"10.1063\/1.470369","article-title":"Expected Significance of Weakly Chaotic Vibrational Motions in Single Molecule Spectroscopy","volume":"103","author":"Hashimoto","year":"1995","journal-title":"J. Chem. Phys."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"1893","DOI":"10.1063\/1.475568","article-title":"Quantum Localization and Dynamic Tunneling ofQuasi-separatrix Wavefunctions for Molecular Vibration","volume":"108","author":"Hashimoto","year":"1998","journal-title":"J. Chem. Phys."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"1515","DOI":"10.1103\/PhysRevLett.53.1515","article-title":"Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits","volume":"53","author":"Heller","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"6057","DOI":"10.1063\/1.470434","article-title":"A novel method to calculate eigenfunctions and eigenvalues in a given energy range","volume":"103","author":"Takatsuka","year":"1995","journal-title":"J. Chem. Phys."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/0021-9991(82)90091-2","article-title":"Solution of the Schr\u00f6inger equation by a spectral method","volume":"47","author":"Feit","year":"1982","journal-title":"J. Comput. Phys."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"2333","DOI":"10.1063\/1.457041","article-title":"Calculation of scattering wave functions by a numerical procedure based on the Mller wave operator","volume":"91","author":"Viswanathan","year":"1989","journal-title":"J. Chem. Phys."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1016\/0021-9991(92)90379-D","article-title":"Extraction of Accurate Frequencies from the Fast-Fourier-Transform Spectra","volume":"102","author":"Takatsuka","year":"1992","journal-title":"J. Compt. Phys."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"8011","DOI":"10.1063\/1.468999","article-title":"Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. I. Theory and application to a quantum-dynamics model","volume":"102","author":"Wall","year":"1995","journal-title":"J. Chem. Phys."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1063\/1.441832","article-title":"Quantum dynamical tunneling in bound states","volume":"75","author":"Davis","year":"1981","journal-title":"J. Chem. Phys."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1063\/1.1577313","article-title":"Dynamical tunneling in molecules: Role of the classical resonances and chaos","volume":"119","author":"Keshavamurthy","year":"2003","journal-title":"J. Chem. Phys."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"8636","DOI":"10.1021\/acs.jpca.8b08626","article-title":"Relevance of the Resonance Junctions on the Arnold Web to Dynamical Tunneling and Eigenstate Delocalization","volume":"122","author":"Karmakar","year":"2018","journal-title":"J. Phys. Chem. A"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1016\/0009-2614(93)89192-K","article-title":"Concept of Phase-Space Large Amplitude Motion. A Classical Study","volume":"204","author":"Takatsuka","year":"1993","journal-title":"Chem. Phys. Lett."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1146\/annurev.pc.32.100181.001411","article-title":"Quasiperiodic and Stochastic Behavior in Molecules","volume":"32","author":"Noid","year":"1981","journal-title":"Ann. Rev. Phys. Chem."},{"key":"ref_92","unstructured":"Heller, E.J. (2018). The Semiclassical Way to Dynamics and Spectroscopy: Structure, Dynamics and Quantum Chaos in Atoms and Molecules under Strong Magnetic Fields, Princeton University Press."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1143\/PTPS.166.19","article-title":"Riemann Zeta function and quantum chaos","volume":"166","author":"Bogomolony","year":"2007","journal-title":"Prog. Theoret. Phys. Suppl."},{"key":"ref_94","doi-asserted-by":"crossref","unstructured":"Schulman, L.S. (1981). Techniques and Applications of Path Integration, Wiley.","DOI":"10.1063\/1.2914703"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"066205","DOI":"10.1103\/PhysRevE.76.066205","article-title":"Finding periodic orbits of higher-dimensional flows by including tangential components of trajectory motion","volume":"76","author":"Koh","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1098\/rspa.1994.0154","article-title":"High orders of the Weyl expansion for quantum billiards: Resurgence of periodic orbits, and the Stokes phenomenon","volume":"447","author":"Berry","year":"1994","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1209\/0295-5075\/9\/6\/002","article-title":"Convergence of the Semi-Classical Periodic Orbit Expansion","volume":"9","author":"Eckhardt","year":"1989","journal-title":"Europhys. Lett."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1063\/1.165964","article-title":"\u210f expansion for the periodic orbit quantization of chaotic systems","volume":"3","author":"Alonso","year":"1993","journal-title":"CHAOS"},{"key":"ref_99","unstructured":"Bohm, D. (1989). Quantum Theory, Dover."},{"key":"ref_100","unstructured":"Messiah, A. (2017). Quantum Mechanics, Dover."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1088\/0034-4885\/35\/1\/306","article-title":"Semiclassical approximations in wave mechanics","volume":"35","author":"Berry","year":"1972","journal-title":"Rep. Prog. Phys."},{"key":"ref_102","doi-asserted-by":"crossref","unstructured":"Maslov, V.P., and Fedoriuk, M.V. (1981). Semi-Classical Approximation in Quantum Mechanics, Reidel.","DOI":"10.1007\/978-94-009-8410-3"},{"key":"ref_103","unstructured":"Blrack, M., and Bhaduri, R. (2003). Semicalssical Physics, Taylor & Francis."},{"key":"ref_104","unstructured":"Tannor, D.J. (2007). Introduction to Quantum Mechanics, University Science Books."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"022110","DOI":"10.1103\/PhysRevA.69.022110","article-title":"Geometrical evaluation of the Maslov index","volume":"69","author":"Takahashi","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"4326","DOI":"10.1103\/PhysRevA.45.4326","article-title":"Dynamics and Quantization of Hamiltonian Chaos. Density of States in Phase-Space Semiclassical Mechanics","volume":"45","author":"Takatsuka","year":"1992","journal-title":"Phys. Rev. A"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"016224","DOI":"10.1103\/PhysRevE.64.016224","article-title":"An Amplitude-free Correlation Function Based on an Algebra for Coordinate Transformation in Semiclassical Integrals","volume":"64","author":"Takatsuka","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_108","doi-asserted-by":"crossref","unstructured":"Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4757-2063-1"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"1544","DOI":"10.1063\/1.430620","article-title":"Time-dependent approach to semiclassical dynamics","volume":"62","author":"Heller","year":"1975","journal-title":"J. Chem. Phys."},{"key":"ref_110","unstructured":"Takatsuka, K. (2023). Huygens-like proliferation of the secondary coherent-state wavelets from a single Gaussian wavepacekt to fit in the length-scale hierarchical Schr\u00f6dinger dynamics. to be published."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"224112","DOI":"10.1063\/1.1924388","article-title":"Extended quantization condition for constructive and destructive interferences and trajectories dominating molecular vibrational eigenstates","volume":"122","author":"Ushiyama","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1103\/PhysRevA.60.112","article-title":"Roles of Phase Cancellation Removing Spuriously Quantized States in Semiclassical Mechanics","volume":"60","author":"Takatsuka","year":"1999","journal-title":"Phys. Rev. A"},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1098\/rspa.1976.0062","article-title":"Closed orbits and the regular bound spectrum","volume":"349","author":"Berry","year":"1976","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1088\/0305-4470\/10\/3\/009","article-title":"Calculating the bound spectrum by path summation in action-angle variables","volume":"10","author":"Berry","year":"1977","journal-title":"J. Phys. A"},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"084112","DOI":"10.1063\/1.2772274","article-title":"Phase quantization of chaos in semiclassical regime","volume":"127","author":"Takahashi","year":"2007","journal-title":"J. Chem. Phys."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1143\/PTPS.166.56","article-title":"Phase quantization of chaos and the role of the semiclassical amplitude factor","volume":"166","author":"Yamashita","year":"2007","journal-title":"Prog. Theor. Phys. Suppl."},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"4785","DOI":"10.1088\/0305-4470\/36\/17\/306","article-title":"Semiclassical quantization of chaos in terms of an amplitude-free quasi-correlation function","volume":"36","author":"Hotta","year":"2003","journal-title":"J. Phys. A Gen. Math."},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"021104","DOI":"10.1063\/1.2431178","article-title":"Energy quantization of chaos with the semiclassical phases alone","volume":"126","author":"Takatsuka","year":"2007","journal-title":"J. Chem. Phys."},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/0003-4916(89)90096-1","article-title":"A numerical evaluation of the semiclassical coherent state path integral","volume":"195","author":"Adachi","year":"1989","journal-title":"Ann. Phys."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1103\/PhysRevLett.74.682","article-title":"Complex classical trajectories and chaotic tunneling","volume":"74","author":"Shudo","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"154102","DOI":"10.1103\/PhysRevLett.109.154102","article-title":"Tunneling effect and the natural boundary of invariant tori","volume":"109","author":"Shudo","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"4353","DOI":"10.1103\/PhysRevA.51.4353","article-title":"Tunneling Solutions of Hamilton\u2013Jacobi Equation for Multi-dimensional Semiclassical Theory","volume":"51","author":"Takatsuka","year":"1995","journal-title":"Phys. Rev. A"},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/S0370-1573(99)00036-8","article-title":"Tunneling Paths in Multi-dimensional Semiclassical Dynamics","volume":"322","author":"Takatsuka","year":"1999","journal-title":"Phys. Rep."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"7023","DOI":"10.1063\/1.473725","article-title":"Semiclassical Study on Multidimensional Effects in Tunneling Chemical Reactions: Tunneling Paths and Tunneling Tubes","volume":"106","author":"Ushiyama","year":"1997","journal-title":"J. Chem. Phys."},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1103\/PhysRevE.53.115","article-title":"Statistical Redistribution of Trajectories from a Torus to Tori by Chaotic Dynamical-Tunneling","volume":"53","author":"Ushiyama","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"9664","DOI":"10.1063\/1.477635","article-title":"Quasi-semiclassical Approach to Tunneling Chemical Reactions","volume":"109","author":"Ushiyama","year":"1998","journal-title":"J. Chem. Phys."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"4561","DOI":"10.1063\/1.1645772","article-title":"Time-dependent probability of quantum tunneling in terms of quasi-semiclassical method","volume":"120","author":"Ushiyama","year":"2004","journal-title":"J. Chem. Phys."},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"11807","DOI":"10.1021\/jp0521205","article-title":"Very fast tunneling in the early stage of reaction dynamics","volume":"51","author":"Ushiyama","year":"2005","journal-title":"J. Phys. Chem. A"},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"2413","DOI":"10.1088\/0305-4470\/17\/12\/013","article-title":"Semiclassical level spacings when regular and chaotic orbits coexist","volume":"17","author":"Berry","year":"1984","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"943","DOI":"10.1103\/PhysRevLett.51.943","article-title":"Distribution of Energy Eigenvalues in the Irregular Spectrum","volume":"51","author":"Pechukas","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1103\/PhysRevLett.54.1883","article-title":"New Approach to the Statistical Properties of Energy Levels","volume":"54","author":"Yukawa","year":"1985","journal-title":"Phys. Rev. Lett."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"1661","DOI":"10.1103\/PhysRevLett.57.1661","article-title":"Complete Integrability in a Quantum Description of Chaotic Systems","volume":"56","author":"Nakamura","year":"1986","journal-title":"Phys. Rev. Lett."},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"2641","DOI":"10.1143\/JPSJ.56.2641","article-title":"Level Spacing Distribution and Avoided Crossing in Quantum Chaos","volume":"56","author":"Shudo","year":"1987","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"3546","DOI":"10.1103\/PhysRevA.47.3546","article-title":"Level dynamics: An approach to the study of avoided level crossings and transition to chaos","volume":"47","author":"Wang","year":"1993","journal-title":"Phys. Rev. A"},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"4859","DOI":"10.1063\/1.479822","article-title":"Femtosecond time-resolved photoelectron imaging on ultrafast electronic dephasing in an isolated molecule","volume":"111","author":"Suzuki","year":"1999","journal-title":"J. Chem. Phys."},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1038\/43410","article-title":"Discerning vibronic molecular dynamics using time-resolved photoelectron spectroscopy","volume":"401","author":"Blanchet","year":"1999","journal-title":"Nature"},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1063\/1.479248","article-title":"Femtosecond time-resolved photoelectron\u2014Photoion coincidence imaging studies of dissociation dynamics","volume":"111","author":"Davies","year":"1999","journal-title":"J. Chem. Phys."},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/S0009-2614(99)00153-0","article-title":"Femtosecond Energy- and Angle-Resolved Photoelectron Spectra","volume":"302","author":"Arasaki","year":"1999","journal-title":"Chem. Phys. Lett."},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"8871","DOI":"10.1063\/1.481534","article-title":"Femtosecond Energy- and Angle-resolved Photoelectron Spectroscopy","volume":"112","author":"Arasaki","year":"2000","journal-title":"J. Chem. Phys."},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1039\/b002739l","article-title":"Probing Wavepacket Dynamics with Femtosecond Energy- and Angl-Resolved Photoelectron Spectroscopy","volume":"115","author":"Takatsuka","year":"2000","journal-title":"Faraday Discuss."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"248303","DOI":"10.1103\/PhysRevLett.90.248303","article-title":"Pump-probe photoionization study of the passage and bifurcation of a quantum wave packet across an avoided crossing","volume":"90","author":"Arasaki","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"7913","DOI":"10.1063\/1.1609397","article-title":"Studies of electron transfer in NaI with pump-probe femtosecond photoelectron spectroscopy","volume":"119","author":"Arasaki","year":"2003","journal-title":"J. Chem. Phys."},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"10392","DOI":"10.1021\/ja904780b","article-title":"Probing ultrafast internal conversion through conical intersection via time-energy Map of photoelectron angular anisotropy","volume":"131","author":"Horio","year":"2009","journal-title":"J. Am. Chem. Soc."},{"key":"ref_144","doi-asserted-by":"crossref","first-page":"174302","DOI":"10.1063\/1.3395206","article-title":"Time-resolved photoelectron imaging of ultrafast S2\u2192S1 internal conversion through conical intersection in pyrazine","volume":"132","author":"Suzuki","year":"2010","journal-title":"J. Chem. Phys."},{"key":"ref_145","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1126\/science.1208664","article-title":"Conical Intersection Dynamics in NO2 Probed by Homodyne High-Harmonic Spectroscopy","volume":"334","author":"Bertrand","year":"2011","journal-title":"Science"},{"key":"ref_146","doi-asserted-by":"crossref","unstructured":"von Conta1, A., Tehlar, A., Schletter, A., Arasaki, Y., Takatsuka, K., and W\u00f6rner, H.J. (2018). Conical-intersection dynamics and ground-state chemistry probed by extreme-ultraviolet time-resolved photoelectron spectroscopy. Nat. Comm., 9, 3162.","DOI":"10.1038\/s41467-018-05292-4"},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"193003","DOI":"10.1103\/PhysRevLett.115.193003","article-title":"Catching Conical Intersections in the Act: Monitoring Transient Electronic Coherences by Attosecond Stimulated X-Ray Raman Signals","volume":"115","author":"Kowalewski","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_148","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1021\/acs.jpclett.5b02476","article-title":"Experimental Detection of Branching at a Conical Intersection in a Highly Fluorescent Molecule","volume":"7","author":"Brazard","year":"2016","journal-title":"J. Phys. Chem. Lett."},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1126\/science.aat0049","article-title":"CF3I conical intersection and photodissociation dynamics with ultrafast electron diffraction","volume":"361","author":"Yang","year":"2018","journal-title":"Science"},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"184305","DOI":"10.1063\/1.4966965","article-title":"Real-time observation of wavepacket bifurcation on nonadiabatically coupled field-dressed potential energy curves by means of spectrogram of induced photon-emission from molecules driven by CW laser","volume":"145","author":"Mizuno","year":"2016","journal-title":"J. Chem. Phys."},{"key":"ref_151","doi-asserted-by":"crossref","first-page":"066221","DOI":"10.1103\/PhysRevE.63.066221","article-title":"Chaos Induced by Quantum Effect Due to Breakdown of the Born-Oppenheimer Nonadiabaticity","volume":"63","author":"Fujisaki","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_152","doi-asserted-by":"crossref","first-page":"3497","DOI":"10.1063\/1.1337801","article-title":"Highly Excited Vibrational Eigenfunctions in a Multimode Nonadiabatic System with Duschinsky Rotation","volume":"114","author":"Fujisaki","year":"2001","journal-title":"J. Chem. Phys."},{"key":"ref_153","doi-asserted-by":"crossref","first-page":"1718","DOI":"10.1063\/1.458054","article-title":"Mode mixing and chaos induced by potential surface crossings","volume":"92","author":"Heller","year":"1990","journal-title":"J. Chem. Phys."},{"key":"ref_154","doi-asserted-by":"crossref","first-page":"2083","DOI":"10.1088\/0305-4470\/10\/12\/016","article-title":"Regular and irregular semiclassical wavefunctions","volume":"10","author":"Berry","year":"1977","journal-title":"J. Phys. A"},{"key":"ref_155","doi-asserted-by":"crossref","first-page":"2338","DOI":"10.1103\/PhysRevA.14.2338","article-title":"Kolmogorov entroyy anti numerical exyeriments","volume":"14","author":"Benettin","year":"1976","journal-title":"Phy. Rev. A"},{"key":"ref_156","doi-asserted-by":"crossref","first-page":"035203","DOI":"10.1103\/PhysRevE.66.035203","article-title":"Quantum chaos induced by nonadiabatic coupling in wavepacket dynamics","volume":"66","author":"Higuchi","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_157","doi-asserted-by":"crossref","first-page":"064111","DOI":"10.1063\/1.2167648","article-title":"Non-Born-Oppenheimer paths in anti-Hermitian dynamics for nonadiabatic transition","volume":"124","author":"Takatsuka","year":"2006","journal-title":"J. Chem. Phys."},{"key":"ref_158","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1103\/PhysRev.115.485","article-title":"Significance of electromagnetic potentials in the quantum theory","volume":"115","author":"Aharonov","year":"1959","journal-title":"Phys. Rev."},{"key":"ref_159","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1039\/df9633500077","article-title":"Intersection of potential energy surfaces in polyatomic molecules","volume":"35","author":"Herzberg","year":"1963","journal-title":"Discuss. Faraday Soc."},{"key":"ref_160","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1098\/rspa.1984.0023","article-title":"Quantal phase factors accompanying adiabatic changes","volume":"392","author":"Berry","year":"1984","journal-title":"Proc. R. Soc. Lond. A"},{"key":"ref_161","doi-asserted-by":"crossref","first-page":"2167","DOI":"10.1103\/PhysRevLett.51.2167","article-title":"Holonomy, the Quantum Adiabatic Theorem, and Berry\u2019s","volume":"51","author":"Simon","year":"1984","journal-title":"Phase. Phys. Rev. Lett."},{"key":"ref_162","doi-asserted-by":"crossref","first-page":"084312","DOI":"10.1063\/1.4976976","article-title":"Lorentz-like force emerging from kinematic interactions between electrons and nuclei in molecules: A quantum mechanical origin of symmetry breaking that can trigger molecular chirality","volume":"146","author":"Takatsuka","year":"2017","journal-title":"J. Chem. Phys."},{"key":"ref_163","doi-asserted-by":"crossref","first-page":"22A520","DOI":"10.1063\/1.4742155","article-title":"Electron wavepacket dynamics in highly quasi-degenerate coupled electronic states: A theory for chemistry where the notion of adiabatic potential energy surface loses the sense","volume":"137","author":"Yonehara","year":"2012","journal-title":"J. Chem. Phys."},{"key":"ref_164","doi-asserted-by":"crossref","first-page":"164304","DOI":"10.1063\/1.4947302","article-title":"Nonadiabatic electron dynamics in densely quasidegenerate states in highly excited boron cluster","volume":"144","author":"Yonehara","year":"2016","journal-title":"J. Chem. Phys."},{"key":"ref_165","doi-asserted-by":"crossref","first-page":"114101","DOI":"10.1063\/1.5094149","article-title":"Chemical bonding and nonadiabatic electron wavepacket dynamics in densely quasi-degenerate excited state manifold of boron clusters","volume":"150","author":"Arasaki","year":"2019","journal-title":"J. Chem. Phys."},{"key":"ref_166","unstructured":"Mandelbrot, B. (1982). The Fractal Geomery of Nature, Freeman."},{"key":"ref_167","doi-asserted-by":"crossref","unstructured":"Arasaki, Y., and Takatsuka, K. (2023). Quantum chaos in nonediabatic electron dynamics. to be published.","DOI":"10.1063\/5.0219345"},{"key":"ref_168","unstructured":"Schiff, L.I. (1968). Quantum Mechanics, McGraw-Hill."},{"key":"ref_169","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1002\/jcc.25557","article-title":"Electronic and nuclear flux analysis on nonadiabatic electron transfer reaction: A view from single-configuration adiabatic Born-Oppenheimer representation","volume":"40","author":"Matsuzaki","year":"2019","journal-title":"J. Comput. Chem."},{"key":"ref_170","doi-asserted-by":"crossref","first-page":"014103","DOI":"10.1063\/1.5066571","article-title":"Electronic and nuclear fluxes induced by quantum interference in the adiabatic and nonadiabatic dynamics in the Born-Huang representation","volume":"150","author":"Matsuzaki","year":"2019","journal-title":"J. Chem. Phys."},{"key":"ref_171","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.cplett.2009.05.077","article-title":"Electron flux in molecules induced by nuclear motions","volume":"476","author":"Okuyama","year":"2009","journal-title":"Chem. Phys. Lett."},{"key":"ref_172","doi-asserted-by":"crossref","first-page":"15240","DOI":"10.1021\/jp905583s","article-title":"Electron-wavepacket reaction dynamics in proton transfer of formamide","volume":"113","author":"Nagashima","year":"2009","journal-title":"J. Phys. Chem. A"},{"key":"ref_173","doi-asserted-by":"crossref","first-page":"11167","DOI":"10.1021\/jp304781m","article-title":"Early-stage dynamics in coupled proton-electron transfer from \u03c0-\u03c0* state of phenol to solvent ammonia clusters: An electron dynamics study","volume":"116","author":"Nagashima","year":"2012","journal-title":"J. Phys. Chem. A"},{"key":"ref_174","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1246\/bcsj.20110237","article-title":"Dynamical electron mechanism of double proton transfer in formic acid dimer","volume":"85","author":"Okuyama","year":"2012","journal-title":"Bull. Chem. Soc. Jpn."},{"key":"ref_175","doi-asserted-by":"crossref","first-page":"29421","DOI":"10.1039\/C5CP03982G","article-title":"Quantum theory of concerted electronic and nuclear fluxes associated with adiabatic intramolecular processes","volume":"17","author":"Bredtmann","year":"2015","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_176","doi-asserted-by":"crossref","first-page":"139462","DOI":"10.1016\/j.cplett.2022.139462","article-title":"Spin current in chemical reactions","volume":"793","author":"Hanasaki","year":"2022","journal-title":"Chem. Phys. Lett."},{"key":"ref_177","doi-asserted-by":"crossref","unstructured":"Takatsuka, K., and Arasaki, Y. (2022). Real-time electronic energy current and quantum energy flux in molecules. J. Chem. Phys., in press.","DOI":"10.1063\/5.0131200"},{"key":"ref_178","unstructured":"Pauling, L. (1960). The Nature of the Chemical Bond and the Structure of Molecules and Crystals, Cornell University Press. [3rd ed.]."},{"key":"ref_179","doi-asserted-by":"crossref","first-page":"3221","DOI":"10.1063\/1.1724989","article-title":"Phase-Space Theory of Chemical Kinetics","volume":"40","author":"Light","year":"1964","journal-title":"J. Chem. Phys."},{"key":"ref_180","doi-asserted-by":"crossref","first-page":"4967","DOI":"10.1063\/1.464952","article-title":"Absolute classical densities of states for very anharmonic systems and applications to the evaporation of rare gas clusters","volume":"98","author":"Weerasinghe","year":"1993","journal-title":"J. Chem. Phys."},{"key":"ref_181","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1063\/1.470822","article-title":"Isomer specific evaporation rates: The case of aniline\u2013Ar2","volume":"104","author":"Parneix","year":"1996","journal-title":"J. Chem. Phys."},{"key":"ref_182","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/S0301-0104(98)00317-6","article-title":"Structure, On the use of evaporation dynamics to characterize phase transitions in van der Waals clusters: Investigations in aniline\u2013(argon)n up to n = 15","volume":"239","author":"Parneix","year":"1998","journal-title":"Chem. Phys."},{"key":"ref_183","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s100530050170","article-title":"Monte-Carlo simulations of rotating clusters","volume":"3","author":"Calvo","year":"1998","journal-title":"Eur. Phys. J. D"},{"key":"ref_184","doi-asserted-by":"crossref","first-page":"2183","DOI":"10.1021\/jp0036487","article-title":"Thermal Stability of the Solidlike and Liquidlike Phases of (C60)n Clusters","volume":"105","author":"Calvo","year":"2001","journal-title":"J. Phys. Chem. B"},{"key":"ref_185","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1063\/1.1577333","article-title":"Statistical evaporation of rotating clusters. I. Kinetic energy released","volume":"119","author":"Calvo","year":"2003","journal-title":"J. Chem. Phys."},{"key":"ref_186","doi-asserted-by":"crossref","first-page":"9469","DOI":"10.1063\/1.1615518","article-title":"Statistical evaporation of rotating clusters. II. Angular momentum distribution","volume":"119","author":"Parneix","year":"2003","journal-title":"J. Chem. Phys."},{"key":"ref_187","doi-asserted-by":"crossref","first-page":"2780","DOI":"10.1063\/1.1633258","article-title":"Statistical evaporation of rotating clusters. III. Molecular clusters","volume":"120","author":"Calvo","year":"2004","journal-title":"J. Chem. Phys."},{"key":"ref_188","unstructured":"Takatsuka, K., and Arasaki, Y. (2023). Intra-molecular nonadiabatic electronic energy redistribution. to be published."},{"key":"ref_189","doi-asserted-by":"crossref","unstructured":"Zewail, A.H. (1994). Femtochemistry: Ultrafast Dynamics of the Chemical Bond, World Scientific.","DOI":"10.1142\/9789814287616"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/63\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:55:37Z","timestamp":1760147737000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/63"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,29]]},"references-count":189,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010063"],"URL":"https:\/\/doi.org\/10.3390\/e25010063","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,29]]}}}