Vibrational Levels Associated with Hydrogen Bonds and Semiclassical Hamiltonian Normal Forms

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Vibrational Levels Associated with Hydrogen Bonds and Semiclassical Hamiltonian Normal Forms George A. Hagedorn and Alain Joye Dedicated to Jean-Michel Combes, in celebration of his 65th birthday Abstract. We describe and extend our recent proposal to model mathemat- ically the vibrational levels associated with hydrogen bonds in symmetric tri- atomic molecules. Our approach is based on modification of the usual Born- Oppenheimer approximation to take into account the lighter mass of the hy- drogen nucleus and the weakness of the hydrogen bond, using special features of the electron energy level surface associated with the hydrogen bond. Ne- glecting bending of the molecule for simplicity, we achieve this by scaling the mass of the hydrogen atoms differently from the heavier atoms, and by using a modified form for the electronic energy surface. As a result, anharmonic effects play a role at leading order in the limit where the nuclear masses go to infinity. Our analysis is based on close exam- ination of the numerical data available for the ground state energy surface of the FHF? ion, and we make a comparison with experimental data for the vibrational levels of that ion. The theory we propose is, however, quite general and can accomodate asymmetric tri-atomic molecules. Moreover, we provide an extension of our results to molecules with nuclei of several different species, where we assume that each of the masses scales differently.

standard born–oppenheimer based

born–oppenheimer approximation

symmetrical hydrogen

hydrogen bond

state electronic

nuclei

born-oppenheimer approximation

adapted ground

surface associated

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VibrationalLevelsAssociatedwithHydrogenBondsandSemiclassicalHamiltonianNormalFormsGeorgeA.HagedornandAlainJoyeDedicatedtoJean-MichelCombes,incelebrationofhis65thbirthdayAbstract.Wedescribeandextendourrecentproposaltomodelmathemat-icallythevibrationallevelsassociatedwithhydrogenbondsinsymmetrictri-atomicmolecules.OurapproachisbasedonmodiﬁcationoftheusualBorn-Oppenheimerapproximationtotakeintoaccountthelightermassofthehy-drogennucleusandtheweaknessofthehydrogenbond,usingspecialfeaturesoftheelectronenergylevelsurfaceassociatedwiththehydrogenbond.Ne-glectingbendingofthemoleculeforsimplicity,weachievethisbyscalingthemassofthehydrogenatomsdiﬀerentlyfromtheheavieratoms,andbyusingamodiﬁedformfortheelectronicenergysurface.Asaresult,anharmoniceﬀectsplayaroleatleadingorderinthelimitwherethenuclearmassesgotoinﬁnity.Ouranalysisisbasedoncloseexam-inationofthenumericaldataavailableforthegroundstateenergysurfaceoftheFHF−ion,andwemakeacomparisonwithexperimentaldataforthevibrationallevelsofthation.Thetheoryweproposeis,however,quitegeneralandcanaccomodateasymmetrictri-atomicmolecules.Moreover,weprovideanextensionofourresultstomoleculeswithnucleiofseveraldiﬀerentspecies,whereweassumethateachofthemassesscalesdiﬀerently.Consideringanadaptedgroundstateenergysurface,wecomputetheleadingtermofthecorrespondingvibra-tionallevelsinthelimitoflargenuclearmassesbymeansofanormalformHamiltonian.1.IntroductionThestandardtime–independentBorn–Oppenheimerapproximation[1]takesad-vantageofthelargemassesofthenucleirelativetothemassofanelectron.Itallowsonetocomputethelow-lyingvibrationalstatesofthenuclearmotionfromknowledgeofthegroundstateelectronenergylevelsurfacenearitsminimumun-dertheassumptionthattheminimumisnon-degenerate.Toleadingorder,thevibrationalenergylevelsarethoseofaharmonicoscillatorassociatedwiththenon-degenerateminimum.1991MathematicsSubjectClassiﬁcation.81V55.Keywordsandphrases.Hydrogenbonds,vibrationallevels,Born-Oppenheimerapproxima-tion,semiclassicalnormalforms.PartiallySupportedbyNationalScienceFoundationGrantsDMS–0303586andDMS–0600944.1

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