g. ramachandran and g. s. ezra- orbiting complex formation in na^+/n2 collisions: a phase space view

8/3/2019 G. Ramachandran and G. S. Ezra- Orbiting Complex Formation in Na^+/N2 Collisions: A Phase Space View

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J. Phys. Chem. 1995 ,99, 2435-2443 2435

ARTICLESOrbiting Complex Formation in Na+/N2 Collisions: A Phase Space View

G. Ramachandran? and G. S. Ezra*Baker Laboratory, Department of Chemistry, Come11 University, Ithaca, New York 14853Received: July 25, 1994; In Final Form: October 20, 1994@

In this paper w e investigate the dynamics of orbiting com plex formation in Na+/N2 collisions using classicaltrajectories. W e first demonstrate that the rigid-rotor approximation is appropriate for computing com plexformation probabilities and lifetimes, as the NZ ibration is weakly co upled to the other degrees of freedom.For a collision ensemble with zero impact parameter an d total angular momen tum J = 0, complex formationprobabilities exhibit oscillatory behavior with E . Several different representations of th e phase space structureare compared, and it is found that the ( y p , , ) urface of section provides most insight into the complex formationdynamics. Oscillatory co mplex formation probabilities and the influence of potential anisotropy on com plexformation rates are analyzed in terms of the ( y , p v ) section.

I. IntroductionComplex formation in ion-molecule collisions is an impor-tant precursor to both reactive and nonreactive processes.Experimentally determined rates of ion-molecule reactions areoften analyzed in terms of a kinetic scheme in which the firststep is the formation of an orbiting complex, which can eitherdecay back to reactants or lead to products.1.2 The rate offormation of orbiting complexes (association) is typicallyestimated using Langevin-type capture the~ries.~-~everaltrajectory studies have shown the inadeq uacy of capture theoriesfor computing the association rate.617 In a study of Kr/02+collisions, for example, we found that the complex formationrate, as opposed to the capture rate (rate of passage over thecentrifugal barrier), depends strongly on the co llisional energy,although this dependence lessens at very low collision energies(see Figure 9 of ref 7 ) . The dynamics of complex formationand dissociation in ion-molecule systems, although of consid-erable fundamental importance, is at present quite poorlyunderstood, despite much theoretical work on collisions insystems with long-range attractive forces6 One key questionconcems the influence of lon g-range versus short-range featuresof the potential surface (e.g., anisotropy) on orbiting complexformation and decay rates.In this paper we study complex formation in collisions ofNa+ with Nz, a system studied by previous wo rkers.s-10 T hereare no strong chemical interactions in this system, and a simple

functional form for the potential is employed. The shape ofthe simple model potential is easily manipulated to study theeffect of various aspects of the potential on the complexformation rate and probability.Low-energy Na+/N2 collisions have been studied previouslyby Schelling and Castleman (SC).8 SC used a planar iodrig id-rotor model with the interaction potential between the ion andthe diatom consisting of an attractive anisotropic iodinduced-dipole term (l/r) and an isotropic repulsive l / r l * erm. Theobject of their study was to determine the effect of the anisotropy

Present address: Courant Institute of Mathematical Science s, 25 1Mercer Street, Ne w York, NY 10012.@ Abstract published in Advance ACS Abstracts, February 1, 1995.

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of the N2 polarizability on the complex formation lifetimes; anincrease in complex lifetime with increase in anisotropy wasreported.8Brass and Schlier (BS)g estimated complex lifetimes for Naf /N2 by sca ling trajectory results previous ly obtained for H+/H2.Their estimated lifetimes were -25 times longer than thosecalculated by SC. Moreover, by computing complex formationcross sections and lifetimes as a function of asymmetryparameter in a simple triple-Morse potential for H +/H*, BSconcluded that the rigid-rotor approximation used by SC wasinadequate and that intermediate storage of translation energyin the vibrational degree of freedom was essential for complexformation, so that the vibrational degree of freedom could not

be omitted from consideration. We discuss this conclusion inmore detail in section 11, where we compare the results ofcomputations of complex formation with and without N2vibration.Rotational and vibrational excitation in high-energy (> 100eV) Na+/N2 collisions has been studied both experimentally andtheoretically by T unama et a1.I0Long-lived scattering trajectories have been w idely discussedin the context of chaotic scattering.* In particular, classicalcollision complex lifetimes have been expressed in terms ofthe properties of the underlying chaotic re~e1ler.l~ he phasespace analysis of complex formation and unimolecular decayhas proved to be very powerful for understanding deviationsfrom statistical theories.I4One of the aims of the present paper is to present a phasespace view of orbiting complex formation in the case thatrotational degrees of freedom are important. We model Na+/N2 as a two degree-of-freedom system (rigid-rotor approxima-tion, total angular momentum J = 0), so that the phase spacestructure can in principle be examined using a 2-dimensionalPoincar6 section.I5 We point out, however, that the presenceof a rotational degree of freedom leads to difficulties with the( r ,pr ) ype of section used previously in the study of complexformation and dissociation in T-shaped van der Waals com-plexes.I6 An alternative representation, the (y,pr) section, istherefore introduced. This representation is found to provideuseful insight into the dynamics of complex formation. A keyrole in the phase space an alysis of com plex formation and decay

0 1995 American Chemical Society


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2436 J . Phys. Chem., Vol. 99, No. 9, 1995 Ramachandran and Ezrain van der Waals m olecules is played by the periodic orbit at r= m . Segments of the stable and unstable manifolds of thisperiodic orbit define the reactive sepratrix bounding the complexregion.I4.l6 We show that an analogous role in the case oforbiting complex formation and decay is played by the orbit-ing periodic orbit, which is located at finite r atop a centrifugalbarrier.In section I1 we present the Hamiltonian and the modelpotential surface for Naf/N2. The trajectory ensembles are alsodescribed. In section I11 we discuss the validity of the rigid-rotor approximation for the purpose of studying complexformation rates and lifetimes in Na+/N2 collisions. In sectionIV we present the results of our calculations of complex lifetimesand formation probabilities. Section V describes several phasespace representations we have used to analyze the trajectoryresults. Section VI presents a summary and conclusions.11. System and Methodsdiatom (Na+/N2) collisions in the plane, with HamiltonianA. Hamiltonian. We treat the classical dynamics of atom-

where s is the NZbond length, r is the radial distance betweenthe Na+ atom and the center of mass of the Nz diatom, and yis the angle between s and r. p s , p r , and p y are the conjugatemomenta, and J is the total angular momentum for rotation inthe plane.For those calculations in which we freeze the NZbond lengthat its equilibrium value seq, the Hamiltonian used is simply

where I = use? is the moment of inertia of the rigid rotor.B. Potential for Na+/Na?. We used a model potentialconsisting of an attractive anisotropic iodindu ced-dipo le inter-action and a short-range exponential repulsion term centeredon each of the nitrogen atoms. This potential is easily modifiedto increase or decrease the anisotropy of either the attractive orrepulsive terms. The general form of V ( r , y , s) is

where a(y) s an anisotropic polarizability and 11 and rz aredistances of the Na+ ion from the two N atoms. The third termin this expression is a M orse oscillator potential describing theN2 vibration; values for the dissociation energy De, the equi-librium bond distance seq.and the range factor a are taken fromref 17. The anisotropic polarizability a(y) s

a(y) 1/2{(c4, + ai>+ c4 ,- a,) COS(2Y)I (4)where ql and aL re the parallel and perpendicular polarizabili-ties of the Nz m olecule, respectively (values taken from ref 18).The constants C and , characterizing the repulsive part of theNa+/N2 interaction are taken from Tanuma et a1.I0 The valuesof the potential parameters are shown in Table 1, and contoursof the potential surface are shown in Figure 1.C. Initial Conditions. We h ave computed classical complexformation probabilities and lifetimes for ensembles of initial

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Figure 1. Contours for the Na+/N2 potential. The NZbond lies alongth e x axis, with the midpoint of the N-N bond at the origin. TheN-N bond length is fixed as s = seq = 2.075 au. Contours are atmultiples of 0.027 21 eV.TABLE 1: Parameters for Na+/N2 Potential (atomic units)

2.08 clll 14.851 a 1.423seq 2.075 ai 10.179 C 55.124De 0.3642P

conditions with fixed total energy E and total angular momentumJ = 0. The ensemble consists of trajectories undergoing head-on collisions with p y ( 0 ) = 0 and impact parameter b = 0.Trajectories are started with r (0 )= 70 au. The only remainingvariable to be specified is the initial rotor ph ase y(O), which isuniformly distributed between 0 and n. ach ensemble consistsof 3000 trajectories. Fixing the impact parameter b defines a1 dimensional ensemb le of initial co nditions; this choice ofensemble facilitates a detailed analysis of the classical dynamics.For our stud y of the validity of the rigid-rotor approximationin section III we use a slightly different ensemble with J f 0.This ensemble is a fixed translational energy (Et ) nsemble withp y ( 0 )= 8h, and the impact parameter b varied between 0 andb,,,, where bmax,he maximal impact parameter beyond whichcomplex formation does not occur, is determined numerically.Each trajectory was started with r(0) = 70.0 au, and theensemble consisted of 3000 trajectories as in the previous case.Vibrational initial conditions for the rotating N Z diatom w eresampled using a procedure described in ref 7.As in previo us work:, we consider a com plex to be formedwhen there are two or more inner tuming points in the radialcoordinate r . The complex lifetime is then defined to be thetime between the first and last radial inner tuming points.111. The Rigid-Rotor Approximation

Schelling and Castleman (SC)8 studied the influence ofpolarizability anisotropy on the lifetimes of com plexes formedin collisions of Na + with N2 and CO2 using classical trajectoriesand found that increasing the aniso tropy of the long-range partof the potential increases the average trajectory collision lifetime.SC treated iodrigid -rotor co llisions in the plane. Brass andSchlier (BS)9 criticized SCs use of a rigid-rotor approximationon the grounds that their own investigations revealed thatintermediate energy storage in the vibrational degree of freedomwas an essential factor determining the complex lifetime. BSbased their arguments on a classical trajectory study using atriple-Morse H+/H2 potential, in which the Morse pairpotentials used to model the H-H6+ interaction were modifiedby changing the equilibrium distance. BS also found thatcollision complex lifetimes depended on the frequency of thefastest normal mode of the system in accordance with RRKh4theory, suggesting complete intemal energy ex~hange.~n


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Orbiting Complex Formation in Na+/N2 Collisions J. Phys. Chem., Vol. 99, No. 9, 1995 24310'40u.32

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Figure 2. Survival fraction as a function of the lifetime comp uted fortrajectory ensembles with and without Nz vibration included. Bothcalculations are carried out at a collisional energy of 0.0329 eV. Solidline, vibration included; dotted line, vibration frozen.contrast, SC observed a nonexponentd decay of the survivalfraction as a function of the lifetime.8 (The survival fraction attime t is defined as the fraction of the trajectory ensemble withlifetimes greater than t . )

In Figure 2 we show the survival fraction as a function ofthe lifetime for trajectory ensembles with and without the N2vibration, for the potential of eq 3, E = 0.0329 eV. It is seenthat inclusion of the vibration makes little difference to the decayrate of the complexes and hence to the lifetime. We haveperformed calculations at several collisional energies (Et) andhave seen the same quantitative agreement between decaybehavior with and without the vibrational degree of freedom.The N2 vibrational degree of freedom is thus found to beineffectively coupled to the translational and rotational degreesof freedom in the Na+/N2 system, as might be anticipated onthe basis of sim ple time scale arguments; Le., the period of theN2 vibration is m uch shorter than either the rotational periodor the collision time at the collisional energies and rotationalenergies studied. Schelling and Castlem an's use of a rigid-rotor approximation for the NZdiatom is therefore justified forthe purpose of determining the lifetimes of complexes formedin our model system. The scaling arguments used by BS toestimate complex lifetimes in Naf/N2 ap pear to be inapplicablein this case, due to the high degree of sym metry still present inthe modified H+/H2 potential used in their trajectory calcula-t i o n ~ . ~his sym metry, inherited from the D3h potential,results in extensive energy transfer due to resonance betweencomplex vibrational degrees of f r e e d ~ m . ~

Our previous work' on the dynamics of Kr/02+ collisionsshowed that, although intermediate energy storage in thevibrational degree of freedom does lead to a slight increase incomplex lifetimes, it is by no means essential for orbitingcomplex formation. In agreement with SC, we find that forNa+/N 2 collisio ns translat ional-ro tationa l (T-R) energy trans-fer is the critical factor in complex formation and decay.Inclusion of the N2 vibration is therefore not necessary foraccurate computation of com plex formation rates and lifetimesin classical Na+/N2 collisions, so we shall henceforth use therigid-rotor approximation.IV. Complex Formation Rates and Lifetimes

A. Complex Formation Probabilities. In Figure 3 we plotthe fraction of complexes formed in the collision ensembledescribed in section 11 as a function of the total energy of thesystem, E . Figure 3 shows that the complex formation

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E k V )Figure 3. Fraction of complexes formed as a function of the totalenergy E. The trajectory ensemble used is the J = 0 ensemble describedin section 11.

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Figure 4. Lifetimes as a function of the total energy E. The trajectoryensemble used is the J = 0 ensemble described in section 11.

probability for this ensemble exhibits an interesting nonmono-tonic dependence on E. Above E = 0.12 eV the complexformation probability smoothly decreases with E . At totalenergies below E = 0.12 eV, however, the complex formationfraction shows a number of oscillations. This behavior isinterpreted in terms of phase space structure in the next section.(For ensembles in which a range of impact parameters issampled, the o scillations wash out [see section V.C.21.)B. Complex Lifetimes. In Figure 4 we plot the lifetimesfor a given collision ensemble (as taken from the best straightline fit to the logarithm of the decay curve of the survivalfraction) as a function of the energy E. At low energy the fitto an exponential decay is often poor. In particular, at 0.03 eVa curious stepwise decay is seen. The decay curve for E =

0.03 eV is plotted in Figure 5 .V. Phase Space Representations of Orbiting ComplexFormation

We now turn to a phase space view of the dynamics oforbiting complex formation. The planar atom rigid-rotor modelfor Na+/N2 collisions is a two degree of freedom system, andmany methods for studying the phase space structure of suchsystems are a~ ai1 able . l~A . Surfaces of Section, Complex Formation, and theReactive Separatrix. The PoincarC surface of section,I5 forexample, has proved valuable in understanding the origin ofnonstatistical behavior in two-mode models of unimoleculardecay of van der W aals complexes such as He/I2.I6 The su rface


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Figure 5. Decay curve (fraction of trajectories surviving for time tversus t ) for E = 0.03 eV . The ensemble used is the J = 0 ensembledescribed in section 11of section construction provides the means to define andvisualize phase spa ce dividing surfaces (bottlenecks) that formthe basis for modified statistical theories for the unimoleculardecay rate.I6 Extension of these methods to higher dimensionalsystems is possible,21 although d i f f i c ~ l t . * ~ -~ ~

The Poincark return map M is obtained by choosing a two-dimensional surface in phase space, 2, ransverse to trajectoriesin the three-dimensional energy shell. Successive iterations ofthe mapping M correspond to successive intersections of atrajectory with the surface 2, Thus, the image of the phasepoint zo E C under the mapping M is the next point on thetrajectory starting at zo that pierces 2. The mapping M issymplectic (area-preserving for two degrees of f re e d ~ m ) . ~uchmappings have been used directly to mode l unimolecular decaydynamics.22.26In the Hen2 system ,I6 for examp le, a sectioning surface wasdefined by setting the I2 bond length equal to its equilibriumvalue and choosing a particular sign for the velocity. The radialcoordinate and momentum of the He atom were then plottedevery time the 12 diatom expanded through its equilibriumposition. The vibration of the 12 bond provided a natural clockfor the dynam ics; that is, trajectory return times (the time takenbetween successive intersections with the surface of section)were almost constant at the energies studied. For such systems,there is a straightforward connection between areas on thesurface of section and volumes in the full phase space thatfacilitates analysis of the kinetics of unimolecular decay.I6Using the surface of section for H e k t is possible to definethe r eact ive separa t r i~ . ~ . ~his surface consists of the unionof segments of the stable and unstable branches of the fixedpoint on the Poin cart section corresponding to the periodic orbitin which the 12 molecule vibrates freely with He stationary atinfinity. The separatrix is a surface which, in the absence ofcoupling between the 12 vibration and the I2-He degrees offreedom, forms an invariant boundary between bound (complex)and unbound trajectories. In the presence of coupling, theseparatrix is no longer invariant (Le., it is b r~k en ~ )nd servesto define the boundary that trajectories must cross to passbetween the bound and unbound regions, or vice versa. Thereactive separatrix has two important properties: all trajectoriesthat cross it from the inside dissociate immediately, and no directscattering trajectories ever cross it. Com plex forming trajec-tories are therefore precisely those trajectories whose intersec-tions with the PoincarB section reside within the region boundedby the separatrix for some time.A key observation is that phase space transport across thereactive separatrix is mediated by regions called tumstiles,28

Ramachandran and Ezra30.0

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Figure 6. The (r ,pr)surface of section. The total energy is 0.25 eV.The dark line is the reactive separatrix.which are areas (lobes) enclosed by segments of the stableand unstable manifold^.*^.^^ The significance of the turnstilesis that all trajectories that cross the separatrix do so by passingthrough the tumstile region. If trajectory return times areapproximately con stant, as is the case for the 12-He systemdiscussed above, the area of the outgoing tumstile lobe is ameasure of thejlux of phase points out of the complexed region.In general, however, trajectory return time s will not be constant,and the relation between decay kinetics and the structure of thesu rfac e of se ction is m ore c o m p l i ~ a t e d . ~ ~ ~ ~ ~n addition to thereactive separatrix just discussed, intramolecular bottlenecksassociated with remnants of invariant tori (called cantori28) avealso been invoked to account for nonstatistical behavior.I6In the present section we ap ply the phase spa ce concepts justdescribed to elucidate orbiting complex formation in Na+/N2collisions. We shall see that this system exhibits complicationsnot seen in the H en2 case.B. The ( r , p r )Surface of Section. In Figure 6 we show thereactive separatrix on the ( r ,p r ) surface of section for Na+/N2at E = 0.25 eV, where the sectioning conditions are y = 0,p v> 0. The reactive separatrix divides direct trajectories fromcomplexed ones.A key role in the phase space analysis of complex formationand decay in T-shaped van der Waals molecules is played bythe periodic orbit at r = w . Segme nts of the stab le and unstablemanifolds of this periodic orbit (fixed point at p r = 0, r = w )define the reactive separatrix bounding the complex r e g i ~ n . ~ . ~An analogous role in the case of orbiting complex formationand decay is played by the orbiting periodic orbit. Theorbiting periodic orbit has r = 0 and sits atop a centrifugalbarrier at a finite value of r = 23.91 au. (Note that, althoughthe total angular momentum J = 0, the diatom and orbitalangular momenta can be nonzero but of opposite sign. Also,due to sym metry, the intersection of the orbiting periodic orbitwith the ( r , p r ) ection occurs at p r = 0). The location of thefixed point on the ( r , pr ) urface of section corresponding to theorbiting trajectory is obtained numerically by a 1D earch alongthe line p r= 0. The reac tive separatrix of F igure 6 is composedof segme nts of the stable and unstable manifolds of the orbitingperiodic orbit.It can be se en that the separatrix has several unusually longprotrusions or tendrils. These tendrils are associated withtrajectories where most of the energy has gone into thetranslational degree of freedom, leaving the diatom to rotateslowly, so that the radial distance between successive pointson a trajectory for which the sectioning condition ( y = 0,p v =-0) is satisfied become s very large. At the tips of the tendrils,the N2 rotor takes longer and longer to return to the surface of


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Orbiting Complex Formation in Na+/N* Collisions J. Phys. Chem., Vol. 99 , No. 9, 1995 24390 0

I 4.5 7.0 9.5 I 4.5 7.0 9.5r r

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Figure 7. The four-map representation for an initial ensembleconsisting of points spread uniformly on the ( y = 0, u > 0) surfaceof section (map b). The sectioning conditions are (a) ( y = 0, pr < 0);(b ) ( y = 0, It> 0); c) ( y = n, r < 0); nd (d) ( y = n, r > 0).Thedark points on map b represent initial conditions which intersect withmap d on the first inters ection, while the lighter points in map brepresent initial conditions that intersect map a at the first intersection.section, so that the sectioning condition is only satisfied as r-00. (We note that sim ilar phase space structu res were seen byFrey, Jensen, and Simons in their work on rotational predisso-ciation of model atom-diatom c ~ m p l e x e s .~ ~ )It is characteristic of a prob lem involving a rotational d egreeof freedom transverse to the sectioning plane that the returntime varies drastically with the amount of energy in rotation.Areas on the (r,pr) surface of section cannot th erefore be usedin a straightforward way to calculate comple x dissociation rates.Moreover, turnstile areas on the (r,pr) surface of section arethemselves difficult to determine, so we seek a more tractablerepresentation of the complex formation dynamics.Another feature of the (r ,pr)surface of section which is notimmediately app arent from Figure 6 is that trajectories in whichthe rotor permanently reverses direc tion (change of sign of p y )never return to the surface of section, even though they maystill be complexed. Such a loss of trajectories from the (r,pr)surface of section is problematic, as trajectories in whichsubstantial rotational-translational energy transfer occurs willbe incompletely depicted in this representation. (There is ofcourse a compensating gain of trajectories that arrive at thesurface of section just after having the sign of th e rotor angularmomentum reversed; the net result is a zero change of phasespace area.)

One way to get around the problem of lost trajectories isto use not one but four surface of section maps to fully sampledifferent regions of phase spa ce. In Figure 7 we show such afour-map* epresentation (cf. ref 32). The total energy of thesystem is negative in this example (-0.0272 eV), and alltrajectoties are bound, so that there are no tendrils at this energy.The four surfaces are sections taken at ( y = 0, p ,, > 0 ) , ( y =0, PY < O) , ( y = n, y > 01, nd ( y = n, y < 0). The initialensemble consists of points u niformly distnbuted on the ( y =0, p y > 0) surface of section. We continu e the trajectories onlyuntil they intersect one of the four surfaces. (Only two willactually be intersected.) The initial conditions in Figure 7 havebeen divided into two typ es; the darker points represent thoseinitial cond itions which end up in the ( y = n, y > 0) surface

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0.0 0.2 0.4 0.6 0.8 1.0Y/2n

Figure 8. Periodic orbits in the Na+/N2 system fo r J = 0, lotted in( r , y ) configuration space: (a) periodic orbit y = d 2 , E = -0.0272eV ; (b) orbiting periodic orbit, E = 0.03 eV .

of sectio n at the first intersectio n, while the lighter points endup in the ( y = 0, p y .c 0) surface of section. Thus , the darkpoints are those initial conditions for which the direction ofrotation is not reversed at the first close encounter, while thelighter points are those for which the direction of rotation ofthe rotor is reversed. At the border between these two types oftrajectories lie initial conditions for trajectories in which therotor is asymptotically stationary, so that the boundary corre-sponds to the intersection with the sectioning plane of the stablemanifold, W , f the periodic orb it with y = n/2. This periodicorbit is shown in Figure 8a.Further work on th e four-map picture could provide insightinto coupling of large-amplitude internal motions and rota tionsin isom erization reactions, for example. The compli cated nature

of this representation at higher energies leads, however, todifficulties analyzing the process of complex formation in ascattering problem analogous to those found for the (r,pr)section. We therefo re present another surface of sectionrepresentation in the next subsection.C. The ( y , ~ , , ) urface of Section. A representation that iseasier to interpret than the (r,pr) section, and which can be usedto understand the factors affecting complex formation prob-abilities, is obtained by defin ing the Poincark section at the innerturning points in the radial coordinate r. The sectioningcondition is (pr= 0, pi- < 0), where pi- is the conjugatemomentum just before the sectioning point. Phase points onthe section are classified according to whether the trajectoryreturns to the surface of section or not. In Figure 9a we showsuch a classificatio n of initial conditio ns on the section at totalenergy E = 0.05 eV. Here, the darker points are points that donot return to the surface of section in forward time. That is, ifa trajectory lands on the d arker band in Figure 9a, it exits thecomplex region, and there are no furthe r radial turning points.

On the boundary of the dark region in Figure 9a aretrajectories that n either escape nor return; they are asym ptoticto the orb iting periodic orbit (Fig ure 8b) and so lie on the stablemanifold of this orbit. Note that the orbiting trajectory itselfdoes not lie in the ( y , p y )section. Note also that the periodicorbit at y = n / 2 shown in Figure 8a no longer exists for E >0 and so does not appear on the ( y , p y ) ection.In Figure 9b we show a similar picture with the differencethat here the darker points are those that do not return to the


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2440 J . Phys. Chem., Vol. 99 , No . 9, 1995 Ramachandran and Ezra40.0

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Figure 9. Geom etry of complex formation. Behavior of trajectoriesinitiated in the ( y , p y ) urface of section. (a) The dark band is composedof phase points that never retum to the surface of section. (b ) Th edark band is composed of phase points that have no preimage on thesurface of section. (c) Areas marked (D ) correspond to directtrajectories; areas marked (C) correspond to trajectories that formcomplexes.surface in negative time. At the boundary of the dark regionlie trajectories on the unstable manifold of the orbiting periodicorbit. All points in a collision ensemble therefore have theirfirst intersection with the surface of section in the band of darke rpoints shown in Figure 9b. The overlap regions of the twobands in Figure 9a,b (regions D in Figure 9c) contain pointsthat do not return in either forward or reverse time, which isthe set of direct (non-complex-forming) collisions.33 The areaslabeled C in Figure 9c contain those points in the collisionensemble that form complexes; these regions therefore form thetumstile for orbiting complex formation. In the subsectionsbelow we show how the ( y , p v )surface of section can be usedto understand complex formation probabilities and lifetimes inthe Na+/N2 system.

1. Complex Lifetimes. In Figure 10 we sho w a sequence ofplots at different total en ergies in which the first intersection ofthe collision en semble with the ( yp , ) surface of section is show nalong with the turnstiles for complex formation. Also shownare the two "bands" containing phase points that never returnin forward or reverse time as explained above and as seen inFigure 9. The band containing points that never retum inforward time removes trajectories in the initial ensemble thatare direct. (For these trajectories the Na+ has only oneencounter with the N2 diatom.) The remaining trajectories areby definition complex forming. Thus, the complex formationprobability depends on both the size of the two bands and theway the initial ensemble intersects these areas. The set ofcomplex forming trajectories is composed of distinct subsetscorresponding to the intersection of successive preimages ofthe outgoing turnstile with the initial ensemble.I6

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Figure 10. Variation of phase space structure and complex formationwith energy. For all plots the dark band is composed of phase pointsthat never retum to the surface of section; the light points correspondto phase points that have no preimage on the surface of section. Thecollision ensemble is the dense line of points superimposed on thebandsand corresponds to the initial conditions J = 0, b = 0. (a ) E = 0.005eV, (b) E = 0.01 eV, and (c) E = 0.1 eV .We define the complex lifetime to be the time between thef i s t and last radial inner tuming points. It is important to note

that there is no necessary correlation between the lifetime andthe number of radial inner turning points. The complex lifetimewill be longer for those initial conditions that fall near the stablemanifold W of the periodic orbit corresponding to orbitingmotion of Na+ about N2 (shown in Figure 8b). The stablemanifold W s of the orbiting periodic orbit forms the bound arybetween initial conditions that return to the surface of sectionand those that do not, and trajectories that pass close to thestable manifold spend a long time in the vicinity of the orbitingperiodic orbit. We the refore expect that trajectory return times,hence lifetimes, will vary greatly according to the proxim ity ofan initial condition to Ws.We are able to explain the unusual appearance of the decaycurve shown in Figure 5 for the ensemble at 0.03 eV using thisidea. In Figure 11, we sho w a plot of the lifetimes of individualcomplexes as a function of the initial angle, yi, nderneath aplot of the first intersections with the ( y , p , ) surface of sectionthat give rise to these complexes, for E = 0.03 eV. There arefour sets of com plex forming points, symmetrically placed abouty i= n/2, o that there are in fact two dyna mically distinct setsof initial conditions that give rise to complexes. The first setof com plex forming points, labeled A in Figure l l a , has adifferent decay curve than the second set, labeled B. Decaycurves for both sets are shown in Figure 12.The a verage complex lifetime of set A is longer than that ofse t B, due primarily to a significant time lag before decay ofsubensemble A begins. (The combined decay curve is shownin Figure 5) . The relation between the lifetime and the


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Orbiting Complex Formation in Na+/Nz Collisions J. Phys. Chem., Vol. 99, No . 9, 1995 2441The intersection of the collision ensembles with the ( y ,p r )surface of section for energies lower than 0.05 eV are loopedin appearance (although of course they are still connected 1-Dsets) and become tightly coiled as the energy is lowered.The looping comes about as follows. For large values of r,the potential V is essentially a long-range iodinduced-dipoleinteraction. The polarizability of the diato m is anisotropic, andend-on collisions are energetically favored. (This changes atsmall r , where T-shaped configurations have lower energies.)Examination of incoming trajectories shows that at the lowerenergies studied they exhibit hindered rotation or librationalmotion in y , oscillating about y = 0 or n. When projectedonto the ( y ,p , ) plane, this oscillatory motion results in a coilingof the initial ensemble. As the collision energy decreases, theanisotrop y of the long-ra nge part of the potential has more tim eto act on the incoming trajectories, resulting in more pronouncedlibration and tighter looping of the initial ensemble about y =0, n. As the energy increases, the long-range part of thepotential has less effect.It can clearly be seen from Figure 10 that the length of thecurve corresponding to the intersection of the collision ensemblewith the surface of section increases as E decreases. Theensemble is therefore stretched, and the density of points per

unit length of the ensem ble curve decreases with E .The behavior of the comp lex formation probability over thewhole range of collision energies may then be understood asfollows: At very high collision energies, the angular dependenceof the potential has little effect on trajectories over the shortduration of the collision; put another way, turnstiles for complexformation are small. As the collision energy decreases, theinfluence of the nonspherical part of the potential increases,resulting in larger turnstiles and increasing complex formationprobabilities. As E decreases still further, the s tretching effectcomes into play. That is, the stretching of the ensem bleovercomes the effect of increase in tumstile size so that thecomplex formation probability begins to decrease. At this stage( E = 0.05 eV in Figure 3), only a single segm ent of the initialensemble intersects the tumstile region upon the first crossingof the ( y , p y ) section. Decreasing E still further, we reach thepoint at which a loop pokes through into the tumstile region(cf. Figure 1 a), with a resulting rapid increase in the complexformation probability. As E is lowered, the stretching effectcomes into play again and decreases the contribution from thenew loo p, until an energy is reached at which a new loo p pokesthrough, and so on .

The constant-energy collision ensemble we use consistsentirely of head-on collisions and has J = 0 (i.e., pb = b = 0),leaving the initial angle y as the only variable to be sampledin the ensemble. Inclusio n of b t initial conditions in the J= 0 ensemble will wash out the observed oscillations in thecomplex formation probability. Although the construction ofa constant E , J = 0 ensem ble that correctly samples the relevantphase sp ace hypersurface is a nontrivial matter,34we note that,when the J = 0 constrain t is removed by, for examp le, allowin gnonzero impact parameters b , the resulting complex formationprobabilities do not exhibit the oscillations seen in Figure 3.

3. Influence of Potential Anisotropy. In Figure 13 show the(y ,pr ) surface of section and turnstiles for complex formationfor three different potentials at the same total energy, E = 0.054eV .The potential described in section 11, which has bothanisotropic attractive and repulsive terms, leads to the phasespace structure of Figure 1 3a. For Figure 13b the potential ismodified to make the attractive l/# term isotropic, by setting

a1= c t l = (a1 ~ 1 ) / 2 . n Figure 13c the potential consists of

2 0.0

I I-40.0 I

0.0 1 o20.0

g. 15.0c

J 10.0svc 5.0

0.00.0 0.2 0.4 0.6 0.8 1.0

Y hFigure 11. Lifetimes and the intersection of the J = 0 collisionensemble with the (y ,pr ) surface of section. The total energy is 0.03eV . (a) The first intersection of the collision ensemble with the surfa ceof section. The dark points are phase points that never return to thesurface of section. (b) Lifetimes as a function of initial phase angle y .

1.2

.ELe 0.6PE5

0.00.0 3.0 6.0 9.0 12.0 15.0 18.0

time (au) (x i o5 )Figure 12. Decay curve for subset A of the collision ensemble (seeFigure 1I), solid line; decay curve for subset B, dotted line.proximity to the stable manifold of the orbiting periodic orbitcan be clearly seen in the comparis on of set A with set B. Eachtrajectory in set A, which consists of initial conditions tightlylooped near Ws,must make at least one pass close the orbitingperiodic orbit before it can dissociate, thus giving rise to theobserved time lag in the decay curve for the subensemb le.The decay of the ensemble at E = 0.01 eV (Figure lob) ismore nearly single exponential. For this ensemble, the loopcorresponding to set A protrudes much further into the tumstileregion, so that the majo rity of comp lex-form ing trajectories inthis subset are not in close proxim ity to the stable manifold ofthe orbiting trajectory .2 . Complex Formation Probabilities. Th e (yp,,) surface ofsection enables us to understand the origin of the nonm onoto nicbehavior of the complex formation probability as a function ofenergy seen in Figure 3. As mentioned above, the complexformation probability is determined not only by the turnstileshape and area but also by the way in which the collisionensemble intersects the turnstile region on the surface of section.


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2442 J. Phys. Chem., Vol. 99, No . 9, 1995 Ramachandran and Ezra40.0

h 0.0

-40.040.0

h 0.0

-40.040.0

h 0.0

-40.0 I0.0 1 .or / .Figure 13. Influence of potential anisotropy on the phase spacestructure. The ( y ,p r ) surface of section shows initial conditions thatnever return to the surface of section (dark band) and initial conditionsthat have no preimage on the surface of section (light band). (a )Anisotropic repulsion and attraction; (b) anisotropic repulsion, isotropicattraction; (c ) isotropic repulsion, anisotropic attraction.the original anisotropic aitragive term, but the repulsive termis an isotropic function Ce-P.. The parameters C and p areobtained by averaging the repulsive part of the potential (3)over y and fitting the resulting function of r to exp onential form.Parts a and b of Figures 13 have a similar qualitativeappearance, although in the isotropic attraction case (Figure 13b)the turnstiles are more angular. Theere is however a strikingdifference between parts a and c; making the repulsive termisotropic drastically reduces the size of the turnstile for complexformation. The insensitivity of the phase space structure to theanisotropy of the long-range attraction and the extrem e sensitiv-ity of the turnstiles to the anisotropy of the repulsive part ofthe potential are consistent with the notion that the energytransfer leading to orbiting complex formation occurs throughimpulsive collisions at the inner turning point?Although it is not o ur intention to provide a detailed critiqueof the work of Schelling and Castleman,s we now com ment onthe relation of their study to our findings. SC employ a potentialin which the only source of coupling between relative translationand internal rotation is the anisotropy of the long-range part ofthe potential.8 In the absence of this anisotropy, all trajectorieshave zero lifetime according to our definition but have nonzerolifetimes (the time spent with r smaller than a given, arbitrary,value) according to SC. Adding long-range anisotropy thenleads to some complex formation and an increase in complexlifetime according to either definition.In the absence of anisotropy in the short-range part of thepotential, the size of the turnstiles is reduced. Hence, comp lexformation probabilities are smaller than with the full potential.On the other hand, trajectory calculations using a potential withisotropic short-range interaction show that complex lifetimes

are comparable to those found for the full potential at giventotal energy.VI. Conclusion and Summary

1. Our trajectory calculations of complex formation prob-abilities and lifetimes in Na+/N2 collisions show that the N2vibrational degree of freedom does not couple effectively tothe rotational and translational degrees of freedom and thereforedoes not play an important role in determining the complexlifetimes. This finding is in agreement with Schelling andCastlema ns study,8 hough at variance with Brass and Schliersscaled H+/H2 study.92. The complex formation probabilities depend nonmono-tonically on the total energy for J = 0, b = 0 ensembles. Thecomplex formation probabilities decrease with increasing energyonly for E > 0.12 eV. At energies below 0.12 eV the complexformation probability exhibits oscillations.3. The (rp, .)surface of section, previously used in the studyof van der W aals comp lexes,I6 does not provide a pa rticularlyuseful representation of phase space structure in the case oforbiting complex formation. Trajectories can disappear fromthe section, and the reactive separatrix and associated turnstileshave a complicated form. The (rp,.) section can however beused in conjunction with a four-map approach (cf. ref 32) tosample the full phase space.4. In contrast to the ( r ,p r ) urface section, the ( y , p r ) urfaceof section provides considerable insight into the dynamics oforbiting complex formation. Using the ( y , py ) urface of section,we are able to explain the low-energy oscillations seen in thecomplex formation probabilities, as well as the unusual decaycurves seen at some energies.5. The ( y , p r ) surface of section shows that varying theanisotropy of the repulsive part of the potential has a much largereffect on complex formation than v arying the anisotropy of theattractiv e term of the poten tial. The insensitivity of the complexformation turnstiles to the anisotropy of the attraction and theextreme sensitivity of the turnstiles to the anisotropy of therepulsion indicate that the com plex formation process is a veryimpulsive one, dependent on the short-range potential. Thesefindings are not in conflict with those of Schelling andCastlema n,x as these authors use both a different definition ofcomplex lifetime and a potential surface with no short-rangeanisotropy.

Acknowledgment. This work was supported by NSF G rantsCHE-9101 357 and CHE-94035 72. Com putations reported herewere performed in part on the Come11 National Su percomputerFacility, which is supported by NSF and IBM Corp.References and Notes

(1) See, for example: Advances in Classical Trajectory Methods; Hase,(2) Ferguson, E. E. J . Phys. Chem. 1986, 90, 731.(3 ) Levine, R. D. ; Bemstein, R. B. Molecular Reaction Dynamics and

W. L., Ed.; JAI Press: London, 1994; Vol. 2 and references therein.Chemical R eactivity; Oxford University Press: Oxford, 1987.(4 ) Chesnavich. W. J.; Bowers, M. T. f r o g . Reac. Kinet. 1982, 1 2 ,137. Troe, J. J . Chem. Phys. 1987, 87, 2113.(5 ) Clary, D. C. Mol. Phys. 1984, 53, 3 ; Mol. Phys. 1985, 54, 605.

(6) See: Hase, W. L.; Darling, C. L. ; Zhu, L. J . Chem. Phys. 1992,(7) Ramachandran, G.; Ezra, G. S. J . Chem. Phys. 1992, 9 7 , 6322.(8 ) Schelling, F. J. ; Castleman, A . W . Chem. Phys. Lett. 1984, 111 ,(9) Brass, 0.; Schlier, C. J . Chem. Phys. 1988, 88 , 936.

96, 8295 an d references therein.

47 .(10) Tanuma, H.; Kita, S . ; Kusunoki, I. ; Shimakura, N . Phys. Rev. A1988, 38, 5053.(11) Gerlich, D.; Nowotny, U. ; Schlier, C. ; Teloy, E. Chem. Phys. 1980,47 , 245. Schlier, C.; Vix, U. Chem. Phys. 1985; 95 , 401; 1987, 113, 211.


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Orbiting Complex Formation in Na+/N2 Collisions(12) See, for example: CHAOS 1993, 3 (4), focus issue on ChaoticScattering.(13) Rice, S . A,; Gaspard, P.; Nakamura, K. Adv. Classical TrajectoryMethods 1992, I , 215.(14) Davis, M. J.; Skodje, R. T. Adv. Classical Trajectory Methods 1992,1, 77 .(15) Lichtenberg, A . J. ; Lieberman, M. A. Regular and StochasticMotion; Springer: Berlin, 1983. Guckenheimer, J. ; Holmes, P. NonlinearOscilla tions, Dynamical Systems and Bifurcations of Vector Fields;Springer: Berlin, 1983.(16) Davis, M. J.; Gray, S. K. J. Chem. Phys. 1986, 84, 5389.(17) Huber, K. P. ; Herzberg, G. Molecular Spectra and Molecular

Structure,N . Constants of Diaromic Molecules; Van Nostrand New York,1979.(18) Bogaard, M. P.; Om, . J. Int. Rev. Sei., Phys. Chem. Ser. 2 1975,2, 149.(19) Robinson, P. J .; Holbrook, K. A. Unimolecular Reactions; Wiley:New York, 1972. Forst, W., Theory of Unimolecular Reactions; Aca-demic: New York, 1973.(20) Gray, S. K.; Rice, S. A, ; Davis, M. J. J . Phys. Chem. 1986, 90,3470.(21) Wiggins, S. Physica D 1990, 44, 47 .(22) Gillilan, R. E. ; Reinhardt, W. P. Chem. Phys. Lett. 1989,156,478.

J. Phys. Chem., Vol. 99, No . 9, 1995 2443(23) Gillilan, R. E. J . Chem. Phys. 1990, 93, 5300.(24) Gillilan, R. E.; Ezra, G. S. J . Chem. Phys. 1991, 94, 2648.(25) Tersigni, S. H.; Gaspard, P.; Rice, S. A . J . Chem. Phys. 1990,92,(26) Gaspard, P.; Rice, S. A. J . Phys. Chem. 1989, 93, 6947.(27) Channon, S. R.; Lebowitz, J. L. Ann. N.Y. Acad. Sci. 1980, 357,108.(28) MacKay, R. S .; Meiss, J. D.; Percival, I. C. Physica D 1984, 13,82 .(29) Wiggins, S. Chaotic Transport in Dynamical Systems; Springer:Berlin, 1992.(30) Binney, J. ; Gerhard, 0. E.; Hut, P. Mon. Not. R. Astron. Soc. 1985,215, 59.(31) Frey, R. F.; Jensen, J. 0.; Simons, J. J . Phys. Chem. 1985, 89,788.(32) Ozorio de Almeida, A. M.; De Leon, N.; Mehta, M.; Marston,C.C. Physica D 1990, 46 , 265.(33) Pollak, E.; Child, M. S. J. Chem. Phys. 1980, 73, 4373. See also:Marston, C. C.; DeLeon, N. J . Chem. Phys. 1989, 91, 3392, 3405.(34) Dumont, R. S. J. Chem. Phys. 1992, 96, 2203.

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g. ramachandran and g. s. ezra- orbiting complex formation in na^+/n2 collisions: a phase space view

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