Craig C. Martens et al- Local Frequency Analysis of Chaotic Motion in Multidimensional Systems:...

8/3/2019 Craig C. Martens et al- Local Frequency Analysis of Chaotic Motion in Multidimensional Systems: Energy Transport a

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Volume 142, number 6 CHEMICAL PHYSICS LETTER S 25 December 1987

LOCAL FREQUENCY ANALYSIS OF CHAOTIC MOTIONIN MULTIDIMENSION AL SYSTEMS: ENERGY TRANSPORTAND BOTnENECKf.3 IN PLANAR OCS

Craig C. MA RTENS ,., Michael J. DAV IS b and Gregory S. EZRA a,3 Department of Chemistry Baker Laboratory, Cornell University, thaca. NY 14853, USAb Chemistry Division, Argonne National Laboratory, Argonne, IL 604 39 , US A

Received 25 September 1987

The m ethod of local frequency analysis for study of chaotic motion in multidimensional systems is described, and applied tothe analysis of intramolecular energy flow in a three-mode model for OCS . The na ture of possible partial barriers responsible forlong-time correlations in multimode systems is discussed.

1. IntroductionRecent research on classical theories of unimolec-

ular and bimolecular processes in two-mode systemshas established the important role of dynamical bot-tlenecks in non-statistical behavior [ l-51. Previouswork on long-time co rrelations in area preservingmap s has identified two types of partial b arrier totransport in chaotic regions of phase space [ 6-81.The first are invariant Cantor sets, called cantori [ 61,which are remnants of invariant tori that have bro-ken up under the influence of non-integrable per-turbations. The second are broken separatrices [ 8 1,which form the boundaries of resonance zones [ 91,or, in the case of dissociative dynamics, the rem-nants of the last bound phase curves [ 21. Transportacross both cantori and broken separatrices [ 9, lo]has been incorporated into statistical mo dels of in-tramolecular vibrational energy relaxation (IVR)[ I], vibrational p redissociation of van der W aalsmolecules [ 21, isomerization [ 31 and bimolecularreaction rates [4,5] in two-degree of freedom (DOF)systems. M athematical Sciences Institute Fellow.a Present address: Department of Chemistry, University of

Pennsylvania, Philadelphia, PA I9 104, USA.3 Alfred P. Sloan Fellow.

The phase space structures associated with trap-ping and long-time correlations in two DOF are mostnaturally characterized in terms of relations betweenthe frequencies of motion. The least permeable can-tori, which form minimum flux intramolecular tran-sition states for IVR, are the remnan ts of invarianttori with the most irrational frequency ratios [6].Physically, these robust tori are those for which thetwo coupled modes are farthest from resonance [ 111.Trapping of trajectories inside a resonance zone isassociated with a locking of the two fundam ental fre-quencies into .a particular rational ratio [ 121.

The nature of the bottlenecks responsible for trap-ping in Na 3-mode systems [ 13,141 is at present un-known. The structure of chaotic phase space for Ng 3DO F is qualitatively different from the two-mo decase, due mainly to the fact that invariant N-tori can-not partition the (2N- 1)-dimensional energy sur-face for N> 2, and relaxation pathway s andmechanisms for long-time correlations are poorlyunderstood. The studies that have led to the presentunderstand ing of the two-mode case have relied ontechniques, such as the Poincare surface of section,[ 111, which are not readily applicable to higher-di-mensional systems. Since real polyatomics have atleast three vibrational modes, an understanding ofphase space structure in higher dimensions is clearlydesirable.

0 009-26 14/87/$ 03.50 0 Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division) 519


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Volume 1 42, number 6 CHEMICAL PHYSICS LETTERS 25 December 1987The present note summarizes our recent work on

the intramolecular dynam ics of polyatomic mole-cules, with a particular focus on the distinctive fea-tures of multidimension al transport exhibited by athree-DOF model for OCS constrained to vibrate ina space-fixed plane (planar OCS) [ 13,141 . We firstdescribe a new methodology for studying chao tic dy-nam ics in multimode systems, based upon local fre-quency analysis of non-quasipetiodic trajectories.The structure of phase space for three-DOF systemsis then briefly considered, with particular attentiongiven to the importance of relations between the lo-cal frequencies. Application of local frequency anal-ysis to planar OCS is discussed, and the resultingpicture of transport and trapping summ arized. A de-tailed account will be given elsewhere [ 151. We notethat parallel work on the phase space structure ofthree-DOF systems has been done by Gillilan andReinhardt [ 161.

2. Local frequency analysis and dynamics infrequency ratio space

To motivate our approach to the analysis of cha-otic dynamics in multimode systems, consider boundmotion described by an integrable HamiltonianHo(Z) [ 111, where {Z} s a set of zeroth-order actionvariables, u) (I) = 6ZZ( )/aZ are the correspondingfrequencies, and Z-Z0 s taken to approxim ate theHam iltonian H of interest. If the relation betweenthe actions and frequencies is invertible, the fre-quencies themselves can be used in place of the ac-tions as phase space coordinates. In a three-modesystem, the energy sh ell can be projected onto a 2Dspace of two independent frequency ratios (w indingnumbers) ; the resulting tune space [ 171 is shownin fig. 1 for planar OC S, and is a key element in ourunderstand ing of three-DOF transport.

Whereas the actions {I}, and hence the frequen-cies o(Z), are constant under the motion gen eratedby Ho, they are time dependent for nonqu asiper-iodic trajectories of the full Ham iltonian H [ 111. Bydefining loculjivquencies of the motion (see below),the time evolution of non-quasiperiodic trajectoriescan be followed in frequency ratio sp ace. Just as fortwo-DO F systems, important phase space structuresin three-DOF (those leading to either enhanced52 0

Fig. 1. Frequency ratio space for planar OCS . Resonance lines(solid) and pairwise noble conditions (dashed) are indicated.

transport or trapping) can be defined by relationsbetween frequencies of motion. A local frequencyanalysis of the motion therefore focuses attention di-rectly on the aspect of central imp ortance fordynamics.

The utility of local frequency analy sis for chaotictrajectories of 3D OCS is based on the em pirical ob-servation that, although the motion may be mani-festly irregular [ 111 when viewed over long-timescales ( k: 50 ps), qu antities such as fundam ental fre-quencies and sho rt-time averages of zeroth-order ac-tions and mode energies are often fairly con stant overtimes ( z I ps) corresponding to many v ibrationalperiods. Trajectories are localized in particular re-gions of phase sp ace characterized by these proper-ties. Although non-quasiperiodic trajectories willeventually pass from one such region to another, theexistence of long-time correlations implies that themean lifetime in a region is typically considerablylonger than either a vibrational period or the localrelaxation rate implied by, say, the maximalLyapunov exponent [ 18,191 (conversely, this maybe taken as an operational definition of a dynam i-cahy significant region of phase space).Local frequency analysis of chaotic motion is ac-complished by performing fast Fourier transforms ofshort (1.4 ps) sequential segm ents of trajectories. Theprocedure used is described in detail elsewhere [ 151.


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Volume 142. number 6 CHEMICAL PHYSICS LETTERS 25 December 1987

The different regions occupied by a trajectory arecharac terized by the local frequen cies, or by their ra-tios (coordinates in frequency ratio space), and theevolution of the system can be studied by followingthe changes of the local frequencies with time. Thetrajectory segments used to define local frequenciesmust be long enough to allow sufficient frequencyresolution, but short enough to ensure that the dy-namical co rrelations of interest occur on a time scalelonger than the seg ment length. This calculation [ 15 1yields time-dependent local frequencies for cha-otic trajectories.Local frequency analysis is a useful method for thestudy of chaotic dynam ics in multimode systems. Forexample, it is straightforward to determine whethera trajectory segment satisfies one or more resonanceconditions of the form R W = 0, or whether a panic-ular local frequency ratio passes through a givenvalue. By simply counting the number of times agiven condition on the local frequencies is satisfiedalong a given trajectory, it is possible to calculatefluxes across dividing surfaces defined in terms ofthe frequencies without having to construct the sur-face itself [ 151. It is thereby possible to inve stigatethe nature of dynamical bottlenecks in multidimen-sional phase space.The calculation of Fourier spectra of short seg-men ts of long trajectories has been previously dis-cussed by McDonald and Marcus [ 201 and Marcus,Hase, and Swamy [21]. Their emphasis, however,was on the time dependence of Fourier amplitudesas an indicator of the dynamics; they did not analyzethe local frequencies.

3. Phase space structure for three degrees offreedom

To help interpret our trajectory results for planarOCS, it is useful to briefly survey the structure ofphase space for three DOF, with reference to the fre-quency ratio spac e of fig. 1. A detailed account isgiven elsewhere [ 15 1.

The portion of frequency ratio space correspond-ing to the dynamically accessible region of planarOC S is shown in fig. 1. Coordinates are the ratios%&bend (x axis), ranging from 1.0 to 2.2, andw,-Jo~~~~ (y axis), ranging from 3.5 to 4.7. For an

integrable three-DOF H amiltonian H o(I), each pointin frequency ratio space represents a n invariant to-rus, specified uniquely by two frequency ratios andthe energy (equivalent to specifying the three actions1).

The solid straight lines in frequency ratio spaceidentify multidimensional resonance zones; that is,families of frequencies satisfying a single resonancecondition of the formk-w =o ) (1)with k= ( I&,~,,,,,CS ,kO) a vector of integers. For ex-ample, vertical solid lines correspond to the pair-wisefrequency locking ~-J~,,~~~=co nstant, while slop-ing solid lines passing through the origin of fre-quency ratio space correspond to ~c&o~s=rational.Sloping solid lines that do not pass through the or-igin correspond to a general ( ternary) resonancecondition with all compon ents of the vector k non-zero. For H o , the resonance condition defines a one-parameter family of resonant tori. For a non-inte-grable Hamiltonian H , condition (1) defines a res-onance zone, a 5D region of phase space withassociated stochastic layer and multidimensionalbroken separatrix (a 4D surface). An important as-pect of dynam ics in Na3-D OF systems is the pos-sibility of transport along resonance zones [ 111, thatis, along the solid lines in fig. 1 (see section 4).

The dense network of resonance lines in the fre-quency ratio plane forms the Arnold web [ 111. Onlya few of the lowest-order resona nces important inplanar OC S are shown in fig. 1. At the intersectionof every p air of resonance lines two independent res-onance conditions are satisfied, k m = O an dk CI.I 0, say, implying an infinity of resonance con-ditions nk-o +nR *w =O (n, n integer). For H o ,the intersection of two resonance lines specifies asingle torus consisting of a two-param eter family ofperiodic orbits. For a non-integrable Hamiltonian H ,the junction of two resonance zones is a dynamicallysignificant region of phase sp ace. Trajectories canturn the comer from one resonance zone to an-other at the resonance junctions, and the cumulativeeffect of the resu lting chang es in direction is a dif-fusive wandering, known as Arnold diffusion[ 11,121 . Extensive trapp ing of trajectories in the vi-cinity of resonance junctions is found in planar O CS

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Volume 142,number 6 CHEMICALPHYSICSLETTERS 25 December 1987(see section 4); this effect appears to have receivedlittle previous attention.

We now consider the n ature of possible bottle-necks to IVR in three DOF. Single tori or cantori,which define 3D invariant or near invariant mani-folds, cannot act as barriers on the 5D energy shell.However, a single constraint on the frequencies,F( o ) =O, defines a 4D surface in phase space (inthe integrable case, a one-parameter family of tori).Such a surface has the correct dimension to dividethe 5 D energy shell into d isjoint regions. In addition,the 4D broken separatrices associated with three-DOF resonance zones have the right dimensionalityto act as partial barriers to transport in and ou t ofresonance zones.

Wh ich conditions F( w ) define dynam ical bottle-necks in three-DOF systems? Is it possible to findsingle constraints that define bottlenecks globally, orare partial barriers best defined piecewise by a set oflocal constraints on the frequencies? Definitive an-swers to these central qu estions cannot be given asyet. The notion that partial barriers to transport areassociated with sets of particularly robust tori sug-gests, however, that any such constraints should re-flect extreme m utua l irrationality of the frequencies.Local frequency analy sis is therefore a mo st effectiveway of identifying such bottlenecks in multimodesystems.

An example of a global constraint is the p&wiseirrationality conditionO,/Uj = Noble . (2)(Noble numbers are highly irrational numbers of theform (ntny)l(mtmy), with ~=f(5~-1), thegolden mean, and nm-nm = ?1 [6].) Eq. (2) isthe simplest possible generalization from the wellstudied two-mode case to three-DOF systems. Foran integrable problem, condition (2) defines a one-parameter family of invariant tori. Under a non-in-tegrable perturbation, this 4D manifold develops aninfinity of resonant holes, corresponding to si-multaneou s satisfaction of conditions (1) and (2).The existence of the resonant holes means that, evenfor arbitrarily sm all couplings, condition (2) doesnot define an absolute barrier to transport. None-theless, since most resonances are high order and/ornot strongly driven by coupling terms in the Ham-iltonian, it is conceivable that noble pairwise bottle-522

necks defined by (2) may impede transport in certainregions of phase space, i.e. lead to long-time corre-lations in N > 3-DO F systems. T his possibility isconsistent with trajectory data on OCS discussed be-low. In the limit that all modes but i and j uncouple,the pair-wise irrationality condition (2) defines acantorus in the two-DOF i,j subspace; this limitingbehavior sugge sts that the 4D bottlenecks defined by(2) may act as barriers to transport in regions ofphase space close to the relevant two-DOF subspace.Pairwise irrational surfaces are represented bystraight da shed lines in fig. 1. For ex ample, the slop-ing dashed line corresponding to the irrational fre-quency ratio wco/wcs = 2 t y falls between the solidlines associated with the resonance vectors (0, 3 , - 1)and (0, 5, -2).Pairwise irrationality conditions are certainly no tthe only candidates for three-DOF bottlenecks. Someother more general possibilities are discussed in sec-tion 4.

4. Energy relaxation and long-time correlations inplanar ocs

We now illustrate the general considerations ofsection 3 by examining representative trajectories for3D OCS at a single energy, E= 20000 cm- ( x 90%of the dissociation energy). At this energy, most tra-jectories are quite chaotic, as indicated by their non-zero Lyapunov exponents [ 13,141. The trajectoriesshown below have been selected from 8 ensembles of50 trajectories each (4 00 total). Each trajectory is 45ps long, and is divided into 128 overlapping seg-men ts, as described above. Careful analysis of all 400trajectories was performed, and it was found thatseveral distinct patterns of dynamical behavior ap-peared repeatedly in the data. These patterns are nowillustrated with some typical trajectories.4.1. Transport along resonance zones

Fig. 2a show s time-dependent local frequency ra-tios and segment-averaged mode energies for a typ-ical chaotic trajectory of planar OCS. The top framegives the CS/bend frequency ratio, the second frameshows the CO/CS ratio, while the third frame givesthe CO/bend value. For certain segments, more than


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Volume 142, number 6 CHEMICAL PHYSICS LETTERS 25 December 1987

i12 51 0_=

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Volume 42,number CHEMICALHYSICS ETTERS 25 December 987one local frequency ratio is plotted; these are am- the local frequency ratios change by a series of smallbiguous segmen ts, having spectra with at least two steps. Detailed examination of the frequency datapeaks of comparable intensity. Also included are show s that the trajectory gets trapped for several seg-horizontal lines, indicating pair-wise resonant con- men ts in the vicinity of a resonance junction, thenditions (dotted) and noble ratios M ? y , M integer rapidly jump s to another, where it remains for sev-(dashed). The bottom three frames show segmen t- eral more segme nts, and so on. Many other examp lesaveraged normal mode energies for CO stretch, CS of localization at resonance junctions can be foundstretch, and bend, respectively. in the full trajectory ensem ble [ 151.The local frequency ratios change with time,sometimes smoothly and continuously, sometimessharply and discontinuously. Chan ges in the fre-quency ratios can be correlated w ith the norm al m odeenergy exchange by noting that, whereas the CO andCS mode frequencies decrease with increasing ex-citation, the bend frequency increases when energyis transferred into bending motion.

4.3. Dy nam i ca l barri er s t o chaot i c t ranspor t

A notab le feature of the trajectory of fig. 2a is theoccurrence of several groups of segments (for ex-ample, from 3.9 to 7 ps) where the CS/ben d fre-quency ratio is fairly constant around the value 1.5.Wh ile w,-~/w~ ~~ executes sm all fluctuations aboutthis value, the other frequency ratios, and the nor-mal mode energies, show substantial changes. Con-figuration space projections [ 15 ] clearly show thetrajectory entering a region of phase space charac-terized by a 3 : 2 resonance between the CS and bendmodes. Between 3.9 and 7 ps, large changes in localfrequencies and mode energies occur, indicating ex-change of energy between the modes. Nevertheless,t he C S and bend m odes remai n l ocked i n a 3 : 2 res-onance t hroughout t he se changes . When plotted infrequency ratio space, the trajectory points move upand down along the vertical line corresponding tothe resonance vector (3, -2, 0) (fig. 2b).

At about 20 ps, the trajectory show n in fig. 3 be-gins a period of more active energy exchange. Theabrupt change in behavior occurs when the COKSfrequency ratio suddenly passes through the value2 + y. Just at the crossing point, the CO /bend fre-quency ratio satisfies a 4 : 1 resonance condition.

This behavior may be interpreted as follows: Lo-cally, the 2+y COK S surface acts as a dynam icalbottleneck in three-DOF OC S. (Globally, however,the 2 + y surface m ay be only part of a larger dividingsurface.) The trajectory wand ers through a region ofresonance junctions localized o n one side of the bar-rier, until it enters the 4 : 1 CO/be nd resonance zone,whereupon it rapidly passes through to a more cha-otic region of phase space. The local division of phasespace into two parts by a 2+y pairwise bottleneck,one more chaotic than the other, is consistent withthe well studied behavior of the limiting collinearconfiguration [ I].

The picture that emerges from application of localfrequency analysis to this and other similar examplesis one of the CSlbend 3 : 2 resonance zone acting asa pathway for facile energy exchange in planar OCS.Transport along the (0,4, - 1) resonance zone is alsofound to be important in 3D OC S.4.2. Long- t i m e correl a t i ons near j unc t i ons o fresonance zones

The above discussion is based on a pairwise viewof the dynamics, in which partial barriers in three-DO F transport are assum ed to consist of families ofthe corresponding two_DOF dividing surfaces, Whilethis picture is appealing as a minimal extrapolationfrom the two-DOF problem, it cannot be valid ingeneral, and new ideas mu st be developed to fullyunderstand three-DOF dynam ics. We now outlinetwo new approaches to the problem of defining bot-tlenecks in three DOF, each based upon a generali-zation of number theoretic ideas that have beensuccessful in understanding two-DOF systems.

We consider now another type of behavior seen inplanar O CS. Fig. 3 show s a trajectory for which themotion is localized for about 20 ps before any sig-nificant energy exchange occurs. During this time,

The first approach focuses on the most robust t or i .Kim and Ostlund have introduced an algorithm forgenerating simultaneously irrational pai r s of irra-tional num bers [23]. They argue that these are theappropriate generalizations of the notion of noblefrequency ratios to three DOF, so that the resulting

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Volume 14 2, number 6 CHEMICAL PHYSICS LETTER S 25 December 1987

jl],0.0 5 . 0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0

T i me (ps)Fig. 3. Local frequency ratios and locally averaged mode energies versus time for a trajectory of planar OCS . This trajectory exhibitslong-time trapping prior to a sudden transition to a more chaotic region of phase space. See caption of fig. 2a for details.

spiral noble pairs of frequency ratios should de-fine the most robust tori in three DOF. Fig. 4a showsa plot of the sp iral noble pairs of frequency ratios inthe region of the frequency ratio plane correspond-ing to the dynam ically accessible range for planarOCS at 20000 cm - , as determined by the algorithmof Kim and O stlund [ 231 ( z 450000 points are plot-ted). It is observed that these points, correspondingto robust tori, avoid the low-order resonance lines.

A picture of transport and trapp ing sugge sts itself,based on diffusion through the maze of the robusttori. Regions where persistent tori are densely dis-tributed trap the trajectory for long times , while theunblocked resonance zones allow rapid flow betweenresonance junctions. Bottlenecks along resonancelines (see below) may result from a pinching ofthe resonance zone by families of robust tori (cf. the3 : 2 resonance zone shown in fig. 4b).

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Volume 14 2, number 6 CHEMICAL PHYSICS LETTERS 25 December 1987

4 . 5

4 . 3

3. 9

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a .I.:,1 i.2 i . A i . 6 1 1 8 i

%h n d

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25c 0 3.5 3.7 3.9 41 4.3 4.5 4.7

%+bandFig. 4. (a) A subset of the 222most irrational pairs derived from the algorithm of Kim and Ostlund [231. There are s 4 50000 points inthe range shown in the figure, which is appropriate for OCS. (b) Expanded view of part of (a) near the 3 : 2 resonance line. (c) This plotshows the number of crossings of frequency ratio surfaces near the 3 :2 resonance line. The surfaces are defined by ~co/~ ~~~= constant,u~cs/c~~ ~~ 1 S + 0.0 1. The num ber of crossings observed for sequential segments of 400 trajectories is plotted versus oco/wbmd Thevertical lines correspond to the values of the we/w bend atio associated with the 16 most irrational pairs of frequency ratios, w ith onepair constrained to be 3 :2. These were derived using the Rim and Ostlund algorithm [231, and are related to the pinching observedin (b).

In the second approach, the dynamics is taken toconsist of a sequence of transitions between neigh-borhoods of resonance junctions, as seen in the first20 ps of the trajectory in fig. 3. Transp ort is expectedto occur along resonance lines joining the junctions.The fundam ental question is then: A re there bar-riers to transport along resonance lines, i.e. bottle-

necks to Arnold diffusion?. We are investigating thepossibility that bottlenecks along resonance zones canbe defined by an irrationality condition involving ingeneral all three frequencies of the mo tion, subjectto a single resonance constraint k,,,+w = 0, where k,,defines the resonance zone connecting the initial andfinal junctions [ 151. It is possible to systematically

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Volume 142, number 6 CHEMICAL PHYSICS LETTERS 25 December 1987

generate appropriate pairs of frequency ratios via thespiral noble construction of Kim an d Ostlund [ 231.Local frequency analysis can be used to calculate theflux along a resonance zone between a pair of reso-nance junctions through a surface defined by an in-dependent frequency ratio, and a comp arison madewith the spiral noble results. This is done in fig. 4c,which shows the flux along the (3, - 2,0) resonanceline as a function of the COK S frequency ratio. It issuggestive that one of the most pronounced minim ain the flux (at oco/wcsz 4.1) occurs in the vicinityof 2 of the 16 mo st irrational frequency ratios; thispoint is currently under study [ 151.

By identifying points of minim um flux lying alongthe dense set of resonance lines emanating from agiven junction, it may be possible to construct a 4Dpartial barrier enclosing the junction. It is not clearwhether the resulting surface in frequency ratio spacewould be smooth and differentiable or fractal.

5. Conclusion

The ideas and results on IVR in multimode sys-tems presented here raise many interesting and un-resolved questions. Local frequency analysis ofchaotic trajectories for planar O CS suggests certainplausible mechanisms for transport and trapping inthree-DOF systems, and the resulting view of thephase space structure in three DOF is shown sche-matically in fig. 5. The H amiltonian used for planarOCS is very complicated, however, and trajectoryintegration takes a large amount of computer time.Planar OCS is also strongly chaotic at the energystudied. Study of simpler, more weakly coupledmodel systems is clearly needed to test our ide as morefully. Preliminary results [ 15 ] on transport in cou-pled standard maps [ 221 lend support to the picturedeveloped here.

Our work establishes the utility of local frequencyanalysis as a method for studying chaotic dynamicsof multimode systems, and application to otherproblems of chemical interest should yield interest-ing results. The quantum manifestations [24] ofclassical barriers in A+ 3 DOF is an open question.

J, Action Space

resonance

Energy SurfaceFig. 5. Schematic view of phase space structure for three-degreeof freedom systems, such as planar OCS.Acknowledgement

We are pleased to acknowledge stimulating dis-cussions with Richard E. Gillilan and William P.Reinhardt. The work carried o ut at Cornell Univer-sity was supported by NSF Grant CHE-84 10685. Thework at Argonne Na tional Laboratory was supportedby the Office of Basic Energy Sciences, Division ofChem ical Sciences, US Department of Energy, un-der contract No. W-31 109-ENG-38. CC M w ishes tothank the Departmen t of Educational Prog ramm esat Argonne N ational Laboratory for mak ing his visitpossible, and the members of the Theoretical Chem-istry Group for their hosp itality during h is stay. CCMacknowledges support of the US Army Research Of-fice, through the Mathematical Sciences Institute ofCornell University.

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(1986) 3470.[31 SK. Gray and S.A. Rice, J. Chem. Phys. 86 (1987) 2020.[4] M.J. Davis, J. Chem. Phys. 86 (1987) 3978.[ 51R.T. Skodje and M.J. Davis, to be published.

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Volume 142, number 6 CHEMICAL PHYSICS LETTER S 25 December 1987[6] R.S. MacK ay, J.D. Meiss and I.C. Percival, Physica 13D

(1984) 55.[7] D. Bensimon and L.P. Kadanoff, Physica 13D (1984) 82.[8] S.R. Channon and J. Lebowitz, Ann. NY Acad. Sci. 357

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chastic motion (Springer, Berlin, 1983).[ 121 CC. Martensand G.S. Ezra, J. Chem. Phys. 86 (1987) 279.[ 131 D. Carter and P. Brumer, J. Chem. Phys. 77 (1982) 4208.[ 141 M.J. Da vis and A.F. W agner, in: Resonances in elec-

tron-molecule scattering, van der Wa als complexes, and re-active chemical dynamics, ed. D.G. Truhlar, Am. Chem. Sot.Symp. Ser. ( 1984).

[ 151 CC. Martens, M.J. Davis and G .S. Ezra, in preparation.[ 161 RX Gillilan and W.P. Reinhardt, Chem. Phys. Letters, to

be published.[ I?] R.D. Ruth, Lecture Notes Phys. 247 (1986) 37.[ 181 Ya.G. Sinai, Acta Phys. Austr. SuppI. 1 0 (1973) 575.[ 191 I. Hamilton and P. Brumer, J. Chem. Phys. 78 (1983) 2682.]20] J.D. McDonald and R.A. Marcus, J. Chem. Phys. 65 (1976)2180.[21] R.A. Marcus, W.L. Hase and K.N. Swamy, J. Phys. Chem.

88 (1984) 6717.[22] K. Kanekoand R.J. Bagley, Phys. LettersAl 10 (1985) 435.1231 S.H. Kim and S. Ostlund, Phys. Rev. A34 (1986) 3426.[241 G. Radons, T. Geisel and J. Rubner, Advan. Chem. Phys.,

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