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

ARTICLES

Orbiting 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 14853

Received: July 25, 1994; In Final Form: October 20, 1994@

In this paper we investigate the dynamics of orbiting complex formation in Na+/N2 collisions using classical trajectories. We first demonstrate that the rigid-rotor approximation is appropriate for computing complex formation probabilities and lifetimes, as the NZ vibration is weakly coupled to the other degrees of freedom. For a collision ensemble with zero impact parameter and total angular momentum J = 0, complex formation probabilities exhibit oscillatory behavior with E. Several different representations of the phase space structure are compared, and it is found that the (yp , , ) surface of section provides most insight into the complex formation dynamics. Oscillatory complex formation probabilities and the influence of potential anisotropy on complex formation rates are analyzed in terms of the (y ,pv) section.

I. Introduction

Complex formation in ion-molecule collisions is an impor- tant precursor to both reactive and nonreactive processes.’ Experimentally determined rates of ion-molecule reactions are often analyzed in terms of a kinetic scheme in which the first step is the formation of an “orbiting” complex, which can either decay back to reactants or lead to products.1.2 The rate of formation of orbiting complexes (association) is typically estimated using Langevin-type capture t h e ~ r i e s . ~ - ~ Several trajectory studies have shown the inadequacy of capture theories for computing the association rate.617 In a study of Kr/02+ collisions, for example, we found that the complex formation rate, as opposed to the capture rate (rate of passage over the centrifugal barrier), depends strongly on the collisional energy, although this dependence lessens at very low collision energies (see Figure 9 of ref 7 ) . The dynamics of complex formation and dissociation in ion-molecule systems, although of consid- erable fundamental importance, is at present quite poorly understood, despite much theoretical work on collisions in systems with long-range attractive forces6 One key question concems the influence of long-range versus short-range features of the potential surface (e.g., anisotropy) on orbiting complex formation and decay rates.

In this paper we study complex formation in collisions of Na+ with Nz, a system studied by previous workers.s-10 There are no strong chemical interactions in this system, and a simple functional form for the potential is employed. The shape of the simple model potential is easily manipulated to study the effect of various aspects of the potential on the complex formation rate and probability.

Low-energy Na+/N2 collisions have been studied previously by Schelling and Castleman (SC).8 SC used a planar iodrigid- rotor model with the interaction potential between the ion and the diatom consisting of an attractive anisotropic iodinduced- dipole term (l/r“) and an isotropic repulsive l / r l* term. The object of their study was to determine the effect of the anisotropy

Present address: Courant Institute of Mathematical Sciences, 25 1 Mercer Street, New York, NY 10012.

@ Abstract published in Advance ACS Abstracts, February 1, 1995.

0022-3654/95/2099-2435$09.00/0

of the N2 polarizability on the complex formation lifetimes; an increase in complex lifetime with increase in anisotropy was reported.8

Brass and Schlier (BS)g estimated complex lifetimes for Naf/ N2 by scaling trajectory results previously obtained for H+/H2.” Their estimated lifetimes were -25 times longer than those calculated by SC. Moreover, by computing complex formation cross sections and lifetimes as a function of asymmetry parameter in a simple triple-Morse potential for H+/H*, BS concluded that the rigid-rotor approximation used by SC was inadequate and that intermediate storage of translation energy in the vibrational degree of freedom was essential for complex formation, so that the vibrational degree of freedom could not be omitted from consideration. We discuss this conclusion in more detail in section 11, where we compare the results of computations of complex formation with and without N2 vibration.

Rotational and vibrational excitation in high-energy (> 100 eV) Na+/N2 collisions has been studied both experimentally and theoretically by Tunama et a1.I0

Long-lived scattering trajectories have been widely discussed in the context of chaotic scattering.’* In particular, classical collision complex lifetimes have been expressed in terms of the properties of the underlying chaotic re~e1ler.l~ The phase space analysis of complex formation and unimolecular decay has proved to be very powerful for understanding deviations from statistical theories.I4

One of the aims of the present paper is to present a phase space view of orbiting complex formation in the case that rotational 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 space structure can in principle be examined using a 2-dimensional Poincar6 section.I5 We point out, however, that the presence of a rotational degree of freedom leads to difficulties with the (r,pr) type of section used previously in the study of complex formation and dissociation in T-shaped van der Waals com- plexes.I6 An alternative representation, the (y,pr) section, is therefore introduced. This representation is found to provide useful insight into the dynamics of complex formation. A key role in the phase space analysis of complex formation and decay

0 1995 American Chemical Society

2436 J. Phys. Chem., Vol. 99, No. 9, 1995 Ramachandran and Ezra

in van der Waals molecules is played by the periodic orbit at r = m. Segments of the stable and unstable manifolds of this periodic orbit define the reactive sepratrix bounding the complex region.I4.l6 We show that an analogous role in the case of orbiting complex formation and decay is played by the “orbit- ing” periodic orbit, which is located at finite r atop a centrifugal barrier.

In section I1 we present the Hamiltonian and the model potential surface for Naf/N2. The trajectory ensembles are also described. In section I11 we discuss the validity of the rigid- rotor approximation for the purpose of studying complex formation rates and lifetimes in Na+/N2 collisions. In section IV we present the results of our calculations of complex lifetimes and formation probabilities. Section V describes several phase space representations we have used to analyze the trajectory results. Section VI presents a summary and conclusions.

11. System and Methods

diatom (Na+/N2) collisions in the plane, with Hamiltonian A. Hamiltonian. We treat the classical dynamics of atom-

where s is the NZ bond length, r is the radial distance between the Na+ atom and the center of mass of the Nz diatom, and y is the angle between s and r. ps , pr , and p y are the conjugate momenta, and J is the total angular momentum for rotation in the plane.

For those calculations in which we freeze the NZ bond length at 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 potential

consisting of an attractive anisotropic iodinduced-dipole inter- action and a short-range exponential repulsion term centered on each of the nitrogen atoms. This potential is easily modified to increase or decrease the anisotropy of either the attractive or repulsive terms. The general form of V(r, y , s) is

where a(y) is an anisotropic polarizability and 11 and rz are distances of the Na+ ion from the two N atoms. The third term in this expression is a Morse oscillator potential describing the N2 vibration; values for the dissociation energy De, the equi- librium bond distance seq. and the range factor a are taken from ref 17. The anisotropic polarizability a(y) is

a(y) = 1/2{(c4, + ai> + (c4, - a,) COS(2Y)I (4)

where ql and aL are the parallel and perpendicular polarizabili- ties of the Nz molecule, respectively (values taken from ref 18). The constants C and ,!3 characterizing the repulsive part of the Na+/N2 interaction are taken from Tanuma et a1.I0 The values of the potential parameters are shown in Table 1, and contours of the potential surface are shown in Figure 1.

C. Initial Conditions. We have computed classical complex formation probabilities and lifetimes for ensembles of initial

8.0

P.

0 v

0.0 0.0 8.0

x ( 4 Figure 1. Contours for the Na+/N2 potential. The NZ bond lies along the x axis, with the midpoint of the N-N bond at the origin. The N-N bond length is fixed as s = seq = 2.075 au. Contours are at multiples of 0.027 21 eV.

TABLE 1: Parameters for Na+/N2 Potential (atomic units) 2.08 clll 14.851 a 1.423

seq 2.075 ai 10.179 C 55.124 De 0.3642

P

conditions with fixed total energy E and total angular momentum J = 0. The ensemble consists of trajectories undergoing head- on collisions with py(0) = 0 and impact parameter b = 0. Trajectories are started with r(0) = 70 au. The only remaining variable to be specified is the initial rotor phase y(O), which is uniformly distributed between 0 and n. Each ensemble consists of 3000 trajectories. Fixing the impact parameter b defines a 1 -dimensional ensemble of initial conditions; this choice of ensemble facilitates a detailed analysis of the classical dynamics.

For our study of the validity of the rigid-rotor approximation in section III we use a slightly different ensemble with J f 0. This ensemble is a fixed translational energy (Et) ensemble with py(0) = 8h, and the impact parameter b varied between 0 and b,,,, where bmax, the maximal impact parameter beyond which complex formation does not occur, is determined numerically. Each trajectory was started with r(0) = 70.0 au, and the ensemble consisted of 3000 trajectories as in the previous case. Vibrational initial conditions for the rotating NZ diatom were sampled using a procedure described in ref 7.

As in previous work:,’ we consider a complex to be formed when there are two or more inner tuming points in the radial coordinate r. The complex lifetime is then defined to be the time between the first and last radial inner tuming points.

111. The Rigid-Rotor Approximation

Schelling and Castleman (SC)8 studied the influence of polarizability anisotropy on the lifetimes of complexes formed in collisions of Na+ with N2 and CO2 using classical trajectories and found that increasing the anisotropy of the long-range part of the potential increases the average trajectory collision lifetime. SC treated iodrigid-rotor collisions in the plane. Brass and Schlier (BS)9 criticized SC’s use of a rigid-rotor approximation on the grounds that their own investigations revealed that intermediate energy storage in the vibrational degree of freedom was an essential factor determining the complex lifetime. BS based their arguments on a classical trajectory study using a triple-Morse H+/H2 potential,” in which the Morse pair potentials used to model the H-H6+ interaction were modified by changing the equilibrium distance. BS also found that collision complex lifetimes depended on the frequency of the fastest normal mode of the system in accordance with RRKh4 theory, suggesting complete intemal energy ex~hange . ’~ In

Orbiting Complex Formation in Na+/N2 Collisions J. Phys. Chem., Vol. 99, No. 9, 1995 2431

0'40 u 0.32

c 2 0.24 e 9 5 0.16

-

0.08

0.00 ' 0.0 8.0 16.0 24.0 32.0

Time (OU) (x 10')

Figure 2. Survival fraction as a function of the lifetime computed for trajectory ensembles with and without Nz vibration included. Both calculations are carried out at a collisional energy of 0.0329 eV. Solid line, vibration included; dotted line, vibration frozen.

contrast, SC observed a nonexponentd decay of the survival fraction as a function of the lifetime.8 (The survival fraction at time t is defined as the fraction of the trajectory ensemble with lifetimes greater than t . )

In Figure 2 we show the survival fraction as a function of the lifetime for trajectory ensembles with and without the N2 vibration, for the potential of eq 3, E = 0.0329 eV. It is seen that inclusion of the vibration makes little difference to the decay rate of the complexes and hence to the lifetime. We have performed calculations at several collisional energies (Et) and have seen the same quantitative agreement between decay behavior with and without the vibrational degree of freedom. The N2 vibrational degree of freedom is thus found to be ineffectively coupled to the translational and rotational degrees of freedom in the Na+/N2 system, as might be anticipated on the basis of simple time scale arguments; Le., the period of the N2 vibration is much shorter than either the rotational period or the collision time at the collisional energies and rotational energies studied. Schelling and Castleman's use of a rigid- rotor approximation for the NZ diatom is therefore justified for the purpose of determining the lifetimes of complexes formed in our model system. The scaling arguments used by BS to estimate complex lifetimes in Naf/N2 appear to be inapplicable in this case, due to the high degree of symmetry still present in the modified H+/H2 potential used in their trajectory calcula- t i o n ~ . ~ This symmetry, inherited from the D3h potential, results in extensive energy transfer due to resonance between complex vibrational degrees of f r e e d ~ m . ~

Our previous work' on the dynamics of Kr/02+ collisions showed that, although intermediate energy storage in the vibrational degree of freedom does lead to a slight increase in complex lifetimes, it is by no means essential for orbiting complex formation. In agreement with SC, we find that for Na+/N2 collisions translational-rotational (T-R) energy trans- fer is the critical factor in complex formation and decay. Inclusion of the N2 vibration is therefore not necessary for accurate computation of complex formation rates and lifetimes in classical Na+/N2 collisions, so we shall henceforth use the rigid-rotor approximation.

IV. Complex Formation Rates and Lifetimes

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

0.6

0.5

1 0.4 1 ?

.s E 0.3 0

E 0.2

0.1

0.0 0.00 0.06 0.12 0.18 0.24 0.30

EkV)

Figure 3. Fraction of complexes formed as a function of the total energy E. The trajectory ensemble used is the J = 0 ensemble described in section 11.

30.0

v - * I ! a 15.0

0.00 0.08 0.12 0.18 0.24 0.30 W )

Figure 4. Lifetimes as a function of the total energy E. The trajectory ensemble 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 complex formation probability smoothly decreases with E. At total energies below E = 0.12 eV, however, the complex formation fraction shows a number of oscillations. This behavior is interpreted in terms of phase space structure in the next section. (For ensembles in which a range of impact parameters is sampled, the oscillations wash out [see section V.C.21.)

B. Complex Lifetimes. In Figure 4 we plot the lifetimes for a given collision ensemble (as taken from the best straight line fit to the logarithm of the decay curve of the survival fraction) as a function of the energy E. At low energy the fit to an exponential decay is often poor. In particular, at 0.03 eV a curious stepwise decay is seen. The decay curve for E = 0.03 eV is plotted in Figure 5 .

V. Phase Space Representations of Orbiting Complex Formation

We now turn to a phase space view of the dynamics of orbiting complex formation. The planar atom rigid-rotor model for Na+/N2 collisions is a two degree of freedom system, and many methods for studying the phase space structure of such systems are a~ai1able.l~

A. Surfaces of Section, Complex Formation, and the Reactive Separatrix. The PoincarC surface of section,I5 for example, has proved valuable in understanding the origin of nonstatistical behavior in two-mode models of unimolecular decay of van der Waals complexes such as He/I2.I6 The surface

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$ ‘5 v)

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J. Phys. Chem., Vol. 99, No. 9, 1995

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-

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-

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L I

I \ I

0.0 3.0 6.0 9.0 12.0 15.0 18.0 time (au) (x 10’)

Figure 5. Decay curve (fraction of trajectories surviving for time t versus t ) for E = 0.03 eV. The ensemble used is the J = 0 ensemble described in section 11.

of section construction provides the means to define and visualize phase space dividing surfaces (bottlenecks) that form the basis for modified statistical theories for the unimolecular decay rate.I6 Extension of these methods to higher dimensional systems 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, transverse to trajectories in the three-dimensional energy shell. Successive iterations of the mapping M correspond to successive intersections of a trajectory with the surface 2, Thus, the image of the phase point zo E C under the mapping M is the next point on the trajectory starting at zo that pierces 2. The mapping M is symplectic (area-preserving for two degrees of f reed~m). ’~ Such mappings have been used directly to model unimolecular decay dynamics.22.26

In the Hen2 system,I6 for example, a sectioning surface was defined by setting the I2 bond length equal to its equilibrium value and choosing a particular sign for the velocity. The radial coordinate and momentum of the He atom were then plotted every time the 12 diatom expanded through its equilibrium position. The vibration of the 12 bond provided a natural “clock” for the dynamics; that is, trajectory return times (the time taken between 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 the surface of section and volumes in the full phase space that facilitates analysis of the kinetics of unimolecular decay.I6

Using the surface of section for H e k it is possible to define the reactive separa t r i~ . ’~ . ’~ This surface consists of the union of segments of the stable and unstable branches of the fixed point on the Poincart section corresponding to the periodic orbit in which the 12 molecule vibrates freely with He stationary at infinity. The separatrix is a surface which, in the absence of coupling between the 12 vibration and the I2-He degrees of freedom, forms an invariant boundary between bound (complex) and unbound trajectories. In the presence of coupling, the separatrix is no longer invariant (Le., it is “br~ken”~’) and serves to define the boundary that trajectories must cross to pass between the bound and unbound regions, or vice versa. The reactive separatrix has two important properties: all trajectories that cross it from the inside dissociate immediately, and no direct scattering trajectories ever cross it. Complex forming trajec- tories are therefore precisely those trajectories whose intersec- tions with the PoincarB section reside within the region bounded by the separatrix for some time.

A key observation is that phase space transport across the reactive separatrix is mediated by regions called “tumstiles”,28

Ramachandran and Ezra

30.0 I

-30.0 ’ I

0.0 10.0 20.0 30.0 40.0 50.0 r

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 stable and unstable manifold^.*^.^^ The significance of the turnstiles is that all trajectories that cross the separatrix do so by passing through the tumstile region. If trajectory return times are approximately constant, as is the case for the 12-He system discussed above, the area of the outgoing tumstile lobe is a measure of thejlux of phase points out of the complexed region. In general, however, trajectory return times will not be constant, and the relation between decay kinetics and the structure of the surface of section is more c o m p l i ~ a t e d . ~ ~ ~ ~ ~ In addition to the reactive separatrix just discussed, intramolecular bottlenecks associated with remnants of invariant tori (called cantori28) have also been invoked to account for nonstatistical behavior.I6

In the present section we apply the phase space concepts just described to elucidate orbiting complex formation in Na+/N2 collisions. We shall see that this system exhibits complications not seen in the Hen2 case.

B. The (r,pr) Surface of Section. In Figure 6 we show the reactive separatrix on the (r, p r ) surface of section for Na+/N2 at E = 0.25 eV, where the sectioning conditions are y = 0, p v > 0. The reactive separatrix divides direct trajectories from complexed ones.

A key role in the phase space analysis of complex formation and decay in T-shaped van der Waals molecules is played by the periodic orbit at r = w . Segments of the stable and unstable manifolds of this periodic orbit (fixed point at pr = 0, r = w )

define the reactive separatrix bounding the complex r e g i ~ n . ’ ~ . ’ ~ An analogous role in the case of orbiting complex formation and decay is played by the “orbiting” periodic orbit. The orbiting periodic orbit has r = 0 and sits atop a centrifugal barrier at a finite value of r = 23.91 au. (Note that, although the total angular momentum J = 0, the diatom and orbital angular momenta can be nonzero but of opposite sign. Also, due to symmetry, the intersection of the orbiting periodic orbit with the (r,pr) section occurs at p r = 0). The location of the fixed point on the (r,pr) surface of section corresponding to the orbiting trajectory is obtained numerically by a 1D search along the line p r = 0. The reactive separatrix of Figure 6 is composed of segments of the stable and unstable manifolds of the orbiting periodic orbit.

It can be seen that the separatrix has several unusually long protrusions or “tendrils”. These tendrils are associated with trajectories where most of the energy has gone into the translational degree of freedom, leaving the diatom to rotate slowly, so that the radial distance between successive points on a trajectory for which the sectioning condition (y = 0, pv =- 0) is satisfied becomes very large. At the tips of the tendrils, the N2 rotor takes longer and longer to return to the surface of

Orbiting Complex Formation in Na+/N* Collisions J. Phys. Chem., Vol. 99, No. 9, 1995 2439

0 0

I 4.5 7.0 9.5 I 4.5 7.0 9.5 r r

% I 1 % I 1

I 4.5 7.0 9.5 I 4.5 7.0 9.5 r r

Figure 7. The “four-map’’ representation for an initial ensemble consisting of points spread uniformly on the ( y = 0, pu > 0) surface of section (map b). The sectioning conditions are (a) ( y = 0, pr < 0); (b) ( y = 0, pIt > 0); (c) ( y = n, pr < 0); and (d) ( y = n, pr > 0). The dark points on map b represent initial conditions which intersect with map d on the first intersection, while the lighter points in map b represent 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 similar phase space structures were seen by Frey, Jensen, and Simons in their work on rotational predisso- ciation of model atom-diatom c~mplexes .~~)

It is characteristic of a problem involving a rotational degree of freedom transverse to the sectioning plane that the return time varies drastically with the amount of energy in rotation. Areas on the (r,pr) surface of section cannot therefore be used in a straightforward way to calculate complex dissociation rates. Moreover, turnstile areas on the (r,pr) surface of section are themselves difficult to determine, so we seek a more tractable representation of the complex formation dynamics.

Another feature of the (r,pr) surface of section which is not immediately apparent from Figure 6 is that trajectories in which the rotor permanently reverses direction (change of sign of p y ) never return to the surface of section, even though they may still be complexed. Such a loss of trajectories from the (r,pr) surface of section is problematic, as trajectories in which substantial rotational-translational energy transfer occurs will be incompletely depicted in this representation. (There is of course a compensating gain of trajectories that arrive at the surface of section just after having the sign of the rotor angular momentum reversed; the net result is a zero change of phase space area.)

One way to get around the problem of “lost” trajectories is to use not one but four surface of section maps to fully sample different regions of phase space. In Figure 7 we show such a “four-map’* representation (cf. ref 32). The total energy of the system is negative in this example (-0.0272 eV), and all trajectoties 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, py > 01, and ( y = n, py < 0). The initial ensemble consists of points uniformly distnbuted on the ( y = 0, p y > 0) surface of section. We continue the trajectories only until they intersect one of the four surfaces. (Only two will actually be intersected.) The initial conditions in Figure 7 have been divided into two types; the darker points represent those initial conditions which end up in the ( y = n, p y > 0) surface

9

0.0 0.2 0.4 0.6 0.8 1.0 Y/2n

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

of section at the first intersection, while the lighter points end up in the (y = 0, p y .c 0) surface of section. Thus, the dark points are those initial conditions for which the direction of rotation is not reversed at the first close encounter, while the lighter points are those for which the direction of rotation of the rotor is reversed. At the border between these two types of trajectories lie initial conditions for trajectories in which the rotor is asymptotically stationary, so that the boundary corre- sponds to the intersection with the sectioning plane of the stable manifold, W, of the periodic orbit with y = n/2. This periodic orbit is shown in Figure 8a.

Further work on the four-map picture could provide insight into coupling of large-amplitude internal motions and rotations in isomerization reactions, for example. The complicated nature of this representation at higher energies leads, however, to difficulties analyzing the process of complex formation in a scattering problem analogous to those found for the (r,pr) section. We therefore present another surface of section representation in the next subsection.

C. The ( y , ~ , , ) Surface of Section. A representation that is easier to interpret than the (r,pr) section, and which can be used to understand the factors affecting complex formation prob- abilities, is obtained by defining the Poincark section at the inner turning points in the radial coordinate r. The sectioning condition is (pr = 0, pi-’ < 0), where pi-’ is the conjugate momentum just before the sectioning point. Phase points on the section are classified according to whether the trajectory returns to the surface of section or not. In Figure 9a we show such a classification of initial conditions on the section at total energy E = 0.05 eV. Here, the darker points are points that do not return to the surface of section in forward time. That is, if a trajectory lands on the darker band in Figure 9a, it exits the complex region, and there are no further radial turning points.

On the boundary of the dark region in Figure 9a are trajectories that neither escape nor return; they are asymptotic to the orbiting periodic orbit (Figure 8b) and so lie on the stable manifold of this orbit. Note that the orbiting trajectory itself does not lie in the (y ,py) section. Note also that the periodic orbit at y = n/2 shown in Figure 8a no longer exists for E > 0 and so does not appear on the (y ,py) section.

In Figure 9b we show a similar picture with the difference that here the darker points are those that do not return to the

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

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Figure 9. Geometry of complex formation. Behavior of trajectories initiated in the ( y , p y ) surface of section. (a) The dark band is composed of phase points that never retum to the surface of section. (b) The dark band is composed of phase points that have no preimage on the surface of section. (c) Areas marked (D) correspond to direct trajectories; areas marked (C) correspond to trajectories that form complexes.

surface in negative time. At the boundary of the dark region lie trajectories on the unstable manifold of the orbiting periodic orbit. All points in a collision ensemble therefore have their first intersection with the surface of section in the band of darker points shown in Figure 9b. The overlap regions of the two bands in Figure 9a,b (regions D in Figure 9c) contain points that do not return in either forward or reverse time, which is the set of direct (non-complex-forming) collisions.33 The areas labeled C in Figure 9c contain those points in the collision ensemble that form complexes; these regions therefore form the tumstile for orbiting complex formation. In the subsections below we show how the (y ,pv) surface of section can be used to understand complex formation probabilities and lifetimes in the Na+/N2 system.

1. Complex Lifetimes. In Figure 10 we show a sequence of plots at different total energies in which the first intersection of the collision ensemble with the (yp,) surface of section is shown along with the turnstiles for complex formation. Also shown are the two "bands" containing phase points that never return in forward or reverse time as explained above and as seen in Figure 9. The band containing points that never retum in forward time removes trajectories in the initial ensemble that are direct. (For these trajectories the Na+ has only one encounter with the N2 diatom.) The remaining trajectories are by definition complex forming. Thus, the complex formation probability depends on both the size of the two bands and the way the initial ensemble intersects these areas. The set of complex forming trajectories is composed of distinct subsets corresponding to the intersection of successive preimages of the outgoing turnstile with the initial ensemble.I6

40.0 ( 0 ) 1

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7 h Figure 10. Variation of phase space structure and complex formation with energy. For all plots the dark band is composed of phase points that never retum to the surface of section; the light points correspond to phase points that have no preimage on the surface of section. The collision ensemble is the dense line of points superimposed on the bands and corresponds to the initial conditions J = 0, b = 0. (a) E = 0.005 eV, (b) E = 0.01 eV, and (c) E = 0.1 eV.

We define the complex lifetime to be the time between the f i s t and last radial inner tuming points. It is important to note that there is no necessary correlation between the lifetime and the number of radial inner turning points. The complex lifetime will be longer for those initial conditions that fall near the stable manifold W of the periodic orbit corresponding to orbiting motion of Na+ about N2 (shown in Figure 8b). The stable manifold Ws of the orbiting periodic orbit forms the boundary between initial conditions that return to the surface of section and those that do not, and trajectories that pass close to the stable manifold spend a long time in the vicinity of the orbiting periodic orbit. We therefore expect that trajectory return times, hence lifetimes, will vary greatly according to the proximity of an initial condition to Ws.

We are able to explain the unusual appearance of the decay curve shown in Figure 5 for the ensemble at 0.03 eV using this idea. In Figure 1 1, we show a plot of the lifetimes of individual complexes as a function of the initial angle, yi, underneath a plot of the first intersections with the (y,p,) surface of section that give rise to these complexes, for E = 0.03 eV. There are four sets of complex forming points, symmetrically placed about y i = n/2, so that there are in fact two dynamically distinct sets of initial conditions that give rise to complexes. The first set of complex forming points, labeled A in Figure l l a , has a different decay curve than the second set, labeled B. Decay curves for both sets are shown in Figure 12.

The average complex lifetime of set A is longer than that of set B, due primarily to a significant time lag before decay of subensemble A begins. (The combined decay curve is shown in Figure 5 ) . The relation between the lifetime and the

Orbiting Complex Formation in Na+/Nz Collisions J. Phys. Chem., Vol. 99, No. 9, 1995 2441

The intersection of the collision ensembles with the (y ,pr) surface of section for energies lower than 0.05 eV are “looped” in appearance (although of course they are still connected 1-D sets) 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-dipole interaction. The polarizability of the diatom is anisotropic, and end-on collisions are energetically favored. (This changes at small r, where T-shaped configurations have lower energies.) Examination of incoming trajectories shows that at the lower energies studied they exhibit hindered rotation or librational motion in y , oscillating about y = 0 or n. When projected onto the (y ,p , ) plane, this oscillatory motion results in a coiling of the initial ensemble. As the collision energy decreases, the anisotropy of the long-range part of the potential has more time to act on the incoming trajectories, resulting in more pronounced libration and tighter looping of the initial ensemble about y = 0, n. As the energy increases, the long-range part of the potential has less effect.

It can clearly be seen from Figure 10 that the length of the curve corresponding to the intersection of the collision ensemble with the surface of section increases as E decreases. The ensemble is therefore “stretched”, and the density of points per unit length of the ensemble curve decreases with E.

The behavior of the complex formation probability over the whole range of collision energies may then be understood as follows: At very high collision energies, the angular dependence of the potential has little effect on trajectories over the short duration of the collision; put another way, turnstiles for complex formation are small. As the collision energy decreases, the influence of the nonspherical part of the potential increases, resulting in larger turnstiles and increasing complex formation probabilities. As E decreases still further, the stretching effect comes into play. That is, the stretching of the ensemble overcomes the effect of increase in tumstile size so that the complex formation probability begins to decrease. At this stage ( E = 0.05 eV in Figure 3), only a single segment of the initial ensemble intersects the tumstile region upon the first crossing of the (y ,py) section. Decreasing E still further, we reach the point at which a loop “pokes through’ into the tumstile region (cf. Figure 1 la), with a resulting rapid increase in the complex formation probability. As E is lowered, the stretching effect comes into play again and decreases the contribution from the new loop, until an energy is reached at which a new loop pokes through, and so on.

The constant-energy collision ensemble we use consists entirely of head-on collisions and has J = 0 (i.e., pb = b = 0), leaving the initial angle y’ as the only variable to be sampled in the ensemble. Inclusion of b t 0 initial conditions in the J = 0 ensemble will “wash out” the observed oscillations in the complex formation probability. Although the construction of a constant E, J = 0 ensemble that correctly samples the relevant phase space hypersurface is a nontrivial matter,34 we note that, when the J = 0 constraint is removed by, for example, allowing nonzero impact parameters b, the resulting complex formation probabilities 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 formation for three different potentials at the same total energy, E = 0.054 eV.

The potential described in section 11, which has both anisotropic attractive and repulsive terms, leads to the phase space structure of Figure 13a. For Figure 13b the potential is modified to make the attractive l/# term isotropic, by setting a1 = c t l = (a1 + ~1) /2 . In Figure 13c the potential consists of

2 0.0

I I -40.0 ’ I

0.0 1 .o

20.0

g. 15.0 c

J 10.0

s h

v c 5.0

0.0 0.0 0.2 0.4 0.6 0.8 1.0

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

1.2

.E L

e 0.6 P ‘E 5

0.0 0.0 3.0 6.0 9.0 12.0 15.0 18.0

time (au) (x io5)

Figure 12. Decay curve for subset A of the collision ensemble (see Figure 1 I), solid line; decay curve for subset B, dotted line.

proximity to the stable manifold of the orbiting periodic orbit can be clearly seen in the comparison of set A with set B. Each trajectory in set A, which consists of initial conditions tightly looped near Ws, must make at least one pass close the orbiting periodic orbit before it can dissociate, thus giving rise to the observed time lag in the decay curve for the subensemble.

The decay of the ensemble at E = 0.01 eV (Figure lob) is more nearly single exponential. For this ensemble, the loop corresponding to set A protrudes much further into the tumstile region, so that the majority of complex-forming trajectories in this subset are not in close proximity to the stable manifold of the orbiting trajectory .

2 . Complex Formation Probabilities. The (yp,,) surface of section enables us to understand the origin of the nonmonotonic behavior of the complex formation probability as a function of energy seen in Figure 3. As mentioned above, the complex formation probability is determined not only by the turnstile shape and area but also by the way in which the collision ensemble intersects the turnstile region on the surface of section.

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

40.0

h 0.0

-40.0

40.0

h 0.0

-40.0

40.0

h 0.0

-40.0

I 0.0 1 .o

r / . Figure 13. Influence of potential anisotropy on the phase space structure. The (y ,pr) surface of section shows initial conditions that never return to the surface of section (dark band) and initial conditions that have no preimage on the surface of section (light band). (a) Anisotropic repulsion and attraction; (b) anisotropic repulsion, isotropic attraction; (c) isotropic repulsion, anisotropic attraction.

the original anisotropic aitragive term, but the repulsive term is an isotropic function Ce-P.. The parameters C and p are obtained by averaging the repulsive part of the potential (3) over y and fitting the resulting function of r to exponential form.

Parts a and b of Figures 13 have a similar qualitative appearance, although in the isotropic attraction case (Figure 13b) the turnstiles are more “angular”. Theere is however a striking difference between parts a and c; making the repulsive term isotropic drastically reduces the size of the turnstile for complex formation. The insensitivity of the phase space structure to the anisotropy of the long-range attraction and the extreme sensitiv- ity of the turnstiles to the anisotropy of the repulsive part of the potential are consistent with the notion that the energy transfer leading to orbiting complex formation occurs through impulsive collisions at the inner turning point?

Although it is not our intention to provide a detailed critique of the work of Schelling and Castleman,s we now comment on the relation of their study to our findings. SC employ a potential in which the only source of coupling between relative translation and internal rotation is the anisotropy of the long-range part of the potential.8 In the absence of this anisotropy, all trajectories have zero lifetime according to our definition but have nonzero lifetimes (the time spent with r smaller than a given, arbitrary, value) according to SC. Adding long-range anisotropy then leads to some complex formation and an increase in complex lifetime according to either definition.

In the absence of anisotropy in the short-range part of the potential, the size of the turnstiles is reduced. Hence, complex formation probabilities are smaller than with the full potential. On the other hand, trajectory calculations using a potential with isotropic short-range interaction show that complex lifetimes

are comparable to those found for the full potential at given total energy.

VI. Conclusion and Summary

1. Our trajectory calculations of complex formation prob- abilities and lifetimes in Na+/N2 collisions show that the N2 vibrational degree of freedom does not couple effectively to the rotational and translational degrees of freedom and therefore does not play an important role in determining the complex lifetimes. This finding is in agreement with Schelling and Castleman’s study,8 though at variance with Brass and Schlier’s scaled H+/H2 study.9

2. The complex formation probabilities depend nonmono- tonically on the total energy for J = 0, b = 0 ensembles. The complex formation probabilities decrease with increasing energy only for E > 0.12 eV. At energies below 0.12 eV the complex formation probability exhibits oscillations.

3. The (rp,.) surface of section, previously used in the study of van der Waals complexes,I6 does not provide a particularly useful representation of phase space structure in the case of orbiting complex formation. Trajectories can “disappear” from the section, and the reactive separatrix and associated turnstiles have a complicated form. The (rp,.) section can however be used in conjunction with a four-map approach (cf. ref 32) to sample the full phase space.

4. In contrast to the (r,pr) surface section, the (y ,pr) surface of section provides considerable insight into the dynamics of orbiting complex formation. Using the (y ,py) surface of section, we are able to explain the low-energy oscillations seen in the complex formation probabilities, as well as the unusual decay curves seen at some energies.

5. The ( y ,p r ) surface of section shows that varying the anisotropy of the repulsive part of the potential has a much larger effect on complex formation than varying the anisotropy of the attractive term of the potential. The insensitivity of the complex formation turnstiles to the anisotropy of the attraction and the extreme sensitivity of the turnstiles to the anisotropy of the repulsion indicate that the complex formation process is a very impulsive one, dependent on the short-range potential. These findings are not in conflict with those of Schelling and Castleman,x as these authors use both a different definition of complex lifetime and a potential surface with no short-range anisotropy.

Acknowledgment. This work was supported by NSF Grants CHE-9101357 and CHE-9403572. Computations reported here were performed in part on the Come11 National Supercomputer Facility, 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 Reactivity; Oxford University Press: Oxford, 1987. (4) Chesnavich. W. J.; Bowers, M. T. f rog . 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, 97, 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 and references therein.

47.

(10) Tanuma, H.; Kita, S . ; Kusunoki, I.; Shimakura, N. Phys. Rev. A 1988, 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.

Orbiting Complex Formation in Na+/N2 Collisions

(12) See, for example: CHAOS 1993, 3 (4), focus issue on Chaotic Scattering.

(13) Rice, S . A,; Gaspard, P.; Nakamura, K. Adv. Classical Trajectory Methods 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 Stochastic Motion; Springer: Berlin, 1983. Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, 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, B. 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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