Gomathi Ramachandran and Gregory S. Ezra- Vibrational deactivation in Kr/O^+-2 collisions: Role of...

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Vibrational deactivation in Kr/O$ collisions: Role of complex formationand potential anisotropyGomathi Ramachandran and Gregory S. EzraBakerLaboratory, epartment f Chemistry, ornellUniversity,thaca,New York 14853(Received 11 May 1992;accepted20 July 1992)We report a theoretical study of vibrational deactivation of the diatomic ion 02 (u= 1) bylow-energy ( < 1 eV) collisions with Kr. It is shown that one-dimensionalcollision modelsare not able to reproduce the experimentally observedminimum in the deactivation rate as afunction of collision energy, even when attractive interactions are included. Classical trajec-tory calculations on an empirical potential surface ead to good agreementwith experiment,confirming the essential ole of rotational degreesof freedom in the deactivation process.Wetind that the upturn in the deactivation rate at low energies s due to the formation of orbit-ing complexes,as suggested y Ferguson. Our results show that the energy dependence f theassociation (complex formation) rate is an important factor in determining that of the deacti-vation rate at low collision energies,whereasorbiting complex lifetimes show relatively littlevariation over the energy range studied. At very low collision energies,our computed deacti-vation rate becomes ndependentof energy, n accord with recent experimentsof Hawley andSmith . We also investigate he role of potential anisotropy in the deactivation process. t issuggested hat the decrease n deactivation rate with increasedpotential well width (i.e., de-creasing anisotropy) is due to the elimination of a transient resonancebetweenhindered rota-tion and diatom vibration.

I. INTRODUCTIONAn understanding of the dynamics of thermal ion-neutral collisions is important for many processesnvolv-ing ions, such as ion-molecule reactions and vibrationalquenching.I* Consider, for example, the deactivation of avibrationally excited diatomic ion AB+ (v) by collisionwith a neutral speciesC. The usual Landau-Teller (LT)mode13 or V-P T energy transfer basedon purely repul-sive interactions predicts a monotonic increase n quench-ing rate with collision energy. Recent experiments usingion flow drift tube techniques**have shown that, for theion-neutral pair Nzf/I-Ie at collision energies arger than-0.25 eV, the vibrational quenching rate is fitted well bystandard LT theory. That is, a plot of ln(kquench) ersusT-3 is linear over almost all of the observed energyrange. For several other i on-neutral pairs studied to date,however, the vibrational deactivation rate is not in accordwith simple LT predictions at any energy. For 0: (v= l)/Kr, the deactivation rate as a function of collisionenergy exhibits a minimum, first decreasingand then in-

creasingwith increasing collision energy. A minimum inthe quenching rate is also seen n drift tube experimentsonNO+ (u = 1 /CH, (Ref. 12) and for very low-temperaturecollisions of NO+( U= 1) with He, Ar, and N, in a free etflow.t3 Many other ion-neutral pairs exhibit vibrational de-activation rates well in excessof those found in typicalneutral-neutral systems. Deviations from LT theory arealso found for neutral-neutral systems such asHCl-HC1,4 where strong attractive forces are present.To explain the anomalous behavior of the vibrationaldeactivation probability found for O$/Kr, the followingmechanismhas been proposedby Ferguson:,

kAB+(v)+C~~ABf(v)...C]*~AB+(U


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In Fergusons mechanism, the enhanced vibrationalquenching probability at low collision energies s a conse-quence of repeated collisions of the partners during thelifetime of the complex, so that each complex-forming col-lision is effectively equivalent to several direct collisions.The initial decrease n k, with relative collision energy isthen ascribed to a rapid decrease in the lifetime of theorbiting complex (i.e., increase in k,) with increasing in-ternal energy. At higher collision energies, deactivation bydirect ion-neutral collisions presumably takes over as thedominant mechanism, although the efficiency of these col-lisions is higher than predicted by simple LT consider-ations for Y+ T energy transfer.Ferguson has correlated the rate of vibrational deacti-vation according to scheme ( 1) with that for three-bodyassociation:% 4AB++Ce[AB+**-C]*+[AB+---Cl. (5)kl M

The steady-state rate coefficient kc3) for three-body associ-ation

k(3)= Usk:+MM 1

becomeskik-p k; k,r;u

(6)

for short-lived complexes (k,+ VT: > k&M]). Assuming7; - 7: and k, - ki gives

G. Ramachandran and G. S. Ezra: Kr/02+ collisions 6323ture rate (that is, the rate of passageover the centrifugalbarrier) in the presenceof long-range attractive forces, butthe dynamics of subsequent complex formation and decayis not fully understood (cf. Ref. 19). The fundamentalprocess involved in formation of orbiting complexes istransfer of energy from relative translation of the ion andneutral species nto and out of internal degreesof freedom.The orbiting complex [AB+ * * *c]* in the proposed mech-anism, Eq. ( 1 , is assumed to be formed by transfer ofenergy from translation to internal rotation; the reverseprocess of complex decay then occurs by rotational predis-sociation. If complex formation is important for vibra-tional deactivation at low energies, then rotational decou-pling approximations such as the infinite order suddenapproximation20,21 IOSA) would not be expected to de-scribe the quenching process accurately.

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Several theoretical studies have addressed Fergusonsmechanism. Tanner and Maricq have made a quantum-mechanical study of vibrational deactivation in N2/Li+,taken as a model for NOf/He.23 Using the IOS/distortedwave (DW) approximations,5 the deactivation cross sec-tion was found to exhibit a minimum as a function ofcollisional energy Ep2* Moreover, Tanner and Maricq con-cluded that the anomalous behavior of the deactivationcross section as a function of coll ision energy seen n Kr/Oz, for example, could be explained in terms of a LTtype expression for V+ T energy transfer modified to in-corporate the effects of attractive interactions, and thatcomplex formation and rotational degreesof f reedom playno role in the deactivation process.22 anner and Maricqsanalysis of vibrational quenching is discussed n more de-tail below. A modified LT treatment has also been given byParson.24Tosi et al. made a classical trajectory study of vibra-An approximately linear relation between k, and kc3 has,in fact, been found by Ferguson. Assuming efficient col-lisional deactivation, measured values of kq and kc3) thenyield values for the vibrational predissociation rate k, inthe range 109-10 s-.I0In the limit of very low collision energies, where theorbiting complex is extremely long-lived, k,#k,, and

k,-kc,, (9)so that the quenching rate is equal to the association rate.If the complex formation rate is approximately equal to theLangevin capture rate at very low coll ision energies, weexpect the vibrational quenching rate to become tempera-ture independent for collisions of ions with polarizablenonpolar neutrals. Such behavior has very recently beenobserved by Hawley and Smith for collisions of NO+ withAr and N2.t3

tional deactivation in Kr/O$, and were able to reproducethe observed dependence of the deactivat ion rate on colli-sion energy.2526 ignificant complex formation was foundat low energies, and an increasing fraction of vibrationaldeactivation occured via orbiting complexes with decreas-ing Ep Nevertheless, direct collisions made the dominantcontribution to deactivation even at the lowest energiesstudied26 (cf. Ref. 27). The preponderance of V-R andV-R, T over V- T energy transfer however suggested heessential mportance of rotational degreesof freedom. Clas-sical trajectory studies of vibrational quenching in colli-sions of N$ (u= 1) with He have been made by Zenevichet al,** while Goldfield has recently made a quantum-mechanical wave-packet study of the Kr/O$ system.

The mechanism of Eq. ( 1) raises several interestingdynamical quest ions. Ion-neutral interactions are typicallyintermediate in strength between weak van der Waals in-teractions and ordinary chemical bonds, and have accu-rately known long-range attractive components (ion/induced-dipole, ion quadrupole, etc.). Several theories ofLangevin type6* are available to calculate the total cap-

Based on scaling arguments and the results of trajec-tory calculations on H+ + H,, Brass and Schlier have sug-gested that energy transfer into diatom vibration is essen-tial for the formation of long-lived complexes.30In the present paper we further examine the role ofvibrational quenching via direct versus complex formingcollisions in the Kr/O$ system. Our study is carried outusing classical trajectories on a potential surface similar tothat used by Tosi et al.* Fergusons mechanism is reexam-ined, in particular, the interplay between orbiting complex

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formation and decay and vibrational predissociation. Therole of potential anisotropy is also explored.27*31In Section II we discuss n some detail the possibility offitting experimental quenching rate data for Kr/O$ usingdeactivation probabilities obtained from one-dimensionalmodels such as Landau-Teller theory modified to incorpo-rate attractive interactions. The Kr/O$ Hamiltonian andthe classical trajectory methodology used in this paper arepresented in Sec. III, while Sec. IV gives our trajectoryresults. Discussion and conclusions are given in Sec. V.

quantum-mechanical coupled channels problem, the valid-ity of the IOSA or related decoupling approximations suchas the coupled states approximation is difficult to judge. Itis nevertheless elatively straightforward to investigate thevalidity of rotational decoupling approximations withinclassical mechanics (see, for example Refs. 2 1 and 33). Astudy of classical rotational decoupling approximations forvibrational deactivation in ion-neutral collisions is cur-rently in progress.34

II. CAN ONE-DIMENSIONAL MODELS FIT THEEXPERIMENTAL DATA?Before turning to a description of our classical trajec-tory studies on the Kr/Oz system, we first address theimportant question: Is it possible to fit the experimentalrate constants for quenching of vibrationally excited 02 byKr as a function of collision energy using LT-type ex-pressions (or, more generally, expressions derived fromone-dimensional models)? That is, can a satisfactory fit tothe experimental data be obtained without invoking the

formation of orbiting complexes? A satisfactory fit to thedata must not only reproduce the observed shape of theexperimental rate curve for physically appropriate param-eter values, but should also yield absolute rate values inreasonableaccord with experiment.

The usual LT expression for the probability of vibra-tional deactivation in a single collinear collision between adiatom (reduced mass p, vibrational quantum &) in theu= 1 state and an atom (atom-diatom reduced mass M)interacting via a purely repulsive exponential interactionwith range parameter Q is,j8n2M2wp::o= 2a ~fi

Tanner and Maricq have given a critical discussion ofFergusons proposed mechanism for vibrational relaxationof diatomic ions via transient ion/neutral complex forma-tion, and have argued that it is possible to fit the Kr/O$data using a modified LT expression.22 he Li+/N,( u= 1)system, for which an ab initio surface is available,32 waschosen as a model for vibrational relaxation in NO+/Hecollisions. Coupled-channel equations for vibrationally in-elastic collisions in Li+/N,( u= 1) were solved using theDWA, and the IOSA was used to decouple rotationalchannels. The DWA/IOSA quantum calculations of Tan-ner and Maricq showed that the cross section for vibra-tional relaxation of N2( u= 1) exhibits a minimum as afunction of energy (Fig. 1 of Ref. 22). Moreover, even thes-wave cross section (Z=O) was found to have a mini-mum.22 On this basis, Tanner and Maricq concluded thattransfer of energy to molecular rotation, which wouldplay a role in stabilizing a transient complex . , thereforedoes not contr ibute to the unusual dependence of crosssection on energy.22

where k,=(2ME,)/fi and kO=[2M(E,+&)]12/fi arethe initial and final wave numbers, respectively. This ex-pression can be derived in a variety of ways;3-5 t is, forexample, the classical limit (2n%cda> 1 and 2n-k,/a+ 1) ofthe quantum-mechanical expression of Jackson andMott,35 which is obtained using the DWA. Tanner andMaricq suggestedan empirical modification to Eq. ( 10) toaccount for the influence of an attractive component of theatom/diatom interaction.22 The modification involves (a)replacing the collision energy Et with E,+E, where E s thedepth of the potential well and (b) replacing a by minusthe logarithmic derivative of the potential relative to thepotential minimum.22 To compute the deactivation rate, itis necessary o calculate a deactivation cross section whichmust then be multiplied by the relative collision velocity.Maricq and Tanner apparently obtain a deactivation crosssection for s-wave scattering by multiplying their modif iedLT probability by r/k:; the resulting cross section has aminimum as a function of Ep It is claimed that this expres-sion (suitably scaled) evaluated for a Morse interactionpotential is able to provide a reasonable it to cross-sectiondata extracted from experimental rate constant results onKr/O$ and to their DWA/IOSA calculations for Li+/N,(cf. Fig. 3 of Ref. 22).Several comments are in order here. We first note thatan expression for the quantum vibrational deactivationprobability with a Morse interaction potential

Furthermore, Tanner and Maricq concluded that theanomalous behavior of the vibrational deactivation crosssection as a function of collision energy for Kr/O$ (Ref.11) could be reproduced both by a suitable scaling of theLi+/N, DWA/IOSA results and by a LT type expressionfor V-t T energy transfer modified to incorporate the ef-fects of attractive interactions, so that complex formationand rotational degrees of freedom play no role in the de-activation process.22

Y(R)=D(~-~K(R-R~)_~~-~(R--R,)) (11)has been derived by Lennard-Jones and Devonshi re withinthe first-order DWA (Ref. 36) (see also Ref. 37):fiPK!G ~~&d)2

sinh ( 2rq, ) sinh ( 2nq,,) (A,* +-+I2 [cosh(2rq,) -cosh(2rqO)12 A,,+, (12)Use of the IOSA to decouple rotational degreesof free-dom of course precludes from the outset an assessmentofthe importance of rotational degreesof freedom for vibra-tional quenching. In the absenceof a solution to the full where

J. Chem. Phys., Vol. 97, No. 9, 1 November 1992

6324 G. Ramachandran and G. S. Ezra: Kr/Oi collisions

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2 lo-.g2 10-2I?Ql0-JP5 10-455 10"

G. Ramachandran and G. S. Ezra: Kr/O$ collisions100

1o-130 10-l

lo-'to-7

0.01 0.1 1 10Collision Energy (eV)

FIG. 1. Vibrational quenching probability as a function of collision en-ergy for one-dimensional models. Solid triangles, Mott-Jackson DWAresult for repulsive exponential (Ref. 36); open circles, Lennard-Jones-Devonshire DWA result for Morse potential (Refs. 37 and 38); opensquares, modified LT expression for Morse potential (Ref. 23).

A,= 1 ( -d+iq+f) 1 (13)and

It is of interest to examine pp-: as a function of Et and tocompare its behavior with the modified LT expression pro-posed by Maricq and Tanner, as the latter expression s anapproximation to the former.

(14) where the sum is over partial waves with orbital angularmomentum I, and collisions with I> &,,, have negligibledeactivation probability pilo( The full deactivationcross section is obtained by averaging Eq. ( 15) over allorientations y. The s-wave (Z=O) contribution to the crosssection (T,,~ is approximated by multiplying the collineardeactivation probability by rr/e [cf. Eq. ( 10) of Ref. 221In Fig. 1 we plot pp+: versus Et for a Morse interactionpotential with parameters appropriate for Kr/O$ (K=I.45 a.u., D=O.33 eV; see Sec. III). The first-orderDWA gives probabilities greater then unity for collisionenergies greater than an eV or so. The LJD deactivationprobability decreasesmonotonically with energy, becomingnearly constant for Et-O.01 eV. We have followed theLJD deactivation probability down to much lower collisionenergies ( - lo- eV), and have verified that it has the E:dependence required by the l/v law in the low-energylimit where scattering is predominantly s wave.4P38 hesemiclassical limit of the LJD expression [cf. Eqs. (6.9)-(6.14) of Ref. 3; Eq. (48) of Ref. 37; Ref. 391 s found toprovide a reasonable approximation to the full LJD resultin the energy range of interest. We also plot the Jackson-Mott deactivation probability35 for a repulsive exponentialinteraction with range parameter a=2~; this probabilitydecreasesmuch more rapidly with Et than the LJD result.The modified LT expression proposed by Tanner and Mar-icq can be seen to have qualitatively the same behavior asthe LJD probability, but is in error by about a factor of 10at the lowest-energy plotted.In order to calculate a quenching rate, it is necessary ocalculate the deactivation cross section al-c. Within theIOSA, the deactivation cross section oieo for collisions atfixed orientation y can be written asz2

6325

I' - ..'.'8*

' """'I 10.01 0.1 1 10Collision Energy (eV)

FIG. 2. Quenching rates as a function of collision energy for Kr/O:.Solid triangles, experimental data (Ref. 11); open squares, DWA deacti-vation probability for Morse potential times Langevin capture rate, Eq.(17); open stars, modified LT deactivation probability times Langevincapture rate, Rq. (17); solid squares, DWA deactivation probability forMorse potential times s-wave collision rate, Eq. (16); solid stars, modifiedLT deactivation probability times s-wave collision rate, Eq. ( 16).

q,o(y) =; ;;; (2Z+ l)Pllo(Y), (15)

n-Gl=qP1- * (16)The full deactivation cross section can be estimated bymultiplying the collinear deactivation probability by theLangevin capture cross section aL= 2rr( a/2E,) 2

&o=2T $ PI-O,( ) (17)twhere a is the polarizability of the neutral species (Kr inthe present case). Classically, this approximation assumesthat all collisions with impact parameter out to b,,, = I,,,/k, lead to equally efficient quenching. The maximum im-pact parameter b,,, a Et- 14.In Fig. 2 we compare quenching rates calculated in theway just described with experimental quenching rate coef-ficients for Kr/O$ as a function of collision energy. Thes-wave deactivation rate curves estimated using either theLJD or Tanner and Maricq deactivation probabilities for aMorse interaction show the same qualitative trend as theexperimental points. In particular, the theoretical curveshave shallow minima in the vicinity of Et-O. 1 eV. Thes-wave deactivation rates are however over 2 orders of mag-nitude smaller than the experimental rates. This is not sur-prising, since many partial waves will contribute to the



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G. Ramachandran and G. S. Ezra: Kr/O$ collisions 6327TABLE I. Potential parameters (atomic units).

BseDabc4I92r0a

1.4714 hohr-2.1089 bohr0.2447 hartree16.74 bohr1.6504 bohr-38.5 hartree7.3502X lo- hartree1.45 bohr-5.0 bohr75.0

ied. In contrast to systems such as He/N;,* it is thereforenot necessary to resort to devices such as the momentmethod43 or extracting vibrational deactivation probabili-ties from classical trajectory calculations.Selection of initial conditions for individual trajectoriesis straightforward. For given relative translational velocityU, Et=Mv2/2, the Kr atom is placed a large distance (R=60 a.u.) from the 0: center of mass. Values for the O$angular momentum p; the orientation yc, and the impactparameter b a re then chosen, and

p;= -MvR9The impact parameter distribution is taken to be uniformin the interval -6,, to b,,, as appropriate for planardynamics. The maximum value of the impact parameter,b ,,=, beyond which the vibrational deactivation probabilityis negligible, is energy dependent, and can be estimatedusing the Langevin formula for the critical impact param-eter in ion/induced-dipole collisions.6 At each energy b,,is, however, determined numerically. The angle p is cho-sen randomly in the interval 0-r (the angle range ishalved due to the symmetry of the diatom). A two-dimensional Boltzmann weighted distribution of (positive)

(DG

3oeq-m>

0d 0.0 3.8 7.6X (4

FIG. 3. Contour plot of the Kr/Oz potential surface, JZq. ( 19). The 0:molecule liea along the x axis, with the midpoint of the O-G bond at theorigin. The G-O bond length is fixed at S=S, Contours are at multiples of0.046 eV (cf. Ref.26).

J. Chem. Phys., Vol. 97, No. 9, 1 November 1992

integral p! values was used or the 0: rotor, correspondingto a rotational temperature of 300 K.For given p$ it is necessary o determine ps and s val-ues corresponding to the quasiclassical action J,= 1.5fi forthe initial vibrational state v= 1 of the rotating Morse O$oscillator. The vibrational action J, is a conservedquantityin the absence f the coupling term Vanisotropic,ven houghrotational and vibrational energies are not individuallyconserved during a cycle of the vibrational motion. A nu-merical method is used to calculate initial conditions forthe rotating Morse oscillator with v= 1 for arbit rary p!With the Kr atom held fixed a large distance from the 0;diatom, for a given value of& the initial vibrational energyis varied until the vibrational action (as determined bynumerical quadrature over ten vibrational periods) is1 XMXi. Once the appropriate value of the vibrational en-ergy is determined, it is fixed and s and ps values output atconstant time (i.e., angle) intervals over one period of thevibration. This procedure yields a set of {s,p,} values uni-formly spaced n vibrational phase angle for a trajectorywith a given py and fixed vibrational action. Sets of s and psvalues are generated n this way and stored for each inte-gral value of pr from ofi to 16%For each value of Ep an ensembleof 3000 trajectoriesis run. The final vibrational and rotational actions arebinned in the usual quasiclassical ashion.43Vibrational de-activation occurs if the final value of the vibrational actionJ, lies between 0.0 and 1.0.

D. Opacity functions and rate coefficientsFor given En the trajectory ensemble averages overvibrational phase,diatom angular momentum, diatom ori-

entation, and impact parameter. Trajectory results are con-veniently represented by computing the opacity function.(b).6*19 In terms of the opacity function, the planar de-activation cross section (cross length) is defined asuplanar - 2 s * dbI-I(b).0 (25)

lI( 6) is a probability distribution for successful colli-sions. If trajectories densely and uniformly sample b, and ifANJ b) is the number of successful (deactivating in thiscase) collisions and AN(b) is the total number of trajec-tories with impact parameter n the range b * (Ab/2), thenII(b) is approximated byAN,(b)--n(b)- AN(b) . (26)

Converting the integral to a summation over bins of widthAb givesANr(bi)

uplanar - 2C~~hN(b). (27)i iIf the trajectories uniformly sample impact parameter andif Max is a suitably large impact parameter such that anytrajectory with an impact parameter larger than b,,, haszero probability of deactivation, then AN(b) is given by



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6328 G. Ramachandran and G. S. Ezra: Kr/O$ collisions

N( AWb,,,), where N is the total number of trajectories.The expression or the cross length then becomes25.F CCLN,,

Hqamar - N i jj ,040 me

where the summation is over bins i, or 9%nax -E2 . .--~planar - N ; xkp (29) $% 10-l :

where the summation is over individual trajectories, and ::the characteristic function x is unity for a deactivatingtrajectory and zero otherwise. The cross length for com-plex formation is defined analogously; complex formingtrajectories are defined to be those with one or more outerturning points in the relative coordinate r.&To compare our planar computations with experimen-tally determined quenching rates, we must obtain an ef-fective three-dimensional cross section using our planarresults. In terms of the opacity function, the three-dimensional cross section is

FIG. 4. Vibrational quenching rate coefficients as a function of collisionenergy. Solid squares, trajectory results, solid triangles, scaled trajectoryresults; open circle-s, experiment (Ref. 11).

03D = 2a s m dbl-I,,(b)b. (30)0In the spirit of the centrifugal suddenapproximation, wenow assume hat

nI,,@) --n(b), (311i.e., the opacity function for three-dimensional collisionscan be approximated by that obtai ned n our planar calcu-lations. In terms of a sum over bins i

.n . .n n 8 0

80 AAem b0@ oAo d.62 A

10-12 1 * I * 0.01 0.1 1Collision Energy (eV)

perimental results of Kriegl et al.; in particular, the up-turn in the rate at the lowest experimental collision ener-gies s well reproduced.Our use of planar trajectory resultsto define an effective three-d imensional cross section over-estimates the rate coefficient by a factor of -6 (the theo-retical results have been scaled to match the experimentalpoint at E=O.147 eV).

where the summation is over bins i, or as a sum over in-dividual trajectories k,U3D- -277% ;x$k.

In terms of the effective three-dimensional cross section,the deactivation rate is estimated as

These results therefore confirm the finding of Tosiet al. 25 hat quasiclassical trajectory calculations with asuitable model potential surface are able to reproduce theobserved dependence f the quenching rate coefficient onrelative translational energy. Moreover, at very low colli-sion energies,our trajectory results show a leveling off ofthe rate coefficient in accord with the recent experiments ofHawley and Smith.13 n the rest of this paper we examinethe mechanism of deactivation in further detail.8. Collisional energy t ransfer

k,=qD . (34)The use of the planar opacity function to calculate a three-dimensional cross section leads to an overestimate of thedeactivation rate (see Sec. IV). A possible explanat ion isthat the planar calculation cannot account for relativelyinefficient collisions in which the rotor is oriented out ofplane.

The averageenergy transfer into or out of translation,rotation, and vibration was calculated separately or deac-tivating and nondeactivating trajectories. These values areplotted as a function of the collision energy in Fig. 5.

IV. RESULTS

Several broad trends can be seen in Fig. 5. At highcollision energies, all trajectories gain rotational energyand lose translational energy. Trajectories for which vibra-tional deactivation occurs lose more translational energyand gain more rotational energy than nondeactivating tra-jectories.

A. The quenching rate coefficientFigure 4 shows our calculated quenching rate coeffi-cient k, as a function of the collision energy Ep The po-tential sur face used is that descr ibed n Sec. III. The plotshows a broad minimum in the region of 0.33 eV, which isthe depth of the chemical well between Kr and 03. Itcan be seen rom Fig. 4 that the behavior of our theoreticalrate coefficient kq is in reasonableagreementwith the ex-

A strong correlation between otational excitation andexcitation of stretching vibrations has previously beenfound by Kreutz and Flynn in collisions of fast H atomswith C02.45For this system, a simple breathing ellipsoidmode14 rovides a useful basis or analysis of experimentalresults. In essence, xcitation of stretching modesof CO2 smost probable or those collisions in which the momentumtransfer has a sign&ant component along the CO, (semi-major) axis. Such collisions, which occur predominantl yJ. Chem. Phys., Vol. 97, No. 9, 1 November 1992



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G. Ramachandr an and G. S. Ezra: Kr/O.$ collisions 63290.04

3-3 0.02$%6&x 0.00e2W

3K-o.029a

-0.0410-2 10-l 100Collision Energy (eV)

.*: A:.:.,. A.j; -g _i-,,~~~.-~~.~~.-,~-:.-.4-1.-*, * * ,..*. *. .* l 4,; 2 ,*

8, L

FIG. 5. Average energy transfer for deactivating and nondeactivatingtrajectories as a function of collision energy for Kr/O$. Open symbols,nondeactivated trajectories; solid symbols, deactivating trajectories. Cir-cl=, A&.,,r; triangles, AE,,,; stars, AE,,.

near the 0 ends of the CO, molecule, also give rise tolarger than average otational excitatation4We have applied a planar breathing ellipse model topredict average energy transfers in high-energy (Et> 0.5eV) Kr-O,f collisions.46 The breathing ellipse results arefound to be in remarkably good agreement with our tra-jectory calculations.&At lower collisional energies he pattern shifts to onewhere deactivating trajectories gain both translational androtational energy at the expenseof vibrational energy. Thetranslational mode gains substantially more energy thanthe rotational mode at the very lowest collision energies.At high collision energy the transfer is thereforeV,T-R, while at lower collisional energies t is V+T,R.Pure V-+ T t ransfer is not observed,as previously noted byTosi et a1.26As also noted by Tosi et al.,26 the crossoverbetween these two patterns of energy transfer occursaround the collision energy where the broad minimum oc-curs in the quenching rate coefficient, and is suggestiveofthe existence of two mechanisms of vibrational deactiva-tion in the Kr/Oz system.26 n fact, the change n energytransfer pattern with decreasingenergy signals the onset ofdeactivation by complex forming collisions.C. Single versus multiple turning point collisions

In Fig. 6 we decompose he total vibrational quenchingrate of Fig. 4 into contributions from trajectories with asingle turning point (STP) in the relative coordinate r andthose with multiple turning points [ (MTP); theseare com-plex forming by aefinitiony. There is a substantial contri-bution to vibrational quenching from STP (i.e., direct)trajectories down to the very lowest energies (cf. Fig. 7 ofRef. 26), so that deactivation occurs both by direct colli-sions and via formation of transient complexes even at lowenergies.The importance of Fergusons mechanism for ex-

0.008

3st;.$.$f 0.0048CJ2B

0.000 L

A I .*.b AA

i *A Bar .

*10-Z

**,, h , *, * I I .,I, * I

10-l 100Collision Energy (eV)

FIG. 6. Quenching rate coefficient for Kr/Ot decomposed into contri-bution from single turning point (open triangles) and multiple turningpoint (open stars) trajectories.

plaining the experimental rate data is, however, apparentwhen we note that the quenching rate due to single turningpoint trajectories alone shows no upturn in rate at lowerenergies cf. Fig. 6)) tending instead to a constant value forenergiesbelow about 0.1 eV. Note that, at the lowest col-lision energy studied (0.005 eV), the MTP contribution tothe quenching rate itself levels off.For potentials that tend asymptotically to isotropicion/induced-dipole form, such as that studied here, therate of capture collisions (i.e., those in which the part-ners are able to pass over the centrifugal barrie& will atvery low collision energies end to a constant value givenby the Langevin rate 2~(a/M) 12.6 t low energies, here-fore, a constant fraction of the trajectories passingover thecentrifugal barrier leads to deactivation of the 0: vibra-tion in a single encounter, resulting in a constant value ofthe direct contribution to the deactivation rate. On theother hand, the contribution of MTP collisions to the rateincreases rapidly as Et decreasesand this increase is as-cribed in Fergusons mechanism to the combination of aconstant (essentially Langevin) rate for complex forma-tion [k, Eq. (2)] with an increasing unimolecular decaylifetime for orbiting complexes [TV, Eq. (4)l.rTo shed further light on the vibrational quenchingmechanism, we have examined opacity functions Il (6) forquenching by different types of trajectory. Figure 7 showsthat the opacity function for STP trajectories is flat out tob and approximately constant for three different ener-g:yin the low-energy regime, whereas he opacity functionfor MTP trajectories is not constant but increases as thecollision energy is decreased. Once over the centrifugalbarrier, then, trajectories at low energieshave an intrinsicprobability of deactivating by a direct collision that is es-sentially independent of collision energy and impact pa-rameter. This lack of dependence f the value of the opac-ity function on Et for STP collisions, although inconsistentwith the standard LT model, is consistent with a modifiedLT picture,22 n which the effective collision kinetic energy



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6330 G. Ramachandr an and G. S. Ezra: Kr/O$ collisions

0.0 3.0 6.0 9.0 12.0 15.0b (4

FIG. 7. Opacity functions at three collision energies. Solid line, E,=O.OleV; dotted line, E,=O.O385 eV; dashed line, &=0.147 eV. (a) MTPtrajectories; (b) STP trajectories; (c) all trajectories.

is gained by rolling into the potential well, For smallenoughE,, this effective kinetic energy s independentof EtAs E,+O, an increasing fraction of those trajectories thatdo not deactivate upon the first close encounter are subse-quently turned back, leading to a larger fraction of MTPcollisions (Fig. 8).D. Capture rates, complex lifetimes, and deactivationrates

We now examine n more detail the dynamics of vibra-tional deactivation of 0: by formation of transient orbit-ing complexes. In Fergusons kinetic model, the rate offormation of orbiting complexes is taken to be approxi-mately equal to the Langevin capture rate, which is inde-pendent of Et for ion/induced-dipole potentials. The ob-served ncrease n deactivation rate with decreasing elativetranslational energy s, in this model, a consequence f theincrease n rotational predissociation lifetime r, with de-creasing complex energy [Eq. (4)].To investigate the validity of this picture, we have cal-culated [Oz * * *Kr]* complex formation rates and complexlifetimes as a function of collision energy EC Recall that wedefine a complex forming trajectory to be one having oneor more outer turning points in the relative coordinate r. InFig. 9 we plot the effective three-dimensional association

m 0.15 -s.$E$ 0.10 -3eR 0.05 - l

0.00.,,,F 1o-2 10-l(4 Collision Energy (eV)

0.08

m 0.06s.9.$bs 0.04

0.00U.3

1o-2 1o-1Collision Energy (eV)

FIG. 8. Deactivating versus nondeactivating collision rates (a.u.). (a)STP trajectories. Solid circles, nondeactivating; open circles, deactivating.(b) MTP t rajectories. Solid triangles, nondeactivating; open triangles,deactivating.

(complex formation) rate derived from our planar trajec-tory calculations as a function of t ranslational energy EC Itcan be seen that, far from being approximately constantover the energy range of interest, the rate increases apidlyas Et decreases efore eveling off (cf. Ref. 19). Moreover,even at the lowest collision energy studied ( 10m2eV), thecomplex formation rate is only about one-half of theLangevin capture rate (cf. Ref. 19).The lifetime of an orbiting complex is defined as thetime between he first and last inner turning points in r.191MDirect trajectories have zero lifetime by definition. Thelifetime distribution for an ensembleof trajectories at fixedEt can often be reasonably well approximated on a times-tale of several picoseconds as an exponential function.Complex lifetimes thus determined are given in Table II. Itis apparent that complex lifetimes do not increase drasti-cally over the range of Et for which orbiting complexes areformed (the range s from 2.0-5.0 ps). Deactivating MTPtrajectories have ifetimes that are noticeably longer on theaverage han nondeactivating MTP trajectories (Table II).Our computations show that the increase n complexrotational predissociation lifetimes (7,) with decreasing



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G. Ramachandran and G. S. Ezra: Kr/O.$ collisions 63310.15

0.109

0.05

0.00

-7 1 rv.1

. . . . .-Longsvin

A AA A

*A

.A, , .l,,*1o-2 -1Collision Eneiiy (eV) 100

PIG. 9. Association (complex formation) rate as a function of collisionalenergy. The dotted l ine is the Langevin capture rate.

collisional energy (Et) is neither the only nor even themost significant factor accounting for the anomalousquenching behavior of the Kr/Oz system. The dominantfactor is the rapid increase n complex formation rate withdecreasing collision energy. Even at the lowest collisionenergies, he complex formation rate coefficient k, is onlyroughly half the value of the Langevin capture rate coeffi-cient. These findings and those of others (Ref. 19 and ref-erences herein) on the translational energy dependence fthe association rate are of some general nterest, as kineticschemes for ion-molecule reaction rates typically assumethat the rate of complex formation is approximately con-stant and equal to the Langevin capture rate.4749At very low collision energies, he ion-neutral associa-tion rate becomes rate determining [cf. Eq. (9)], and thequenching rate will then show the same temperature de-pendence as the association rate. An approach of thequenching rate to a constant value at very low collisionenergieshas recently been seenexperimentally by Hawleyand Smith.13We have also determined the effect of the 02 vibra-tional degree of f reedom on complex formation by calcu-TABLE II. Orbiting complex unimolecular decay lifetimes (picoseconds)and formation probabilities obtained using classical trajectories with andwithout inclusion of diatom vibration. rtot, complex lifetime for wholeensemble; rW lifetime for trajectories that undergo vibrational deactiva-tion; rW lifetime for nondeactivating trajectories. At high collision ener-gies (marked by an asterisk), lifetime estimates are unreliable due to poorstatistics. Complex formation probability pe is the ratio of the effective 3Dcomplex formation cross section to rrtim, (cf. Sec. II).

E, (eV) rtotVibration included Vibrationless

7-Q r PC 7, PC0.250* 2.02 2.02 0.019 1.31 0.0130.143* 2.22 2.13 2.27 0.088 1.77 0.0470.080 2.96 3.39 2.74 0.190 2.69 0.1100.036 4.12 4.12 3.71 0.312 3.25 0.2560.020 4.10 4.91 3.78 0.429 4.20 0.3330.010 5.08 6.01 4.63 0.502 6.17 0.397

u) 80.0 - jl.g ;I::::tj ::: :.% 60.0 - ; j-! j j: :% i 1b 40.0 - ; ;f

: :2 20.0 - :; ;fy,,

0.0 0.2 0.4 0.6 0.8 1.0Y/2K

PIG. 10. Trajectory funnelling and deactivation. The number of STPtrajectories versus the angle y at the inner turning point for E,=O.O2 eV.Dotted l ine, nondeactivated trajectories; solid line, deactivating trajecto-ries.

lating formation rates and lifetimes for trajectory ensem-bles in which the 02 vibration is frozen. Freezing thevibrational degree of f reedom results in a decrease n thecomplex formation rate and slight changes in lifetimes(Table II). Although complex lifetimes do change n thepresenceof vibration, neither the directions nor the mag-nitudes of the changes support the view that storage ofenergy n diatom vibration is essential or the formation oftransient complexes. oThis conclusion is consistent withour computations on the ion-molecule system Na+/N2,34where the presenceor absence f vibration is found to haveno effect on the complex lifetime. (In a trajectory study ofCl-/CH,Cl association, Vande Linde and Hase havefound that excitation of CH3Cl modes has only a minoreffect on lifetimes of Cl- * * *CH3C1complexes.50)The Kr/0; complex formation rate does,however, ncreasenotice-ably in the presence of vibration. This increase may beunderstood by noting that, for complex formation (multi-ple turning points in r) to occur at low collision energiesE, it is necessary o transfer only a small amount of energyout of translation into internal degreesof freedom. Classi-cally, even a very small amount of energy transfer intodiatom vibration facilitates complex formation.

E. Rotor orientation at inner turning pointand trajectory funnelingFor STP trajectories, there is a strong correlation be-tween vibrational deactivation and the orientation of thediatom (y) at the inner turning point. Trajectories thatdeactivate are more likely to approach such that the Kratom is at or near the chemical well (Fig. 10). All STPtrajectories are funneled either into one of the deepchemical wells or into one of the two shallow troughs at 90and 270, but deactivation occurs solely for those trajecto-ries that access he chemical wells. Trajectory funneling is

most pronounced at low collision energies (Fig. 11).J. Chem. Phys., Vol. 97, No. 9, 1 November 1992



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TABLE III. Complex formation rates (a.u.) and lifetimes (picoseconds)for potentials I, II, and III. At high collision energies (marked by anasterisk), lifetime estimates are unreliable due to poor statistics.

E, (WPotential I Potential II Potential IIIkc Ttot kc rtot kc Ttot

0.250* 0.0034 2.02 0.0028 1.16 0.0036 1.280.143* 0.0131 2.22 0.0096 2.22 0.0086 1.410.080 0.0271 2.96 0.0264 3.31 0.0169 3.290.036 0.0475 4.12 0.0473 4.36 0.0338 5.140.020 0.0548 4.10 0.0576 4.53 0.0474 6.610.010 0.0630 5.08 0.0625 5.44 0.0575 9.30

neutral association rates and orbiting complex lifetimescannot account for the decrease n deactivation rate at lowenergieswith decreasingpotential anisotropy.Our trajectory results show that at low collision ener-gies all deactivating trajectories (both STP and MTP)have at least one turning point in the angle y. We havetherefore examined the effect of potential anisotropy on thenature of the librational or rocking motion. In Fig. 14(a)we show a distribution of bend ing frequencies, where the

0.10

0.08

c 0.06.s8

t 0.04

0.02

0.00

G. Ramachandran and G. S. Ezra: Kr/O$ collisions 6333latter were estimated by taking the inverse of twice thetime between wo successive urning points in y (changesin sign of p,). Care was taken to ensure that the timeintervals corresponded o genuine ibration in a particularwell, rather than motion from one well to another. Thelibrational frequencieswere then binned and the fraction ofoscillations at a given frequency plotted. Fig. 14(a) showsthe distribution of bending frequencies or potentials I, II,and III, where the collisional energy or all three ensemblesis 0.01 eV. There is a marked trend towards lower frequen-cies as the potential well is widened. In Fig. 14(b) we showthe distribution of frequencies or deactivating trajectoriesfor potentials I and II; that is, we plot the distribution oflibrational frequencies estimated for the half cycle justprior to deactivation. It can be seen hat deactivating tra-jectories have higher averagebending frequencieson aver-age, approximately i to $ the zeroth order vibrational fre-quency for Or (u = 1) Our results suggest hat particularvibration-libration resonancesmay be responsible or deac-tivation at low collision energies. The softening of thebending vibration upon reduction of the potential anisot-ropy then eliminates the resonances, educing the quench-ing rate. More work is, however, required to elucidate thedetailed dynamics of the quenching process.V. CONCLUSION AND SUMMARY

The conclusions of our classical trajectory study ofvibrational quenching in Og/Kr collisions are as follows.( 1) We were unable to obtain a satisfactory fit to ex-perimental rate data using one-dimensional transitionprobabilities, such as the Lennard-Jones/DevonshireJ6ormodified Landau-Teller22expressions,even when the inter-action potential has a minimum (e.g., Morse potential).These results confirm the importance of rotational degreesof freedom in the quenching process.

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010(a) Frequency (au)

0.15

0.12

(2) Our quasiclassical trajectory calculations are ableto reproduce the observed behavior of the vibrationalquenching rate coefficient kq as a function of collision en-ergy Et in Kr/O$ collisions, in particular, the existenceofa minimum, thereby confirming the results of Tosi et al.25,26

c 0.09.ozeL 0.06

(3) At high energies,Kr/O,f collisions transfer energyfrom translation into rotation, whereasat low collision en-ergies both translation and rotation gain energy from vi-bration.(4) The upturn in the quenching rate at low energies sstrictly due to the increasing contribution of complex form-ing (multiple turning point) trajectories with decreasingED as suggested y Ferguson. Removing the contributionto the quenching rate from MTP trajectories eliminates theminimum.0.0000 0.0002 0.0004 0.0006 0.0008 0.0010Frequency (au)

PIG. 14. Distribution of approximate librational frequencies at E,=O.OleV. Solid line, potential I; dashed line, potential II; dotted line, potentialIII. (a) All trajectories; (b) deactivatingrajectories.

(5) A major factor governing the translational energydependence f the quenching rate via complex formation isthe marked Et dependenceof the ion-neutral associationrate in the energy range of interest. Even at the lowestcollision energy studied (5 X 10v3 eV), the associationcross section is only one-half of the Langevin value (cf.Ref. 19). This finding is in contrast to the picture of Fer-



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6334 G. Ramachandran and G. S. Ezra: Kr/Og collisions

guson,o in which only the energy dependence f the orbit-ing complex unimolecular decay lifetime is considered.(6) Ion-neutral complex lifetimes are of the order ofseveral picoseconds, and vary by a factor of 3 over theenergy range of interest. Complex lifetimes are insensitiveto the presence or absence of the diatomic vibrationalmode.(7) The quenching rate at low collision energies s verysensitive to variations in potential anisot ropy (i.e., the an-gular width of the potential well). Decreasing he potentialanisotropy greatly reduces he quenching rate. This reduc-tion in quenching rate is due neither to a decrease n ion-neutral association rate nor to a decrease n complex life-time. Our results suggest that the softening of thelibrational or rocking mode upon widening of the potentialwell greatly reduces the possibility of resonancewith thediatom vibration, but further work is required on this as-pect of the problem.

ACKNOWLEDGMENTSThis work was supported by Nat ional Science Foun-dation Grant No. CHE-9101357. Computations reportedhere were performed in part on the Cornell National Su-percomputer Facility, which is supported in part by theNSF and IBM Corporation. We thank Dr. Evi Goldfieldand Professor R. Parson for helpful discussions.

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Lindinger, T. D. Mark, and F. Howorka (Springer-Verlag. Wein,1984).16J. Tree, J. Chem. Phys. 87, 2773 (1987).D. C. Clary, Mol. Phys. 54, 605 (1985).*W. J. Chesnavich and M. T. Bowers, Prog. React. Kinet. 11, 137(1982).19(a) Part I, W. L. Hase and D.-F. Feng, J. Chem. Phys. 75,783 (1981);(b) Part II, K. N. Swa my and W. L. Hase, J. Chem. Phys. 77, 3011(1982); (c) Part III, K. N. Swamy and W. L. Hase, J. Am. Chem. Sot.106, 4071 (1984); (d) Part IV, S. R. Vande Linde and W. L. Hase,Comp. Phys. Comm. 51, 17 (1988); (e) Part V, W. L. Hase, C. L.Darling, and L. Zhu, J. Chem. Phys. 96, 8295 (1992).A. S. Dickinson, Comp. Phys. Comm. 17, 51 (1979).2T. Mulloney and G. C. Schatz, Chem. Phys. 45, 213 (1980).22J. J. Tanner and M. M. Maricq, Chem. Phys. Lett 138, 495 ( 1987).23W. Federer, W. Dobler, F. Howorka, W. Lindinger, M. Durup-Ferguson, and E. E. Ferguson, J. Chem. Phys. 83, 1032 (1985).21R. Parson (private communication).25P. Tosi, M. Ronchetti, and A. Lagana, Chem. Phys. Lett. 136, 398(1987).26P. Tosi, M. Ronchetti, and A. Lagana, J. Chem. Phys. 88,4814 (1988).27M. K. Osbom and I. W. M. Smith, Chem. Phys. 91, 13 ( 1984).28V. A. Zenevich, W. Freysinger, S. K. Pogrebnya, W. Lindinger, I. K.Dmitrieva, P. I. Porshnev, and P. Tosi, J. Chem. Phys. 94,7972 ( 1991).29E. Goldtield, J. Chem. Phys. 97, 1773 ( 1992).0. Brass and C. Schlier, J. Chem. Phys. 88, 936 (1988).F. J. Schelling and A. W. Castleman, Chem. Phys. Lett. 111,47 (1984).

32V. Staemmler, Chem. Phys. 7, 17 (1975).J. A. Harrison and H. R. Mayne, Chem. Phys. Lett. 158, 356 ( 1989).G. Ramachandran and G. S. Ezra, Chem. Phys. Lett. (in press).jJ M. Jackson and N. F. Mott, Proc. R. Sot. London Ser. A 129, 146(1930).36J E Lennard-Jones and A. F. Devonshire, Proc. R. Sot. London Ser.i 1;6, 253 (1937).A. Miklavc, J. Chem. Phys. 78, 4502 ( 1983).sL. D. Landau and E. M. Lifshitz, Quantum Mechanics (Addison-Wesley, New York, 1958).39Miklavcs result [Eq. (48) of Ref. 371 agrees with Cottrells semiclas-sical treatment &s. (6.9)-(6.14) of Ref. 31 except for an extra factorof 4 in the argument of the arctan in Miklavcs expression, which ap-pears to be due to an algebraic error. Cottrells quantum mechanicalexpression does not tend to his (correct) semiclassical expression, as hisexpression for the quantum-mechanical transition probability [Eq.(6.54) of Ref. 31 is incorrectly reported from a paper of Blythe [Eq.(1.6) of Ref. 511.mM. F. Jarrold, L. Misev, and M. T. Bowers, J. Chem. Phys. 81, 4369(1984). A. A. Radzig and B. M. Smimov, Reference Data on Atoms, Moleculesand Ions (Springer-Verlag, New York, 1985 ) .42W. H. Miller, Adv. Chem. Phys. 25, 69 ( 1974).Atom-Molecule Collision Theory, edited by R. B. Bernstein (Plenum,New York, 1979).L. M. Babcock and D. L. Thompson, J. Chem. Phys. 78, 2394 (1983).45T. G. Kreutz and G. W. Flynn, J. Chem. Phys. 93, 452 (1990).&G. Ramachandran and G. S. Ezra (unpublished).K. F. Lim and J. I. Brauman, J. Chem. Phys. 94, 7164 ( 1991).48W. E. Fameth and J. I. Brauman, J. Am. Chem. Sot. 98,789l (1976).49W. N. Olmstead and J. I. Brauman, J. Am. Chem. Sot. 99, 4219(1977).OS. Vande Linde and W. L. Hase, J. Chem. Phys. 93, 7962 (1990).A. R. Blythe, T. L. Cottrell, and A. W. Read, Trans. Faraday Sot. 57,935 (1961).

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Gomathi Ramachandran and Gregory S. Ezra- Vibrational deactivation in Kr/O^+-2 collisions: Role of...

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