%% -*- coding: latin-1; fill-column: 79; comment-column: 0; -*- @string{DASad = "Sadovski\'{\i}, Dmitri\'{\i} A."} @string{BIZhi = "Zhilinski\'{\i}, Boris I."} @string{NNNek = "Ne\t{kh}oro\t{sh}ev, Nikola\'{\i} N."} @String{RHCus = {Cushman, Richard H.}} @String{FFas = {Francesco Fass{\`o}}} @String{RCD = {Reg. \& Chao. Dyn.}} @String{PLA = {Phys. Lett. A}} @String{CPL = "Chem. Phys. Lett."} @String{PRL = {Phys. Rev. Lett.}} @String{PRA = "Phys. Rev. A"} @String{RMP = {Rev. Mod. Phys.}} @String{JoPA = {J. Phys. A: Math. Theor.}} @String{JCP = "J. Chem. Phys."} @String{JGPh = "J. Geom. Phys."} @String{TMPh = "Mol. Phys."} %% articles without DOI are not entered to ORCID %% https://orcid.org/0000-0002-4611-6469 @Unpublished{DS-tutorial, author = DASad, title = {Classical and quantum {H}amiltonian monodromy: a case study of the hyperbolic circular billiard}, note = {This manuscript, intended as a popular chapter for physicists, was largely written in 2008}, month = {Jul}, year = 2010, OPTurl = {/home/dima/LaTeX/delos/scatter/tutorial.tex} } %% see /home/dima/Maple/CPn/chbot.txt %% canonical rotation by pi/4 which makes L a diagonal 1:-1 harmonic oscillator @Unpublished{DS-twocentre, author = DASad, title = {Resonant integrable approximations of a general perturbation of the hydrogen atom by electric and magnetic fields}, note = {This manuscript, written in 2006--2008, is both a preparation for and a followup of the discovery of resonance zones \citep{Efstathiou2007DS-zones-letter} and most of its results were incorporated into \citep{KEDS-hatom-RMP2010}}, month = {Jan}, year = 2008, OPTurl = {/home/dima/LaTeX/monodromy/twocentre/twocentre.tex} } @Unpublished{DS-resonances, author = DASad, title = {On the geometry of resonant oscillator systems}, note = {Started in 2000 as an intended chapter in the collection of papers edited by Louis Michel \citep{Michel20017}. Under development until 2010. Results which remain largely unpublished concern the geometry of the singular reduction of two-oscillator systems in \(m_1{:}m_2\) resonance, the Poisson morphism map of the reduced phase space $P_m^{m_1{:}m_2}$ to a smooth $\setS^2$ (theorem 5.2). There are also some interesting results on 3-oscillator systems with and without additional \SO2 symmetry, and local (non-compact) analysis of two-mode systems in the presence of $C_k$-symmetry}, abstract = {We reduce the $\bS^1$-symmetry of resonant $2$- and $3$-oscillators and relate reduced systems obtained for different resonances. In particular, we map explicitly the reduced system of the $m_1{:}m_2$ oscillator (with $m_1$ and $m_2$ arbitrary positive integers) to that of the regular $1{:}1$ oscillator.}, month = {Sept}, year = 2010, OPTurl = {Maple/CPn/tex/resonances.tex} } @Unpublished{DS-snfHcf, author = DASad, title = {Complete analysis of the hydrogen atom in crossed fields based on the second normal form}, note = {Preparation for \citep{Cushman1999DS-monodromy-cf}, with Appendix~B.2 discussing for the first time the projection of the second reduced phase space $P_m$ onto a smooth $2$-sphere, a never published result that continued into \citep{DS-resonances,DS-twocentre}}, abstract = {The problem of the hydrogen atom in orthogonal electric and magnetic fields of sufficiently small strength is analyzed using a two-step normal form transformation which reduces theapproximate Keplerian and ${\mathbf S}^{1}$ symmetry of the initial problem. Group actions, invariant polynomial bases, and topology of reduced spaces are discussed in detail. A complete geometric analysis of the problem is given, and characteristics of all possible dynamically invariant subspaces are obtained from the second normal form.}, month = {Feb}, year = 1999, OPTurl = {LaTeX/monodromy/unpublished/main.tex} } @Unpublished{HH112, author = DASad, title = {Symmetric {H}amiltonian {H}opf bifurcation of a relative equilibrium of the $1{:}1{:}2$ resonant oscillator}, note = {Most likely a preparation for \citep{Cushman2004DS-CO2-letter,Zhilinskii2007DS-HHopf-112}}, abstract = {We consider the $1{:}1{:}2$ oscillator system with dynamical oscillator symmetry $S_1$ and additional specific \(\SO(2)\) symmetry, such as present in the \emph{swing spring\/} or \emph{linear molecules\/}, notably the CO$_2$ molecule. After reducing of the \(S^1\times\SO(2)\) symmetry, we study relative equilibria and their bifurcations during the transition to the Fermi limit.}, month = {Jul}, year = 2004, OPTurl = {Maple/CPn/tex/HH112.tex} } @Unpublished{DS-dpnotedirac112-2021, author = {D. A. Sadovski\'{\i} and B. I. Zhilinski\'{\i}}, title = {Universal local form of quaternionic slow-fast systems}, note = {in preparation}, month = {Nov.}, year = 2021, pages = 67, OPTurl = {http://purple.univ-littoral.fr/~sadovski/downloads/dpD112.pdf}, abstract = { We construct a slow-fast dynamical system with co-dimension $1{+}4$ (one general formal parameter and two slow degrees of freedom) degeneracy of its fast semi-quantum eigenvalues on the example of a fast spin-reversal spin-$\frac32$ subsystem coupled to two slow oscillators in spin-oscillator resonance \RES112. This is the most elementary dynamical equivalent of the spin-$\frac32$ system with a five-parameter quadrupole Hamiltonian which was introduced by Avron, Sadun, Segert, and Simon, [Commun.\ Math.\ Phys. \textbf{124}(4), 595--627 (1989)] as a time-reversal generalization of the Berry paradigm of systems exhibiting geomeric phases. The spectrum of the quantum universal local form consists of superbands which correspond to the semi-quantum eigenvalues, and the sole remaining formal parameter governs the redistribution of quantum states between these superbands. The equivalence to the formal quadrupole system is uncovered through the equality of the numbers of redistributed states and Chern numbers $c_2$ computed by Avron \emph{et al}. } } @Article{dpnote-Dosc112-lett, author = {D. A. Sadovski\'{\i} and B. I. Zhilinski\'{\i}}, title = {Quaternionic {D}irac oscillator}, journal = JoPA, note = {submitted to \href{https://iopscience.iop.org/journal/1751-8121}{\emph{J. Phys. A: Math. \& Theor.}}, ms no.~\href{http://purple.univ-littoral.fr/\%7esadovski/downloads/dpD112lett.pdf}{JPhysA-117217} on 21 Feb 2022; accepted Aug 5, 2022}, OPTurl = {http://purple.univ-littoral.fr/\%7esadovski/downloads/dpD112lett.pdf}, month = {Aug}, day = "3", year = 2022, doi="10.1088/1751-8121/ac8708", abstract = {We construct an elementary quaternionic slow-fast Hamiltonian dynamical system with one formal control parameter and two slow degrees of freedom as half-integer spin in resonance 1:1:2 with two slow oscillators. Invariant under spin reversal and having a codimension-5 crossing of its fast Kramers-degenerate semi-quantum eigenvalues, our system is the dynamical equivalent of the spin-quadrupole model by Avron, Sadun, Segert, and Simon [Commun. Math. Phys. 124(4), 595--627 (1989)], exhibiting non-Abelian geometric phases. The equivalence is uncovered through the equality of the spectral flow between quantum superbands and Chern numbers $c_2$ computed by Avron \emph{et al.}} } @Article{dpnote-S2xS2res112-Sym2022, author = {D. A. Sadovski\'{\i} and B. I. Zhilinski\'{\i}}, title = {Rearrangement of energy levels between energy super-bands characterized by second {C}hern class}, journal = {Symmetry}, year = 2022, month = {Jan.}, volume = 14, number = 2, pages = {183/17}, msid = {symmetry-1475277, received November 6, 2021, accepted January 4, 2022, published January 18, 2022}, OPTurl = {https://www.mdpi.com/2073-8994/14/2/183}, OPTnote = {as part of the Special Issue ``Symmetry and control of discrete and continuous systems'', Academic Editors: Axel Pelster, Marco Favretti, Franco Cardin, and Andrea Giacobbe}, doi = {10.3390/sym14020183} } @Comment HTML: https://www.mdpi.com/2073-8994/14/2/183/htm @Comment PDF: https://www.mdpi.com/2073-8994/14/2/183/pdf @Comment https://www.mdpi.com/journal/symmetry/special_issues/symmetry_control_discrete_continuous_systems @Article{IwaiDSBZ-dpnotedirac-JGeomMech2020, author = {T. Iwai and D. A. Sadovski\'{\i} and B. I. Zhilinski\'{\i}}, title = {Angular momentum coupling, {D}irac oscillators, and quantum band rearrangements in the presence of momentum reversal symmetries}, journal = {J. Geom. Mech.}, journalFULL = {Journal of Geometric Mechanics}, year = 2020, month = {Jul}, volume = 12, number = 3, pages = {455--505}, publisher = {American Institute of Mathematical Sciences}, address = {Department of Physics,Universit\'e du Littoral---C\^ote d'Opale, 59140 Dunkerque, France}, doi = {10.3934/jgm.2020021}, OPTurl = {http://purple.univ-littoral.fr/~sadovski/downloads/dpD2019.pdf}, OPTnote = {ms 190927-Zhilinskii; submitted September 27, 2019; revised February 27, 2020}, issue = {special issue in honor of James Montaldi}, editor = {Luis Garc\'{\i}a-Naranjo}, abstract = {We investigate the elementary rearrangements of energy bands in slow-fast one-parameter families of systems whose fast subsystem possesses a half-integer spin. Beginning with a simple case without any time-reversal symmetries, we analyze and compare increasingly sophisticated model Hamiltonians with these symmetries. The models are inspired by the time-reversal modification of the Berry phase setup which uses a family of quadratic spin-quadrupole Hamiltonians of [Mead, Phys. Rev. Lett. \textbf{59}, 161--164 (1987); Avron \emph{et al}, Commun. Math. Phys. \textbf{124}(4), 595--627 (1989)]. An explicit correspondence between the typical quantum energy level patterns in the energy band rearrangements of the finite particle systems with compact slow phase space and those of the Dirac oscillator is found in the limit of linearization near the conical degeneracy point of the semi-quantum eigenvalues.} } @Article{benpol-mono, author = {Dmitri\'{\i} A. Sadovski\'{\i} and Boris I. Zhilinski\'{\i}}, title = {Monodromy in non-integrable systems on certain compact classical phase spaces}, journal = {Phys. Lett. A}, journalFULL = {Physics Letters A}, year = 2019, volume = 383, number = 5, pages = {452--457}, publisher = {Elsevier}, issn = {0375-9601}, doi = {10.1016/j.physleta.2018.11.008}, abstract = {On the example of bending vibrational polyads of the acetylene molecule (C2H2) in the approximation of the 1:1:1:1 resonant oscillator with axial symmetry, whose geometry is similar to the n-shell approximation of the perturbed hydrogen atom, we show how remaining invariant tori of the underlying classical non-integrable system form a nontrivial continuous family with monodromy. We read this monodromy off the quantum energy spectrum which was observed experimentally by spectroscopists, and we uncover its origins through the particular topology, geometry, and symmetry. We explain how monodromy characterizes the chaotic region surrounded by the tori. We detail the explicit correspondence between the bending polyads of C2H2 and the n-shells of the hydrogen atom, and uncover the dynamical SO(3) symmetry of the bending polyads and the corresponding spherically localized vibrational states.}, OPTnote = "Submitted September 9, 2018 as PLA-D-18-01874, revised October 25, 2018, in final form November 1, accepted November 7, online November 12, 2018; Communicated by A. Eisfeld", OPTurl = "http://purple.univ-littoral.fr/~sadovski/downloads/bpmono.pdf", } @Article{benpol-C2H2, author = {Dmitri\'{\i} A. Sadovski\'{\i} and Boris I. Zhilinski\'{\i}}, title = {Qualitative models of intramolecular dynamics of acetylene: relation between the bending polyads of acetylene and perturbed {K}eplerian systems}, journal = {Mol. Phys.}, journalFULL = {Molecular Physics}, volume = {116}, number = {23-24}, pages = {3564--3601}, year = 2018, month = 5, publisher = {Taylor & Francis}, issue = {HRMS2017 / Michel Herman Festschrift}, editor = {Fr\'ed\'eric Merkt and Stefan Willitsch}, doi = "10.1080/00268976.2018.1473649", OPTnote = "Submitted on January 29, 2018, revised April 5, 2018, accepted April 8, 2018, available online May 2018", OPTurl = "http://purple.univ-littoral.fr/~sadovski/downloads/benpol-C2H2.pdf", abstract = { We analyse qualitatively the bending vibrational polyads of the acetylene molecule (C2H2) in the approximation of the resonant oscillator with axial symmetry using an effective vibrational Hamiltonian which reproduces bending vibrational energy levels computed by Michel Herman and coworkers [M. Herman, A. Campargue, M.I.E. Idrissi, and J.V. Auwera, J. Phys. Chem. Ref. Data 32(3), 921 (2003). doi: 10.1063/1.1531651]. We explain how the classical limit of this quantum system for the total vibrational angular momentum is equivalent to a reduced perturbed Keplerian system on the classical phase space, such as the hydrogen atom in external electric and magnetic fields in the Kustaanheimo-Stiefel formalism. In particular, bending vibrational n,l=0-polyads of C2H2 correspond to the n-shells of the perturbed hydrogen atom. Within this approach, using the techniques developed for the Keplerian systems and methods of the qualitative theory, we account concisely for all series of bifurcations of the classical nonlinear normal modes and their manifestations in the quantum energy level spectrum described by our predecessors [V. Tyng and M.E. Kellman, J. Phys. Chem. B 110 (38), 18859 (2006). doi: 10.1021/jp057357f]. In addition to local oscillator approximations near stable equilibrium points, notably the local modes discussed by Rose and Kellman [J. Chem. Phys. 105 (24), 10743 (1996). doi: 10.1063/1.472882] and Jacobson et al. [J. Chem. Phys. 109 (1), 121 (1998). doi: 10.1063/1.476529; J. Chem. Phys. 111, 600 (1999). doi: 10.1063/1.479341; J. Chem. Phys. 110, 845 (1999). doi: 10.1063/1.478052], we introduce two new global integrable approximations, and confirm them by constructing respective two regular complementary lattices of quantum states within one n,l=0-polyad. From the stratification of the phase space, we uncover the geometrical meaning of the corresponding new good quantum numbers and define new kind of wavefunction localisation in the neighbourhood of two spheres in the four-dimensional space. Furthermore, we use our two approximate lattices to uncover quantum monodromy of the system through the evolution of an elementary quantum cell along a closed path in the images of the nearly integrable energy-momentum maps. } } @Article{DF-csTsSO3, author = {Daniele Fontanari and Dmitrií A. Sadovski\'{\i}}, title = {Coherent states for the quantum complete rigid rotor}, journal = {J. Geom. Phys.}, journalFULL = {Journal of Geometry and Physics}, year = 2018, month = "Jul", pages = {70--89}, volume = 129, doi = {10.1016/j.geomphys.2018.02.021}, OPTurl = {https://www.sciencedirect.com/science/article/pii/S039304401830113X}, key = {Coherent states, Rigid body, Rotation group, Euler top, Symplectic structure, Husimi distribution, Resolution of identity, Quantum mechanics in phase space}, OPTnote = {Submitted Feb 23, 2017; available online March 7, 2018}, abstract = {Motivated by the possibility to describe orientations of quantum triaxial rigid rotors, such as molecules, with respect to both internal (body-fixed) and external (laboratory) frames, we go through the theory of coherent states and design the appropriate family of coherent states on T SO(3), the classical phase space of the freely rotating rigid body (the Euler top). We pay particular attention to the resolution of identity property in order to establish the explicit relation between the parameters of the coherent states and classical phase-space variables, actions and angles.} } @Article{kolya-foreword-2016, author = {Alekseí Vl. Borisov and Alekseí V. Bolsinov and Anatolí I. Nejshtadt and Dmitrií A. Sadovskií and Boris I. Zhilinskií}, title = {Nikola\'{\i} {N}. {N}ekhoroshev}, journal = {Reg. \& Chao. Dyn.}, journalFULL = {Regular and Chaotic Dynamics}, year = 2016, doi = {10.1134/S1560354716060010}, volume = 21, number = 6, pages = {593--598}, month = 12, note = {On the 70th Birthday of Nikola\'{\i} N. Nekhoroshev}, OPTurl = {http://rcd.ics.org.ru/upload/iblock/366/RCD593.pdf} } @Article{DS-nnmethod-2016, author = {Dmitri\'{\i} A. Sadovski\'{\i}}, title = {Nekhoroshev's approach to {H}amiltonian monodromy}, journalFULL = {Regular and Chaotic Dynamics}, journal = {Reg. \& Chao. Dyn.}, year = 2016, issn = {1560-3547}, doi = {10.1134/S1560354716060113}, volume = 21, number = 6, pages = {720--758}, month = {Nov}, URL = {http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rcd&paperid=221&option_lang=eng}, publisher = {Pleiades Publishing, Ltd}, annote = {Using the hyperbolic circular billiard, introduced in [31] by Delos et al. as possibly the simplest system with Hamiltonian monodromy, we illustrate the method developed by N. N. Nekhoroshev and coauthors [48] to uncover this phenomenon. Nekhoroshev's very original geometric approach reflects his profound insight into Hamiltonian monodromy as a general topological property of fibrations. We take advantage of the possibility of having closed form elementary function expressions for all quantities in our system in order to provide the most explicit and detailed explanation of Hamiltonian monodromy and its relation to similar phenomena in other domains.} } @Article{DS-nnafterw-2016, author = {Dmitrií A. Sadovskií}, title = {Afterword to the paper by {N. N.} {N}ekhoroshev}, journal = {Rus. J. Nonlin. Dyn.}, journalFULL = "Nilenejnaya Dynamika", year = 2016, volume = 12, number = 3, pages = {543-553}, doi = {10.20537/nd1603008}, OPTurl = {http://nd.ics.org.ru/upload/iblock/ef6/nd1603008p.pdf}, abstract = {This afterword accompanies the last, previously unpublished work by N. N. Nekhoroshev (pp. 413--541 of this volume). In order to explain Nekhoroshev's motivations and objectives, we describe briefly the background of his involvement in the theory of Hamiltonian monodromy during the last years of his life and his role in the discovery of fractional monodromy which is the subject of this last paper. We tell the history of the manuscript. In order to further the understanding of Nekhoroshev's point of view include excerpts from Nekhoroshev's exchanges with the editors of ``Izvestia RAN''. We end with a brief biographic commentary which accompanies photographs and materials from the archive of N. N. Nekhoroshev. Together with the manuscript, they were kindly provided by Irina Vasil'eva, the late widow of N. N. Nekhoroshev.} } @Comment Article reference: PLA2397, Final version published online: 9-AUG-2016 @Article{DF-lettS2xS2-PhLA2016, author = {Daniele Fontanari and Francesco Fass{\`o} and Dmitri\'{\i} A. Sadovski\'{\i}}, title = {Quantum manifestations of {N}ekhoroshev stability}, journal = {Phys. Lett. A}, journalFULL = {Physics Letters A}, publisher = {Elsevier}, year = 2016, volume = 380, pages = {3167--3172}, month = {Aug}, abstract = {We uncover quantum manifestations of classical Nekhoroshev theory of resonant dynamics using a simple quantum system of two coupled angular momenta with conserved equal magnitudes which corresponds to a perturbed classical integrable anisochronous Hamiltonian system}, key = {anisochronous system, coupled angular momenta, Nekhoroshev theorem, joint expectation value spectrum, resonant normal form, exponentially long time stability}, OPTurl = {http://authors.elsevier.com/a/1TWQv1LUy9Bfj0}, doi = {10.1016/j.physleta.2016.07.047} } @Article{Senent-DMSOabinitio-JPC2015, author = {M. L. Senent and S. Dalbouha and A. Cuisset and D. A. Sadovskii}, title = {Theoretical spectroscopic characterization at low temperatures of dimethyl sulfoxide: the role of anharmonicity}, journal = {J. Phys. Chem. A}, year = 2015, doi = {10.1021/acs.jpca.5b06941}, volume = 119, number = 37, pages = {9644--9652}, month = {August 27}, annote = {Published as part of The Journal of Physical Chemistry A virtual special issue "Spectroscopy and Dynamics of Medium-Sized Molecules and Clusters: Theory, Experiment, and Applications".} } @article{Fasso-hatomNekh-2015, author={Fass{\`o}, Francesco and Fontanari, Daniele and Sadovski{\'{\i}}, Dmitri{\'{\i}} A.} , title={An application of {N}ekhoroshev theory to the study of the perturbed hydrogen atom}, ALTjournal={Mathematical Physics, Analysis and Geometry}, journal={Math. Phys., Analysis and Geom.}, OPTeid=30, volume=18, number=1, pages=30, year=2015, OPTissn={1385-0172}, publisher={Springer Netherlands}, abstract = {We return to the Keplerian or n-shell approximation to the hydrogen atom in the presence of weak static electric and magnetic fields. At the classical level, this is a Hamiltonian system with the phase space S2xS2. Its principal order Hamiltonian $H_0$ was known already to Pauli in 1926. $H_0$ defines an isochronous system with a linear flow on S2xS2 and with frequencies depending on the external fields. Small perturbations of $H_0$ due to higher order terms can be studied by further normalization, either resonant or nonresonant. We study the question, raised previously, of how to decide for given parameters of the fields what normalization should be used and with regard to which resonances. We base this analysis on the Nekhoroshev theory--a branch of the Hamiltonian perturbation theory that complements the Kolmogorov-Arnold-Moser theorem. Our answer depends on the a priori choice of the maximal order $N$ of resonances that are going to be taken into account (a cutoff). For any given N, there is a decomposition of the parameter space into resonant and nonresonant zones, and a normal form with a remainder of order \(\exp (-N)\) may be consistently constructed in each of such zones}, keywords={perturbed hydrogen atom; n-shell; resonances; Normal form; Nekhoroshev theory; 37J40; 70H08; 70F05; 81V45; 37J40}, language={English}, OPTurl = {https://link.springer.com/article/10.1007/s11040-015-9200-y}, doi="10.1007/s11040-015-9200-y" } @Article{Fontanari-inhomfields-2015, author = {Fontanari, Daniele and Sadovski\'{\i}, D.A.}, title = {Perturbations of the hydrogen atom by inhomogeneous static electric and magnetic fields}, journalFULL = {Journal of Physics A: Mathematical and Theoretical}, journal = {J. Phys. A: Math. Theor.}, year = 2015, volume = 48, number = 9, pages = 095203, doi = {10.1088/1751-8113/48/9/095203}, month = 3, abstract = {The hydrogen atom is a specific, and physically relevant, Keplerian two-body system with one of the bodies assumed infinitely heavy. Isochronous integrable n-shell approximations for the perturbations of the hydrogen atom, known as Stark and Zeeman effects in the presence of homogeneous electric and magnetic fields, respectively, have been widely studied. In order to go beyond the set of linear systems, we considered perturbations by inhomogeneous electromagnetic fields. In particular, we found that the perturbation by a generic inhomogeneous magnetic field results in an approximately integrable anisochronous system which can be described as two rigid Euler tops coupled through higher order terms.} } @InCollection{RHC-signexch-2013, author = {Cushman, R. H. and Robbins, J. M. and Sadovski{\'{\i}}, D. A.}, title = {The sign exchange bifurcation in a family of linear {H}amiltonian systems}, booktitle = {Nonlinear physical systems: spectral analysis, stability, and bifurcations}, chapter = 3, pages = {41--65}, publisher = {Wiley}, year = 2013, editor = {Kirillov, O. N. and Pelinovsky, D. E.}, address = {New York}, isbn = {978-1-84821-420-0}, doi = {10.1002/9781118577608}, OPTurl = {https://onkirillov.files.wordpress.com/2015/03/ch3.pdf}, OPTurl = {http://purple.univ-littoral.fr/~dima/preprints/RHC-signexchange-nps2013-pp41-65-106.pdf}, month = {20 Dec} } @article{Cuisset-DMSOmw-2013, author = "Arnaud Cuisset and Marie-Aline Martin Drumel and Francis Hindle and Gaël Mouret and Dmitrií A. Sadovskií", title = "Rotational structure of the five lowest frequency fundamental vibrational states of dimethylsulfoxide", journalFULL = "Chemical Physics Letters", journal = "Chem. Phys. Lett.", volume = 586, number = "Suppl. C", pages = "10--15", year = 2013, issn = "0009-2614", doi = "10.1016/j.cplett.2013.09.029", OPTurl = "http://www.sciencedirect.com/science/article/pii/S0009261413011846", abstract = "Abstract We report on the successful extended analysis of the high-frequency (200-700GHz) part of the gas phase (sub)mm-wave spectra of dimethylsulfoxide (DMSO). The spectrum was recorded at 100kHz resolution using a solid state subTHz spectrometer. The five lowest energy fundamental vibrational states of DMSO with frequencies below 400cm-1 were observed as sidebands along with the main 0<-0 band. Neglecting the internal rotation of methyls, our rotational Hamiltonian reproduced the spectrum to the subMHz accuracy. We have found that the asymmetric bending state nu23 is the only low frequency fundamental vibrational state with the <> rotational structure uncovered in Cuisset et al. [1]. dmsomw 2013-09-04 15:03 (c) 2013 Elsevier" } @Article{Cuisset-DMSOnu23-JCP, author = {Cuisset,Arnaud and Sadovski{\'{\i}}, D. A.}, title = {Gyroscopic destabilisation in polyatomic molecules. Rotational structure of the low-frequency bending vibrational states {$\nu_{23}$} and {$\nu_{11}$} of dimethylsulfoxyde ({DMSO})}, journal = {J. Chem. Phys.}, year = 2013, doi = {10.1063/1.4809738}, volume = 138, pages = {234302/1--18}, month = {Jun}} @article{Cuisset-DMSOnu23-letter, title = {Gyroscopic Destabilization of Molecular Rotation and Quantum Bifurcation Observed in the Structure of the {$\nu_{23}$} Fundamental of Dimethylsulfoxyde}, author = {Cuisset, Arnaud and Pirali, Olivier and Sadovski{\'i}, Dmitri{\'i} A.}, OPTnote = {short account of the spectroscopic observation of the gyroscopic destabilisation in the $\nu_{23}$ state of DMSO}, journal = {Phys. Rev. Lett.}, volume = {109}, issue = {9}, pages = {094101}, numpages = {5}, year = {2012}, month = {Aug}, doi = {10.1103/PhysRevLett.109.094101}, OPTurl = {http://link.aps.org/doi/10.1103/PhysRevLett.109.094101}, publisher = {American Physical Society} } @Article{KEDS-hatom-RMP2010, author = {Efstathiou,K. and Sadovski{\'i},D. A.}, title = {Normalization and global analysis of perturbations of the hydrogen atom}, journal = {Rev. Mod. Phys.}, year = 2010, abstract = {The hydrogen atom perturbed by sufficiently small homogeneous static electric and magnetic fields of arbitrary mutual alignment is a specific perturbation of the Kepler system with three degrees of freedom and three parameters. Normalization of the Keplerian symmetry reveals that the parameter space is stratified into resonant zones of systems, each zone with an internal dynamical stratification of its own (Efstathiou, Sadovskií, and Zhilinskií, 2007, Proc. R. Soc. London, Ser. A 463, 1771). Based on the fully integrable approximation, the bundle of invariant tori of individual systems within zones is characterized globally and the qualitative dynamical stratification is uncovered. The techniques involved in this analysis are illustrated with the example of the 1:1 resonance zone (near orthogonal fields) whose structure is known at present. Applications in the corresponding quantum system are also described.}, publisher = {American Physical Society}, doi = {10.1103/RevModPhys.82.2099}, OPTurl = {http://link.aps.org/doi/10.1103/RevModPhys.82.2099}, volume = 82, number = 3, pages = {2099--2154}, month = "Aug"} @Article{DAS-CH4polyads-PRA2010, author="Sadovski{\'i}, D. A. and Kozlov, D. N. and Radi, P. P.", title="Direct absorption transitions to highly excited polyads 8, 10, and 12 of methane", journal="Phys. Rev. A", year="2010", month="Jul", day="06", publisher="American Physical Society", volume="82", issue="1", pages="012503/1--17", doi="10.1103/PhysRevA.82.012503", OPTurl="http://link.aps.org/doi/10.1103/PhysRevA.82.012503" } @Article{Cuisset-DMSOnu11-CPL2010, author="Cuisset, Arnaud and Nanobashvili, Lia and Smirnova, Irina and Bocquet, Robin and Hindle, Francis and Mouret, Ga{\"e}l and Pirali, Olivier and Roy, Pascale and Sadovski{\'i}, D. A.", title="Far-infrared high resolution synchrotron {FTIR} spectroscopy of the $\nu_{11}$ bending vibrational fundamental transition of dimethylsulfoxyde", journal="Chem. Phys. Lett.", year=2010, month="May", day=26, volume=492, issue="1-3", pages="30--34", doi="10.1016/j.cplett.2010.04.042" } @Article{Efstathiou-1:1-JPhA2009, author="Efstathiou, K. and Lukina, O. V. and Sadovski{\'i}, D. A.", title="Complete classification of qualitatively different perturbations of the hydrogen atom in weak near orthogonal electric and magnetic fields", journal="J. Phys. A: Math. Theor.", year="2009", volume="42", issue="5", pages="055209/1--29", doi="10.1088/1751-8113/42/5/055209" } @Article{DelosAPhy2009-dynmon, author="Delos, J. B. and Dhont, G. and Sadovski{\'i}, D. A. and Zhilinski{\'i}, B. I.", title="Dynamical manifestations of {H}amiltonian monodromy", journal="Ann. Phys.", year="2009", month="Sep", volume="324", issue="9", pages="1953--1982", keywords="Hamiltonian monodromy, Energy-momentum map", abstract="Monodromy is the simplest obstruction to the existence of global action-angle variables in integrable Hamiltonian dynamical systems. We consider one of the simplest possible systems with monodromy: a particle in a circular box containing a cylindrically symmetric potential-energy barrier. Systems with monodromy have nontrivial smooth connections between their regular Liouville tori. We consider a dynamical connection produced by an appropriate time-dependent perturbation of our system. This turns studying monodromy into studying a physical process. We explain what aspects of this process are to be looked upon in order to uncover the interesting and somewhat unexpected dynamical behavior resulting from the nontrivial properties of the connection. We compute and analyze this behavior.", doi="10.1016/j.aop.2009.03.008", issn="0003-4916", OPTurl="http://www.sciencedirect.com/science/article/B6WB1-4VXB8Y2-4/2/3b16857078ab21b8c6505456a7cfe8c8" } @Article{EfstathiouPRL2008-typ12, author="Efstathiou, K. and Lukina, O. V. and Sadovski\'{\i}, D. A.", title="Most Typical {$1{:}2$} Resonant Perturbation of the Hydrogen Atom by Weak Electric and Magnetic Fields", journal="Phys. Rev. Lett.", year="2008", month="Dec", day="19", publisher="APS", volume="101", issue="25", pages="253003/1--4", doi="10.1103/PhysRevLett.101.253003", OPTurl="http://link.aps.org/abstract/PRL/v101/e253003" } @Article{Delos2008-pearl, author="Delos, John B. and Dhont, Guillaume and Sadovski{\'i}, D. A. and Zhilinski{\'i}, B. I.", title="Dynamical manifestations of {H}amiltonian monodromy", journal="Europhys. Lett.", year="2008", month="Jul", day="04", volume="83", pages="24003/1--6", abstract={Hamiltonian monodromy - a topological property of the bundle of regular tori of a static Hamiltonian system which obstructs the existence of global action-angle variables - occurs in a number of integrable dynamical systems. Using as an example a simple integrable system of a particle in a circular box with quadratic potential barrier, we describe a time-dependent process which shows that monodromy in the static system leads to interesting dynamical effects.}, doi="10.1209/0295-5075/83/24003", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/pearl-EPL83-2008-24003.pdf" } @Article{Efstathiou2007DS-zones-letter, author="Efstathiou, K. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Classification of perturbations of the hydrogen atom by small static electric and magnetic fields", journal="Proc. Roy. Soc. London, Ser. A", year="2007", month="Jul", day="08", volume="463", issue="2083", pages="1771--1790", keywords="perturbed Kepler system", keywords="singular reduction", keywords="energy-momentum map", keywords="monodromy", abstract="We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1:1 resonance corresponds to the orthogonal field limit, studied earlier by Cushman \& Sadovski{\'i} 2000 (Physica 142, 166-196). We describe the structure of the 1:1 zone, where the system may have monodromy of different kinds, and consider briefly the 1:2 zone.", OPTnote="online 2007/05/15 (http://www.pubs.royalsoc.ac.uk/proceedingsa); received 2007/01/17, accepted 2007/03/13; ms.07PA0028", doi="10.1098/rspa.2007.1843", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/zones.pdf" } @Article{Crogman2007DS-LMRESi, author="Crogman, H. and Boudon, V. and Sadovski\'{\i}, D. A.", title="Local modes of silane within the framework of stretching vibrational polyads", journal="Europ. Phys. J. D", FULLjournal="European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics", year="2007", publisher="EDP Sciences, Societ{\`a} Italiana di Fisica, Springer-Verlag", volume="42", issue="1", pages="61--72", keywords="relative equilibrium, vibrational polyad, local mode, complex projective space", abstract="We define stretching relative equilibria (RE) of silane and other similar tetrahedral molecules in terms of the dynamical polyad symmetry which assumes the resonance condition 1:1 between the two stretching vibrational modes $\nu_1$ and $\nu_3$ of the molecule. Exploiting symmetry and topology arguments and reducing the dimension of the classical mechanical system, we find these RE. One of them, with local symmetry C3v and minimal energy within a polyad, corresponds to the local modes. We give the upper energy limit of the local mode localization within a polyad.", doi="10.1140/epjd/e2007-00011-3", OPTnote="33.15.Mt, 33.20.Vq", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/silanelm.pdf" } @Article{Zhilinskii2007DS-HHopf-112, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Hamiltonian systems with detuned {$1{:}1{:}2$} resonance: Manifestation of bidromy", journal="Ann. Phys.", year="2007", month="Jan", publisher="Academic Press", volume="322", issue="1", pages="164--200", keywords="Fermi resonance", keywords="energy-momentum map", keywords="monodromy", keywords="lattice defect theory", keywords="Nonlinear dynamical systems and bifurcations", keywords="Vibrational analysis molecular spectra", abstract="We consider a generalization of the 1:1:2 resonant swing-spring [see H. Dullin, A. Giacobbe, R.H. Cushman, Physica D 190 (2004) 15] which is suggested both by the symmetries of this system and by its physical and in particular molecular realizations [see R.H. Cushman, H.R. Dullin, A. Giacobbe, D.D. Holm, M. Joyeux, P. Lynch, D.A. Sadovski\'{\i}, B.I. Zhilinski\'{\i}, Phys. Rev. Lett. 93 (2004) 024302-1-024302-4]. Our generic integrable system is detuned off the exact Fermi resonance 1:2. The three-dimensional (3D) image of its energy-momentum map EM consists either of two or three qualitatively different non-intersecting 3D regions: a regular region at low vibrational excitation, a region with monodromy similar to that studied for the exact resonance, and in some cases-an intermediate region in which the 3D set of regular values of EM is partially self-overlapping while remaining connected. In the presence of this latter region, the system has an interesting property which we called bidromy. We analyze monodromy and bidromy for a concrete integrable classical Hamiltonian system of three coupled oscillators and for its quantum analog. We also show that the bifurcation involved in the transition from the regular region to the region with monodromy can be regarded as a special resonant equivariant analog of the Hamiltonian Hopf bifurcation. [All rights reserved Elsevier]", doi="10.1016/j.aop.2006.09.011" } @Article{Efstathiou2007DS-fracmono-konst, author="Efstathiou, K. and Cushman, R. H. and Sadovski\'{\i}, D. A.", title="Fractional monodromy of the {$1{:}(-2)$} resonance", journal="Adv. Math.", year="2007", month="Feb", day="15", publisher="Academic Press (Elsevier)", volume="209", issue="1", pages="241--273", keywords="Liouville integrable system, Singular toric fibration, Monodromy", abstract="We give an analytic proof of the fractional monodromy theorem for the 1:-2 oscillator system with S^1 symmetry formulated by N.N. Nekhoroshev, D.A. Sadovski{\'i}, and B.I. Zhilinski{\'i} in C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985\backslashu2013988. Our proof is based on an analytic description of the Hamiltonian flow on the fibers of the integral map of this system.", doi="10.1016/j.aim.2006.05.006", issn="0001-8708" } @Article{DS-BIZ-mphmon-review-2006, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Quantum monodromy and its generalizations and molecular manifestations", ALTjournal="Molecular Physics", journal="Mol. Phys.", year="2006", month="Aug", day="20", publisher="Taylor {\&} Francis", volume="104", issue="16-17", pages="2595--615", keywords="Fermi resonance", keywords="quantum monodromy", keywords="Specific calculations and results for atoms and molecules", keywords="Vibrational analysis molecular spectra", abstract="Quantum monodromy is a non-trivial qualitative characteristic of certain non-regular lattices formed by the joint eigenvalue spectrum of mutually commuting operators. The latter are typically the Hamiltonian (energy) and the momentum operator(s) which label the eigenstates of the system. We give a brief review of known quantum systems with monodromy, which include such fundamental systems as the hydrogen atom in external fields, Fermi resonant vibrations of the CO/sub 2/ molecule, and non-rigid triatomic molecules. We emphasize the correspondence between the classical Hamiltonian monodromy and its quantum analogue and discuss possible generalizations of this characteristic in classical integrable Hamiltonian dynamical systems and their quantum counterparts", doi="10.1080/00268970600673363" } @Article{Nekhoroshev2006DS-fracmono-big, author="Nekhoroshev, N. N. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Fractional {H}amiltonian monodromy", journal="Ann. H. Poincar{\'e}", journalFULL={Annales Henri Poincar{\'e}}, year=2006, publisher={Birkh\"{a}user {V}erlag}, volume=7, issue=6, pages="1099--211", keywords="classical mechanics, matrix algebra, nonlinear dynamical systems, oscillators, quantum theory, Foundations, theory of quantum measurement, miscellaneous quantum theories, Nonlinear dynamical systems and bifurcations, Algebra, set theory, and graph theory, Classical mechanics of discrete systems: general mathematical aspects", abstract="We introduce fractional monodromy in order to characterize certain non-isolated critical values of the energy-momentum map of integrable Hamiltonian dynamical systems represented by nonlinear resonant two-dimensional oscillators. We give the formal mathematical definition of fractional monodromy, which is a generalization of the definition of monodromy used by other authors before. We prove that the 1:(-2) resonant oscillator system has monodromy matrix with half-integer coefficients and discuss manifestations of this monodromy in quantum systems", doi="10.1007/s00023-006-0278-4" } @Article{Sanrey2006DS-CO2-monodromy, author="Sanrey, M. and Joyeux, M. and Sadovski\'{\i}, D. A.", title="Classical and quantum-mechanical plane switching in {CO}{$_2$}", ALTjournal="Journal of Chemical Physics", journal="J. Chem. Phys.", year="2006", month="Feb", day="21", publisher="AIP", volume="124", issue="7", pages="74318-1--12", keywords="carbon dioxide, Fermi resonance, quantum monodromy, vibrational states and polyades, Specific calculations and results for atoms and molecules, Molecular rotation, vibration, and vibration rotation constants", abstract="Classical plane switching takes place in systems with a pronounced 1:2 resonance, where the degree of freedom with the lowest frequency is doubly degenerate. Under appropriate conditions, one observes a periodic and abrupt precession of the plane in which the doubly degenerate motion takes place. In this article, we show that quantum plane switching exists in CO/sub 2/. Based on our analytical solutions of classical Hamilton's equations of motion, we describe the dependence on vibrational angular momentum and energy of the frequency of switches and the plane switching angle. Using these results, we find optimal initial wave-packet conditions for CO/sub 2/ and show, through quantum-mechanical propagation, that such a wave packet indeed displays plane switching at energies around 10000 cm$^{-1}$ above the ground state on time scales of about 100 fs", doi="10.1063/1.2167747", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/switchCO2.txt" } @Article{Sadovskii2005DS-O3-vib-RE, author="Kozin, I. N. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Assigning vibrational polyads using relative equilibria: application to ozone", journal="Spectrochim. Act. A", ALTjournal="Spectrochimica Acta, Part A Molecular and Biomolecular Spectroscopy", year="2005", month="Oct", publisher="Elsevier", volume="61", issue="13-14", pages="2867--85", keywords="ozone, nonlinear normal mode, relative equilibrium, vibrational polyade, Vibrational analysis molecular spectra, Molecular rotation, vibration, and vibration rotation constants, Specific calculations and results for atoms and molecules", abstract="We demonstrate how relative equilibria of a vibrating molecule, which are families of principal periodic orbits otherwise known as nonlinear normal modes, can be used to describe the global polyad structure of vibrational energy levels. The classical action integral n (E) computed along these orbits at different energies E corresponds to the polyad quantum number n so that the energy E (n) of different relative equilibria describes the splitting of n-polyads. Further information on the internal polyad structure can be driven from the stability analysis of relative equilibria. We use the ozone molecule as a concrete example where n-polyads or ``hyperpolyads'' should be distinguished from the well-known polyads of the 1:1 stretching mode resonance; the stretching polyads are structural elements of hyperpolyads. We give dynamical interpretation of the relation between relative equilibria and n-polyads based on the normal form reduction in the limit of small vibrations near the equilibrium. [All rights reserved Elsevier]", doi="10.1016/j.saa.2004.10.039", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/ozone.ps.gz" } @Inbook{Efstathiou2005DS-lnotes, author="Efstathiou, K. and Sadovski\'{\i}, D. A.", chapter="No Polar Coordinates. Lectures by {R}ichard {C}ushman", title="Geometric Mechanics and Symmetry. The {P}eyresq Lectures", year="2005", month="Jun", publisher="Cambridge University Press", address="Cambridge, UK", pages="211--302", dpi="10.1017/CBO9780511526367.005", abstract="MASIE Summer School, September, 2000. Contents: 1. Stability in Hamiltonian systems: applications to the restricted three-body problem by Meyer,K.R., Rink,B. and Tuwankotta,T.; 2. A crash course in geometric mechanics: Ratiu,T.S., Sousa Dias,E., Sbano,L., Terra,G. and Tudoran,R.; 3. The Euler-Poincar{\'e} variational framework for modeling fluid dynamics by Holm,D.D.; 4. No polar coordinates by Cushman,R.H., Sadovski\'{\i},D.A. and Efstanthiou,K.; 5. Survey on dissipative KAM theory including quasi-periodic bifurcation theory by Broer,H., Ciocci,M.-C. and Litvak-Hinenzon,A.; 6. Symmetric Hamiltonian bifurcations by Buono,P.-L., Laurent-Polz,F. and Montaldi,J. 412 pages.", isbn="978-0521539579", OPTurl="https://www.cambridge.org/core/books/abs/geometric-mechanics-and-symmetry/no-polar-coordinates-rh-cushman/FDBEDAFF59AC2E840B08257719DDCC06", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/lnotes-Cushman.ps.gz" } @Article{Giacobbe2004DS-qswing-spring, author="Giacobbe, A. and Cushman, R. H. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Monodromy of the quantum $1{:}1{:}2$ resonant swing spring", ALTjournal="Journal of Mathematical Physics", journal="J. Math. Phys.", year="2004", month="Dec", publisher="AIP", volume="45", issue="12", pages="5076--100", keywords="resonant harmonic oscillators, lattice defect theory, quantum monodromy, energy-momentum map, singular reduction, Algebraic methods in quantum theory, Group theory", abstract="We describe the qualitative features of the joint spectrum of the quantum 1:1:2 resonant swing spring. The monodromy of the classical analogue of this problem is studied in Dullin et al. [Physica D 190, 15-37 (2004)]. Using symmetry arguments and numerical calculations we compute its three-dimensional (3D) lattice of quantum states and show that it possesses a codimension 2 defect characterized by a nontrivial 3D-monodromy matrix. The form of the monodromy matrix is obtained from the lattice of quantum states and depends on the choice of an elementary cell of the lattice. We compute the quantum monodromy matrix, that is the inverse transpose of the classical monodromy matrix. Finally we show that the lattice of quantum states for the 1:1:2 quantum swing spring can be obtained-preserving the symmetries-from the regular 3D-cubic lattice by means of three ``elementary monodromy cuts''", doi="10.1063/1.1811788", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/qsspring.pdf" } @Article{Efstathiou2004DS-HHopf-Hatom, author="Efstathiou, K. and Cushman, R. H. and Sadovski\'{\i}, D. A.", title="Hamiltonian {H}opf bifurcation of the hydrogen atom in crossed fields", journal="Physica D", year="2004", month="Jul", day="15", publisher="Elsevier", volume="194", issue="3-4", pages="250--74", keywords="perturbed neutral hydrogen atom, perturbed Kepler system, quantum and classical monodromy, energy-momentum map, singular reduction, normal form, Nonlinear dynamical systems and bifurcations, Stark effect in atoms, Zeeman effect in atoms", abstract="We consider the hydrogen atom in crossed electric and magnetic fields. We prove that near the Stark and Zeeman limits the system goes through two qualitatively different Hamiltonian Hopf bifurcations. We explain in detail the geometry of the bifurcations", doi="10.1016/j.physd.2004.03.003", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/Hatom-HH-PhysD194-2004p250.txt" } @Article{Cushman2004DS-CO2-letter, author="Cushman, R. H. and Dullin, H. R. and Giacobbe, A. and Holm, D. D. and Joyeux, M. and Lynch, P. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="{CO}$_2$ molecule as a quantum realization of the $1{:}1{:}2$ resonant swing-spring with monodromy", journal="Phys. Rev. Lett.", year="2004", month="Jul", day="09", publisher="APS", volume="93", issue="2", pages="024302/1--4", keywords="carbon dioxide, monodromy, energy-momentum map, singular reduction, elastic pendulum, Fermi resonance, vibrational states and polyades, Vibrational analysis molecular spectra, Classical mechanics of discrete systems", abstract="We consider the wide class of systems modeled by an integrable approximation to the 3 degrees of freedom elastic pendulum with 1:1:2 resonance, or the swing-spring. This approximation has monodromy which prohibits the existence of global action-angle variables and complicates the dynamics. We study the quantum swing-spring formed by bending and symmetric stretching vibrations of the CO$_2$ molecule. We uncover quantum monodromy of CO$_2$ as a nontrivial codimension 2 defect of the three dimensional energy-momentum lattice of its quantum states", doi="10.1103/PhysRevLett.93.024302", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/letterCO2.ps.gz" } @Article{Efstathiou2004DS-HCN-monodromy, author="Efstathiou, K. and Joyeux, M. and Sadovski\'{\i}, D. A.", title="Global bending quantum number and the absence of monodromy in the {HCN$\leftrightarrow$CNH} molecule", journal="Phys. Rev. A", year="2004", month="Mar", publisher="APS through AIP", volume="69", issue="3", pages="32504-1--15", keywords="spherical pendulum, energy-momentum map, vibrational states and polyades, Vibrational analysis molecular spectra, Classical mechanics of discrete systems: general mathematical aspects, Isomerization and rearrangement, General molecular conformation and symmetry", abstract="We introduce and analyze a model system based on a deformation of a spherical pendulum that can be used to reproduce large amplitude bending vibrations of flexible triatomic molecules with two stable linear equilibria. On the basis of our model and the recent vibrational potential [J. Chem. Phys. 115, 3706 (2001)], we analyze the HCN/CNH isomerizing molecule. We find that HCN/CNH has no monodromy and introduce the second global bending quantum number for this system at all energies where the potential is expected to work. We also show that LiNC/LiCN is a qualitatively different system with monodromy", doi="10.1103/PhysRevA.69.032504", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/HCN-paper.pdf" } @Article{Efstathiou2004DS-HH-3D, author="Efstathiou, K. and Sadovski\'{\i}, D. A.", title="Perturbations of the $1{:}1{:}1$ resonance with tetrahedral symmetry: a three degree of freedom analogue of the two degree of freedom {H}\'enon-{H}eiles Hamiltonian", journal="Nonlinearity", year="2004", month="Mar", publisher="IOP Publishing; London Math.Soc", volume="17", issue="2", pages="415--46", keywords="resonant harmonic oscillator, perturbation theory, singular reduction, Nonlinear dynamical systems and bifurcations", abstract="We study a class of three degree of freedom (3-DOF) Hamiltonian systems that share certain characteristics with the 2-DOF Henon-Heiles Hamiltonian. Our systems represent a 1:1:1 resonant three-oscillator whose principal nonlinear perturbation is the cubic potential term xyz with tetrahedral symmetry. After normalizing and reducing the 1:1:1 oscillator symmetry, we show that near the limit of linearization all our systems can be described as a one-parametric family. Such reduced systems have been suggested earlier by Hecht (1960 J. Mol. Spectrosc. 5 355) and later by Patterson (1985 J. Chem.Phys. 83 4618) to model triply degenerate vibrations of tetrahedral molecules. We describe relative equilibria (RE) of these systems, classify all qualitatively different family members, and discuss bifurcations of RE involved in the transitions from one region of regular parameter values to the other", doi="10.1088/0951-7715/17/2/003", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/3DHH-cp2.ps.gz" } @Article{Efstathiou2004DS-A4-vibrot-theory, author="Efstathiou, K. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Analysis of rotation-vibration relative equilibria on the example of a tetrahedral four atom molecule", ALTjournal="SIAM Journal on Applied Dynamical Systems (SIADS)", journal="SIAM J. Appl. Dyn. Syst. (SIADS)", year="2004", volume="3", issue="3", pages="261--351", keywords="small vibrations, vibration-rotation of molecules, spherical top, relative equilibria, 1:1:1 resonant oscillator, normalization, reduction, bifurcations, orbit space, finite group action, reversing symmetry, Molien generating function, integrity basis", abstract="We study relative equilibria (RE) of a nonrigid molecule which vibrates about a well defined equilibrium configuration and rotates as a whole. Our analysis unifies the theory of rotational and vibrational RE. We rely on the detailed study of the symmetry group action on the initial and reduced phase space of our system and consider the consequences of this action for the dynamics of the system. We develop our approach on the concrete example of a four-atomic molecule $A_4$ with tetrahedral equilibrium configuration, a dynamical system with six vibrational degrees of freedom. Further applications and illustrations of our results can be found in [Van Hecke, et al, Eur.Phys.J.D 17, 13 (2001)]. Supported by the EU project Mechanics and Symmetry in Europe (MASIE), contract HPRN-CT-2000-00113.", doi="10.1137/030600015", OPTnote="submitted 23 January 2003, accepted 3 October 2003; ms{\#}060001; 37J15, 37J35, 37J40, 81V55, 58D19", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/cp2s2.pdf" } @Article{Efstathiou2003DS-CP2-local-study, author="Efstathiou, K. and Sadovski\'{\i}, D. A. and Cushman, R. H.", title="Linear {H}amiltonian {H}opf bifurcation for point-group-invariant perturbations of the {$1{:}1{:}1$} resonance", ALTjournal="Proceedings of the Royal Society of London, Series A Mathematical, Physical and Engineering Sciences", journal="Proc. Roy. Soc. London, Ser. A", year="2003", month="Dec", day="08", publisher="Royal Soc. London", volume="459", issue="2040", pages="2997--3019", keywords="linear stability, relative equilibrium, perturbation theory, Nonlinear dynamical systems and bifurcations, Group theory", abstract="We consider GxR-invariant Hamiltonians H on complex projective 2-space, where G is a point group and R is the time-reversal group. We find the symmetry-induced stationary points of H and classify them in terms of their linear stability. We then determine those points that can undergo a linear Hamiltonian Hopf bifurcation", doi="10.1098/rspa.2003.1158", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/lHHb.ps.gz" } @Article{Joyeux2003DS-LiNC-monodromy, author="Joyeux, M. and Sadovski\'{\i}, D. A. and Tennyson, J.", title="Monodromy of the {LiNC/NCLi} molecule", ALTjournal="Chemical Physics Letters", journal="Chem. Phys. Lett.", year="2003", month="Dec", day="05", publisher="Elsevier", volume="382", issue="3-4", pages="439--42", keywords="lithium compounds, quantum monodromy, energy-momentum map, spherical pendulum, potential energy surfaces, vibrational states and polyades, General molecular conformation and symmetry, Molecular rotation, vibration, and vibration rotation constants, Molecular bond strengths, dissociation energies, hydrogen bonding", abstract="Using the potential surface of Essers, Tennyson, and Wormer, we show that the system of bending vibrational states of the isomerizing molecule LiNC/NCLi has monodromy. On the basis of a deformed spherical pendulum model, we explain dynamical and geometric reasons of this phenomenon and of its absence in the similar system HCN/CNH", doi="10.1016/j.cplett.2003.10.099", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/LiNC-letter.ps.gz" } @Article{Nekhoroshev2002DS-drobmono, author="Nekhoroshev, N. N. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Fractional monodromy of resonant classical and quantum oscillators", journalFULL="Comptes Rendus de l'Academie des Sciences, Serie I Mathematique", ALTjournal = {Comptes Rendus Acad. Sci. Paris, S\'er. I}, journal = {C. R. Acad. Sci. Paris, S\'er. I}, year="2002", month="Dec", day="01", publisher="Elsevier", volume="335", number="11", pages="985--8", doi = "10.1016/S1631-073X(02)02584-0", keywords="resonance, resonant harmonic oscillator, energy-momentum map, lattice defect theory, quantum monodromy, Classical mechanics of discrete systems: general mathematical aspects, Nonlinear dynamical systems and bifurcations, Semiclassical theories and applications in quantum theory", abstract="We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(-1) and 1:(-2)", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/drobmono-cras-preprint.ps.gz" } @Article{Grondin2002DS-threeL, author="Grondin, L. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Monodromy as topological obstruction to global action-angle variables in systems with coupled angular momenta and rearrangement of bands in quantum spectra", journalFULL="Physical Review A Atomic, Molecular, and Optical Physics", journal=PRA, year="2002", month="Jan", publisher="APS through AIP", volume="65", issue="1", pages="012105/1--15", keywords="angular momentum, energy-momentum map, quantum monodromy, theory of lattice defects, Group theory, SO(2) and SO(3) groups, Semiclassical theories and applications in quantum theory", abstract="We continue to study the relation between the redistribution of levels in quantum energy spectra and monodromy of the corresponding classical system [see Phys.Lett.A 256, 235 (1999)]. A system with two coupled angular momenta and SO(2) symmetry is compared to a system with three momenta and SO(3) symmetry", doi="10.1103/PhysRevA.65.012105", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/lucien.ps.gz" } @Article{van-Hecke2001DS-P4rovib, author="Van-Hecke, C. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I. and Boudon, V.", title="Rotational-vibrational relative equilibria and the structure of quantum energy spectrum of the tetrahedral molecule {P}$_4$", journal="European Physical Journal D", year="2001", month="Oct", publisher="EDP Sciences; Springer-Verlag", volume="17", issue="1", pages="13--35", doi="10.1007/s100530170033", keywords="molecular force constants, phosphorus, rotational vibrational states and polyades, Vibrational rotational analysis molecular spectra", abstract="We find relative equilibria (RE) of the rotating and vibrating tetrahedral molecule P$_4$ and study the correspondence of these RE's to the extremal quantum states in the vibration-rotation multiplet and to the extrema of the semi-quantum rotational energy surfaces obtained for a number of excited vibrational states. To compute the energy of RE's we normalize the full rotation-vibration Hamiltonian H of P$_4$ in the approximation of nonresonant modes $\nu_2$/sup E/ and nu /sub 3/(F$_2$) and find the stationary points of the resulting normal form (known as reduced effective Hamiltonian H/sub eff/) which is defined on the reduced phase space CP$_2$ x CP/sub 1/ x S$_2$. Most of these points are fixed points of the symmetry group action on CP$_2$ x CP/sub 1/ x S$_2$. To explain our results in more detail we introduce numerical values of the parameters of H, such as the cubic force constants, using an atom-atom harmonic potential with one adjustable parameter. This simple model gives correct qualitative description of the rotational structure of the lowest excited vibrational states $\nu_2$, nu /sub 3/ and $\nu_2$ + nu /sub 3/ of P$_4$", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/P4rovib.ps.gz" } @Inbook{Sadovskii2001DS-SPT2001, author="Sadovski\'{\i}, D. A.", chapter="Normal forms, geometry, and dynamics of atomic and molecular systems with symmetry", title="Symmetry and Perturbation Theory (SPT2001)", year="2001", month="Oct", publisher="World Scientific", address="Singapore", pages="191--205", abstract="Erratum: eq.(1) must read \backslash(H = H_0 + \backslashepsilon V(q) = \backslashtfrac12(p_x^2+q_x^2) + \backslashtfrac12(p_y^2+q_y^2) + \backslashepsilon (\backslashfrac13 q_x^3 - q_x q_y^2)\backslash)", note="Proceedings of the international conference, Cala Gonone, Sardinia, Italy 6 - 13 May 2001", isbn="981-02-4793-1", doi="10.1142/9789812794543_0026", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/spt2001-talk.ps.gz" } @Article{Cushman2000DS-monodromy-cf-detailed, author="Cushman, R. H. and Sadovski\'{\i}, D. A.", title="Monodromy in the hydrogen atom in crossed fields", journal="Physica D", year="2000", month="Aug", day="01", publisher="Elsevier", volume="142", issue="1-2", pages="166--96", doi="10.1016/S0167-2789(00)00053-1", keywords="atomic structure, hydrogen neutral atom, Zeeman and Stark effect, Kustaanheimo-Stiefel regularization, SO(4) dynamical symmetry group, normal form, singular reduction, energy-momentum map, quantum monodromy, quantisation quantum theory, General mathematical and computational developments for atoms and molecules, Semiclassical theories and applications in quantum theory", abstract="We show that the hydrogen atom in orthogonal electric and magnetic fields has a special property of certain integrable classical Hamiltonian systems known as monodromy. The strength of the fields is assumed to be small enough to validate the use of a truncated normal form H/sub snf/ which is obtained from a two step normalization of the original system. We consider the level sets of H/sub snf/ on the second reduced phase space. For an open set of field parameters we show that there is a special dynamically invariant set which is a ``doubly pinched 2-torus''. This implies that the integrable Hamiltonian H/sub snf/ has monodromy. Manifestation of monodromy in quantum mechanics is also discussed", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/mndr-PhysicaD.ps.gz" } @Article{Dhont2000DS-MoCO6, author="Dhont, G. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I. and Boudon, V.", title="Analysis of the `unusual' vibrational components of triply degenerate vibrational mode $\nu_6$ of {Mo(CO)}$_6$ based on the classical interpretation of the effective rotation-vibration Hamiltonian", journal="Journal of Molecular Spectroscopy", year="2000", month="May", publisher="Academic Press", volume="201", issue="1", pages="95--108", keywords="molybdenum compounds, rotational vibrational states, Vibrational rotational analysis molecular spectra", abstract="Rotational structure of the triply degenerate vibrational state nu /sub 6/(F/sub 1u/) of the octahedral molecule Mo(CO)/sub 6/ is analyzed qualitatively on the basis of classical mechanics. We show that the energy level redistribution between the vibrational components of nu /sub 6/(F/sub 1u/) occurs due to rotational excitation and is related to the formation of singular points of classical rotational energy surfaces. The singularity is stable under small variations of parameters of the effective rovibrational Hamiltonian. Parameters responsible for the persistence of this phenomenon are specified. Comparison with quantum calculations demonstrates the high qualitative and quantitative accuracy of our classical analysis", doi="10.1006/jmsp.2000.8070", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/MoCO6paper.ps.gz" } @Article{Van-Hecke1999DS-vanhecke, author="Van-Hecke, C. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Qualitative analysis of molecular rotation starting from inter-nuclear potential", ALTjournal="European Physical Journal D", journal="Europ. Phys. J. D", year="1999", month="Sep", publisher="EDP Sciences; Springer-Verlag", volume="7", issue="2", pages="199--209", doi="10.1007/s100530050366", keywords="potential energy functions, potential energy surfaces, rotational vibrational states and polyades, Semiclassical theories and applications in quantum theory, Vibrational rotational analysis molecular spectra", abstract="We study how qualitative features of the molecular rotational dynamics can be derived directly from the internuclear (vibrational) potential. This approach is presented on the example of a tetrahedral molecule A$_4$ using several increasingly elaborated models of the potential", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/A4RES.ps.gz" } @Article{Cushman1999DS-monodromy-cf, author="Cushman, R. H. and Sadovski\'{\i}, D. A.", title="Monodromy in perturbed {K}epler systems: hydrogen atom in crossed fields", journal="Europhys. Lett.", year="1999", month="Jul", day="01", publisher="Eur. Phys. Soc. by EDP Sciences and Soc. Italiana Fisica", volume="47", issue="1", pages="1--7", doi="10.1209/epl/i1999-00341-6", keywords="hydrogen neutral atom, many body problems, perturbation theory, quantum theory, energy-momentum map, Stark effect in atoms, Zeeman effect in atoms, Semiclassical theories and applications in quantum theory, Classical mechanics of discrete systems: general mathematical aspects", abstract="We demonstrate that an integrable approximation to the hydrogen atom in orthogonal electric and magnetic fields has monodromy, a fundamental dynamical property that makes a global definition of action-angle variables and of quantum numbers impossible. When the field strengths are sufficiently small, we find our integrable approximation using a two step normalization procedure. One of dynamically invariant sets of the resulting integrable system is a doubly pinched torus whose existence proves the presence of monodromy", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/mnote-A4.ps.gz" } @Article{Sadovskii1999DS-dpnote2, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Monodromy, diabolic points, and angular momentum coupling", ALTjournal="Physics Letters A", journal="Phys. Lett. A", year="1999", month="Jun", day="07", publisher="Elsevier", volume="256", issue="4", pages="235--44", doi = {10.1016/S0375-9601(99)00229-7}, keywords="angular momentum theory, quantum monodromy, energy-momentum map, algebraic methods in quantum theory", abstract="Monodromy, or the most basic obstruction to global action-angle coordinates is shown to be present in the well known problem of two coupled angular momenta. It is also shown that in the corresponding quantum problem monodromy manifests itself as the redistribution of energy levels between different multiplets of the quantum spectrum", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/dpmnote-A4.ps.gz" } @Article{Sadovskii1998DS-crossed, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Tuning the hydrogen atom in crossed fields between the {Z}eeman and {S}tark limits", journal="Phys. Rev. A", journalFULL="Physical Review A Atomic, Molecular, and Optical Physics", year="1998", month="Apr", publisher="APS through AIP", volume="57", issue="4", pages="2867--84", doi="10.1103/PhysRevA.57.2867", keywords="Geometry, differential geometry, and topology, Group theory, SO(4) dynamical symmetry group, hydrogen neutral atom, perturbed Kepler system, Kustaanheimo-Stiefel regularization, normal form, perturbation theory, relative equilibrium, Stark and Zeeman effect in atoms", abstract="We consider the hydrogen atom in the orthogonal electric and magnetic fields whose strength is assumed to be small enough for the Coulomb n-shell perturbation theory to apply. Appropriate scaling of the two fields leads to a uniform parametrization of the problem by S, the combined strength of the two fields, and alpha , the ratio of the two field strengths. The initial six-dimensional phase space R/sub 6/ is lifted to the standard Kustaanheimo-Stiefel eight-dimensional space and then reduced explicitly to the S$_2$xS$_2$ reduced space of the n shell using the Lie transformation to the third order in S. At fixed S the system is uniformly tuned between the Zeeman and the Stark limits using the analytic formulas of the perturbation theory. The approach requires application of the invariant theory, group theory, and topology to the analysis of the dynamics on the reduced space S$_2$xS$_2$ and subsequent explicit transition to the original R/sub 6/. In particular we follow the evolution of the four principal periodic orbits (nonlinear normal modes) and corresponding four relative equilibria on S$_2$xS$_2$", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/csfield.ps.gz" } @Article{Dabrowski1997DS-KrD-f, author="Dabrowski, I. and Sadovski\'{\i}, D. A.", title="Spectra of rare gas hydrides. {VI}. The $4f$ complex and the electronic structure of {KrD}", journal="Journal of Chemical Physics", year="1997", month="Dec", day="01", publisher="AIP", volume="107", issue="21", pages="8874--85", doi="10.1063/1.475179", keywords="molecular Rydberg electronic states, rotational structure electronic molecular states, Infrared molecular spectra, Rotational analysis electronic molecular spectra, Molecular rotation, vibration, and vibration rotation constants", abstract="The complete rotational analysis of the 4f(/sup 2/ Phi , /sup 2/ Delta , /sup 2/pi, /sup 2/ Sigma /sup +/)to4d/sup 2/ Delta (6600 cm$^{-1}$), 4f(/sup 2/ Phi , /sup 2/ Delta , /sup 2/pi, /sup Sigma +/)to4d/sup 2/pi (8500 cm$^{-1}$), and 4d/sup 2/ Delta to5p /sup 2/pi (7100 cm$^{-1}$) Rydberg electronic transitions of KrD is reported. The two newly observed states, the doublet 4f complex and 4d/sup 2/ Delta together with the np(/sup 2/pi, /sup 2/ Sigma /sup +/), nd /sup 2/ Sigma /sup +/, ns /sup 2/ Sigma /sup +/, and nd /sup 2/pi studied earlier by Dabrowski et al., [Mol. Phys. 63, 269 (1988)], and by Dabrowski and Sadovski\'{\i} [Mol.Phys. 81, 291 (1994)], form a set of states which permits the whole Rydberg electronic structure of the molecule to be determined from the experimental data" } @Article{Sadovskii1996DS-nf-PT, author="Sadovski\'{\i}, D. A. and Delos, J. B.", title="Bifurcation of the periodic orbits of {H}amiltonian systems: an analysis using normal form theory", journalFULL="Physical Review E Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics", journal="Phys. Rev. E", year="1996", month="Aug", publisher="APS / AIP", volume="54", issue="2", pages="2033--70", doi = "10.1103/PhysRevE.54.2033", keywords="Lie algebra, Lie transform, perturbation theory, normal form, Nonlinear dynamical systems and bifurcations", abstract="We develop an analytic technique to study the dynamics in the neighborhood of a periodic trajectory of a Hamiltonian system. The theory begins with Poincare and Birkhoff; major modern contributions are due to Meyer, Arnol'd, and Deprit. The realization of the method relies on local Fourier-Taylor series expansions with numerically obtained coefficients. The procedure and machinery are presented in detail on the example of the ``perpendicular'' (z=0) periodic trajectory of the diamagnetic Kepler problem. This simple one-parameter problem well exhibits the power of our technique. Thus, we obtain a precise analytic description of bifurcations observed by J.-M. Mao and J.B. Delos [Phys.Rev. A 45, 1746 (1992)] and explain the underlying dynamics and symmetries", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/nf-PT.ps.gz" } @Article{Sadovskii1996DS-collapse, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I. and Michel, L.", title="Collapse of the {Z}eeman structure of the hydrogen atom in an external electric field", journal="Phys. Rev. A", journalFULL="Physical Review A Atomic, Molecular, and Optical Physics", year="1996", month="Jun", publisher="APS via AIP", volume="53", issue="6", pages="4064--4067", doi="10.1103/PhysRevA.53.4064", keywords="Group theory, hydrogen neutral atom, Rydberg states, Zeeman effect in atoms, Stark effect in atoms", abstract="We analyze the transition from a Zeeman to a Stark structure of a weakly split Rydberg $n$ multiplet of the H atom in parallel magnetic and electric fields. The use of classical mechanics, topology, and group theory provides a detailed description of the modifications of dynamics due to the variation of the electric field. We focus on the point where the collapse of the Zeeman structure occurs, give the sequence of classical bifurcations responsible for the transition between different dynamic regimes, and compare it to the quantum energy-level structure", url="http://purple.univ-littoral.fr/~dima/preprints/Collapse.ps.gz" } @Article{Sadovskii1995DS-polyads, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Counting levels within vibrational polyads: generating function approach", journal="J. Chem. Phys.", journalFULL="Journal of Chemical Physics", year=1995, month="Dec", day=22, publisher="AIP", volume=103, issue=24, pages="10520--36", doi="10.1063/1.469836", keywords="density of states, resonant harmonic oscillator, vibrational polyades, invariant theory, generating fumction, General mathematical and computational developments for atoms and molecules, Vibrational analysis molecular spectra", abstract="Simple analytical formulas for the number of energy levels in the vibrational polyads are given. These formulas account for the resonances between the vibrational modes, and for the symmetry of the problem, so that the number of states of a particular symmetry type can be computed. The formulas are used to estimate the differential and integral densities of states from the minimum initial information about the molecule. Examples of the vibrational structure of triatomic molecules A/sub 3/, tetrahedral molecules AB$_4$, and linear molecules AB$_2$ are considered. The analytical formulas are compared to the ab initio results for H$_3^+$ [J.R. Henderson et al., J.Chem.Phys. 98, 7191 (1993)]", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/Polyads.ps.gz" } @Article{Sadovskii1995DS-PT-lett, author="Sadovski\'{\i}, D. A. and Shaw, J. A. and Delos, J. B.", title="Organization of sequences of bifurcations of periodic orbits", journalFULL="Physical Review Letters", journal="Phys. Rev. Lett.", year=1995, month="Sep", day=11, volume=75, issue=11, pages="2120--2123", doi="10.1103/PhysRevLett.75.2120", keywords="perturbation theory, normal form, Nonlinear dynamical systems and bifurcations", abstract="Numerical calculations have shown that bifurcations of periodic orbits of Hamiltonian systems often occur in organized groups. Normal-form theory provides an explanation", OPTurl="http://purple.univ-littoral.fr/~dima/preprints/PRL75p2120-1995.djvu" } @Article{Majewski1994DS-H3pHIA, author="Majewski, W. A. and McKellar, A. R. W. and Sadovski\'{\i}, D. A. and Watson, J. K. G.", title="New observations and analysis of the infrared vibration-rotation spectrum of {{H}$_3^+$}", journal="Can. J. Phys.", journalFULL="Canadian Journal of Physics", year="1994", month="Nov", volume="72", issue="11-12", pages="1016--27", doi="10.1139/p94-133", keywords="triatomic hydrogen, positive molecular ions, rotational vibrational states, Infrared molecular spectra, Vibrational rotational analysis molecular spectra, Molecular rotation, vibration, and vibration rotation constants, Molecular lifetimes, absolute and relative line and band intensities", abstract="The coverage of the infrared spectrum of the H$_3^+$ molecular ion is extended by further measurements of the emission from a small high-pressure hollow-cathode electrical discharge cell, using the pressure dependence of the intensities to distinguish the lines of H$_3^+$ from the many lines due to electronic transitions of H$_2$ in the same region. The lines were assigned by means of calculations with a large vibration-rotation basis, using a Morse-based discrete variable representation for the vibrations. Eight coefficients in the Meyer-Botschwina-Burton ab initio potential were adjusted in order to fit 621 lines of H$_3^+$ from this and previous work with a standard deviation of 0.118 cm$^{-1}$" } @Article{Amano1994DS-D3pHIA, author="Amano, T. and Chan, M. C. and Civis, S. and McKellar, A. R. W. and Majewski, W. A. and Sadovski\'{\i}, D. A. and Watson, J. K. G.", title="The infrared vibration-rotation spectrum of the {D}$_3^+$ molecular ion: extension to higher vibrational and rotational quantum numbers", journal="Canadian Journal of Physics", year="1994", month="Nov", volume="72", issue="11-12", pages="1007--15", doi="10.1139/p94-132", keywords="deuterium, positive molecular ions, rotational vibrational states, Infrared molecular spectra, Molecular rotation, vibration, and vibration rotation constants", abstract="The assigned infrared spectrum of the D$_3^+$ molecular ion is extended from 84 to 529 lines by three types of spectroscopic measurements on hollow-cathode electrical discharges through deuterium gas: (i) emission from a small hollow cathode, in which the D$_3^+$ lines were distinguished from the electronic transitions of D$_2$ by the pressure dependence of their intensities; (ii) absorption in the same hollow cathode with discharge amplitude modulation; and (iii) absorption in a larger hollow cathode cell with discharge amplitude modulation. The observed lines being mainly to the $\nu_2$/sup 1/-0 fundamental and its hot bands 2 $\nu_2$/sup 0/- $\nu_2$/sup 1/, 2 $\nu_2$/sup 2/- $\nu_2$/sup 1/ and nu /sub 1/+ $\nu_2$/sup 1/- nu /sub 1/ in the 1330-2530 cm$^{-1}$ region in experiments (i) and (ii), and to the first overtone band 2 $\nu_2$/sup 2/-0 in the 3160-4000 cm$^{-1}$ region in experiment (iii). The lines were assigned by means of calculations with a large vibration-rotation basis, using a Morse-based discrete variable representation for the vibrations. This method overcomes the convergence problems for the centrifugal terms in the effective Hamiltonian approach. Six coefficients in the Meyer-Botschwina-Burton ab initio potential were adjusted in order to fit these data with a standard deviation of 0.059 cm$^{-1}$" } @Article{Dabrowski1994DS-KrD-p, author="Dabrowski, I. and Sadovski\'{\i}, D. A.", title="Spectra of the rare gas hydrides. {V}. The {$np$} {R}ydberg series of {KrD}", ALTjournal="Molecular Physics", journal="Mol. Phys.", year="1994", month="Feb", day="10", volume="81", issue="2", pages="291--326", doi="10.1080/00268979400100201", keywords="krypton compounds, molecular excited electronic states, molecular Rydberg states, spin orbit interactions, visible spectra of diatomic inorganic molecules", abstract="The 23920 cm$^{-1}$, 21950 cm$^{-1}$, and 17840 cm$^{-1}$ emission bands of KrD have been analysed successfully as transitions from the 8p, 7p, and 6p Rydberg complexes to the lowest bound state, /sup 2/ Sigma . It was found that the structure of these states corresponds to a special case of l uncoupling with a Coriolis-like linear dependence of the case (d) electron-rotation energies on the quantum number R. The structure of the lowest member of the series, the 5p(/sup 2/ Sigma ,/sup 2/pi) complex, has been obtained from the simultaneous analysis of the 5050 cm$^{-1}$ and 5800 cm$^{-1}$ emission bands as transitions from the 4d/sup 2/pi case (b) upper state to the 5p/sup 2/ Sigma and 5p/sup 2/pi components of the complex. The latter are widely split, E/sub pi/-E/sub Sigma /=735 cm$^{-1}$ due to electronic interactions and are further distorted by strong spin-orbit interactions. The 5p(/sup 2/ Sigma ,/sup 2/pi) complex corresponds to the p complex of ArD studied recently by Dabrowski et al. (1992)" } @Article{Sadovskii1993DS-CP2, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Qualitative study of a model three-level {H}amiltonian with {$\mathrm{SU}(3)$} dynamical symmetry", journalFULL="Physical Review A: Atomic, Molecular, and Optical Physics", journal="Phys. Rev. A", year=1993, month="Aug", volume=48, issue=2, pages="1035--44", doi="10.1103/PhysRevA.48.1035", keywords="bifurcation, Algebraic methods in quantum theory, Geometry, differential geometry, and topology, complex projective space, Group theory, SU(3) group", abstract="Bifurcations found by Leboeuf and Saraceno [Phys.Rev. A 41, 4614 (1990)] in a model Hamiltonian system with SU(3) dynamical symmetry are analyzed in terms of a topological, symmetry-adapted approach. The topological structure of the reduced phase space of the problem, a complex projective space in two dimensions, CP$^2$, is used explicitly in the analysis. The study of the action of the finite (additional) symmetry group of the problem leads to a complete classification of stationary solutions and their bifurcations" } @Article{Sadovskii1993DS-H3plus-vib, author="Sadovski\'{\i}, D. A. and Fulton, N. G. and Henderson, J. R. and Tennyson, J. and Zhilinski\'{\i}, B. I.", title="Nonlinear normal modes and local bending vibrations of {H$_3^+$} and {D$_3^+$}", journal="Journal of Chemical Physics", year="1993", month="Jul", day="15", volume="99", issue="2", pages="906--18", doi="10.1063/1.465355", keywords="positive triatomic hydrogen and deuterium ions, Nonlinear normal modes, relative equilibria of molecules, molecular energy level calculations, molecular vibration, Specific calculations and results for atoms and molecules, Molecular rotation, vibration, and vibration rotation constants", abstract="The structure of bending overtones of the H$_3^+$ and D$_3^+$ molecular ions at the energies below the barrier to linearity is analyzed using energies and wave functions from full three-dimensional discrete variable representation calculations. The lowest-in-energy states of the vibrational polyads $\nu_2$=4,5,6 are shown to follow the localization pattern of local bending modes, three equivalent-by-symmetry principal periodic trajectories of the corresponding classical two-mode system near the equilibrium" } @Article{Sadovskii1993DS-gt-analysis, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Group-theoretical and topological analysis of localized rotation-vibration states", journal="Phys. Rev. A", journalFULL="Physical Review A Atomic, Molecular, and Optical Physics", publisher = {American Physical Society}, year="1993", month="Apr", volume="47", issue="4", pages="2653--71", keywords="Group theory, molecular rotation vibration calculations, Vibrational rotational analysis molecular and atomic spectra", abstract="A general scheme of qualitative analysis is applied to molecular rovibrational problems. The classical-quantum correspondence provides a description of different classes of localized quantum rotation-vibration states associated with localized classical motion. A description of qualitative features, such as localized motion, and of qualitative changes, such as localization phenomena, is based on the concept of the simplest Hamiltonian. It uses only the topological properties of the compact reduced phase space and the action of the symmetry group on this space. The qualitative changes of the simplest Hamiltonian are analyzed as bifurcations caused by rotational or vibrational excitation. The relation between the stationary points of the classical Hamiltonian function on the reduced phase space and the principal periodic trajectories in the coordinate space is analyzed for vibrational Hamiltonians. In particular, the relation between the nonlinear normal modes, proposed by Montaldi, Roberts, and Stewart [Philos. Trans.R.Soc. London, Ser.A 325, 237 (1988)], and normal- and local-mode models widely used in molecular physics is discussed. Along with a general consideration of localized rotational and vibrational states a more detailed analysis of the vibrational dynamics of an X/sub 3/ molecule with the D/sub 3h/ symmetry, such as the H$_3^+$ molecular ion, is given", doi = {10.1103/PhysRevA.47.2653}, OPTurl = {https://link.aps.org/doi/10.1103/PhysRevA.47.2653} } @Article{Krivtsun1992DS-31, author="Krivtsun, V. M. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Study of the {$\nu_1,\nu_3$} bands of {{$^{120}$SnH$_4$}}. {II}. {S}emiclassical qualitative analysis: critical phenomena and diabolic points", journal="Proceedings of the SPIE The International Society for Optical Engineering", year="1992", volume="1724", pages="165--76", keywords="critical phenomena, molecular rotation vibration calculations, tin compounds, Vibrational rotational analysis molecular spectra", abstract="For part I see ibid., vol.1724, p.148-64 (1992). The qualitative modifications of the rovibrational energy-level structure under rotational excitation are studied for the $\nu_1$, $\nu_3$ bands of $^{120}$SnH$_4$. Two types of qualitative changes exist: the critical phenomena corresponding to the modification of the cluster structure and the so-called diabolic points associated with the redistribution of energy levels between different branches as the rotational quantum number increases. A simple model interpretation of energy-level redistribution is given to show that such a phenomenon is typical for tetrahedral molecules" } @Article{Krivtsun1992DS-32, author="Krivtsun, V. M. and Sadovski\'{\i}, D. A. and Snegirev, E. P.", title="Study of the $\nu_1$, $\nu_3$ bands of $^{120}${SnH}$_4$. {I}. {D}iode laser spectroscopy", journal="Proceedings of the SPIE The International Society for Optical Engineering", year="1992", volume="1724", pages="148--64", keywords="infrared spectra of polyatomic inorganic molecules, molecular electronic states, molecular rotation vibration, molecular vibration, tin compounds, Infrared molecular spectra, Vibrational analysis molecular spectra, Vibrational rotational analysis molecular spectra", abstract="An absorption spectrum of the SnH$_4$ molecule has been measured with Doppler limited resolution in the region 1903-60 cm$^{-1}$ by means of a tunable diode laser spectrometer. IR transitions of the Q- and R-branches of allowed $\nu_3$(F$_2$) and `forbidden' $\nu_1$(A$_1$) bands were observed. These two states were strongly mixed by the rotational-vibrational interaction, and as a consequence the $\nu_1$ band appears in absorption. Wavenumbers of transitions to the $\nu_1$, $\nu_3$ energy levels with the quantum number of angular momentum J=1-15 have been obtained with an accuracy of about 0.001 cm$^{-1}$. More than 230 transitions of the $^{120}$SnH$_4$ isotopic species were used for simultaneous analysis of the $\nu_1$, $\nu_3$ resonance states. In this way 21 spectroscopic parameters of the upper states have been determined using a model Hamiltonian, which explicitly takes into account the resonance interaction. The parameters outlined reproduce the experimental spectrum with r.m.s. discrepancy of 0.002 cm$^{-1}$" } @Article{Dabrowski1992DS-ArD-p, author="Dabrowski, I. and DiLonardo, G. and Herzberg, G. and Johns, J. W. C. and Sadovski\'{\i}, D. A. and Vervloet, M.", title="Spectra of the rare gas hydrides. IV. Three new bands of argon deuteride involving a low-lying `$p$' Rydberg state", journal="Journal of Chemical Physics", year="1992", month="Nov", day="15", volume="97", issue="10", pages="7093--110", doi="10.1063/1.463535", keywords="argon compounds, infrared spectra of diatomic inorganic molecules, molecular excited states, molecular rotation calculations, Infrared molecular spectra, Rotational analysis molecular spectra, Excited states of atoms and molecules", abstract="For part III see Mol.Phys. 65(5), p.1217-25 (1988). Three fairly strong emission bands of ArD at 6100, 6900, and 10200 cm$^{-1}$ have been studied at high resolution. They show very different structures: the first exhibits only Q-form branches, the second P, R, O, and S-form branches, and the third Q, S, and O-form branches. The rotational analysis of each of these bands is not trivial, mainly because the upper state of the first which proved to be also the lower state of the second and third is strongly affected by l uncoupling [i.e., transition to Hund's case (d)] and as a result the usual method of combination differences cannot be applied in their analysis. However, more sophisticated combination differences have been found between the three bands which lead to unambiguous assignments of rotational quantum numbers and thus to rotational constants. It was found that the upper state of the 6100 cm$^{-1}$ band can be most conveniently represented as a $p(^2\backslashSigma, ^2\backslashPi)$ complex. This assumption, together with the theoretical selection rules for such a case, led to a full understanding of the structure of the three bands" } @Article{Krivtsun1991DS-34, author="Krivtsun, V. M. and Sadovski\'{\i}, D. A. and Snegirev, E. P.", title="Rotational structure of $^{120}${SnH}$_4$ molecule: $\nu_1$, $\nu_3$ bands", journal="Optics and Spectroscopy", year="1991", month="May", volume="70", issue="5", pages="590--2", keywords="infrared spectra of polyatomic inorganic molecules, molecular rotation vibration, tin compounds, Infrared molecular spectra, Vibrational rotational analysis molecular spectra, Molecular rotation, vibration, and vibration rotation constants", abstract="The absorption spectrum of the $^{120}$SnH$_4$ molecule has been measured in the 1903-1960-cm$^{-1}$ region with {\textasciitilde}10/sup -3/-cm$^{-1}$ resolution using an automated tunable diode laser spectrometer. More than 230 transitions of the $\nu_1$, $\nu_3$ bands for the quantum number values J=1-15 were recorded and their assignment was given by solving the inverse spectroscopic problem. The values of the twenty-one model Hamiltonian parameters thus obtained reproduce the experimental spectrum with a standard deviation of 2x10$^{-3}$ cm$^{-1}$" } @Article{Sadovskii1991DS-tensor-prods, author="Sadovski\'{\i}, D. A.", title="Standard decomposition of the product of spherical tensor equivalent operators", journal="Physics Letters A", year="1991", month="Jul", day="01", volume="156", issue="7-8", pages="377--80", doi="10.1016/0375-9601(91)90710-P", keywords="SO groups, Other internal and higher symmetries in particle physics", abstract="A method is proposed to express a product of SO(3)-tensor equivalents in terms of an operator basis which comprises all the equivalents, both diagonal and nondiagonal in angular momentum, multiplied by powers of the number-of-spins scalar n=2j" } @Article{Davarashvili1990DS-CF4-exp, author="Davarashvili, O. I. and Zhilinski\'{\i}, B. I. and Krivtsun, V. M. and Sadovski\'{\i}, D. A. and Snegirev, E. P.", title="Experimental study of a sequence of quantum bifurcations leading to the crossover of a rotational multiplet", journal="JETP Letters", year="1990", month="Jan", day="10", volume="51", issue="1", pages="18--21", OPTurl="http://jetpletters.ru/ps/1135/article_17172.pdf", keywords="molecular rotational vibrational structure, spherical top molecules, tetrahedral molecules", abstract="in Russian (Pis'ma v Zh.Eksp.Theor.Phyz.) pp.17--19" } @Article{Sadovskii1990DS-manifestations, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I. and Champion, J. P. and Pierre, G.", title="Manifestation of bifurcations and diabolic points in molecular energy spectra", ALTjournal="Journal of Chemical Physics", journal="J. Chem. Phys.", year=1990, month="Feb", day=01, volume=92, issue=3, pages="1523--37", doi = {10.1063/1.458083}, keywords="molecular rotation vibration, Vibrational rotational analysis molecular spectra, Quantum theory, spherical top molecules", abstract="The general scheme of the qualitative analysis of finite-particle quantum problems is discussed. Theoretical methods for the qualitative analysis of energy spectra in quantum problems are applied to the study of the rovibrational energy levels of spherical top molecules. The existence of modifications in the rotational cluster structure and of redistributions of the energy levels between different branches under the rotational excitation is shown for the $\nu_2$/ $\nu_4$ dyad of the /sup 12/CH$_4$ and /sup 28/SiH$_4$ molecules. An interpretation of the qualitative features of energy spectra in terms of bifurcations and formations of conical intersection points (diabolic points) on the energy surfaces of the corresponding classical problem is proposed" } @Article{Krivtsun1990DS-SnH4-theor, author="Krivtsun, V. M. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Critical phenomena and diabolic points in rovibrational energy spectra of spherical top molecules", journal="Journal of Molecular Spectroscopy", year="1990", month="Jan", volume="139", issue="1", pages="126--46", doi="10.1016/0022-2852(90)90246-M", keywords="critical phenomena, molecular rotation vibration calculations, organic compounds, tin compounds, Vibrational rotational analysis molecular spectra", abstract="The qualitative modifications of the rovibrational energy-level structure under rotational excitation are studied for spherical top molecules. The $\nu_2$, $\nu_4$ bands of /sup 12/CD$_4$ and $^{120}$SnH$_4$ and the $\nu_1$, $\nu_3$ bands of $^{120}$SnH$_4$ are treated as examples. Two types of qualitative changes are shown to exist: the critical phenomena corresponding to the modification of the cluster structure and the so-called diabolic points associated with the redistribution of energy levels between different branches as the rotational quantum number increases. A simple model interpretation of energy-level redistribution is given to show that such a phenomenon is typical for tetrahedral molecules. Manifestation of critical phenomena and diabolic point formation in high-resolution infrared spectra are discussed" } @Article{Krivtsun1990DS-SnH4-exp, author="Krivtsun, V. M. and Sadovski\'{\i}, D. A. and Snegirev, E. P. and Shotov, A. P. and Zasavitski\'{\i}, I. I.", title="Diode laser study of the $\nu_1$ and $\nu_3$ bands of the $^{120}${SnH}$_4$ molecule", journalFULL="Journal of Molecular Spectroscopy", journal="J. Molec. Spectros.", year="1990", month="Jan", volume="139", issue="1", pages="107--25", doi={10.1016/0022-2852(90)90245-L}, keywords="infrared spectra of polyatomic inorganic molecules, molecular rotation vibration, tin compounds, tetrahedral molecules, spherical top molecules, Infrared molecular spectra, Vibrational rotational analysis molecular spectra", abstract="The absorption spectrum of the SnH$_4$ molecule has been measured with Doppler limited resolution in the region 1903-1960 cm$^{-1}$ by means of a tunable diode laser spectrometer. IR transitions of the Q and R branches of allowed $\nu_3$(F$_2$) and `forbidden' $\nu_1$(A$_1$) bands were observed. These two states were found to be strongly mixed by the rotational-vibrational interaction, and as a consequence of the $\nu_1$ band appears in absorption. Wavenumbers of transitions to the $\nu_1$, $\nu_3$ energy levels with the quantum number of angular momentum J=1-15 have been obtained with an accuracy of about 0.001 cm$^{-1}$. More than 230 transitions of the $^{120}$SnH$_4$ isotopic species were used for simultaneous analysis of the $\nu_1$, $\nu_3$ resonance states. In this way 21 spectroscopic parameters of the upper states have been determined using a model Hamiltonian which explicitly takes into account the resonance interaction. The parameters obtained reproduced the experimental spectrum with r.m.s. discrepancy of 0.002 cm$^{-1}$" } @Article{Zhilinskii1989DS-40, author="Zhilinski\'{\i}, B. I. and Sadovski\'{\i}, D. A.", title="Qualitative changes in the structure of rovibrational levels. {D}yad $\nu_2$, $\nu_4$ of spherical top molecules", journal="Bulletin of the Academy of Sciences of the USSR, Physical Series", year="1989", volume="53", issue="59", pages="80--83", keywords="molecular vibration rotations states, Rovibrational interactions molecular spectra, Rotational analysis molecular spectra, tetrahedral molecules, spherical top molecules", abstract="in Russian (Izvestiya Akademii Nauk SSSR, Seriya Fizika) pp.1736--1740" } @Article{Sadovskii1989DS-ct-mu, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Contact transformations in tensor formalism. Effective {H}amiltonian and dipole moment for the $\nu_2$, $\nu_4$ dyad of tetrahedral molecules", journal="Journal of Quantitative Spectroscopy and Radiative Transfer", year="1989", month="Dec", volume="42", issue="6", pages="575--83", doi={10.1016/0022-4073(89)90050-2}, keywords="spherical top molecules, molecular vibration and rotations, Vibrational and rotational analysis molecular spectra, Molecular electric and magnetic moments and derivatives, polarizability, and magnetic susceptibility", abstract="Simultaneous unitary transformations of the effective Hamiltonian and the effective dipole moment are studied via an irreducible tensor formalism for the $\nu_2$, $\nu_4$ dyad of tetrahedral molecules. Large unitary transformations relating resonance and isolated band models are treated by taking into account the non-leading contributions and multiple commutators. The unitary equivalence of different parameter sets of the effective dipole moment is explicitly shown for the $\nu_2$, $\nu_4$ bands of /sup 12/CH$_4$" } @Article{Pierre1989DS-crossover-note, author="Pierre, G. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Organization of quantum bifurcations: crossover of rovibrational bands in spherical top molecules", ALTjournal="Europhysics Letters", journal="Europhys. Lett.", year="1989", month="Nov", day="01", volume="10", issue="5", pages="409--14", doi={10.1209/0295-5075/10/5/004}, keywords="molecular rotation vibration calculations, quantum theory, Vibrational rotational analysis molecular spectra, Algebraic methods in quantum theory", abstract="Qualitative changes in the rotational structure of a finite particle quantum system are studied. The crossover phenomenon is explained from the point of view of consecutive quantum bifurcations. The generic organization of bifurcations is related to the stratification of the space of dynamical variables imposed by the invariance group of the Hamiltonian" } @Article{Sadovskii1988DS-q-vibrot-MP, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Qualitative analysis of vibration-rotation {H}amiltonians for spherical top molecules", ALTjournal="Molecular Physics", journal="Mol. Phys.", year="1988", month="Sep", volume="65", issue="1", pages="109--128", doi = "10.1080/00268978800100891", keywords="spherical top molecules, Vibrational rotational analysis molecular spectra, Molecular rotation, vibration, and vibration rotation constants, Rotational analysis molecular spectra", abstract="A new approach to the qualitative analysis of the rotational structure of a group of degenerate or quasidegenerate vibrational states is proposed. The approach is based on a quantum description of resonant vibrational states and on using classical methods when describing the rotational structure of these states. The approach is used for studying the rotational structure of two- and three-fold degenerate vibrational states of the tetrahedral AB$_4$ molecule as well as studying the $\nu_1$, $\nu_3$ diad of /sup 28/SiH$_4$. The general scheme of the analysis is realized by computer" } @Article{Pavlov-Verevkin1988DS-dp-note, author="Pavlov-Verevkin, V. B. and Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="On the dynamical meaning of the diabolic points", journalFULL="Europhysics Letters", journal="Europhys. Lett.", year="1988", month="Aug", day="01", volume="6", issue="7", pages="573--8", doi = {10.1209/0295-5075/6/7/001}, keywords="Quantum theory, spherical top molecules", abstract="Using a simple exactly soluble quantum model, it is shown that the diabolic points may be associated with the qualitative phenomenon of the redistribution of the energy levels between different branches in the energy spectra" } @Article{Zhilinskii1986DS-q-vibrot-OS, author="Zhilinski\'{\i}, B. I. and Sadovski\'{\i}, D. A.", title="Qualitative changes in the rotational structure of vibrational states with increasing angular momentum", journal="Optics and Spectroscopy", year="1986", month="Sep", volume="61", issue="3", pages="301--304", keywords="spherical top, tetrahedral molecules, molecular rotation vibration, rotational vibrational analysis molecular spectra", abstract="The article proposes an effective method for describing the qualitative characteristics of the rotational structure of a group of vibrational states, based on a study of the stationary points of a system of classical surfaces of rotational energy. A scheme for constructing the energy surfaces for effective Hamiltonians with known spectroscopic parameters is described. The rotational structure is analyzed by means of energy diagrams showing the dependence of the stationary points on the rotational quantum number. The general scheme of the analysis was realized with a computer. An effective rotational Hamiltonian for the doubly degenerate vibrational state of methane is investigated as a specific example" } @Article{Zhilinskii1986DS-nu1nu3-Vestnik, author="Zhilinski\'{\i}, B. I. and Sadovski\'{\i}, D. A.", title="Invariant combinations of spectroscopic parameters of tetrahedral molecules", journal="Moscow University Bulletin - Chemistry", year="1986", volume="27", pages="145--148", keywords="spherical top, tetrahedral molecules, molecular rotation vibration constants", abstract="in Russian (Vestnik Moskovskogo Universiteta, Khimiya)" } @Article{Sadovskii1986DS-nu2nu4, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Theoretical analysis of spectroscopic constants for spherical tops: {$\nu_2$}, {$\nu_4$} bands of {AB$_4$} molecules", journal="Journal of Molecular Spectroscopy", year="1986", month="Feb", volume="115", issue="2", pages="235--257", doi="10.1016/0022-2852(86)90045-7", keywords="spherical top, tetrahedral molecules, isotope effects, molecular rotation vibration constants, rotational vibrational analysis molecular spectra", abstract="Extended analysis of the different sets of spectroscopic constants for $\nu_2$, $\nu_4$ bands of CH$_4$, CD$_4$, SiH$_4$ is performed taking into account the possibility of the unitary transformation of the corresponding effective Hamiltonian. Refined relations between different notations of tensor operators and spectroscopic constants are presented. The unitary equivalence between several different sets of constants is shown up to the lambda /sup 4/ order" } @Article{Sadovskii1985DS-nu1nu3, author="Sadovski\'{\i}, D. A. and Zhilinski\'{\i}, B. I.", title="Comparative analysis of spectroscopic constants of tetrahedral molecules: $\nu_1$ and $\nu_3$ bands of silane and germane", journal="Optics and Spectroscopy", year="1985", month="Mar", volume="58", issue="3", pages="344--347", keywords="spherical top molecules, germanium compounds, silicon compounds, molecular rotation and vibration, rotational vibrational analysis molecular spectra, molecular rotation, vibration, and vibration rotation constants", abstract="Explicit transformation formulas are obtained for reduction of the model effective Hamiltonian used to describe the vibrational quasi-degenerate $\nu_1$ and $\nu_3$ states in molecules with tetrahedral symmetry. Using the example of SiH$_4$ and /sup 74/GeH$_4$ molecules, employing a reduction transformation, a comparative analysis of processing performed by various authors is conducted, and the stability of spectroscopic parameters is investigated on a change to reduced-model effective Hamiltonians" }