richard e. gillilan and gregory s. ezra- transport and turnstiles in multidimensional hamiltonian...

8/3/2019 Richard E. Gillilan and Gregory S. Ezra- Transport and turnstiles in multidimensional Hamiltonian mappings for unim

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Transport a d turnstiles in multidi ensional Hamiltonian mappings forunimolecular fragmentation: Application to van der Waals predissociationRichard E. Gillilat?) and Gregory S. Ezrab)Baker Laboratory, Department of Chemistry, Cornell University, Ithaca, New York 14853(Received30 August; accepted15 November)A four-dimensional symplectic (Hamiltonian) mapping of the type studied by GaspardandRice is used o mode l the predissociationof the van der Waals complex He-I,. Phasespacestructure and unimolecular decay n this mapping are analyzed n terms of a generalapproachrecently developedby Wiggins. The two-dimensionalareapreservingmap obtainedbyrestricting the 4D map to the T-shapedsubspaces studied first. Both the Davis-Gray theoryand the analog of the alternative RRKM theory of Gray, Rice, and Davis for discretemaps areapplied o estimateshort-time decay ates. A four-state Markov model involving threeintramolecular bottlenecks (cantori) is found to give a very accuratedescription of decay nthe 2D map at short to medium times. The simplest version of t he statistical Davis-Graytheory, in which only a single ntermolecular dividing surface s considered, s then generalizedto calculate he fragmentation rate in the full 4D map as the ratio of the vol ume of a four-dimensional urnstile lobe and a four-dimensional complex region enclosedby amultidimensional separatrix. Good agreementwith exact numerical results s found at shorttimes. The alternative RRKM theory is also applied, and is found to give a level of agreementwith the Davis-Gray theory comparable o the 2D case.When the height of the barrier tointernal rotation in the van der Waals potential is increased, owever, t is found that volume-enclosing urnstile no longer exist in the 4D phasespace,due to the occurrenceof homoclinictangency.The implications of this finding for transport theories n multimode systemsarebriefly discussed.

1. INTRODUCTIONThere s continued interest in the possibility of nonsta-tistical behavior n elementary chemical processes.Muchtheoretical work has focused on understanding he condi-tions for applicability of conventionalstatistical approachessuch as ransition state theory, RRKM theory,3and phasespace heory,4which a re widely used o predict and correlatethe rates of chemical reactions. Many researchers ave ap-plied the conceptsand methods of nonlinear dynamic? todevelopa deeperunderstanding of the dynamical origin ofnonstatistical behavior of molecules.6 he fundamental pic-ture of phasespace ransport in terms of flux acrosspartialbarriers developed y MacKay, Meiss, and Percival, Bensi-mon and Kadanoff, and Channon and Lebowitz has beenparticularly useful; such a picture makes igorous the often-invoked concept of bottlenecks n molecular phasespace.It is well established hat conventional RRKM theory

overestimateshe classicalvibrational predissociation ate ofweakly bound van der Waals complexes uch as He-I, by atleast an order of magnitude. Davis and Gray havehow-ever shown that accuratevalues or the unimolecular decayrate of a two-dimensional classical model of He-I, can beobtained from a statistical theory incorporating transportbetweena small number of phasespace egionsboundedbydynamically determined bottlenecks. Multiexponential be-havior in rate curves on short to med ium time scales an besuccessfullymodeled, although power law decay typically

Present address: Department of Chemistry, University of California, SanDiego, La Jolla, CA 92093.b, Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar.

found at long times requiresan nfinite hierarchy of bottle-necksdescribing rapping around sticky islands. 3*i4Sta-tistical approaches n which it is assumed hat randomiza-tion is rapid within phasespace egionsbounded by partialbarriers therefore etain their utility evenwhen the system snonstatistical according to conventional criteria. Other ap-plications of t hese deashavebeenmade n two-dimensionalmodels for IVR in OCS, s unimolecular decay of van derWaals complexes,16somerization,* and bimolecular re-actions. 9,*0A major outstanding problem is the extension of themethodologyoutlined above o systemswith more than twodegrees f freedom. Such an extension s of course essentialfor application to realistic modelsof polyatomic molecules,but there have been ew attempts to date. In the so-calledalternative RRKM theory of Gray, Rice, and Davis,i theunimolecular decay ate s determinedby Monte Carlo eval-uation of the outgoing flux across an approximate phasespace ividing surface.Although reasonably ccuratevaluesfor the dissociation rate are obtained n this way for a two-mode model of He-I,, results or a three-dimensionalmodelincorporating the van der Waals bending mode were disap-pointing. Wozny and Gray definedan approximate separ-atrix for three-dimensional Ne-Cl2 by decomposing theproblem into a family of effective wo-mode Hamiltonians,and havepresented vidence hat trajectoriesare confinedbythis multidimensional separatrix. Tersigni and Rice haveexamined he robustnessof cantori in a two-mode systemunder perturbation due to a third mode, and conclude thatthe goldenmeancantorus emainsa significant bottleneck nthe full system.23 he possible mportance of pairwise fre-quency incommensurabilities n multidimensional systems

2648 J. Chem. Phys. 94 (4), 15 February 1991 0021-9606/91/042648-21$03.00 @I 1991 American Institute of PhysicsDownloaded 03 Jan 2001 to 128.253.229.100. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.


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FL E. Gillilan and G. S. Ezra: Mappings for unimolecuiar fragmentation 2649

was also emphasized y Martens, Davis, and Ezraz4 (cf. alsoRef. 25).The recent development of a formal theory of phasespace ransport in multidimensional systemsby Wiggins26,27[reviewed below; for related recent work seealso Refs. 28and 29) provides a firm foundation for extension of the Da-vis-Gray theory to polyatomic systems. n the presentpaperwe apply Wiggins approach o a four-dimensional Hamilto-nian (that is, symplectic) mapping representing he vibra-tional predissociation of the complex He-I, with both thevan der Waals stretch and bending modes ncluded (overallrotation is however excluded). Iteration of such mappings scomputationally much more efficient than integration ofcontinuous-time trajectories for investigation of phase-spacestructure. The four-dimensional Hamiltonian map providesa computationally tractable yet reasonably ealistic examplewith which to investigate the generalization of the Davis-Gray approach to unimolecular decay n a multimode sys-tem.

This paper s organized as ollows: In Sec. I we discussgeneral properties of Hamiltonian maps, and introduce theparticular system treated here. Section III contains a discus-sion of phasespace ased heories of unimolecular decay.Wefirst review the approach for two-mode systems developedby Davis and Gray. We then discuss he generalapproach otransport in multidimensional systems due to Wiggins,26which generalizes he turnstile-based theory of MacKay,Meiss, and Percival to higher dimensional systems. In Sec.IV, after preliminary study of a two-dimensional restrictionof the four-dimensional map to the T-shape subspace,weexamine the dissociative dynamics of the 4D van der Waalsmap from the standpoint of Wiggins theory. We compareexact fragmentat ion rates with thoseobtained using a simplestatistical approachwhich is the generalization of the Davis-Gray theory to multidimensional systems, n which only asingle, intermolecular bottleneck is taken into account. Thealternative RRKM theory is also applied to the 4D map.Interesting topological problems that arise n attempting todefine urnstiles in higher dimensions are discussed.SectionV concludes with a discussion of some points arising fromthe calculations of Sec. V. The Appendix contains details ofthe algorithms for finding invariant manifolds in the multi-dimensional case.II. HAMILTONIAN MAPPINGS FOR MOLECULARFRAGMENTATION

The use of symplectic (Hamiltonian) mappings in thestudy of nonlinear classical dynamics has a ong history.30-38Symplectic mappings provide a computationally efficientmeans of exploring the long-time behavior and the fine de-tails of phase-space tructure for dynamical systems. Thesimplest such maps are area-preserv ingmaps of the plane,which may be thought of as generatedby the PoincarCsur-face of section for a two degree of freedom autonomousHamiltonian system or a periodically forced one degreeoffreedom system.5 Many of the recent advances n under-standing transport in Hamiltonian systemshave come fromthe study of the area-preserving tandard map.7,30v40Recently, Gaspard and Rice3 have studied Hamilto-

nian mappings that mimic the behavior of classical modelsfor unimolecular dissociation of van der Waals molecules.,They explored the phase spacestructure of both two- andfour-dimensional maps and ound behavior n reasonable c-cord with previous classical trajectories studies of van derWaals predissociation. Most notably, both 2D and 4Dmaps were found to exhibit multiply exponential decay onshort to medium time scales, ndicating the presence f bothinter and intramolecular bottlenecks.3g n 2D maps, short-time decay is known to be governed by an intermolecularbottleneck that divides directly scattered rajectories (thosewith only a single inner radial turning point) from thoseassociatedwith complex formation. Decay on longer timescales n 2D mappings can be modeled quite accurately bykinetic schemes nvoking intramolecular bottlenecks asso-ciated with cantori,7 which are the remnants of invariantcurves having particularly irrational winding numbers (cf.Sec. IV).4o On very long times scales, here is typically aninvariant set of nonzero measureconsisting of points in thequasiperiodic region that are bound for all time (in additionto the fractal repellor composedof all bound unstable peri-odic and aperiodic orbits ) .39Trapping near he sticky boun-daries of the quasiperiodic region leads to power law decayof population at very long times. 2 The effectsof resonanceson the measureof the invariant set in t he 2D map were alsostudied by Gaspart and Rice.3g One major differencebetween he 2D and 4D casesnoted by Gaspart and Rice wasthe existence of a quasi-invariant set for the latter; that is,although the invariant set is apparently of zero measure orthe 4D case, there exists a quasi-invariant set of nonzeromeasure hat persists for very long times. Due to the exis-tenceof the quasi-invariant set, he decay at long times (pre-sumably governed by Arnold diffusions727~41is not wellcharacterized.3gIn the present paper the computational convenienceofHamiltonian mappings is exploited to investigate in detailthe dissociation kinetics and phasespacestructure in a 4Dsymplectic map modeling the well-studied van der Waalscomplex He-I,.A. General properties of Hamiltonian mappings

There s a close elation betweencontinuous-time trajec-tory evolution and discrete Hamiltonian (symplectic) map-pings. For a continuous-time, autonomous system with Ndegreesof freedom, the Poincarh surface of section of the(2N - 1 -dimensional constant energy hypersurface inphasespace defines a mapping of a (2N - 2)-dimensionalslice of the spaceonto itself.s*27A three degreeof freedomcontinuous-time system can therefore in principle be re-duced o a four-dimensional surface of section mapping. Be-cause he mapping is the result of Hamiltonian time evolu-tion, it is a member of the special class of symplectictransformations.5.42Symplectic transformations preservephasespacevolume (Liouvilles theorem) as well as a set oflower-dimensional invariants.5*42Although the symplectic mapping associatedwith a giv-en continuous-time evolution cannot in generalbe obtainedexplicitly, a periodically kicked integrable system is an ex-ceptionalcase.Considera freeparticle moving n onedimen-

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2650 R. E. Gillilan and G. S. Ezra: Mappings for unimolecular fragmentationsion, periodically kicked by a delta-function potential withHamiltonian

H=HO + TV(q) nd&*,...Sit-nnyf&J?-, (2.1)

whereT is the period of the kick and V(q) the potential. Theparticle momentum s constant betweenkicks. If the coordi-nates ust before a kick are (p,q), integrating Hamiltonsequationsover one interval between kicks determines hecoordinates p,q) just before the next kick by the mapp+p=p-TE, aqqfqrL-q+~. (2.2)

It is readily verified that this map is area preserving, andsymplectic n the multidimensional case.When T = 1 andV= sin 0, Eq. (2.2) is ust the familiar standardmap.30Ter-signi, Gaspard,and Rice havealsostudied he casewhereHois the Hamiltonian for a Morse oscillator.The relation betweenHamiltonian mappings and con-tinuous-time trajectory integration can be urther illuminat-ed as ollows: Consider he N degree f freedomcontinuous-time Hamiltonian system (massm = 1)

4 dp dV-= --,dt = dt ap (2.39Newtons equations of motion expressedn finite differenceform yield the Verlet algorithm3 for the positions at timet + At in terms of those at times t and t - At:q(t+At)=2q(t)--(t-At) -At?. acl (2.49SettingP(t + I&) N qit + At9 - nit)At (2.59

and rearranging yields the Hamiltonian form of the Verletalgorithm (the leapfrog algorithm),p(t+&At) =p(t-$At) -At% [aitqit+A.t) =s(t) +Atpit+@t), (2.69

which is equivalent to the multidimensional version of Eq.(2.2) with Ts At. For small T, the Hamiltonian mapping sthereforea (third-order) symplectic rajectory integrator.44Thereare hen two interpretations one can give to a particu-lar 2Ndimensional symplectic mapping: first, as the Poin-care return map for a periodically driven N degreeof free-dom (such a system s often referred o asan N + 1 degree ffreedomsystem); second,as an N degreeof f reedom sym-plectic trajectory integrator.44 The first interpretation isusedhere.B. Hamiltonian map for He+1. The HamiltonianThe classical Hamiltonian for planar He-I, is

(2.7)Here, ,u is the reduced massof I,, m is the &-He reducedmass, is the rotational constantof the I, molecule,s s the I,bond ength, Y he distance rom the He atom to the I2 centerof mass,y the anglebetween he diatom axis and the He to I,vector, andp,, p,, andp, the respective onjugatemomenta.I is the orbital angular momen tum (J - p,, ), withJ the otalangular momentum or rotation in the plane. V(s,r,y) is thepotential energy.Before converting the continuous-time HamiltonianEq. (2.7) into a map, we invoke the classical centrifugalsudden (CCS) approximation;45 hat is, we take I to be con-stant. Mulloney and Schatz have ound the CCS approxi-mation to be quite accurate or calculation of rotationallyinelastic cross sections or He/I, collisions. The CCS ap-proximation is convenient or our purposesas t leads o anexplicit symplectic map. When running trajectories the val-ue of 1 s usually taken to be fixed by the initial conditions.45For simplicitly, we take I = 0 (zero impact parameter), sothat the Hamiltonian is

H=&+Pf+&+ V((sry)2p 2m 21 ) * (2.8)

2. Sympectic mapping for He-/2Now consider he model nonautonomousHamiltoniancorresponding o a free particle (He) plus free rotor (I,)describedwithin the zero impact parameter CCS approxi-mation periodically kicked by a potential V( r, y) :H=&+$+TV(r,y) 2 S(t--nT). (2.9)n =o, * I....Integrating over one period T, we obtain a symplectic mapacting in the four-dimensionalspace pr,p,,,r,y) :pr-P:=P,--$:,pr-+p; =pu - Tz,ayr4r f zp;,m

rp;Y-Y+ I mod(2a).The potential V(r,y) is the full three-dimensionalpotentialevaluatedwith the IZ bond length s fixed at its value at thekick (for example,at equilibrium). A mapping of this typewas studied by Gaspard and Rice.39In addition to the obviousperiodicity in the angley, thephasespace p,,pY,r,y) is also periodic in the rotor angularmomentump,,, so that the (y,p, 9 subspaces a 2 torus (cf.the usual standard map3 9. The period Ap, is given by

TA&, _~ - 5-,rwhere the absence f a factor of 2 on the right-hand side ofEq. (2.11) is due to the potential V being periodic in y with

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R. E. Gillilan and G. S. Ezra: Mappings for unimolecular fragmentation 2651

period r, rather than 27r. For T = 8000, the p,, period is1485.27a.u.By symmetry, the map (2.10) has two-dimensional in-variant subspaces orresponding o y = + z-/2 or y = 0, vwith P,, = 0. In these nvariant subspaces, he four-dimen-sional map acts as an area-preservingmap n the two-dimen-sional subspace r,p, 1:p,-tp: =pr - T$,r-tr+ fPi. (2.12)The kicking period Tshould in principle equal he vibra-tional period of the I, molecule n its initial vibrational state.In practice, we find it necessary o use T values smaller thanthe I2 vibrational period in order to obtain dynamics qualita-tively similar to the continuous-time case.The reason s thatthe free motion between kicks allows the helium atom topenetratemore closely into regions of high potential duringoneperiod than it is able o do in the continuous-time systemat physically relevant energies.Those He atoms that havepenetrated near the repulsive core of the potential subse-quently receivevery hard kicks, with the result that the over-all dynamics is more chaotic than the corresponding contin-uous-time system. To the extent that the dissociationdynamics of the driven system Eq. (2.9) are similar to thoseof the full Hamiltonian (2.7), the resulting 4D symplecticmap is a useful tool for the study of van der Waals predisso-ciation. Irrespective of its ability to quantitatively reproducethe results of trajectory calculations, the driven system (2.9 9provides a computationally tractable way to study the phasespacestructure of three degreeof fr eedom systemsundergo-ing dissociation.

3. The po ten tia/We use a simple form of the type proposedby Gaspardand Rice,39consisting of a Morse-type potential in the r co-ordinate with a y-dependent epulsive portion:

V(r,y) ==D{[l +a:cos(2y)]e-2-C -2e-Pr-rc)].(2.13)This potential has he very useful property that the asympto-tic (r,p,.) motion (r--t M) is independentof y. The importantparameter Q (OO):

B/Wa= 2-(B/WID= W(1 -a), (2.14)r, = rmin - [ln( 1 - Cr) //?.

A realistic set of parameter values is given in Table I. Forreasonsexplainedbelow, we shall mainly report results for avalue of the barrier height B = 4.0X 10e--s au.), which isabout half of the physical value. Figure 1 s a contour plot ofthe model potential V(r,y). The potential profile alongy = q/2 is Morse-like in appearance.HI. TRANSPORT AND UNIMOLECULAR DISSOCIATIONA. Two degrees of freedom

Davis and Gray studied a two degree f freedom mod-el for the vibrational predissociation of He-I,,

8 y=j

0.0 0.3 0.5 0.8 1.07/n

FIG. 1. Contour plot of V(r,y), the model potential energy function for theHe+ map.J. Chem. Phys., Vol. 94, No. 4,15 February 1991Downloaded 03 Jan 2001 to 128.253.229.100. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.


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He-I,(v)+I,(v) + He, UCU. (3.1) 0cdIn the conventional RRKM approach, the critical surfacedividing reactant from product is defined in configurationspace.2,3 he rate for the dissociation reaction (3.1) thus 9mobtained is overestimated by orders of magnitude. The lo-cation of the transition state r = 6 is somewhat arbitrary fora processsuch as Eq. (3.1) , in which there is no centrifugal LOati

1 3;o 6.0 9.0 12.0 15.0 18.0r (Bohr)

barrier with zero orbital angular momentum 1.The approach of Davis and Gray is based upon the 9properties of the reactive separatrix, which is a surface in :phase space dividing the bound complex region from un-bound trajectories. Technically, the separatrix is formed bytaking the union ofsegments of the stableand unstable mani-folds of a fixed point of the Poincare return map.5 (For He-I,, the relevant fixed point is associated with the periodicorbit at infinity, corresponding to oscillation of I, in the pres-ence of stationary He at infinity,) The separatrix partitionsphase space nto disjoint regions.To illustrate, consider the motion of an unperturbedone-dimensional oscillator with the potential [cf. Eq.(2.13) J V(P) = V(~,Y)[~=~,~. A phase point undergoingoscillatory motion near the bottom of the potential welltraces out a closed invariant curve in the (r,pr) plane. Theseparatrix, defined here by the zero energy condition

FIG. 3. Phase portrait for the 2D van der Waals map in the invariant sub-space (pu = 0, y = n/2). Orbits near the center are regular (concentric in-variant curves) while those closer to the edge are chaotic. Construction ofthe separatrix yields a pair of homoclinical ly oscillating branches [the sta-ble and unstable manifolds W&(&) ] that do not join smoothly .

and the relative translation of the helium atom. As T-+0, thephase portrait of Fig. 3 becomes dentical to the integrablephase spaceof Fig. 2.39H, ( >=t =g+ V(r) ==o, (3.2)is a smooth curve enclosing all bound trajectories (Fig. 2).The phase portrait for the 2D mapping (2.12), -withy=n-/2,p,=O, T==8000,andB=4~10- stronglyre-semblessurface of section plots for T-shaped He-I, (Fig.3). The separatrix is no longer a smooth curve. Instead, thestable and unstable manifolds of the fixed point at infinity(r = 00,pr = 0), which are the oscillatory curves plotted inFig. 3, intersect transversely at so-called homoclinicpoints. The resulting homoclinic oscillations are a mani-festation of energy transfer between he I, vibrational mode

2652 R. E. Gillilan and G. S. Ezra: Mappings for unimolecular fragmentation

The separatrix, which is formed by taking the union ofsegmentsof the stable and unstable manifolds, is asurface ofno return. That is, it defines an exact transition state forfragmentation; once a trajectory crosses he separatrix fromthe inside to the outside, it cannot return. Phasepoints crossthe separatdx through the turnstile, which is the region ofphase spacemediating transport across the separatrix. Fig-ure 4 shows the path of a typical phase point crossing the

a@- /-A 4 /..WsP4/\

' 4.0 7.0 10.0 13.0 16.0r (Bohr)

10.0 15.0r (Bohr)

FIG. 2. Phase plane for the van der Waals bond stretch at y = 7r/2. Thedashed line marks the separatrix dividing bound orbits from scattering or-bits.

FIG. 4. Several homoclinic oscillations of the manifolds W(M) (Fig.3) are shown together with the turnstile lobes A and B. The complex regionis enclosed by solid lines, excluding lobe A, which contains points that enterthe complex region upon the next iteration of the map. Lobe B containspoints leaving the complex region on the next iteration. Dots mark the pri-mary intersection points (pips) of the separatrix branches and dashed linesshow thecontinuation ofthe branches. Success ive iterates ofsingle point areplotted (stars) illustrating energy loss from the van der Waals stretch modeleading to capture.



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separatrix. The turnstile consists of two (in general,an evennumber) of lobes, eachcontaining points that cross n/out ofthe complex region on the next iteration of the mapping.Exact region-to- region dynamics .on the surface of sectionmay be formulated in terms of the intersections of pre/postimages of lobes (seeRefs. 50 and 5 1)In practice, simple statistical approximations for theunimolecular decay ate are ound to be quite accurate.Letpbe the probability of escapeper iteration of the map, A (L)the areaof the outgoing obe on the surfaceof section, and etii(S) denote he area of the complex region enclosedby theseparatrix, excluding those phasepoints in the invariant set(e.g., trapped forever on invariant tori). The simplest statis-tical approximation top is then

A(L)p=r4(s)* (3.3)This approximation assumes apid randomization of phasepoints in the stochastic region 5 n a single iteration of themap. The unimolecular decay constant is

kuni = -+ln (1 -p)+. (3.4)(For a continuous-time system, T is the mean return timefor the PoincarCmap.52 It is important to point out that, fora 13 degree of f reedom system, A(L) is actually t he flux(phase-space olumeper unit time) between he two homo-clinic trajectories at the intersection of the boundary seg-ments of the lobe. The constant T convertsA (S) into a phase-spacevolume. A detailed d.iscussion f this relationship maybe found in the Appendix of Ref. 29.The randomization assumption mplies the oss of a con-stant fraction of complexes per iteration of the map, i.e.,exponential decay. Such exponential decay s f ound numeri-cally on short to intermediate time scales,even n the pres-enceof significant regular (quasiperiodic) regions of phasespace.1Davis and Gray obtained improved results by solvingkinetic schemes aking into account intramolecular bottle-necks, associatedwith cantori (seenext section). The intra-molecular bottlenecks introduce addi tional time scales ntothe description of the dissociation kinetics, and phasepointstrapped inside such bottlenecks are responsible or slow de-cay at longer times. Multi.state models ncorporating one ortwo important intramolecular bottlenecks were found togive accurate esults for unimolecular decayof He-I,. *I Thestatistical theory basedon dynamically calculated turnstileshas also beenapplied to Ar-I,. l6In the so-calledalternative RRKM approach,2 flux iscalculated across an approximate phasespacedividing sur-face basedon a zeroth-order separatrix. The surface corre-sponds o the condition that the zeroth-order Hamiltonianfor the dissociative mode be zero [seeEq. ( 3.2) ] . The alter-native RRKM theory was applied both to T-shaped He-I,and to a three-mode model including the van der Waalsbend. Although results are much better than the convention-al RRKM approach, significant discrepanciesbetween hetheoretical and numerical simulation results still remain,especiaily or the 3D case. t-is therefore mportant to gener-alize the Davis-Gray approach using dynamically defined

separatrices o systems with more than two degreesof free-dom.B. Generalization to many degrees of freedom

It is well known that, for systems with N>3 degreesoffreedom, N-dimensional invariant tori cannot divide the(2N - 1 -dimensional energy shell into disjoint regions.S,41This fact, which leads o the possibility of Arnold diffusionin multimode systems,41 ppears o be a serious mpedimentto the developmentof a general heory of transport in multi-dimensional phasespacebasedon partial barriers. For ex-ample, patching up the holes n the multidimensional analogof a cantoruss in an N degree of freedom system woulddefine an N-dimensional object, which has too small a di-mensionality to be a barrier to transport. Severalnumericalinvestigations, however, have ndicated the presenceof bot-tlenecks to energy transfer in three-mode systems. Pfeiferand Brickmann, for example, observed ong-term confine:ment of trajectories in the chaotic regime of a polynomialpotential.s4 Carter and Brumer and Wagner and Davis56observedslow relaxation in a model for OCS. Martens, Da-vis, and Ezra subsequentlystudied the dynamics of OCS inthe frequency domain, and found evidence or trapping in-side resonancechannels and in the vicinity of periodic or-bits.24 n addition, certain irrational pairwise frequency ra-tios, important in the vibrational dynamics of collinearOCS,. appear o be associatedwith trapping in 3D OCS.24Tersigni, Gaspard, and Rice have also noted the importanceof pairwise frequency ratios in the multidimensional dynam-ics 25 a

To facilitate generalization of the Davis-Gray ap-proach, we extract the formal essentialsof the two degreeoffreedom continuous-time problem. Our treatment here owesmuch to the paper of Wiggins26 (cf. also the work of Eastonand McGehee ) . The abstract concepts ntroduced in thissubsectionare llustrated for the 4D van der Waals Hamilto-nian mapping in the next section.For two degreesof freedom, the phase space s four-dimensional, and the energy shell is three-dimensional. Amicrocanonical transition-state dividing surface must dividethe energy shell into disjoint regions, and so must be two-dimensional. Such a surface is said to be codimension one,being of dimension one less than the space n which it isembedded the energy shell in this case). How may such asurface be obtained n general?Consider an unstableperiodic orbit, denoted &. A peri-odic orbit is one-dimensional,or codimension wo in the en-ergy shell. An unstable (hyperbolic) periodic orbit has wo-dimensional (codimension one) stable and unstablemanifolds W(d), consisting of all points that tendasymptotically to JZ in forward (stable) or reverse (unsta-ble) time. In a nonintegrablesystem, such as a typical molec-ular Hamiltonian, the two-dimensional stable and unstablemanifolds of the periodic orbit intersect transversely (that is,the stable and unstable manifolds do not join smoothly) .The intersection set WS(&>f7 W(J) consists of allpoints that are asymptotic to 4 in both forward and reversetime, and s thereforecomposedof homoclinic trajectories.7Each homoclinic trajectory is 2 + 2 - 3 = 1 dimensional.



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[The stableand unstable manifolds of two distinct periodicorbits J and J? intersect in so-called heteroclinic orbits.)If the continuous-time two degreeof freedom system sreduced o an areapreserving map R by examining intersec-tions of trajectories on a particular energyshell with a Poin-care surface of section Z;, a period-one (ass,umedwithoutloss of generality) periodic orbit & becomesa ixed pointJZ, = JnZ of R (provided the periodic orbit intersectsX). The manifolds VU(,@) intersect B in one-dimensionalsets WF(J) nB. If t he stable and unstable manifolds in-tersect transversely, they must intersect on the surface ofsection at an infinite number of homoclinic points corre-sponding to the,successive ntersections of homoclinic tra-jectories with Z (cf. Fig. 4).

knowledge, here are in fact no algorithms available or thecomputation of arbitrary NHIMs. In many systems, how-ever, such invariant manifolds may be found by symmetry.An important example is a three-mode system with asaddle, .e., potential barrier. At the saddle, et there be twostable and one unstable directions, let the potential by sym-metric in the unstablecoordinate,q,, and et the kinetic ener-gy be a diagonal quadratic form with constant coefficients.The system then has an NHIM &, which is the symmetrydetermined nvariant subspace:

P = {q, = 0, p, = OIH = El. (3.5)

It is essential o note that, in the area preserving case, fthe stable and unstable manifolds intersect transversely,then Fig. 4 shows the only possible way (apart from theexceptionalcaseof homoclinic tangency) in which they maydo so. As we shall seebelow, the topological possibilities inthe multidimensional caseare much richer and far from be-ing fully understood.

,&a therefore has the topology of the phasespaceof a twodegreeof freedom system with energyE. If the Hamiltonianfor the (ql = O,p, = 0) subspace orresponds o bound mo-tion of two coupled oscillators, ,K is an invariant 3-sphereY3.28.2vSucha surfacewill survive arbitrary, nonsymmetricperturbation of theHamiltonian.27s58 ccording to Wiggins,codimension two NHIMs in N degreeof f reedom systemsare n general2N - 3 spheres.16Segments f the stable and unstable manifolds emanat-ing from a fixed point at infinity A; and meeting.at a given(primary ) homoclinic point define he boundary or separ-atrix enclosing the complex region in phase space.On thefull energy shell the corresponding pieces of W and Wfrom the two-dimensional (codimension one) boundary ofthe three-dimensionalcomplex region. n the caseof van derWaals predissociation, the periodic orbit dm is not hyper-bolic but is a marginally stable orbit at infinity (that is, botheigenvalues f the linearization of t he map about the periodicorbit at infinity are unity). There is however no essentialdifferenceas far as our present application of the theory isconcerned.

If the Hamiltonian restricted to .JO is integrable, thenJo consists of a one-parameter amily of 2 tori. Such anobject was recognizedas he appropriate analog of an unsta-ble periodic orbit in higher dimensionsby Pollak and Pechu-kas5- (cf. also Refs. 60-62). Families of so-called reduced-dimensionality tori have recently been postulated asdynamically important componentsof the phasespace n theinterpretation of the SEP spectrum of Na3.h3

Consider the separatrix consisting of segmentsof Wg(J) and WS, (&) that intersect at a homoclinic point(which is a so-calledprimary intersection point or pip > asshown in Fig. 4. Iteration under the area-preservingmap Rthen generates wo (in general, an even number of). lobes,also shown in Fig. 4. In the two lobe case, there are twodistinct homoclinic points at the intersection of the piecesofW and W defining the boundaries of the lobes, corre-sponding o two distinct homoclinic trajectories in the con-tinuous time system.

In the continuous three-modevan der Waals case, hereexists an invariant 3D manifold A consisting of all pointswith r = ~13,, z 0. At constant total energy E, this mani-fold is parametrized by the rotor angular momentump,, therotor orientation y, and the vibrational phaseangle 19,. hemanifold ..& is not normally hyperbolic, but is marginallystable (just as the periodic orbit at infinity in the two-modecase is not hyperbolic). Nevertheless, the theory carriesthrough in any case.It is natural to conjecture that in multidimensional sys-tems there exist codimension-two NHIMs associatedwithall single resonance onditions of the form

To general ize his picture, consider first the continuoustime case. n A$3 degreeof freedom systems, the naturalgeneralizationof an unstableperiodic orbit is a codimension-two normally hyperbolic invariant manifold (NHIM), de-notedJ.26*27958or N = 3, the manifolds J are 5 - 2 = 3-dimensional invariant manifolds. The property of normalhyperbolicity means that expansion and contraction ratesnormal to the manifold ,& under the flow linearized aboutJ dominate those tangent to ...&.27An important questionin dynamics concerns he persistence f invariant setsunderperturbations. The fundamental theorem on NHIMs en-sures hat normally hyperbolic invariant manifolds, if theyexist n a system, survive arbitrary perturbati on (that is, theyaresrructurally stable) .58 We shall assume hat such objectsexist (although finding them may be difficultl). To our

m-o =0, (3.6)where m is an &dimensional vector of integers, not all zero.[This conjecture has recently been proved by S. Wiggins(private communication).] Moreover, it is plausible thatthesesurfacesmay be determined by a variational principle,by analogy with periodic orbits in two degree of freedomsystems.A further consequence f normal hyperbolicity is thatNHIMs ,# have stable and unstable manifolds that arethemselvesstructurally stable.* Let the stable (unstable)manifolds of&? be denoted W(A); Ws((.A) consistof all points that tend asymptotically to ..& in forward (sta-ble) or reverse (unstable) time. These manifolds are codi-mension one, hat is, they are of dimension one ess han theenergyshe11.26or example, n the caseof a three-modesys-tem with a potential barrier, t he stable and unstable mani-folds ofJo are 4D manifolds with topology y3 X R I. These



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manifolds have beencalled cylinder man$olds by Ozorio deAlmeida et a1.64The three-dimensional invariant manifold at infinity.J in the three-modevan der Waals system, although notnormally hyperbolic, nevertheless as our-dimensional sta-ble and unstablemanifolds consisting of all points for whichpI -+Oas Y-, 00 n forward or reverse ime. Computation ofpoints on the stable and unstable manifolds is discussed nthe next section,In multimode systems, the stable and unstable mani-folds of a particular NHIM & may enclosea region of phasespace n a way analogous o that for two degrees f freedom.That is, segments of WS(A) and W(A) may form aboundary such that any point entering or leaving the en-closed region must do so by entering the multidimensionalanalogof a turnstile. For two degrees f freedom, f the stableand unstable manifolds intersect transversely, then theymust enclosea region of phasespace n the above-mentionedsense.This is not necessarily he case or multidimensionalsystems, however, as seenbelow.Successive ntersections of trajectories on the energyshell in N degreeof freedom autonomous systems with asurface of section Z generatea (2N - 2) -dimensional sym-plectic mapping. For N= 3 the surface of section is fourdimensional, and the four-dimensional symplectic map in-troduced in the prev ious section s a very convenient substi-tute for the computationally intensive calculation of the sur-facesof section for a three-modesystem. The intersection ofcodimension- two NHIMs & with S is, in general, a(2N - 4) -dimensional invariant manifold &ZZ. The stableand unstable manifolds WS(A) intersect I: in (2N-- 3)-dimensional sets WY(l) = W(l) nx. Thesehave di-mension one less than the (2iV- 2)-dimension surface ofsection, and so may serve as components of the boundarybetween egions of phasespacedefined n the surface of sec-tion. For N = 3, then, JZ is 2D with 3D stableand unstablemanifolds WY(A). WS, and Wg will in general ntersecttransversely.In the multidimensional case, he analog of the homo-clinic point in the surface of section is a (2N - 3)+ (XV-- 3) - (2N- 2) * (2N-- 4)-dimensional trans-verse homoclinic manifold. 26The analog of a primary inter-section manifold is 2D, and consists of a two-parameter am-ily of points biasymptotic to &. By straightforwardgeneralization of the two degreeof freedom case,one mightanticipate the existenceof two distinct two-dimensional pri-mary intersection manifolds, corresponding to the pair ofdistinct primary homoclinic points. The primary intersec-tion manifolds, each consisting of points that map asymp-totically in both forward and reverse time onto AX, aremoreover expected o be continuously deformable nto (ho-motopic to) -4, .26 his expectation s confirmed for certainparameter egimes n the 4.Dvan der Waals map asdiscussedin the next section. For other parameter regimes, however,the intersection manifold doesnot have such a simple struc-ture.

In the case hat there are two distinct primary intersec-tion manifolds homotopic to JZ, the definitions of lobesand turnstiles can be carried over directly to describe rans-

port in the multidimensional case. That is, segments ofW(AT)~ and W(A). meeting at a primary intersectionmanifold enclose he complex region of phase space. Iter-ation of the map leads o the formation of two or more lobescomprising the turnstile. Each lobe is boundedby segmentsof W(V&)z and W(k)= meeting at transverse homo-clinic manifolds. Phase points enter or leave the complexregion only by mapping nto the appropriate obe. If the vol-umesof the escaping obe, he complex region and the invar-iant set can be calculated, a simple statistical estimate of theescape ate can be made ollowing Davis and Gray (seenextsection). Ifthesegmentsof WS(.M)x and W(A), do notenclose a region of phasespace,however, it is not obvioushow to develop a turnstile-based heory of transport.

IV. PHASE SPACE STRUCTURE AND DYNAMICS OFTHE VAN DER WAALS MAPIn this section we study the phasespacestructure andunimolecular f ragmentation dynamics of the 4D symplecticmap for the van der Waals molecule He-I, described n Sec.II. We first consider the dissociation of points confined tothe 2D invariant subspace y = 0, y = r/2. We apply thestatistical theory of Davis and Gray* to this case,and showthat, for physically relevant parameter values, a multistatemodel basedon one ntermolecular bottleneck (the separa-trix) and a small number of intramolecular bottlenecks (de-fined by periodic orbit approximants to cantori ) gives anaccuratedescription of the decaykinetics at short to mediumtimes. The analog of the alternative RRKM theory!. for

discrete maps is also applied, and found to give reasonablyaccurate predictions for short-time decay rates.We then turn to the 4D map. The general concepts n-troduced in the previous section are illustrated by applica-tion to the particular caseof the 4D van der Waals map forHe-I,. A method for computation of points on invariantmanifolds in the 4D phasespace s discussed (cf. also theAppendix), and sections through the multidimensionalmanifolds examined to confirm the existence of higher di-mensional analogs of turnstiles. The generalized Davis-Gray statistical theory for unimolecular decay is then ap-plied. A kinetic scheme ncorporating a single intermolecu-lar bottleneck (reactive separatrix) is used to predict theshort-time decay rate, and compared with both numericalsimulation results and the alternative RRKM theory formaps.The exis tence of multidimensional turnstiles is centralto the extensionof the MacKay, Meiss, and Percival theoryof phase space ranspor t to systems with many degreesoffreedom proposedby Wiggins26 and applied in the presentpaper o the 4D van der Waals map. An important observa-tion made here s that turnstiles may fail to form in the 4Dvan der Waals map, even n physically relevant parameterregimes. The reasons or this failure (which was predictedby Wiggins and also noted by us in a study for coupledstandard maps ) and its implications for generalization ofthe Gray-Davis theory to multimode molecules are brieflydiscussed.



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A. Two-dimensional mappingConsider the invariant subspace of the 4D van derWaals mapping defined by the condition p,, = 0, y = n-/2.The 2D map obtainedby the action of the full 4D map n thisinvariant subspacedescribesa periodically kicked van derWaals stretch mode [cf. Eq. (2.12) ] in the T-shapedconfig-uration (bending angle fixed at y = r/2).

I. Fragmentation dynamicsA phaseportrait for the 2D map with T = 8000 is shownin Fig. 3. To study dissociation rates, ensemblesof initialconditions (r,p, ) were chosen n three different ways. First,a quasiclassicaldistribution6 corresponding o the groundvibrational state of the I,-He van der Waals bond was used,where he harmonic approximation was made for the bondpotential V(r,y = r/2). The resulting ensembleof pointslies completely within the reactive separatrix dividing sur-face. Second,a Wigner distribution66 corresponding o the

ground vibrational wave function for the van der Waalsbond was used,again making a harmonic approximation forthe potential. The Wigner function for the harmonic oscilla-tor ground state is a Gaussian distribution in both Pandpr,and there s therefore a small probability that points in theinitial ensemble ie outs ide the reactive separatrix. Third, auniform distribution of (r,p, ) points was chosen,subject tothe constraint that H,(r,p,) be less than - e, wheree/W= 0.37. The constraint ensures hat no points in theensemble re nitially outside the separatrix. Each ensemblecontains at least 1000 points, up to a maximum of 20 000points for those values of T for which decay s most rapid.

' 0.0 1.5 3.0 4.5 6.0 0.0 123 3.0 4.5 6.0t (x IO' am.) t (x 10' au.)

FIG. 5.Populationecay curves for the uniform ensemble (solid line), thequasiclassical ensemble (dotted line), and the Davis-Gray predictioh forshort-time decay rate (dashed line) in the 2D submap. The functionfit) =In{[N(n) --N(n,-)]/[N(O) -N(q)]). (a) T=9000. (b)T= 10000. (c) T= 11000. (d) TS 12000.

A trajectory is defined to have dissociated f r > 20 a.u.and the van der Waals bond energyH, ( y = r/2) is greaterthan zero. The lifetime is then the time (number of iterationstimes T) up to the last inner turning point before dissocia-tion, i.e., the point at which the final change of sign of p,occurs. Let N(0) be he number ofpoints in the ensemble,etN(n) be the number of points remaining undissociatedafter12terations, and let N( nJ) be the number of points remain-ing after a large number of iterations nf (corresponding tothe fixed time T, = lo7 a.u.). The ratio N(n/)/N(O) is ap-proximately the fraction of the initial ensemble n the invar-iant set; N( n/> is nonzero when the phasespaceconsistsof atypical mix of chaotic and quasiperiodic regions, n whichcase he invariant set is of nonzero measure.Decay curvesare obtained by plotting IdiN -N(q) J/[N(O) - N( nf) ] >, and are inear for simple exponentialde-cay.

the other two ensembles. he short-time decay rate for theWigner ensemble s consistently lower t han that for the uni-form ensemble.The GaussianWigner distribution is peakedat the origin Y = r,,,, pr = 0, so that the majority of phasepoints are concentrated close to the central quasiperiodicregion and associated ticky boundary.14r3 t longer times,decay n the uniform and Wigner ensemblesappearsbi- ormultiexponential, asnoted by Gaspard and Rice. (At verylong times systemswith mixed phase pacestructure typical-ly exhibit power law decay. )2. Phase space structure and unimoiecular decay

As discussed n Sec. I, the simplest version of the statis-tical theory of Davis and Gray for the unimolecular decayrate givesN(n) - N(n,) = (1 -p)[N(O) - N(nf)], (4.1)

wherep is the probability of escape er iteration of the map,and s the ratio of the areaof the turnstile lobe [A(L) ] to theareaof the complex region minus the areaof the invariant set[A (S) 1. The separatrix and associated obe or T = 8000 areshown in Fig. 3. Points (r,p,> on the stable and unstablemanifolds of the fixed point at infinity (Am) are obtainedby noting that, for r--t CO,hey satisfyi*39

Figure 5 shows decay curves for two of the three differ-ent ensembles f initial conditions (results for the Wignerensembleare similar to those for the constrained uniformensemble,and are not shown) for four T values. At shorttimes (up to about 150 iterations, ignoring initial tran-sients), the decay curves are linear and can be fit to yieldshort time decay rate constants. Our results show a pro-nouncedensemble ependence f the short-time decay rate.The quasiclassicalensembledecay s markedly slower thaneither the uniform or Wigner ensemblesor all T values,andis on a time scale characteristic of the longer time decay of

pr = f 2(m&-~-~)/2 (4.2)Points seededn this fashion are terated backwards (p, > 0)or forwards (p, ~0) in time to form the stable or unstablemanifolds, respectively.3gThe areasof the lobe and he com-plex region are found by numerical quadrature.The short-time decay curves corresponding to the Da-vis-Gray prediction are plotted in Fig. 5. The plots showthat the short-time decay rates for the constrained uniformensemble re predicted quite accurately by a statistical theo-ry with a single ntermolecular bottleneck. Numerically de-termined short-time decay ates are compared with those ofthe statistical theory in Table II. The overall level of agree-ment between he simplest Davis-Gray theory for the short-



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TABLE II. Predissociation in two-dimensional van der Waals map, Eq.(2.12).T P Pob PwC Pa/ Pm P;

8oQo 0.0946 0.0286 0.0255 0.0032 0.0384 0.14399ooo 0.0621 0.0435 0.0380 0.0168 0.0572 0.161910000 0.0445 0.0748 0.0653 0.0416 0.0681 0.180711cQO 0.0022 0.0657 0.0529 0.0485 0.0815 0.194312000 0.0183 0.1059 0.0912 0.0695 0.0989 0.2173

Fraction of complex region occupied by invariant set, estimated from thefraction of trajectories in the uniform ensemble undissociated after T,- 10 atomic units.bShort-ti me escape probability per iteration for constrained uniform ensem-ble with H,< - 6.0X 10w5.

Short-ti me escape probability per iteration for ground state harmonicWigner ensemble.dShort-ti me escape probability per iteration for ground state quasiclassi calensemble.Probability of escape per iteration calculated using Davis-Gray statisticaltheory with a single (intermolecular) bottleneck.Probability of escape per iteration calculated using alternative RRKMtheory for discrete maps.

time decay rate and the numerical results for the uniformand Wigner ensembless good. As noted above,decay ratesfor the quasiclassical ensembleare markedly lower thanthose for either the uniform or Wigner ensembles.For allvaluesof T, a significant fraction of points in the quasiclassi-cal ensemblecorresponding to the van der Waals stretchground state s located either in the central invariant regionor close o the boundary of the quasiperiodic region. Decayof the quasiclassicalensemble s therefore governedby theflux through intra- rather than intermolecular bottlenecks(seebelow).There s a noticeablediscrepancybetween he numericaland statistical values of the short-time decay rate for theuniform ensemble t T = :l 1000. The reason or this appar-ent anomaly s discussed n more detail below.Whereasgood agreementbetween he Davis-Gray sta-tistical theory and numerical simulation is obtained using adynamically determinedseparatrix and turnstile, it is also ofinterest to apply the analog of the alternative RRKM ap-proach of Gray, Rice, and Davis to dissociation n the 2Dmap, where transport through an approximate separatrix sused o define he rate. The zero-order separatrix s definedby the condition H, (r,p,) = Cl cf. Ref. 2 1 , and the result-ing phasecurve is shown in Fig. 6. To determine he trans-port through the approximate separatrix per teration of themap, the zero-energycurve is iterated once (Fig. 6). Sincethe zero-order separatrix s not an nvariant curve, ts iterateintersects he original curve at isolated points, producing aset of five lobescontaining points that cross n or out of theapproximate separatrix. The outward flux of phasepoints isthen given by the total area of the two lobes hat lie outsidethe zero-energy curve (as the map is area preserving, theoutgoing area s equal to the in-going area, eading to zeronet flux). Note that the outward flux through the approxi-mate separatrix s not localized n a single obe as t is for thedynamically determined separatrix. In the alternative

R.E. Gillilan and G. S. Ezra: Mappingsforunimolecular fragmentation 2657

' 4.0 7.0 10.0 13.0 16.0r (Bohr)

FIG. 6. Zero-order separatrix for the 2D van der Waals map (solid line),together with its iterate under the map (dashed line).

RRKM theory it is assumed hat all points passingout of theapproximate separatrix proceed to dissociate without re-crossing the i ntermolecular dividing surface, so that theprobability of escape er iteration isA,(L)PO=-, A IO\AOW)where A,(L) and A,(S) are analogous to the quantitiesA (L) and A (S), but are defined for the zero-order separa-trix. We have calculatedA,(L) and A,(S) for a number of Tvalues,and the resulting valuesof p. are shown in Table II.Agreement with the numerical results and the Davis-Graystatistical theory is reasonable,within a factor of 2-5. Theratios of the alternative RRKM and Davis-Gray statisticalescape robabilities as a function of Tare plotted in Fig. 7.We now return to the discrepancybetween he statisti-cal short-time rate and the numerical dissociation ate noted

10404

8000 9000 10000 11000 12000T (a.u.)

FIG. 7. Ratio of alternative RRKM rate to the Davis-Gray rate for both the2D map (circl es) and the full 4D map (tri angles) as.a function of T. Thedependence of the ratio on Tis similar in both cases, although the alterna-tive RRKM result is consistently higher for the 4D mapping.J.Chem.Phys.,Vol.94,No.4,15 February1991Downloaded 03 Jan 2001 to 128.253.229.100. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.


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above or the case T = 11 000. Consider the stability of thecentral fixed point of Eq. (2.12) , Y= rmi,, p, = 0. Diagonali-zation of the Jacobian of the mapping Eq. (2.12) at the cen-tral f ixed point yields a pair of eigenvalues hat determine the 0linear stability character of the tied point. For T< 15 000,the eigenvalues ie on the unit circle in the complex plane,indicating that the fixed point is stable. Very close to LOT = 11 000, the eigenvaluespass hrough f i, which is ust

ati

the condition for a 4:1 resonance.39As noted by Gaspardand Rice,39 he area of the invariant (quasiperiodic) region 9decreasesmarkedly at resonance.At T = 12 000, the system 7is no longer resonant, the area of the invariant region in-creases and the short-time rate decay rate increases. Al- 9C-Jthough there s a significant decrease n the area of the quasi-periodic region at resonance, leading to a largerdenominator in the rate expressionof Eq. (3.3), the statisti-cal rates do not show a dip at T = 11 000. The slow decayrate at T = 11 000 s most likely due to the influence of intra-molecular bottlenecks responsible for trapping of phasepoints in the vicinity of the 4:1 resonance; urther study ofthe effect of resonanceson the decay rate is in progress.6s

I 4.0 7.0 10.0 13.0 16.0r (Bohr)

FIG. 8. Inter- and intramolecular partial barriers actually used in the ratecalculation discussed in Sec. IV. The innermost ring is (approximately) theIast KAM torus bounding the quasiperiodic region. Only the two most im-portant catori are plotted: [6,2,1,1,1,.,. 1 and [7,1,1,1,... 1. Trapping in thetiny region between the last KAM torus and the cantorns [6,2,1,1,1,...]mainly determines the decay rate at long times.To describe he decay at longer times, it is necessary odevelop a kinetic model including intramolecular bottle-necks. Following Davis and Gray, we associate he intra-molecular bottlenecks with cantori 7*40corresponding toparticular irrational winding numbers (essentially the ratioof the van der Waals bond vibration frequency to that of theI2 bond). Recall that invariant curves of area-preservingmaps are smooth and continuous only up to a certain valueof the strength of the nonintegrable perturbation (which is sgeneral different for each nvariant curve), after which theybreak up, becoming invariant cantor sets (cantori) with aninfinite number of holes.40 Nevertheless, as found byMacKay, Meiss, and Percival,7 the cantori, although nolonger able to act as absolute barriers to transport, may con-tinue to restrict the flow of phasepoints and so act as partialbarriers. It is found empirically that cantori correspondingto certain special frequency ratios are the most effective par-tial barriers (that is, correspond to local minima of theflux) .7 These so-called nobZe requency ratios are irrationalnumbers that are the most poorly approximated by ration-als,7having continued fraction expansions

turnstile lobe (alternatively, the flux is given by the differ-ence n actions of the two orbits homoclinic to the associatedunstable periodic orbit with the given rational winding num-ber.7Z52Partial barriers defined in this way enclose regionsof phasespace,as shown in Fig. 8, and the statistical proba-bility of passing rom region i to regionj in one iteration ofthe map is thenA ( L,J >

PCi=A(S,)ywhere A (L, ) is the area of the lobe L, mediating transportfrom region i to regionj,50 and A (Si ) is the phase volume(excluding the invariant set) of region i. As shown by Da-vis and Davis and Gray, such ifitramolecular bottle-necks may be used to make accurate statistical models ofIVR and unimolecular decay.A model consisting of a reactive separatrix togetherwith several concentric partial barriers associatedwith peri-odic orbit approximants to cantori is used o model dissocia-tion in the 2D van der Waals map for T = 8000. The phasespace structures are shown in Fig. 8. We assume that themeasure of quasiperiodic phasespace ying outside the cen-tral region is negligible.

1aI +

1 = al,a,,a3,...,a,,l,1,1,...,j, (4.4)1a2 + ~a3 + -.*

terminating in an infinite sequenceof 1s. The cantorus cor-responding to a given irrational winding number may be ap-proximated by a sequenceof periodic orbits with rationalwinding numbers (p/q) obtained by truncation of the con-tinued fraction expansion at successively higher levels.7There are in general two periodic orbits associated with agiven rational winding number; one is hyperbolic (unsta-ble), while the other is either elliptic (stable) or hyperbolicwith reflection (unstable). 4o n practice it is useful to definethe partial barrier by taking the union of segments of thestable and unstable manifolds of the hyperbolic points; theflux across this barrier is given by the area of the associated


(4.5)

Let N, (n) be the number of phasepoints in region aftern iterations ofthe map, wherej = l,...,M labels he region. Inthe statistical (Markov) model for transport the populationsevolve under the difference equations:Nib2 + 1)

=N,Cn) +Pi-l,iNi-l(n) -(pg+1 +Pj.~-1)fVf(fZ)+ Pi+ I,i N, 1 (n>, 2>i>M- 1,

N,(n + 1) =N,(n) +p,lN,(n) -p,,2N[(n),Nw(n + 1) =N&f(n) +p,- ,,,N,- * (n)

(4.6)J. Chem. Phys., Vol. 94, No. 4,15 February 1991

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R. E. Gillilan and G. S. Ezra: Mappings for unImolecular fragmentation 2659

where pdisf s the statistical probability of dissociation periteration from the outermost region M. Numerical solutionof this set of discrete evolution equationsyields the popula-tion of bound complexes:Ne(n) = 2 iVi(n>, (4.7)i- 1

as a function of iteration number n. [We have verified thatintegration of the set of coupled differential equationscorre-sponding to the continuous time version of Eq. (4.6) givesan essentially dentical decay curve.]In the 2D He-II mapping with T = 8000we have denti-fied three cantori that are mportant in the decay kinetics. Acantorus with frequency ratio [ 7,1,1,1. ] z 3/23 governsde-cay at intermediate times, while those with frequencies[6,2,1,1,1,... ~55/32 and [6,1,1,1,... =:8/53 influence be-havior on longer time scales.Two of these bottlenecks aresuperimposed n Fig. 8. The decay curves shown in Fig. 9shows the excellent quality of the fit obtained or the result-ing four-region model for the constrained uniform ensemblewith T = 8000.In general the phase space of a nonintegrable systemcontains an infinite number of cantorLm As more and morecantori are ncluded in a kinetic model of the type describedabove, he underlying assumption of randomization withinregions betweenuncorrelated barrier crossingsbecomes essand ess useful. The model moreover gnores he influence ofpartial barriers leading to trapping around the infinite hier-archy of island chains. 3 Our results and those of oth-ers1*5,7show, however, that simple models involving asmall number of cantori can give a reasonably accurate de-scription of the decay kinetics for short to intermediatetimes. We note that in the context of unimolecular decayvery long time scales are of limited physical interest due toquantum effectsand the influence of external perturbations.

0.0 2.0 4.0 6.0Time (atomic units) (x 109

FIG. 9. Decay curve for the uniform ensemble (solid line) in the 2D van derWaals map at T= 8000 compared with the four-state Markov model(dashedine).

6. Four-dimensional mapping1. Fragmentation dynamics

To study the fragmentation dynamics of the 4D van derWaals map, we select initial conditions at random from aquasiclassical ensemble@j orresponding to the ground vi-brational state of both the van der Waals stretching andbending modes.A harmonic approximation is made or eachmode. Within the harmonic approximation, and using thephysical value of the barrier height B = 8.72X lo-, theground state energy s calculated to be - 20.65 cm- withrespect to infinitely separated He and I,, as compared to- 20.63 cm - for the theoretical ground state calculatedfor the full pairwise potential.46With B = 4 X 10 , the cal-culated ground state energy n the harmonic approximationis - 21.16 cm - . We also use an ensembleof initial condi-tions uniformly distributed in (r,p,.,y,p,, , subject o the con-straint H, (r,p,;y) < - E. We consider the potential withbarrier height B = 4x 10 5 a.u.As for the 2D case, rajectories are defined o have disso-ciated whenever > 20 a.u. and H, (r;p, ) > 0 [recall that thepotential V( r,y) becomes ndependentas yas Y--Py-) . Decaycurves for the quasiclassical and uniform ensemble areshown in Fig. 10 or four values of T. The curves display thecharacteristic biexponential behavior found above n the 2Dmap (also noted by Gaspard and Rice3 ). Short-time decayrates extracted from decay curves for a number of T valuesare given in Table III.

2. Multidimensional phase space structureIn order to apply the Davis-Gray theory to the short-time decay n the 4D map, it is first necessary o compute thestable and unstable manifolds W(d) of the invariantmanifold at infinity, .JYm. The manifolds Ws~U(utdm are ofcodimension one hi the 4D phase space, so that points onthem may in principle be ound by a one-dimensionalsearch.To visualize the manifolds W(dm ), which are 3D

FIG. 10. Population decay curves for the uniform ensemble (solid line), thequasiclassical ensemble (dotted line), and the Davis-Gray prediction forshort-time decay rate (dashed line) in the full 4D map. (a) T= 9COO. b)T= 10 000. (c) T= 11000. (d) T= 12 000.J. Chem. Phys., Vol. 94, No. 4,15 February 1991Downloaded 03 Jan 2001 to 128.253.229.100. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.


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2660 R. E. Gillilan and G. S. Ezra: Mappings for unimolecular fragmentationTABLE III. Predissociation in four-dimensional van der Waals map, Eq. (2.10).

T f P(W ,u(LY fd%* P(L)"= PoC PQCS POG Poi8 000 0.0758 29 452 970.2 34 460 4330.8 0.0245 0.0045 0.0310 0.1289Mx) 0.0465 26 691 1207.8 30 626 4381.6 0.0376 0.0179 0.0436 0.14510 000 0.0253 24 283 1419.8 27 567 4481.6 0.0538 0.0418 0.0574 0.16411000 0.0069 22 262 1599.6 25061 4534.4 0.0715 0.0483 0.0715 0.18112 000 0.0054 20 252 1749.3 22 973 4579.1 0.0984 0.0694 0.0860 0.200

.-Fraction of complex region occupied by quasi-invariant set, estimated from the fraction of trajectories in theuniform ensemble undissociated after T, = 10 atomic units.h Volume of complex region enclosed by dynamical defined separatrix minus volume of quasi-invariant set.Volume of lobe.Volume of complex region enclosed by zeroth-order separatrix.Phase volume passing out of zeroth-order separatrix per iteration of map.Short-time probability of escape per iteration for constrained uniform ensemble.*Short-time escape probability per iteration for ground state quasiclassical ensemble.h Probability of escape per iteration = A(L)/A(S), Davis-Gray theory.Probability of escape per iteration = A(L),/A(S),, alternative RRKM theory.

objects embedded n a 4D space,we examine a seriesof 2Dsections hrough the phasespace,defined by fixing the val-ues of the variables ( y,pu ). The problem is then to findpoints (#,pf) at fixed (/itpk) that lie on V(J>, i.e.,that satisfyHi- [R (dpt,Yi,p:)] -,O (4.8)

as 12-+CO stable) or II -) - OD unstable), where R n repre-sentsn-fold iteration of the initial point under the symplecticmap. Details of the algorithm used to find such points aregiven n the Appendix. Points on the homoclinic intersectionmanifold must satisfy the condition Eq. (4.8) for n -, CO ndn-t - ~0,~~ nd so must be found by a 2D search at fixed( y,pr 1 (cf. Appendix 1.A series of 2D (r,~, ) sections through the 4D phasespace or T = 8000 s shown n Fig. Il. Recall that the phase

FIG. 11. Series of 2D (r,p,) sections of the 3D manifolds W~(.,N~) em-bedded in 4D phase space for T = 8000 and B = 4~ 10 - 5. The numbers inparentheses give the (y,py) values of the particular sections in units of(7r/2,771/7). It can be seen that the manifolds Wc(,H*) enclose a 4Dcomplex region.

space s doubly periodic, of period v in y and period Apv= n-I/T in p,,. The intersection of the manifoldsW(.M ) with eachsection s in general one-dimensional.In the uncoupled case (LX 0, no barrier to internal rota-tion), Fig. 11 would consist of a set of 2D sections, eachidentical with Fig. 4. The 4D complex region and turnstilewould then be direct products of the corresponding objectsfor the 2D map with the two-dimensional ( y,p,,) space.The2D sections of Fig. 11 show the influence of couplingbetween he van der Waals stretch and bending modes (non-zero CL, = 4 X 10 ) on this zero-order picture. It is appar -ent that for these parameter values the structure of the 4Dcomplex region and associated turnstile formed byW(Jm) is topologically identical with the uncoupledcase.The 30 stable and unstable manifolds W(J) in-tersectso as o enclose he complex region precisely as n theuncoupled case, and each of the 2D sections in Fig. 11 isdeformable onto the 2D picture of Fig. 4. As discussed nSec. II, a straightforward generalization of the pair of dis-tinct primary homoclinic points of Fig. 4 to a multidimen-sional system is a pair of disjoint two-dimensional homo-clinic intersection manifolds. The segments of themanifolds wS(dLQ) and W(,k) shown in Fig. 11 nter-sect at two points in each of the 2D sections; each of thesepoints s asymptotic in forward and reverse ime to drn, andonepoint of every pair lies on eachof the distinct intersectionmanifolds. Projections of these homoclinic intersectionmanifolds onto the (pr,pr ) plane are shown in Fig. 12.Eachintersection manifold is continuously deformable onto theinvariant manifold LB (a 2 torus).26It is important to note hat the 2D slices shown n Fig. 11are not invariant planes, as both y and pv will generallychangeunder the mapping. Although a point starting insidethe separatrix on a particular (y,pI ) p lane must eventuallyenter an outgoing obe n order to dissociate, t will in generaldo so for valuesof (y,pu ) different from the initial values. talso follows that the incoming and outgoing lobes on a par-ticular (y,p, ) plane neednot have the same area; t is onlynecessary hat the areas ntegrated over (y,py ) be the same.

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03da

o!0

-+dI

gLd0 300 600 900 1200 1500p7

FIG. 12. projection of the two disjoint homoclinic intersection manifoldsonto the (p,,~?,) plane at T= 8000 and B = 4 x 10 - 5. Points of closest ap-proach of two projections correspond to (y,p,,) values where the turnstilelobes are smallest [cf. Fig. 11 at (i/5,3/5) 1.

3. Statistical theory for short-time decay rateIn Sec. V A, the Davis-Gray statistical theory for theshort-time decay ate n the 2D map was calculated as a ratioof phasespaceareas divided by the kicking per iod T) . Hav-ing established hat the multidimensional turnstile exists inthe4DvanderWaalsmapforB=4x10-5, T=8000-12 000 a.u., the natural generalizat ion of t he statisticalexpression o the 4D case s simply a ratio of the four-dimen-sional phase space volumes of the outgoing lobe and thecomplex region minus the invariant set

(4.9)

The required volumes are evaluated by multidimensionalnumerical quadrature. In principle, it should be possible ocompute lobe volumes by evaluating an action-like flux twoform on the homoclinic manifolds (cf. Ref. 7). To date, how-ever, we have not beenable to derive such a flux form. Thenumerical difficulties involved in implementing the multidi-mensional lux form formalismz8@ for continuous-time sys-tems are discussed n Ref. 29.The problem naturally breaks down into a two-dimen-sional integration over ( y,pr ) of two-dimensional areas nthe (r,p,) planes. For each pair (y,pr ), the points onW*U(Joo) obtained in the separatrix calculation are suffi-ciently close together that the simple trapezoid formulaA--C (PC,+ 1 + Pr,, ),1 2 -(Yn+) -rn) (4.10)

may be used to find the areas nside the separatrix and theturnstile lobe. A 25 x 21 grid on the (y,pr) plane coveringthe ranges [0,7r/2] x [O,lrr/T] is then used in a doubleSimpsons ntegration to obtain the 4D volumes.68The re-sulting volumes and rates are given in Table III, and thedecay curves predicted by the Davis-Gray theory are in-cluded in Fig. 10.

Table III shows hat the statistical rates basedon a sin-gle ntermolecular bottleneck are overall in very good agree-ment with short-time decay rates extracted from decaycurves for the constrained uniform ensemble.As in the 2Dmap, the decay ates or the quasiclassicalensemble re con-siderably smaller than either the uniform ensemble r statis-tical theory values. There is however no anomalous dip inthe rate at T = 11 000, and all rates ncreasemonotonicallywith T.The logarithm of the four-dimensional turnstile vol-umes for B = 4x 10e5 a.u. are plotted vs l/T in Fig. 13.The nearly linear plot shows that the turnstile volumes ap-proximately obey the samescaling relation as hat found byTersigni, Gaspard, and Rice in the caseof a 2D map

p(L) ccc -? (4.11)We havealso applied the discrete mapping version of thealternative RRKM theory to estimate the short-time decayrate in the 4D map, with the zero-order separatrix taken tobe the three-dimensional surface defined by the conditionH,(r,p,;y) = 0. Figure 14 gives a series of 2D (r,p,) sec-tions through the 4D phase space showing the zero-orderseparatrix. To calculate the alternative RRKM decay ate, tis necessary to compute the four-dimensional outgoingphasespace volume enclosedbetween he approximate se-

paratrix and its iterate under the van der Waals map, and totake the ratio of this volume to that of the complex regionminus the invariant set [cf. Eq. (4.3) 1. Sections hrough theiterated approximate separatrix are also shown n Fig.. 14.Asthe 2D sections are defined at fixed (y,pr), points on theiterated separatrix in a 2D section corresponding o a given

03cdLnCD8.0 9.0 10.0 11.0 12.0 13.0

l/T (x IO+)FIG. 13. Four-dimensional turnstile volumes (squares) compared with fitto scaling law of Fq. (4.11) (dashed line).



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FIG. 14. Series of 2D (p,p,) slices through the zero-order separatrix and itsiterate at T= So00 and b = 4x IO- 5. The numbers in parentheses give the(y,~~) values of the particular sections in units of (?r/2,~-1/7).

pair (y,p,,) have preimages on the zero-order separatrixwith (in general)d@rent (y,p,) coordinates.The details ofthe procedure used o find points on the iterated separatrixwith specified (y,p,, ) coordiri&tes are discussed n the Ap-pendix. The 4D volumes needed or the calculation of thealternative RRKM rate are computed using the same nu-merical quadrature method discussedabove n connectionwith the Davis-Gray statistical rate.Alternative RRKM escape robabilities for the 4D vander Waals map are shown in Table III. The level of agree-ment with the numerical results for the uniform ensemble scomparable o that for the 2D map; that is, within a factor of2-5. Ratios of the alternative RRKM escape robabilities tothe Davis-Gray statistical values or the 4D map are shownin Fig. 7, and are seen o be very similar to those or the 2Dcase.To obtain a more accurate description of the decay atlonger times, multistate statistical models ncorporating in-tramolecular bottlenecks in the 4D phasespacemust be de-veloped. Trajectory studies suggest hat bottlenecks asso-ciated with pairwise irrational frequency ratios areimportant in three-mode systems,23-25o that it would beuseful to be able to define ntramolecular bottlenecks asso-ciated with particular pairwise frequency ratios to provide arigorous test of this idea. A natural generalization o the 4Dmap of the (unstable) periodic orbit approximant to a can-tors in the 2D map would be a 2D NHIM A,,, associatedwith a pairwise resonancecondition Eq. (3.6). The stableand unstablemanifolds of J,,, would be three-dimensional ,and segments hereof could be used o define a 3D dividingsurface n 4D phasespace epresenting he multidimensionalgeneralizationof the partial barrier associatedwith a period-ic orbit approximant to a cantorus. The major technical ob-staclecurrently preventing mplementation of this scheme sthe lack of an algorithm for computation of arbitraryNHIMs.

13.E. Gillilan and G. S. Ezra: Mappings for unimolecular fragmentation0hi

9i

2662

9N

LOQ6

9i

5.0 7.0 9.0 11.0 13.0 15.0r

5.0 7.0 9.0 11.0 13.0 15.0r

FIG. 15. Passage through homoclinic tangency. At T= 8000,B= 2.5x lo-* (a) andB= 4.0~ 10W5 (b), themanifolds Ws(&K) in-tersect in such a way that 2D (r,pr) sections at (y = 1.309,~~ = 556.0) canbe deformed onto the 2D phase portrait of Fig: 4. At B = 5.5X 10m5 (c),however, t he stable and unstable manifolds no longer intersect at the lowerleft-hand corner of the (r,p,) section. This phenomenon precludes the con-struction of space-enclosing turnstiles.

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4. Nonexistence of turnstilesAll results up to this point have been presented or the4D van der Waals map corresponding to a potential withbarrier height B = 4.0X 10 , approximately one half thephysical value.46 To show what happens when the barrierheight (and hence the strength of the stretch-bend cou-pling) increases, we display in Fig. 15 a sequenceof 2D

(r,p,> sections at the same (y,p,) point for increasing Bvalues: 2.5X10m5 [Fig. 15(a)]; 4.0X lo- [Fig. 15(b)];5.5~10~~ [Fig. 15(c)].ForB= 5.5X10-,thestableandunstable manifolds of -&, no longer intersect in the samemanner as the manifolds for smaller B values. On the otherhand, sections at different (y,pr) points (not shown here)for B = 5.5X lo- are of the same orm as hose or smallerB values. The fact that the 2D section of Fig. 15 c) isqualitatively different from that of Fig. 15 a), whereas2Dsections taken at different values of (y,pr ) for the same Bvalue (5.5 X lo- ) are similar to Fig. 15(a), means hat forthis value of B 40 volume enclosing turnstiles no longer exist.We refer to the transition shown n Figs. 15 a)-1 5 (c) asa passagehrough homoclinic tangency, by analogy with thecorresponding phenomenon n area-preservingmaps. Pri-or to the appearanceof a homoclinic tangency in the 4Dphasespace, here exist two disjoint 2D primary homoclinicintersection manifolds, eachsmoothly deformableonto 4 m(as discussed above, cf. Fig. 12). At the B value at whichhomoclinic tangency irst occurs, the two intersection mani-folds have a single common point. That is, there is a single(y,pr ) point at which the areaof one of the turnstile lobes nthe (r,pr) section shrinks to zero. For larger B values, the

two previously disjoint intersection manifolds merge. Thisfundamental change n topology was predicted by Wiggins2band observedby us in a study of transport in coupled stan-dard maps.65After the appearance f homoclinic tangency,the homoclinic manifold is no longer homotopic to theNHIM V&.26Although a full mathematical analysis of suchchanges n topology of the homoclinic manifold has yet to begiven, it is apparent that the onset of homoclinic tangencyprecludes he complete enclosureof a 4D complex region byW(&) and WU(y&), and prevents a straightforwardgeneralization of turnstile-based transport theories. Figure16 llustrates the essentialdifficulty preventing enclosureofa 4D complex region when homoclinic tangencyoccurs. It ispossible to find continuous paths in 4D phasespace (note:not orbits under the map) connecting points nominally out-side (lower left of Fig. 16) the complex region to pointsinside (top left of Fig. 16), which cross neither the stablenor the unstable manifold of ,ti m.The fact that all lobes aregeneratedby forward or backward iteration of the primarylobes mplies that any lobe may be reached rom any otherwithout crossing the separatrix branches.As shown by Davis in his study of the collinear H + H,reaction, several unstable fixed points may be nvolved inthe definition of a complex region, dependingon energy.Ourresults on the 4D vau der Waals map do not rule out theexistenceof turnstiles associatedwith someother nontrivialNHIM, or the possibility that such turnstiles might be im-portant in unimolecular dissociation. Investigation of thispossibility will, however, have to await the development of

5.0 6.0 7.0 8.0 9.0 5.0 6.0 7.0 8.0 9.0r rFIG. 16. For the barrier height B = 5.5X lo-, there exist continuouspaths in 4D phase space connecting lobes nominally inside and outsidethe complex region, which cross neither the stable nor the unstable manifoldof&? *. Such a path is represented schematically by the arrows. The arrowsconnecting (a) and (b) correspond to varying y at fixed pr = 557. (a)(r,p,) section at (y= 0.72,~~ = 557). (b) (r,p,) sectionat (y= 1.18,pr= 557).

more general algorithms for computing NHIMs.Figure 17 shows the location of the boundaries n pa-rameter spacebetween he region in which 4D turnstiles arewell defined and that in which they are not. The trends inparameter values at which tangency first occurs may bereadily understood. Qualitatively, factors which tend to de-crease he size of turnstile lobes n the (r,p,) plane sectionsalso make it easier for homoclinic tangency to occur. Forexample, as he massm of the adatom s increased rom thatof helium, the van der Waals stretch and bend frequenciesdecrease, esulting in weaker coupling (better adiabatic sep-aration) to the I, kick (smaller lobes), so that homoclinictangency occurs at smaller B values [Fig. 17 a) 1. Decreas-

2$lo04dYm+

$

:: J3.0 3.5 4.0 4.5 5.0 0.0 1.0 2.0 3.0 4.0 5.0m I

z 32 0d;;8-s? z0 -!25*m moi +N 44 26000 7000 6000 9000 -0.53 0.80 0.67 0.74 0.61

T B

FIG. 17. Boundaries of regions in parameter space corresponding to theonset of homoclinic tangency. Below the lines, space-enclosing turnstilescan be constructed. Qualitatively, factors which tend to increase turnstileamplitude also favor space enclosure at higher B values.J. Chem. Phys., Vol. 94, No. 4,15 February 1991Downloaded 03 Jan 2001 to 128.253.229.100. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html.


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ing the kicking period T also decreases he size of turnstilelobes [Fig. 17 c) 1, as does decreasing he Morse parameter,L?smaller van der Waals stretch frequency, Fig. 17 d), andthe rotor moment of inertia I [smaller rotational frequency,Fig. 17(b)].Although the onset of homoclinic tangency prevents m-plementation of the Davis-Gray statistical theory based ondynamically determined separatrices and turnstiles, the al-ternative RRKM theory may still be applied. Table IV com-pares alternative RRKM short-time decay rates with nu-merical rates calculated for T = 8000 and a seriesof B valuesspanning a range in which passage hrough homoclinic tan-gency occurs. It can be seen that there is no significantchange n the level of agreement between the numerical re-sults and the alternative RRKM results over the full range ofB. Interestingly, examination of (r,p, ) sections (notshown) through the zero-order separatrix and its iterate fora B value at which homoclinic tangency occurs shows thatthe number of ingoing and outgoing lobes changes as the(y,py ) coordinates vary, so that the change in topology ofthe intersection set discussedabove s reflected in the behav-ior of the zero-order separatrix under iteration of the map.V. DISCUSSION AND CONCLUSIONS

Hamiltonian or symplectic mappings provide a compu-tationally efficient means of studying the phasespacestruc-ture of multidimensional systems. For example, a four-di-mensional symplectic map is a very convenient substitute forthe PoincarC eturn map in the four-dimensional surface ofsection obtained by integrating trajectories in a three-modecontinuous time molecular Hamiltonian. In the presentpaper, we have studied a four-dimensional Hamiltonian mapmodeling vibrational predissociation of the van der Waalsmolecule He& Our primary aim has been o generalize heDavis-Gray statistical theory of unimolecular dissocia-tion to systemswith more than two degreesof freedomWe began by pointing out the close relation betweensymplectic maps and symplectic trajectory integrators.* Asthe kick period T tends to zero, a symplectic map generatesapproximate trajectories for a continuous time systemwith anumber of degreeof freedom equal to half the dimension ofthe map phasespace (the relation between he discrete mapdynamics and that of the corresponding continuous time

TABLE IV. Alternative RRKM and numerical short-time decay rates forthe uniform ensemble in the 4D van der Waals map Eq. (2.10). Z= 8000.B PO P

4.0x10-5 0.1279 0.02455,0x10-* 0.1229 0.02666.0x10- 0.1223 0.02897.0x 10-s 0.1232 0.03098.72x10- 0.1297 0.0332Height of barrier to internal rotation.Short-time escape probability per iteration. Alternative RRKM theory.Short-time escape probability per iteration. Numerical.

2664 R. E. Gillilan and G. S. Ezra: Mappings for unimolecular fragmentationsystem will be discussed n more detail elsewhere ). A 4Dvan der Waals map was then derived starting from the fullHamiltonian for planar He-I,. The classical centrifugal sud-den approximation5 was nvoked, and the continuous oscil-lation of the I, bond replaced by a S-function kick actingupon the van der Waals stretch and bend degreesof freedom.After a review of the phasespace heory of unimoleculardecay in two degree of freedom systems due to Davis andGray, the theory of transport in multidimensional systemsrecently developedby WigginsZbwas outlined. Wiggins the-ory provides a rigorous extension of turnstile-based ap-proaches o phasespace ransport7*5051o multimode prob-lems, and enables the Davis-Gray statistical theory to begeneralized to multidimensional systems. An essential ele-ment of Wiggins approach is the observation that the multi-dimensional analog of an unstable periodic orbit in a twodegree of freedom system is a so-called codimension-twonormally hyperbolic invariant manifold,27~58 enoted J?.These manifolds have appearedpreviously in various guisesin the chemical physics literature: families of quasiperiodicorbits embedded in the continuum;5960 center mani-folds (cf. also Refs. 28, 29, 62, and 64); reduced dimen-sionality torib The stable and unstable manifolds of an in-variant manifold J are of dimension one less than theenergy shell, and so have the correct dimensional ity to act ascomponents of the boundary of a phase space egion.In preparation for our study of the full 4D map, we firstexamined the dissociation dynamics in a 2D area-preservingmapping acting upon the invariant subspacecorrespondingto the T-shape van der Waals complex. Significant ensembledependenceof unimolecular decay curves was noted, withthe quasiclassical ensemble or the ground state of the vander Waals stretch mode decaying much more slowly than anensemble of initial conditions uniformly distributedthroughout the complex region. Following the methods ofRefs. 11 and 15, kinetic schemeswere developed for vibra-tional predissociation on short to intermediate time scales,which incorporated both intermolecular (reactive separa-trix) and intramolecular (periodic orbit approximants tocantor-i) bottlenecks. The Davis-Gray statistical predictionfor the short-time decay rate using a single intermolecularbottleneck gave on the whole good agreement with exactnumerical results for the uniform ensemble. An anomalywas however apparent at T = 11 000, and was ascribed o a4: 1 resonanceat the central fixed point. A four-state kineticscheme ncorporating three intramolecular bottlenecks gaveexcellent agreement with the exact numerical decay curvefor short to intermediate times. In addition to applying theDavis-Gray theory basedon dynamically determined separ-atrices and turnstiles, the discrete map analog of the alterna-tive RRKM approach of Gray, Rice, and Davis was alsoused to predict short-time decay rates for the 2D map. Theresulting estimates or the decay constant were a factor of 2-5 larger than the Davis-Gray results, with better agreementobtained at larger T values.We next studied the 4D van der Waals map. As for the2D case, numerical simulation results showed marked dif-ferencesbetween he decay behavior of the uniform and thequasiclassical ensembles.For a 4D map with a dissociative

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coordinate, such as hat investigated here, the relevant codi-mension- two invariant mani fold at infinity, ,drn, may beidentified by inspection. Moreover, points on the three-di-mensional stable and unstable manifolds W+ (-4 m may befound relatively easily, as shown in Sec. IV. For physicallyreasonable parameter values, the manifolds W(JY~)were found to intersect in the 4D phase space n such a wayas to enclose a well-defined four-dimensional complex re-gion. In this case he multidimensional analogs of turnstilesexist and mediate transport in and out of the complex region,Le., complex formation and unimolecular decay.The Davis-Gray estimate of the escapeprobability periteration of the 4D van der Waals map

richard e. gillilan and gregory s. ezra- transport and turnstiles in multidimensional hamiltonian...

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