Gregory S. Ezra, Craig C. Martens and Laurence E. Fried- Semiclassical Quantization of Polyatomic Molecules: Some Recent Developments

8/3/2019 Gregory S. Ezra, Craig C. Martens and Laurence E. Fried- Semiclassical Quantization of Polyatomic Molecules: Some

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J. Phys. Chem. 1987, 9 1 , 3721-3730 3721

FEATURE ARTICLESemiclassical Quantization of Polyatomic Molecules: Some Recent Developments

Gregory S . Ezra,*+ Craig C. M artens,*and Laurence E. FriedBaker Laboratory, Department of Che mistry , Cornell University, Zthaca, New Yor k 14853(Received: January 27 , 1987)Recent progress in the semiclassical quantization of multidimensional molecular potentials is reviewed. We discuss quantizationof multimode and resonant systems using the Fourier transform method and operator-based Lie transform perturbation theory,quantization in the m ultiresonance and chaotic regime using algebraic quantization and resum mation techniques, and applicationof unitary group methods to the description of resonant dynamics.

I. IntroductionRecent experimental advances a re providing information onthe properties of highly excited vibration-rotation stat es of

polyatomic molecules in unprecedented detail, an d are revealingnew and poorly understood spectral regimes (see, for example,ref 1 ). It is now possible to probe experimentally the dynamicsof unimolecular decay* of state-selected molecules. Th e inter-pretation of such experiments to elucidate the rate and extent ofintramolecular energy flow and its role in dynamical processesposes many challeng ing problems for theory. In light of thecorrespondence pr i n~ ip le ,~e expect that semiclassical approx-imations will be useful in characterizin g highly excited states inthe large quantum number regime.4Th e traditional appro ach to the vibration-rotation problem isbased on application of perturbation theory to a zeroth-orderharmonic oscillator plus rigid rotor model appropriate fo rsmall-amplitude o~cillations.~Highly excited states typicallyinvolve large-amplitude nuclear excursions from equilibrium, asituation which renders low-order perturbation treatme nts inad-equate. In addition, generalized Fermi resonances between modessignal the local failure of nondegenerate per turbation theory andrequire special treatment.5In principle, a variational solution of Schrodingers equationfor coupled anharmonic nuclear motion (in the Bom-Oppenheimerapproximation6) will provide accu rate energies and w ave functionsfor highly excited statesS7 In practice, it is extremely difficult,if not impossible, to obtain accurate quan tum d ynamics or highlyexcited energy levels and eigenstates for anharmonic multimodemolecules, due to the extremely large number of basis functionsrequired, which increases exponentially with the number of modesN .7 Moreover, i f vibration-rotation levels for large values of thetotal angular momentum J ar e required, the size of the app ropriatesecular m atrix increases linearly with J . Practical difficulties withconventional variational app roaches ar e appare nt, for example,in the case of triatomics such as H,+ and its isotopomers,* whichexhibit large-amplitude nuclear motions and strong rotation-vi-brat ion coupl ing for moderate vibra tional ex ci ta t i ~ n .~ .~t istherefore imperative to develop other approaches for the calculationof highly excited vibration an d vibration-rotation levels in poly-atomics.As alternatives to exact quantum variational method^,^ dy -namical approximations such a s the self-consistent field approach loor adiabatic approximation have been investigated. In the pastfew years, there ha s also been intense interest in the applicationof semiclassical quantization techniques to determine bound statesin nonseparable multidimensional systems.* These methods

Alfred P. Sloan Foundation Fellow.*Ma them atical Sciences Institute Fellow.

utilize information from classical mechanics (e.g., trajectories,generating functions for canonical transformations) to obtainquantum mechanical quantities such as energy levels and wavefunctions. (W e note also tha t semiclassical methods are usefulfor defining ensembles of initial trajectories and final quantumnumber bins in quasi-classical trajectory studies of inelastic andreactive collisions involving anharm onic polyatomics.) The presentarticle describes some of ou r recent work in this active field.Much of the recent activity in semiclassical quantization ofmultidimensional systems can be viewed as a fulfillment of theprogram of the Old Q uan tum Th eory i4 in the light of recentfunda menta l advances in classical mechanics an d nonlinear dy-n a m i c ~ . ~ istorically,I6 the first quantization condition appearedin Bohrs treatmen t of the hydrogen ato m, where quantization(1) Dai, H. L.; orpa, C. L.; Kinsey, J. L.; Field, R.W . J . Chem. Phys.1985,82, 1688. Abrahamson, E.; Field, R. W.; Imre, D.; Innes, K. K.; Kinsey,J.L. J. Chem. Phys. 1985,83,458. Sundberg, R. L.; Abraha mson, E.; Kinsey,J. L.; Field, R. W. J . Chem. Phys. 1985, 83, 466. G arland, N. G.; Lee, E.K. C. J. Chem. Phys. 1986, 84, 28 .(2) Crim, F. F. Annu. Rev. Phys. Chem. 1984, 35 , 657. Moore, C . B.;Weisshar, J. C. Annu. Rev. Phys. C hem. 1983, 34, 525. Lawrance, W. D.;Moore, C. B.; Hrvoje, P. Science 1985, 227, 895. Beulow, S.; Noble, M.;Radhakrishnan, G.; Reisler, H.; Wittig, C.; Hancock, G. J . Phys. Chem. 1986,90, 1015. Janda, K. C. Adu. Che m. Phys. 1985,60, 201. Miller, R. E. J .Phys. Chem. 1986, 90, 3301.(3) Landau, L. D.; Lifshitz, E. M. Quantum Mechanics; Pergamon: NewYork, 1977.(4) Heller, E. J. Faraday Discuss. Chem. SOC. 983, 7 5, 141. Miller, W.H . Science 1986, 233, 171.(5) Papousek, D.; Aliev, M . R. Molecular Vibrational-Rotational Spectra;Elsevier: New York, 1982.(6) Koppel, H.; Domcke, W.; Ced erbaum, L. S. Adu. Chem . Phys. 1984,57 , 59 . Whetten, R. L.; Ezra, G. S.; rant, E. R. An n u . Rev. Phys. Chem.1985, 36, 211.(7) Carney, G . D.; Sprandel, L. L. ; Kern, C. W. Adu. Chem. Phys. 1978,37, 305.(8) Tennyson, J.;Sutcliffe, B. T. Mol . P h ys . 1986, 58 , 1067.(9) Carrington, A. J. Chem. SOC.,Faraday Trans. 2 1986, 8 2, 1089.

Tennyson, J.; Sutcliffe, B. T. J . Chem.Soc.,Faraday Trans. 2 1986,82, 1151.(10) Ratner, M. A.; Gerber, R. B. J . Phys. Chem. 1986,90, 20. Bowman,J. M. Ace. Chem. Res. 1986, 19 , 202.(11) Ezra, G . S. Chem. Phys. Lef t . 1983, 101, 259. Johnson, B. R.;Reinhardt, W . P. J . Chem:Phys. 1986, 85 , 4538 and references therein.(12) (a ) Percival, I. C. Adu. Che m. Phys. 1977, 36, 1. (b) Noid, D. W.;Koszykowsky, M. L.; Marcus, R. A. Annu. Rev. Phys. Chem. 1981, 32, 267.(c) Reinhardt, W. P., In The Mathe mbfical Analysis of Physical Sysrems;Mickens, R., Ed.; Van Nostrand: New York, 1984.(13) Davis, M. J.; Heller, E. J. . Chem. Phys. 1980, 75 , 3916. Knudson,S. K.; Delos, J. B.; Bloom, B. J . Chem. Phys. 1985, 83, 5703. Knudson, S.K.; Delos, J. B.; Noid, D. W . J. Chem. Phys. 1986, 84, 6886.(14) ter Haar, D. The Old Quan fum Theory;Pergamon: Oxford, U.K.,1967.(15) Lichtenberg, A. J.; Lieberman, M. A. Regular and StochasticMotion; Springer: New York, 1983.(16) Jammer, M . Conceptual Foundations of Quantum Mechanics;McGraw-Hill: New York, 1966. M ehra, J.; Rechenberg, H. The HistoricalDeuelopment of Quantum Mechanics; Springer: New York, 1982; Vol. I.0022-3654/87/2091-3721$01.50/0 0 987 American C hemical Society


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3722 The Journal of Physical Chemistry, Vol. 91 , N o. 14 . 1987 Ezra et al.of electronic angular momentum around the nucleus was invokedto derive the hydrogenic spectrum . A generalization of Bohrsprescription was sought by Ehren festi7 who, arguing tha t reversible(adiabatic) transformations transformed allowed motions intoallowed motions, suggested that the so-called adiabatic invariantsor actions were appro priate quantities for quantization. Such aprocedure could be straightforwardly implemented only forone-dimensional or separable multidimensional systems, leadingto the well-known Bohr-Wilson-Sommerfeld quantizatio n con-ditions.I6 The latter quantization conditions were subsequentlyjustified after the emergence of wave mechanics through theshort-wavelength WKB approximation to solutions of theSchrodinger equation for separable syste ms 3 T he quantizationof nonseparable multidimensional systems (for example, mul-tielectron atoms ), however, clearly presented fu ndam ental dif-ficulties for the Old Qu antu m Theory. Practical procedures forthe determination of good action variables in multidimensionalsystems were proposed by Born,l* who adopted the metho ds ofcanonical perturbation theory from celestial mechanics. Never-theless, in their highest state of development these methods wereunable to effect a quantization of the helium atom,I9 the well-known failure of the Old Q uan tum Theory (for recent work onthis venerable problem, see ref 20).In the rem arkable analysis of Einstein,2i it was recognized that,for quasi-periodic trajectories, quantization of .fp dq along to-pologically distinct paths on the invariant trajectory manifold(invariant torus22)constituted a properly coordinate-independentquantization pro cedure for multidim ensiona l systems. Einsteinalso noted that the existence of non-quasi-periodic, chaotic motions(as found in the 3-body problem by P ~ i n c a r e ~ ~ )endered thecanonical quantizatio n of good action variables inapplicable. Thedeep problem of the nature of quantum-clas sical correspondencein the irregular regime remains essentially unsolved up to thepresent,24 lthough th ere have been successful efforts to quantizeergodic2 s and m ixing26 systems. (Both classical perturbationtheory2-28 nd th e method of ad iabatic switching29have been usedto quantize coupled oscillator potentials in the irregular regime,cf. section IV.)The work of K eller,30 n which Einsteins canonical quantizationconditions for quasi-periodic motions were rederived by imposinga single-valuedness condition on the multidimensional analogueof the WKB wave function, rekindled interest in the problem (fora historical account, see ref 31). It is now customary to referto t he EBK (E inst ein,2i B r i l l~ u in ,~ ~eller30) quantization con-ditions for good action variables. Implem entation of EBKquantization fo r multidimensional sy stems became feasible withthe availability of computers to provide exact numerical solutionsto the classical equations of motion, and Marcu s an d co-workerswere the first to apply th e method to potentials of chemical i n -

( 1 7) Ehrenfest, P. Philos. Mag. 1917, 33 , 500(18) Born, M. Mechanics of the Atom; Ungar: New York, 1960.(19) Kramers, H. A . Z . Phys. 1923, 13 , 312.(20) Leopold, J . G.; Percival, I. C.; Tworkowski, A. S. J . Phys. B 1980,13 , 1028. Leopld , J . G.; Percival, I. C. J . Phys. B 1980, 13 , 1037. Coveney,P. V. : Child, M. S. J . Phys. B 1984, 17 , 319. Solovev, E. A. Sou. Phys. 1985,6 2 , 1148. Wesenberg, G . E.: Noid, D. W .; Delos, J. B. Chem. Phys. Lett.1985, 118, 72 .(21) Einstein, A. Verh. Drsch. Ges. 1917, 19 , 82. An English translationby C. Jaffe is available as JILA Report No. 116, University of Colorado,Boulder, CO .(22) Arnold, V . I . Mathematical Methods of Classical Mechanics;Springer: New York, 1978.(23) Poincare, H. Les Methodes Nouuelles de la Mechanique Celeste;Dover: New York, 1957: Vol. 1-111.(24) Berry, M. V. In Chaotic Behavior of Dererministic System s; Gerard,

I. , Helleman, R. H . G., Stora, R., Eds.; North-Holland: New York, 1983.(25) Gutzwiller, M. C . Physica D 1982, 5 , 183.(26) Berry, M . V. Ann. Phys. 1980, 131, 163.(27) Swimm, R. T. ; Delos, J . J . Chem. Phys. 1979, 71 , 1706.(28) Jaffe, C.; Reinhardt, W . P. J . Chem. Phys. 1979, 7 1 , 1862.(29) (a ) Solovev, E. A. Sou. Phys. J E TP 1978, 48, 635. (b) Skodje, R.T.; Borondo, F.; Reinhardt, W. P. J . Chem. Phys. 1985, 82, 46 1 1. (c )Johnson, B. R. J . Chem. Phys. 1985, 83 , 1204.(30) Keller, J. B. Ann. Phys. 1958, 4 , 180.(31) Keller, J. B. S IAM Reo. 1985, 27 , 485.(32) Brillouin, M . L. J . Phy.r. 1926, 7 , 353.

te re d 3 (for historical comments, see ref 34) . There has sub-sequently been an enormous amount of work in this area.3sThe m odern ma thematical formulation of semiclassical theoryfor multidimensional systems is due to mas lo^,^^ and accessibleaccounts a re available in the reviews of PercivallZaand De10s.~(See also the work of Littlejohd8 on symplectically invariantwavepacket methods.)While it is apparent that semiclassical methods hold greatpromise for the calculation of properties of excited states ofpolyatomic molecules, there a re formidab le technical difficultiesto be overcome, which are intimately associated with the need fora deeper understanding of the phase space str ucture of classicalnonlinear systems. The focus of the present article is on somerecent developments in the semiclassical quantization of multi-dimensional systems of chemical interest. We d iscuss quantizationof multimode and resonant systems using the Fourier transforma nd o pe ra to r-b as ed p er tu rb atio n t h e ~ r y , ~ ~ - ~ ~uan-tization in the multiresonant a nd chaotic regimes using algebraicq ~ a n t i z a t i o n ~ ~ , ~ ~nd resummat ion m e th 0d s ,4 ~ -~ ~nd applicationof unitary group techniques to the description of resonant dy-n a m i c ~ . ~ ~n keeping with t he spirit of these featu re articles, wemake no attempt to be comprehensive (extensive references toearlier work ar e provided in ref 35 , 39 , and 40) but focus insteadon the contributions of our own research grou p. In particular,we shall not discuss th e many recent applications of the methodof adi aba tic switching,29 which is the p ractical realization ofEhrenfests idea described above, or associated fundamentalstudies of the validity of the principle of adiabatic i n v a r i a n ~ e . ~ ~This topic has recently been reviewed by Reir~hardt.~ e note,however, that a comprehensive review of semiclassical methodsfor calculation of molecular properties is in pr e p a r a t i ~ n . ~ ~To provide some background and motivation, we briefly discussthe significance of classical resonances for molecular spectra anddynamics in section 11, together with the problems resonant motionposes for semiclassical quantizatio n. In section I11 we review theFourier transform quantization approach and its application tomultimode and resonant systems. Section IV discusses the useof operator-based classical perturbation theory as an efficientquantization method in the multimode case. Combining per-turbation theory with algebraic quantization and /or resummationtechniques, it is possible to quan tize systems in the multiply re-sonant and/or chaotic regimes. In section V, we consider recentprogress in the application of unitary group m ethods to coupledoscillator problems. Concluding rem arks ar e given in section VI.

(33) Marcus, R. A. Discuss. Faraday Soc. 1973, 55 , 34. Eastes, W.;Marcus, R. A. J . Chem. Phys. 1974,61, 4301. Noid, D. W.; Marcus, R. A.J . Chem. Phys. 1975, 62 , 21 19. Se e also: Chapman, S.;Garrett, B. C.;Miller,W . H . J . Chem. Phys. 1976, 64 , 502.(34) Marcus, R. A. J . Phys. Chem. 1986, 90 , 3460.(35) Ezra, G. S.; Martens, C. C.; Fried, L. E. Chem. Reu., manuscript inpreparation.(36) Maslov, V . P.; Fedoriuk, M. V . Semiclassical Approximation rnQuantum Mechanics; Reidel: Dordrecht, 198 1.(37) Delos, J. B. Adu. Chem. Phys. 1986, 65 , 161.(38) Littlejohn, R. Phys. Rep. 1986, 138, 193.(39) Martens, C. C.; Ezra, G. S . J . Chem. Phys. 1985, 83 , 2990.(40) Martens, C . C.; Ezra, G. S. J . Chem. Phys. 1987, 86 , 279.(41) Eaker, C. W.; Schatz, G. C.; DeLeon, N.; Heller, E. J. J . Chem. Phys.(42) Eaker, C. W.; Schatz, G. C. J . Chem. Phys. 1984, 8 1 , 2394.(43) Fried, L. E.; Ezra, G . S. J . Compur. Chem., i n press.(44) Fried, L. E. ; Ezra, G. S. J . Chem. Phys., in press.(45) Sibert 111, E. L. Chem. Phys. Leu. 1986, 128, 287.(46) Robnik, M. J . Phys. A 1984, 17 , 109.(47) Fried, L. E. ; Ezra, G . S., manuscript in preparation.(48) Farrelly, D.; Uzer, T. J . Chem. Phys. 19S6, 85 , 308. See also: S hirts,(49) Martens, C. C.; Ezra, G. S . J . Chem. Phys., in press.(SO) Dana, I. ; Reinhardt, W . P. preprint. Tennyson, J . L.; Cary, J . R.;Escande, D. F. Phys. Rev. Let t . 1986, 56 , 21 17. Hannay, J . H . J . Phys. A1986, A19, L1067.

1984, 81 , 5913.

R. B.; Reinhardt, W. P. J . Chem. Phys. 1982, 7 7 , 5204.

(51) Reinhardt, W. P. Adu. Chem. Phys., i n press.


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Feature Article The Journal of Physical Chemistry, Vol. 91 , No. 14, 1987 37232 ,

2 i I l , /I -8.0 -4.0 0.0 4.0 8.0X

b

-8.0 -4.0 0.0 4.0 tX

0

Figure 1. Quasi-periodic trajectories for th e 2-mode Barbanis potentialtreated in ref 40, 53, and 54: (a) nonresonant trajectory at energy E =16.504; (b ) 3:4 resonant trajectory a t E = 16.343.11. On the Importance of Resonances

In the present context, the term resonance refers to a localfrequency comm ensurability condition of the form muo= 0 (ageneralized classical Fermi resonances2),where, for a system ofN degrees of freedom, oo(I) = dH,(I)/aZ is the vector of Nzeroth-order fundamental frequencies, I is a vector of N zeroth-order action variables, Ho is the unperturbed reference Hamil-tonian (e.g., for t he case of molecular vibrations, usually a set ofuncoupled normal mode harm onic oscillators), and m s a vectorof N integers. It has long been known that resonances have aprofound effect on the phase space structure of nonseparablesystems, rendering resonant zeroth-order invariant tori unstablewith respect to small perturbation^.'^ At the most fundamentallevel, the ubiquity of resonances in phase space is responsible forthe divergence of canonical classical perturbation theory ( thesmall-denominator problemlS). Qualitatively , resonances lead togross changes in the ap pearance (topology) of configurationspace projections of invariant to ri (if they existlS ),as illustratedin Figure 1. Figure l a shows a trajectory for th e 2-mode modelHamiltonian treated in ref 40, 53, an d 54; it is continuouslydeformable into a rectangular boxlike33 rajectory characteristicof an uncoupled harmon ic oscillator and is therefo re nonresonant.The trajectory of Figu re 1b, on the other hand, clearly cannot bedeformed into a boxlike trajectory and is thus recognized as the

(52) Herzberg, G. Molecular Spectra andStructure; Van Nostrand: New(53) Sorbie, K.; Ha ndy, N. Mol. Phys. 1976, 32, 1327; 1977, 33, 1319.(54) DeLeon, N.; Davis, M. J.; Heller ,E. J. J. Chem. Phys. 1984,80,794.

York, 1945; Vol. IT.

configuration space projection of a resonant torus. From F igurelb, it can be seen that a point on the resonant trajectory m akes(approximately) three oscillations in the x direction for every fourin the y direction, implying an underlying frequency commen-surability 4w, = 3w,. We therefore refer in this case to a 3:4resonance and in general to m: n resonances for 2 degree of freedomsystems. No te that both trajectories shown in Figure 1 arequasi-periodic.Why are low-order, i.e., small m and n, resonances such as thatshown in Figure 1b of importan ce for the sem iclassical study ofmolecu lar spectra and dyna mics ? From the point of view ofnonlinear mechanics, isolated classical resonances lead to theappearance of localized stochastic motions associated with theexistence of homoclinic oscillations,s while the criterion ofoverlapping primary resonances has been used to give a roughestimate of the onset of global classical chaos.ss For resonan ttori, zeroth-order (e.g., normal mode) actions are no longer evenapproximately conserved along the trajectory,lS even when t helatter is quasi-periodic. Resonan t tori therefore correspond tostable patterns of motion for which there is extensive energy flowbetween the zeroth-order modes. Impo rtant examples abound (seeref 40 and references therein). For instance, the transitio n betweennormal- and local-mode behavior in ABA triatomic~,~~ihalo-me thane^,^^ and formaldehydes8 has been studied as th e resultof an isolated non linear resonance: when viewed from a local-modeperspective, normal-mode motions correspond to pronouncedenergy flow between the zeroth-order bond modes. A 2:lbend/stretc h (Ferm is2) resonance between the CH bond stretchand the C H wag has been identified as the primary mechanismfor rapid energy flow out of excited C H bonds in ben zene andis an essential first step in th e explanation of apparent h omoge-neous experimental overtone line widthss9proposed by Sibert eta1.60 (Recent studies have shown that initial rapid C H energydecay rat es are sensitive to th e form of potential coupling includedin the Hamiltonian$). 2: 1 bend-rotation resonances have beenshown to lead to extensive vibration-rotation energy flow inclassical trajectory studies on rigid bender models62 nd realistict r i a t o m i c ~ . ~ ~o-called sequential nonlinear resonances providea mechanism for irreversible energy decay in multimode systems.64Cantori,ts which are invariant sets characterized by highly irra-tional or antiresonant frequenc y are the vestiges of themost robust invariant tori of the uncoupled system and have beenfound to be bottlenecks to intramo lecular energy flow in 2-modemodels of collinear OCS67 and He12.68Resonant regions of phase space can be large enough to supportquantum states (see Figure 2a), and the underlying classicalresonances can dominate the morphology of the associatedquantum wave functions.s4 The correspondence can be seenparticular clearly by using a quantum mechanical phase spacerepresen tation, such as the Hu simi transform69 cf. Figure 2b,c).

( 5 5 ) Chiriko v, B. V. Phys. Rep. 1979, 52 , 263.(56) Sibert 111, E. L.;Hynes, J. T.; Reinhardt, W. P. J. Chem. Phys. 1982,77,3583,3585. Kellman, M. E. Chem. Phys. Lett. 1985,113,489; J . Chem.Phys. 1985,83, 3843.(57) Jaffe, C.; Brumer, P. J. Chem. Phys. 1980, 73, 5646.(58) Gray, S. K.; Child, M. S. Mol. Phys. 1984, 53, 961.(59) Reddy, K.V.; Heller, D. F.;Berry, M. J . J . Chem. Phys. 1982, 76,2814.(60) Sibert, E. L.;Reinhardt, W. P.; Hynes, J. T. J. Chem. Phys. 1984,81,1115. Sibert, E. L.; H ynes, J. T.;Reinhardt, W. P. J. Chem. Phys. 1984,81, 1135.(61) (a) Lu, D.-H.; Hase, W. L.; Wolf, R. J. J . Chem. Phys. 1986, 85,4422. (b) Garcia-Ayllon,A.; Santamaria, J.; Ezra, G. S., ork in progress.(62) (a) Frederick, J.; McClelland, G. M. J. Chem. Phys. 1986,84,4347.(b) Ezra, G. S. Chem. Phys. Lett. 1986, 127,492 .(63) Frederick, J.; McClelland, G. M.; Brumer, P. J. Chem. Phys. 1985,83, 190.(64) Hutchinson, J. S.; Reinhardt, W. P.;Hynes, J. T. J. Chem. Phys.1983, 79,4247.(65 ) (a) MacKay, R. S.; Meiss, J. D.; Percival, I. C. Physica D ( A m -sterdam) 1984, 130, 55 . (b) Bensimon, D.; Kadanoff, L. P. Physica D( Amsterdam) 1984, 130,82.(66) Greene, J. M. J. Math. Phys. 1979, 20, 1183.(67) Davis, M. J. J. Chem. Phys. 1985, 83, 1016.(68) Davis, M. J.; G ray, S. K. J. Chem. Phys. 1986,84, 5389. Gray, S.K.; Rice, S. A.; Davis, M. J. J. Phys. Chem. 1986, 90, 3470.


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3724 The Journal of Physical Chemistry, Vol. 91 , No. 14, 1987 Ezra et al.903

8

I I 1 1 I-80 -40 00 4 0 80

xv

b

aI I

-8 0 -40 0 0 4 0 8.0Y

I3

-80 -4 0 00 4 0 80Y

Figure 2. (a) Q uantum mechanical wave function for state number 62of the 2-m ode Barbanis potential treated in ref 40 , 53 , and 54 . Theeigenstate ha s energy E = 16.351 and corresponds to the resonantquantizing trajectory of Figure 1b. (b) Surface of section for the resonantquantizing trajectory of Figure I b. (c) Section through the quantumphase space density (Husimi transform) of the eigenstate shown in Figure2a . This plot should be compared w i th Figure 2b.Resonant quantum states of the type shown in Figure 2a can actas bottlenecks to multiphoton p umping up a vibrational ladder:70transition moments between nonresonant and resonant stateshaving different nodal structures are in general smaller than thosebetween nonresonant states, and zeroth-order selection rules areviolated.Nonr esonan t classical systems with a small number of degreesof freedom can be semiclassically quantized by using any of theseveral method s developed during the past decade.'* In contrast,the complicated topology of resonant as compared to boxliketrajectories poses some severe practical difficulties for the im-position of the EB K quantization conditions. Methods recentlydeveloped in ou r laboratory, which are described below, enableresonan t motion of arb itrar y topology to be quantized . For

~~ ~~

(69) Martens, C. C.; Davis, M . J .; Ezra, G. S., manuscript in preparation.(70) Brown, R . C ; Wyat t , R. E. J . Chem. Phys. 1985.82, 4777.

discussion of the many other approaches to quantization of res-onances, we refer to ref 35 and 40.111. Fourier Method for EBK QuantizationThe major practical difficulty in implementing semiclassicalquantiz ation for multidimensional systems involves the calculation(and possible none~istence'~)f good action variables and im-position of the appro pria te quantizatio n conditions.12 In thissection we describe recent progress on this problem using theFourier approach.3942 This method has proved to be a generaland efficient technique for quantizing both nonresonant and re-sonant states of multimode Hamiltonians and has provided con-siderable insight into the phase space structu re of such systems.The Fourier transform approach to EBK quantization is basedon the fact that, for motion in quasi-periodic (regular) regionsof phase space, coordinates and momenta can be expanded asconvergent Fourier series in the angle variables 0

q(0,J) = &(J) e'"' ( l a )p(t9,J) = Cpk(J) ( I b )

k

kwhere the angle variables evolve with time as follows

8 = w,t 4- 8 CY = 1, ..., N (2 )( 3 )

Note that the J are good action variables for the full Ham il-tonian H and are assumed to exist , so that H = H ( J ) (at leastlocally). The o re then the actual (as opposed to zeroth order)fundamental frequencies of the motion: for an N-mode system,there are N fundamentals w,, CY = 1, ..., N , but these are notnecessarily in direct correspondence with the zeroth-order fun-damentals wo . This is the case for resonant trajectories, as willbe seen below.

with w, the ath fundamental frequencyw, = dH (J) / d J ,

By virtue of the expansion of eq 1 , the action integralsJ , = 1 ( 2 ? r ) ~ * = d O , (df7/d~,) (4)

can be expressed directly in terms of th e F ourier coefficients (~k,pk)J , = xik,pk*'qk ( 5 )k

This result simplifies in the case that the H amiltonian H i s of theformH = + V(q) ( 6 )J

(for example, harmonic oscillators coupled by an anharmonicperturbation) toJ , = CG,,.w,. (7a)a'

withGed = Ek,hklzkd (7b)k

A 1-dimensional version of this remarkable formula was givenby H e i ~ e n b e r g . ~ 'The multidimensional case was derived byPerciva17* n the con text of a variational theory of inv ariant toriand has since been rederived several time^.^'^'^ Percival proposedan iterative scheme for determination of the Fourier coefficients(qk,pk).74 However, the result takes on its full significance whenit is r e a l i ~ e d ~ ~ , ~ 'hat the required Fourier coefficients can bedetermined directly by Fourier transformation of trajectories.(71) Heisenberg, W . Z . Phys. 1925, 33 , 879.(72) Percival, I. C. J . Phys. A 1974, 7, 794.(73) (a ) Binney, J.; Spergel, D. Asrrophys. J . 1982, 252, 308; Mon . N o t .R . Astron. SOC. 984, 206, 159. (b) Klein, A,; Li, C.-t. J . M a t h . Phys. 1979,20 , 572.(74) Percival, I. C.; Pomphrey, N . Mol . Phys. 1976, 31 , 97 ; J . Phys. B1976, 9, 3131.


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Feature Article The Journal of Physical Chemistry, Vol. 91, No. 14, 1987 3725TABLE I: Energy Eigenvalues For the Four-Dimensional Hamiltonian of ref 29a

Ni N2 N3 N4 EWb ESPSCC ESd EASe E n / EQM~0 0 0 0 2.2565 2.2533 2.2565 2.2566 2.2567 2.2562 2.25581 0 0 0 2.9479 2.9445 2.9479 2.9484 2.9479 2.9492 2.947 10 0 1 0 3.2455 3.2422 3.2455 3.2463 3.2465 3.2465 3.24390 1 0 0 3.5347 3.5313 3.5347 3.5356 3.5350 3.5359 3.52082 0 0 0 3.6358 3.6323 3.63580 0 0 1 3.7805 3.7773 3.7805 3.7808 3.7809 3.7808 3.63711 0 1 0 3.9365 3.9331 3.93651 1 0 ,O 4.2141 4.2105 4.21400 0 2 0 4.2298 4.2263 4.22990 1 1 1 6.0353 6.03 16 6.03530 3 0 0 6.05 11 6.0470 6.051 12 2 0 0 6.1299 6.1 257 6.12972 0 1 1 6.1440 6.1403 6.14401 1 2 0 6.1658 6.1618 6.16580 0 4 0 6.1826 6.1787 6.18274 1 0 0 6.2301 6.2258 6.22990 0 1 2 6.2807 6.2773 6.28073 0 2 0 6.2905 6.2865 6.29060 2 0 1 6.3244 6.3206 6.3244

Eighth-order normal form, using Robniks approximation (ref 44). * Weyl quantization of normal form in (a) (ref 44). CSymmetrypreservingsemiclassical (SPSC) quantization normal form of ( a ) (ref 44). dSemiclassicalquantization of a second-order nonresonant normal form by Saini (ref93). e Adiabatic switching (ref 29b). /Fourier transform method (ref 39). #Quantum mechanical eigenvalues, using a finite difference technique (refEquation 7 therefore sugges ts the following quantization scheme(we consider Hamiltonians of the form eq 6 , although it isstraightforward to implement the general result ( 5 ) for systemswi th non t r iv i a l G-ma t r i ce~~~) :(a) Run a trajectory to obtain coordinate time series ( q ( t ) ) .(b) Using efficient fast Fourier transform routines, obtain theFourier coefficients ( q k ) .This involves (i) identification of thefundamental frequencies (w,), and (ii) accurate determination ofthe amplitudes lqikl.(c) Calculate N actions (J , ) .(d) Either, (i) update the initial conditions until the EBKquan tization conditions

ar e satisfied (n is a vector of integer quan tum num bers, p is avector of Maslov indices36),or (ii) linearly extrapolate to obtainthe energy of the qu antum state76

E(JQ)- Etraj+ w(JQ - J) (9 )This scheme Th e key step needed to make it a practicalmethod is a sta ble and ac cur ate means of calculating frequenciesand Fourier amplitudes using fast Fourier transforms: this nu-merical prob lem4] has been solved by using a special windowingtechnique.39An important point concerning the Fourier quantization methodis that the result (7 ) giving the good actions in terms of coordinateFourier components holds regardless of trajectory topology andso can be app li ed to both n~ nr es o n an t~ ~nd resonant40 motions.The relation is also invariant with respect to canonical transfor-mations of coordinates (q ,p) to (Q,PJ,ay), provided tha t the newcoordinates and momenta {Q,P) re periodic functions of the goodangles.For nonresonant trajectories, there a re by definition no com-mensurabilities b etween normal-mode frequencies, and assignmentof the classical power spectrum is straightforw ard. In Figure 3a,we show coordinate power spectra obtained for the 2-dimensionalnonresonant trajectory of Figure la . The largest peaks in theindividual coordinate spectra are each assigned to nonresonantfundamentals, w l n r = 0.865 and w2 = 1.221, whose values areclose to the zeroth-order frequencies L = 0.949 and wyo= 1.265,respectively. Since the x an d y modes are coupled by a pertur-bation, combination and overtone peaks appea r in the spectrum

(75) Garcia-Ayllon, A. ; Martens, C. C . ; Santamaria, J.; Ezra, G. S.,(76) DeLeon, N. ; Heller, E. J . J . Chem. Phys. 1983, 78, 4005.submitted for publication.

0 0 10 2 0n I R0 0, I - 1 I

0 0 1 0 2 0 0 0 1 0 2 0R b n

Figure 3. Coordinate power spectra for the tra jectories shown in Figure1: (a) nonresonant; the nonresonant assignment of fundamental fre-quencies is shown; (b) resonant; the resonant assignment of fundamentalfrequencies is shown.at frequencies il = k w , where k is a vector of integers.The Fourier method has been applied to quantize a variety ofnonresonant systems with up to 4 degrees of freedom.39 In TableI, we give a representative set of energy eigenvalues for the 4-modesystem treated in ref 29b; quantal, adiabatic switching, andperturbation theory (see below) results a re given for comparison.The FFT EBK energies are very accurate. (In fact, this systemis resonant:39 in order to apply the Fourier quantization m ethod,it is necessary to resort to a n interpolation or resonance avoidanceprocedure, cf. ref 29b.) Th e quantum eigenvalues of Neub ergerand N ~ i d , ~ ~btained by using a grid method, are apparently notconverged with respect to grid size.For resonant motions, the normal-mode frequencies are lockedinto a fixed integer ratio, as can be seen in the coordinate spectrashown in Figure 3b for the 3:4 resonant trajectory of Figure lb.A detailed analysis of coordinate power spectra for resonancesin 2 degrees of freedom has been given.40 In essence, for an m :nresonance two new fund amen tal frequencies a i r nd w ; appear

~~

(77) Neuberger, J . W .; Noid, D.W ., preprint


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3126in place of the nonresonant fundamentals, so that w l r - w x 0 / m- w;/n is the frequency of the m :n periodic orbit at the centerof the resonance, while utr s the frequency characteristic of theslow rotation of the trajectory around the resonant torus, and a tthe center of t he resonance is just the stability frequency78of theperiodic orbit. For the trajectory of Figure lb , w l r = 0.292 andw i = 0.040. Th e assignme nt of spectral peaks using the resonantfundame ntals is given in Figure 3b. It can be seen tha t the peaksfall into char acte ristic clumps centered around multiples of thefundamental w I r , reflecting a time scale separation of motionalong and around the resonant torus. The Fourier methodhas been used to quan tize systems with 1 1, 2:1, and 3:4 resonances;details a re given in ref 40.At present very little is known concerning the phase spacestruc ture of resonances in N 1 mode H a m i l t o n i a n ~ . ~ ~ . ~ ~heFourier approach promises to be of great utility in studying andquantizing such systems.Since it provides a direct route from trajectories to actions viathe Fourier components, the Fourier method has great potentialfor obtaining highly excited vibrational state s of small polyatomicmolecules described by realistic potential surfaces.80 Work onthis problem is in progress. (Ea ker and Sch atz have combinedthe Fourier quantization scheme with the Sorbie-Hand y dynamicalapproximation to obtain energy levels for several triatomic

On e caveat should be borne in mind, however: the Fourierapproach for calculating good action variables will not work ifthe trajectory is chaotic. In tha t case, the Fourier transforms oflong (several hundred periods) trajectories are grassy,81a andit is not possible to assign individual lines as combinations of aset of N fund ame ntal frequencies. However, examination ofcoordinate power spectra of shorter segments (approximately 50periods) of chaotic trajectories can provide valuable insight intothe long-time dynamics (cf. ref 82). Typically, a variety ofbehavior is found. Suc h spectra ca n be grassy with many peaks(Figure 4 a), reflecting a lack of regularity in t he motion over thesegment, or relatively clean and almost quasi-periodic in ap-pearance (Figure 4b), corresponding to correlated or trappedmotion (inside a resonance zone, for example). In this way, anintrinsic characterization of the phase space regions visited by anon-quasi-periodic trajectory can be obtained and long-timec o r r e l a t i o n s i n t h e d y n a m i c s e l u ~ i d a t e d . ~ ~ ~ * ~The Fourier method is also being applied to the sem iclassicalquan tiza tion of rotation-vibration states.85 Frederick andMcClelland have considered the EBK and uniform quantizationof rotation-bending states for a rigid bender model of H 20 ,8 6 utno treatm ent of a full rotating-vibrating triatomic ha s been given.Such calculations for highly excited rotational states would bevery helpful in interpreting spectra in systems such as H3+ ,9 orwhich quantum variational calculations are currently unable toobtain ac cu rat e rotation-vibration levels in th e energy regime ofinterest.8 Insights gained from classical trajectory studies ofvibration-rotation interactions of model systems62are essentialfor successful quan tizat ion of vibration-rotation states. Use oft he A ~ g u s t i n - M i l l e r ~ ~epresentation of rotational motion leadsto the full vibration-rotation problem having only one degree of

The Journal of Physical Chemistry, Vol. 91 , No. 14 , 1987

4309 -

(78) Whittaker, E. A Treatise on the Analytical Dynamics of Particles andRigid Bodies; Dover: New York, 1944.(79) Neshyba, S.; DeLeon, N ., preprint.(80) Murrell, J . N.; Carter, S.;Farantos, S.;Huxley, P. ; Varandas, A. J.C . Molecular Potential Energy Functions; Wiley: New York, 1984.(81) (a) Noid , D. W.; Koszykowski, M. L.; Marcus, R. A . J . Chem. Phys.1977.67.404. (b) Koszykowski, M . L.; Noid, D . W.; Marcus, R. A. J . Phys.Chem. 1982.8 6, 21 13. (c) Wardlaw, D. M. ; Noid, D. W .; Marcus, R . A . J .Phy s . C he m. 1984,88, 536.(82) McDonald, J. D.; Marcus, R. A. J . Chem. Phys. 1976, 65, 2180.(83 ) Carter, D.; Brumer, P. J . Chem. Phys. 1982, 77,420 8. Davis, M . J. ;Wagner, A . A C S S y m p . S e r. 1984, 263, 337.(84) Martens, C. C.; Davis, M . J.; Ezra, G. ., work in progress.(85) Sumpter, B. G.; Ezra, G. S., work in progress.(86) Frederick, J.; McClelland, G . M . J. Chem. Phys. 1986, 84 , 976.(87) Augustin, S. D.; Miller, W . H . J . Chem. Phys. 1974, 61, 3155.(88) Schatz, G. C . In Molecular Collision Dynamics; Bowman, J . M ., Ed.:

Frederick, J. Chem. Phys. Let t . 1986, 131, 60 .Springer: New York, 1983.

t-Q0 I ,t--

0 I

Ezra et al.0

0

Fi-equency

,._I I I I I I 28Fi-equencyFigure 4. Coordinate power spectra for segments (1.38 ps) of a trajectoryof the 3-dimensional OCS model of ref 83 with E = 20000 cm-I. (a)A segment with a grassy appearance, indicating a lack of correlatedmotion. (b ) A segment with a relatively clean, almost quasi-periodicappearance, indicating trapping.freedom more than the rotationless case. For example, a triatomicca n be treated as a 4 degree of freedom system whose quantizationis quite feasible.In addition to providing an efficient route to calculation ofactions, determination of the Fourier com ponents of trajectoriesprovides, via ( l) , the numerical transform ation to good action andangle variables, the so-called topological paths on tori (cf. ref40). Using th e numerical transformation to good angle variables,it is possible to define an ensemble of points (initial conditions)uniformly distributed on a qua ntizing torus, as required, for ex-ample, for quasi-classical trajectory studie s of molecular colli-sionsF8 Moreover, the transformation to angle variables is a direc tparametrization of the Lagra ngia n manifold (torus) requiredto construct semiclassical wave functions in the theory of M aslov(ref 36, cf. also ref 37).Th us far we have only considered th e calculation of primitivesemiclassical eigenvalues using th e EBK quantization condition(8 ) . A uniform quantization of the Henon-Heiles system has beencarried out using the Fourier method to determine an effectiveresonance Hamiltonian; for details, see ref 89.It should be noted that the work of Marcus and co-workers81ainitiated the use of coordinate power spectra as a diagnostic forquasi-periodic and cha otic motions, while approxima te determi-nation of transition frequencies an d intensities8bsc as also beenmade using Fourier transforms of the dipole moment autocor-relation function for trajectories run a t mean actions. However,

(89) Martens, C. C.; Ezra, G . S. In Tunneling; Jortner, J . , Pullman, B. ,Eds.; Reidel: Dordrecht, 1986.


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Feature Article The Journal of Physical Che mistry, Vol. 91, No. 14 , 1987 3727Thes e methods avoid the m ixed variables (one old set, one newset) of conventional canonical perturbation theory'* and th e as-sociated functional inversion required to express new variablesin terms of the old (cf. ref 28). Lie transform method s arecurrently being applied to a variety of physical proble msg 5One very powerful algorithm we have applied in ou r studiesis due to D ragt a nd Finn.96 This version of classical perturbationtheory uses the exponential of a Lie operator (hence the term"operator-based" perturbation theory) to generate canonical co-ordinate transformations

Z(z) = eF z (10)where z denotes the pair of old variables (1,4), Z denotes the pairof new variables (J,O), an d F is the Lie operator associated withthe function f(z), defined byF(*) K I (1 1)

where { ,) is the Poisson bracket. It is most importan t to note tha tth e inverse relationship is also given directly in terms of Lieoperators:z(Z) = zFz (12)

A key result is that a transformation of the form (12) leaves thePoisson brack et invariant% and is therefore canonical. Moreover,any canonical transformation continuously connected to theidentity can be written as a (possibly infinite) sequedce oftransformations of the form (12).96 The Hamiltonian ih the newcoordinate system, K , is given directly in terms of Lie operatorsas follows:

K(O,J) = eFH(+,I) (13)In order to solve the problem of normalizing the Hamiltonianperturbatively, H must be written as a power series in a smallparameter c

the idea of using Fourier transforms to calculate actions directlywas not discussed in these papers.IV. Operator-Based Perturbation Theory

In this section we discuss a second, trajectory-independent,appro ach that has been developed as a practical tool for semi-classical quantization of multidimensional systems based on theuse of high-order operator-based classical perturbation theo-ry.43-45*47348he basic aim of the m ethod is to find a canonicaltransformation to new variables in terms of which the full Ham-iltonian H, onsidered to be perturbed with respect to the referenceHamiltonian Ho, is (to given order in the perturbation strength)in as simple a form as possible. Th e problem of transforming aHam iltonian into normal form, Le., eliminating th e effect of aperturbation, is of course the fundamental problem of mechanics.23In the absence of resonances, th e normal fo rm depends only ongood actions, so that the equations of m otion are trivially solved.T h e n or ma li za tio n al go ri th m of B i r k h ~ f f , ~ ~ ~s modified byGustavsongob to tr eat the resonant case, has been applied tosemiclassical quantization of molecules following the work ofSwimm and De10s.~' O nce the transformation to good actionshas been found, semiclassical energy levels can be calculated ina straightforward fashion by substituting quantizing values of thegood actions (eq 8) into the Hamilto nian. For systems havingexact or near resonances between zeroth-order frequencies, thetransformed H amilton ian will be a function of a m inimal set ofangle variables in addition to the good actions. A more elaborateuniform quantization procedure or matrix diagonalization is thennecessary (see below).Qua ntization via classical perturbation theory is in many re-spects complementary to th e Fourier app roach discussed in theprevious section. On e virtue of perturbation theory is th at it yieldsglobal, analytical results for the transformed Ham iltonian as afunction of the new actions. Th e method has achieved someunforseen successes: in the uniform quantiza tion of the Henon-Heiles system by J affe a nd Reinh ardt28 using the B irkhoff-Gus tavson normal form:0 it was found th at accura te eigenvaluesfor highly ex cited vibrational states of a 2-m ode system could beobtained even when the underlying classical motion is chaotic.Th e applicability of classical perturbation theory to obtain accu ratesemiclassical eigenvalues in the irregular regime is of both fun-dame ntal an d practical significance, since it overcomes the re-striction to the regular regime of many trajectory-based m ethodssuch as the Fourier quan tization appro ach. As discussed byR e i ~ ~ h a r d t , ~ 'se of finite-order perturbation theory effectively"smooths over" chaotic regions of phase space, if they are suf-ficiently small.From a practical point of view, it is clear that application ofthe perturbation theory quantization method to realistic systemswith 3 or more degrees of freedom would be desirable. Furth erstudy of the perform ance of the method in the presence of chaoswould also be of interest. However, the need to keep track of thehuge number of terms generated when carrying out classicalperturbation theory to high order in multidimensional systems hashampered previous work along these lines.92 Low-order results?3on the other hand, a re unlikely to be accur ate at high energiesor for very anharmonic systems.

To implement high-order classical perturbation theory formultidimensional systems efficiently, a special purpose numeri-cal/algebraic manipulation program called PERTURB43 has beendeveloped. Details of the program have been given elsewhere.43Here we note only that it is written in C, has a very flexiblemodular s tructur e, and makes extremely efficient use of CPU timeand memory by dynamical allocation of storage and reuse ofmemory (dumping of inter media te expressions).The program PERTURB is capable of implementing a variety ofLie-operator-based perturbation algorithms94 o very high order.(90) (a ) Birkhoff, G. D. Dynamical Systems; A.M.S. Colloqium: NewYork, 1927; Vol. IX . (b) Gustavson, F. G. Astron. J . 1966, 72 , 670.(91) Reinhardt, W. P. J. Phys. Chem. 1982, 86, 158.(92) Williams, R . D.; Koonin, S. E. Nucl . Phys. 1982, A3 9 1 , 72 .(93) Saini, S . Chem Phys. Lett. 1986, 125, 194.

H = CH,,t"where the reference Ham iltonian Hodepends only on I. A seriesof canonical transformations is then iritroduced, each of whicheliminates the dependence of the new Ham iltod an K on fl to onehigher order in e:

(15)

Hk exp(ekFk).. exp(tF1)H (16)H k = CHknc" (17)

K = ...exp(c3F3) exp(c2F2)exp(cF,)HDefining Hkby

and expanding in powers of c

enables explicit equation s for the Hk, o be obtained in terms ofthe g e n e r a t o r ~ f ~ . ~ ~ . ~ ~he generators a re chosen to simplify thetransformed H amiltonian to the max imum possible extent; in theabsence of exact resonances, Hkkcan be chosen to be the angle-independent part of Hk-'k.The above procedure can be implemented to provide an efficientroute to the calculation of semiclassical eigenvalues for 2- and3-mode systems for which th e frequencies u are sufficiently farfrom r e ~ o n a n c e , ~ ~ * ~ttempting to make the transformed H am-iltonian completely independent of the angles results in thepresence of denom inators of the form kw0 in the expressions forthe generators fk . This is the famous "small denominatorproblem", which is responsible for the eventual failure of theclassical perturb ation scheme. Resonances or near-resonancesbetween zeroth-order frequencies cause the perturbation expansionultimately to diverge;" it is found tha t high-order terms oscillatewildly while increasing in magnitude (Figure 5) and a re of little

(94) Cary, J. Phys. Rep. 1981, 79 , 129.(95) Steinberg, S . Lect. Notes Phys. 1986, 251, 45 .(96) Dragt, A. J.; Finn, J. M. J . Math. Phys. 1976, 17 , 2215; 1979, 20 ,2649.


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/ I

(97) Baker, G. A. Essentials of Pade Approximants; Academic: Ne w(98) Zinn-Justin, J . Phys. R e p. 1981, 70, 109.(99) Silverman, J . N. Phys. R e v . A 1983, 28 , 498.(100) deGroot, S. R. ; Suttorp, L. G. Foundations of Electrodynamics;

York, 1975.

North Holland: Amsterdam, 1972.

TABLE II: Energy Eigenvalues for the Near 1:l ResonantHamiltonian of ref 48N , N2 ER' Ewb ESPSC' ER-~ EQM'0 0 0.99960 1 1.98521 0 1.99270 2 2.94151 1 2.96232 0 2 . 99640 3 3.87431 2 3.89272 1 3.96453 0 3.99550 4 4.77501 3 4.79052 2 4.89083 1 4.95634 0 4.9982

~0.99731.98271.99002.93862.95922.99333.87083.88893.96073.99174.77054.78564.88624.95134.9937

0 . 99961.9853 1.991 311.9926 1.998 812.9417 3.134072.9622 2.966 642.9964 3.008 923.8745 3.884973.892 6 3.941 873.9644 3.923 063.9954 4.014594 . 7752 4 . 859 634.7903 4.902 604.8909 4.883 604.9560 4.901 594 . 9984 5.01 08

1.00081.98681.99382.94352.96382.99733.87683.89463.96573.99594.77814.793 14 . 89244.95684.9981

'Algebraic quantization of a 12th-order normal form, using Rob-nik's rule (ref 44) . bAlgebraic quantization the same form as above,using the Weyl rule (ref 44) . ' A Q of the normal form in ( a ) ,using th eSPSC rule (ref 44) . dResummation of a 12th-order nonresonant nor-mal form using the epsilon algorithm (ref 48) . eQuantum mechanicalvariational calculation (ref 44) .may possess a number of invariants, Le., mutually commutingoperators that com mute with it. For example, in the case of asingle exact resonance, there a re N - 1 invariants linear in th eactions (num ber operators). Sinc e the basis functions must besimultaneous eigenfunctions of these invariants, the resultingHamiltonian mat rix will be block diagonal with finite blocks. Thisprocedure has been discussed in some detail for the case of anexact resonance in a 2-dimensional system by R ~ b n i k ~ ~using theBirkhoff-Gustavson normalizationg0), who has also introducedan approxim ate version of the W eyl quantization rule to simplifycalculations. W e have generalized the method to treat nearlyresonant terms and systems with more than two modes and havegiven a prescription for determining the linear invariants that labeleigenstates result ing f rom the d iag~nalizat ion .~~e have alsointroduced a hybrid quantization rule that gives accurate valuesfor quantum mechanical splittings while reducing to the EBKexpression for nonresonant systems.44As the number of resonant or nearly resonant terms in theHamiltonian increases, the num ber of invariants decreases untilfinally none remain. In tha t case, we ar e left with the problemof diagonalizing the transform ed Ham iltonian in an infinite basis.Nevertheless, progress toward simplifying the H amiltonian hasbeen made even in this limit, as the transformed Hamiltonian isan effective operator in which all nonresonant interactions ar eeliminated to given order.

In Table 111 we give semiclassical eigenvalues for a 5-modemodel system.29b One exact and three near-resonances wereincluded explicit ly in the transformed Ha m i l t ~ n i a n ; ~ ~here wastherefore a single invariant, resulting in a block diagonal Ham-iltonian matrix . Even after inclusion of nearly resonant couplings,eighth-order perturbation theory results still exhibit a slight di-vergence. The eigenvalues were therefore Pade resumm ed by usingthe epsilon al g~ ri th m .~ ' he resulting energies for the lowest fewstates are in good agreement with the adiabatic switching results,29bwhile there are no other excited-state eigenvalues available fo rcomparison.The PERTURB package in combination with algebraic quanti-zation and resum mation methods is clearly a powerful tool forstudy of the semiclassical mechanics of multimo de anharmonicpotentials. Although PERTURB is at present limited to treatingperturbed harmonic oscillator or Morse oscillator systems, v i -bration-rotation interactions could be included by using a mappingof the asymmetric rotor Hamiltonian onto a 2-mode harmonicoscillator. o'It has been noted previously that Dragt-Finn perturba tiontheory is the classical limit of Van Vleck quantum perturbation

(101) Martens, C. C. , unpublished work.


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Feature Article The Journal of Physical Chemistry, Vol. 91 , No. 14, 1987 3129

C

TABLE III: Energy Eigenvalues for the Five-Dimensional System of Ref 29aNl N2 N3 N4 NS E, E W b ESPSCC EASe0 0 0 0 0 2.6897 2.6800 2.6897 2.6909 2.6896

m d l

I0020103I00

00001000000

00100000101

00000010000

010d010d021

3.33303.55863.67873.91673.96704.15804.21384.27644.32164.42534.5477

3.28153.54733.66891.38133.95304.15764.20414.88984.27344.40164.5361

3.33303.55863.67873.91673.96704.15804.21384.27644.32164.42534.5477

3.3489 3.33053.5630 3.56003.6807 3.67853.98673.9700 3.96694.18734.2152 4.21 394.60434.33864.43384.5527a R e ~ u m m e dighth order energies, using Robniks quantization rule (ref 44). bA s in ( a ) ,except with the Weyl quantization rule (ref 44). CA s n(c), except with the SPSC quantization ru le (ref 44). dQu anti zati on of a resonant second order normal form, using the SPSC quantization rule (ref44) . Adiaba tic switching energies with minimum energy dispersion (ref 14) (ref 29b). Resummed results showed poor convergence.

t h e ~ r y . ~ ~ . ~ ~hy not, then, simply apply high-order quantumperturbation theory to obtain eigenvalues for multimode vibrationalHamiltonians? Low-order Van Vleck perturbation theory is ofcourse used in the traditional contact transformation approachto v ibration-ro tat ion ~ p e c t ra ,~s mentioned in the Introduction.The modular structure of PERTURB is such that replacement ofthe Poisson bracket operation by a c omm utator enables Van Vleckperturbation theory to be carried out with the same degree ofefficiency as Dragt-Finn. For the same problem treated to thesam e order th e classical perturbation theory generates fewer terms,however, and is therefore simpler, since there a re no difficultieswith operator ordering. Using the M oyal bracketlo2 (the Weylsymbol3*of the commutator), we have derived convergent cor-rections to semiclassical pert urba tion theory in powers of h Io 3It is thereby possible to compa re the relative accuracy vs. efficiencyof the two approaches and to examine the passage from semi-classical to fully quantal eigenvalues. Stu dy of the convergenceproperties of both types of perturbation theory might also throwsome much needed light on the fundamental problem of quantumvs. classical integrability,l@ an d could point th e way to a rigorousquantum KA M theorem.Io5V. Unitary Group Approach to Multimode ResonantDynamics

In this section we discuss unitary group approa ches to the studyof resonant dynamic s in multimode systems. W e briefly describesome recent work on 2 degree of freedom m:n resonant systems,49indicate some applications of these results currently under in-vestigation, and discuss their possible extension to N I modesystems.I t is well-known that the Hamiltonian of the N degree offreedom isotropic oscillator has t he sym metry group SU(N), th eunitary group in N dimensions, both in classical and quantummechanics.Io6 Th e unitary group includes transformations thatmix positions an d momenta a nd so represents a higher degree ofsymmetry than that of the potential energy alone, SO(N). In 2degrees of freedom, for example, the symm etry group is SU (2) ,which is homomorphic onto the rotation group SO(3).lo6 Oneconsequence is that the degeneracy pattern of the quantum levelsof the 2-mode isotropic oscillator corresponds to the deg eneraciesof the irreducible representations of SU (2): 1,2,3,4 .... Underlyingthe global symmetry group SU(N) is the Lie algebra su(N),defined by the commutator (or Poisson bracket) relations satisfiedby a set of generators, which ar e quantities bilinear in coordina tesand momenta that commute with th e oscillator Hamiltonian.lMEach generator is therefore a constant of the motion: since,however, the generators do not mutu ally comm ute (have vanishingPoisson bracket), they cannot all be simultaneously diagonal (in

(102) Moyal, J. E. Proc. Camb. Phi los . SOC.1949, 4 5 , 99 .(103) Fried, L. E.; Ezra, G. S., ubmitted for publication.(104) Robnik, M. J . Phys. A 1986, 19, L841. Hietarinta, J. J . Math. Phys.(105) Hose, G. ; Taylor, H. S . Phys. Reu. Lett . 1983, 51 , 947.(106) Wybourne, B. Classical Groups or Physicists; Wiley: New York,

1984, 25 , 1833.1974.

0ma bI 4

-60 -3 0 0 0 30 60 -60 -30 0 0 30X X .o

0 00 - m -

*:- *:-0 0m - 0 -

-60 -30 0 0 3 0 6 0 - 6 0 -30 0 0 30 6 0x XFigure 6. Angle paramettizations of invariant tori for the isotropicoscillator (see ref 49). (a) Ang le paths for normal-mode action-anglevariables. (b ) Angle paths for the A (normal mode) representation. (c)Angle paths for the K (local mode) representation. (d) Angle paths forth e L (angular momentum) representation.involutionzz). In su( 2), for example, the classical generators are(w = 1)

(18a)= 1 / 2 ( p $ + X - pvz - y z )

Each generator corresponds to a particular representation or choiceof basis: normal mode (A), local mode ( K ) , and precessing normalmode (t)see Figure 6 ) . The work of KellmanIo7has shown thatthe SU(2) symmetry analysis provides powerful global insightsinto the interrelations between t he local- and normal-mode motionsin 1 resonant oscillator systems. T hus, transformation coeffi-cients relating normal- and local-mode quantu m states can bewritten directly in terms of Wigner d-matrice~.~ he classicalsu(2) algebra of the isotropic oscillator also turns out t o be usefulin the problem of semiclassical quantization via perturbationtheory: in the Henon-Heiles problem, for example, a choice ofdiagonal generator L rather than A renders the Hamiltoniandiagonal to higher order in the perturbation strength than in theCartesian representation.28*lw Farrelly has recently extended these

(107) Kellman, M. E. J . Chem. Phys. 1982,76,4528; Chem. Phys. Let t .(108) Uzer, T.; Marcus, R. . J . Chem. Phys. 1984, 81, 5013.1983, 99,43 7; J. Chem. Phys. 1985,83, 3843.


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3730 The Journal of Physical Chemistry, Vol. 91 , No. 14, 1987

s -* 8-:-I

Ezra et al.

sI

% , I

I 1 I

9 a /

I --4.0 -2.0 0.0 ' 2.0 4X04 b..O

X

-4.0 -2.0 0.0 2.0 4.0XFigure 7. Angle parametrization of m:n resonant tori, as determinedanalytically in re f 49: (a ) 2:l; (b) 3:4; (c ) 3: l .considerations to t he case of a 2:l resonance" and has outlinedhaw to proceed in the general m:n resonant case.In recent work on the SU(2) symmetry of 2 degree of freedomsystems inspired by the work of Kel lmanIo7 and F a r r e l l ~ ' ~ ~mentioned above, we have obtained explicit expressions for thecanonical transformations from normal mode (A) to K an d L toriand have verified that these correspond to the classical limit ofthe gnalogous quantum mechanical t ran~format ions.~~s animportant extension, it has been shown that these results can beused to generate tQri of appropriate topology for general m:nresonant systems . Thus, by m apping the (anisotropic) m:n re -sonant Zm ode oscillator onto an isotropic osci1lator,II0 an explicit

(109) Farrelly, D. . Chem. Fhys. 1986,85, 21 19.(110) Stehle, P.;Han, M. Y. hys. Rev. 1967, 159, 1076.

transformation from normal-mode variables to resonant action-angle variables has been obtained. Analytically generated m:nresonant tori are shown in Figure 7 (2:1, Figure 7a; 3:4, Figure7b; 3:1, Figu re 7c). This work provides a solution to a problemin the implementation of th e adiab atic switching approach tosemiclassical quantizatio n of multidimensional systems, namely,how to defin e an en semble of initial conditions lying on a reso nanttorus of the correct topology. Th e qua ntal version of the classicaltransformation defines a resonant basis set that can be used ins tud ie s o f quan tum loca l i~a t ion .~~~Of g reat in terest is the possibility of extending the analysis ofclassical Lie algebras for isotropic and resonant anisotropic os-cillators to the N 2 3 mode case. For N = 3,degrees of freedom,for example, the Lie algebra su (3)lo6defines an appropriate setof actions and associated tori for describing resonant motion. Ifcanonical transforma tions analogous to those obtained for SU(2)could be found, we would have an analytical tool for study ofdyna mics in 3 degrees of freedom. Th e complexity of the classicalmechanics of the N 2 3 mode case is such that any insightsobtain ed by using un itary group m ethods would be of considerablevalue.VI. Summary and Prospect

In this article we have surveyed some recen t developments inthe semiclassical theory of multimode molecular bound states.Powerful methods are now available for EB K and uniform al-gebraic quantization of excited vibrational and vibration-rotationlevels of small polyatomics described by realistic anharmonicpotential surfaces. We conclude by indicating briefly someprospects for future developments and app lications.A recent major advance in the classical theory of intramolec ulard y n a m i d 7 and u nimolecular decay68has been the application ofideas from the nonlinear dynamics l 1 concerning theorigin of long-time correlations in the chaotic regime. Su chlong-time correlations are associated with the existence of phasespace bott lenecks, which may be c a n t ~ r i ~ ~i.e., in variant cantorsets which are rem nants of invariant tori that have broken up u ndera perturbation) or broken separatrices."' Can tori act as leakybarriers to mode-mode energy flow in 2-mode systems, the leastpermeable barriers being associated with the most irrational(antiresonant) frequency rati0s.6~As discussed in section 111, trajectory Fourier analysis can beextended to non-quasi-periodic ra je ct ~r ie s. ~* *~ ~he power spectraof short segments reveal directly the existence of trapped orcorrelated motion extending over several periods and c an be usedto char t the passage of a trajectory from one trapping region toanother as it wanders in phase space. This approach is currentlybeing applied to investigate the na ture of p hase space bottlenecksin N = 3 mode systems.84We note finally that af ixedfrequency version (as opposed tothe usual fixed action constrain t of can onical perturbatio n theory15)of classical perturba tion the ory would be very useful for findinganalytical approximations to phase space dividing surfaces (seealso ref 112). An efficient high-order implementation such asPERTURB is required, as previous attempts to define approximateintramolecular bottlenecks have been unsuccessful.68 We arecurrently investigating adaptatio n of PERTURB to the fixed fre-quency case.Acknowledgment. We thank the participants of the TellurideSummer Research Institute for many helpful conversations.Particular thanks are due to M. J. Davis for inv aluable discussionsand stimu lating collaboration. T his work was supported by N S FGran t CHE-8 410685 . G.S.E. acknowledges the partial supportof the Alfred P. Sloan Foun dation. C.C.M. acknowledges supportof the U.S. Army Research Office, through the MathematicalScience s Institute of Cornel1 Univ ersity.

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(1 11) Channon, S. R.; Lebowitz, J. Ann. N.Y. cad. Sci. 1980,357,108.(112) Jaffe, C., work in progress.

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