A quasiclassical trajectory study of molecular energy transfer in H2He collisions

Chemical Physics 50 (1980) 175-194 0 North-Holland Publishing Company

A QUASICLASSICALTRAJECTORY STUDY OF MOLECliAii IN H2-He COLLISIONS John E. DOVE, Susanne RAYNOR Department

of Chemjstry,

* and Heshel TEITELBAUM

-.

ENERG-YTRANSFER. : **

..

.-

University of Toronio. Toronto. Ontarfo,,.CanadaMS.9 IA1

Received 23 July 1979 ,

I Molecuhubnergy transfer in ~ollisicnr of Hz with He was investigated by quasiclassical trajectory methods. us&, an ab initio interaction potential. AU of the initial parameters were Monte Carlo selected except for the vibrational and rotational quantum numbers (u. J). the translational energy, and the mitral separation of atom and molecule. A total of i50000 trajectories were calculated for 40 different (u, J) levels of para-Hz, in many cases at a number of different translational energies. For low-lying initial states, rotational excitation is the predominant process. Direct vibrational excitation of non-rotating molecules has a high classical threshold energy and a rehrtively low cross section above threshold. However, molecular rotation strongly enhances the probability of collisional excitation of vibration. For hi:-hly excited molecules, the domiIliUltprocess iS CO&SiOnal interconversion of rotational and viirational energy, for which huge cross sections are found. These tindings indicate that the rotational degree of freedom participates in both vibrational relaxation and dissociation at high temperatures- The mechanism of these processes is discussed.

1. Introduction

That they do play a role is indicated both by experimental measurements of the density changes [19,20] during relaxation and by previous theoretical studies [19-25]_ Questions of interest include the following. Does this effect of rotationally excited molecules mean that vibrational and rotational energy are readily interconverted during collisions? Or is the role of rotational excitation simply to enhance the interconversion of translational and vibrational energy? Are rotationally excited molecules important in.the dissociation process? What is the detailed mechanism of dissociation, and what transitions are rate:determining? These and other questions could be answered definitively if detailed state-to-state rate constants were available, for all signiiicant processes, and were used in a master equation calculation_

The kinetic behavior of a diatomic gas in an inert diluent at high temperatures is an important prototype for molecular relaxation and dissociation processes, and has been studied very extensively, both experimentally [l-7] and theoretically [4,7-13]_ The Hz -He system is of special interest for such studies because of the availability of ab initio quantum mechanical interaction potentials [14-181. In this paper, we rep0rt.a quasi-classical trajectory study of rotational and vibrational energy transfer in collisions of Hz with He. This study is part of a larger investigation of therates and detailed mechanisms of vibrational

relaxation

and dissociation

in this system

at shock tube temperatures. In the case of the relaxation process, there is considerable interest in the role that rotationally excited molecules appear to play.

The study reported here had two main objectives. Firstly, we wished to gain insight into the dynamics of energy transfer collisions between an atom and a diatomic molecule. Secondly, we wanted to obtain energy transfer cross sections for a number of key processes, especially in cases where accurate quantum mechanical calculationswould_ be very dif&mlt

* Present address: Department of Chemist, -Harvard University, Cambrid&, Massachusetts 02138, USA. ** Resent address: Department of Chemistry, University of Ottawa, Ottawa, Ontario, Canada. 175

176

J.E. Dove et al_ /Molecular energy transfer in HT-He

because of the large number of open channels. Such a study would also provide data for comparison with future measurements of state-to-state cross sections or rate constants using, for example, molecular beam techniques. In the present paper, we emphasize transitions among bound states, or from bound to quasibound states. Transitions to the continuum, i-e_ collision-induced dissociation processes, have already been discussed by us [26], and will be further considered in a future paper [27] on dissociation kinetics.

2. Methods We used the ab imtio Hz -He surface of Wilson et al. [ 171 because it covers the wide range of hydrogen internuclear distances needed to describe the energy transfer and Ha dissociation processes accurately. We combined their three-body interaction term, describing the deviation of the potential from painvise additivity, with data from Kolos and Wolniewicz [28] and Bishop and Shih [29] on the H-H pair potential; for the H-He potential we used the functions given by Gengenbach and Toermies [30]_ The agreement of our resultant (u, J) levels with those calculated for H, by Le Roy [31] is good, as described previously [26]. The Wilson potential includes data at large H-H distances, which we need to include when studying the behavior of highly excited molecules. Moreover, the potential agrees well with the recent surface of Raczkowski and Lester [18] at the short He-H distances which we expect to be lmportant in modelling energetic collisions_ However, it does not reproduce the very small van der Waals well [16,18] at longer distances, but this well would be insignificant at the energies involved in our calculations. Fuller details of the surface and fitting procedure have been given previously [26]. The trajectory calculations were carried out as descnied previously 1261, using a modification of a program developed by Schreiber [32] _All of the initial collision parameters were Monte Carlo selected except for the molecular quantum numbers u and J, the relative translational energy Eua,u, and the initial Ha-He distance which was always taken to be 7 A; at this distance, the force between atom and molecule is negligible. The fmal rotational and vibrational quantum numbers, Jr and ur, from each trajectory were calcu-

lafed from Jr-(& + 1) = lr x p/E I2

(1)

and h(ur + 4) = (8$‘*

X

1 ,

J’(f i- 1) fi* I’* dr 2u’

where r and p are the final position and momentum vectors of the H atoms, ,u and E are the reduced mass and total internal energy of the H, molecule, and r, and r> are the inner and outer turning points of the oscillator on the potential curve V(r). Since quasi-classical trajectories lead to non-Integer foal quantum numbers Jr and uf, the histogram technique of IaBudde and Bernstein [33] was used to quantize the product states (u’, J’): u’=IFIX(ur++),

(3a)

J’ = 21FIX ((Jr + 1)/2) ,

(3b)

where IFIX truncates a real number to an integer. Since we only considered para-Ht. J takes even values only; Previous work has shown that this method of quantizing trajectory results is quite reliable for transitions which are classically strongly allowed [33-39]Cross sections for energy transfer at a given initial relative translational energy, Ems, were calculated from

where Af = n(b& 1 - bf ) and Pi is the fraction of trajectories in the ith range of the impact parameter b which gave a product in (u’, J’) with foal relative energy EL_ Normally, we ran 1000 trajectories for each initial state at each initial energy Em. The resulting statistical uncertainty in the cross section for scattering into a given (u’, 5’) depended not only on the number of trajectories ending in that state but also, to some extent, on the way in which those particular trajectories were distributed across the impact. parameter b. The standard deviations for the foliowing values of the cross section in A2 then usually

LE. Dove et al. /

Mole&larenergy transfer in &-He

fell within the range indicated

in brackets: u = 0.003 it*(A0.=0.002-0.003);0_010(0.005-0_010);0.030 (O.Oll-0_026);0.10(0.03-0.07);0.30(0.05-OJS); LOO(O.lZ-0.33);3.00(0.26-0.55);10.0(0.6).The larger value in each range is generally a pessimistic estimate of the sandard deviation and occurred only rarely in our calculations. These cross sections can then be used to calculate state-to-state rate constants at a chosen temperature T by an appropriate integration over a Maxwell-Boltzmann distribution of relative translational energies.

3. Results

Table- 1 Quasi-classicaltrajectories.

Listof initial Hz &es and initial HZ-He relative translational e&&es studied in this work

InitiaLstate

for the s?ates chosen in this .way. Other calculations listed in table 1 were made in order to look for specific trends or to fmd the threshold energy for a process of interest. In the present paper, detailed consideration will be given to the following initS states: the ground state (40); (1,6), a state having moderate amounts of both vibrational and rotational energy; (1,O) and (12,O) which are rotationally unexcited but have respectively moderate and high v&rational energies; (OJ6) and (0,34) which are in the vibrational ground state but have intermediate and high rotational energies; and (5,24) which has substantial amounts of both vibrational and rotational energy. (0,34) and (5,24) are quasi-bound molecules. The energies of these seven selected states are listed in table 2. It will be realised that vibration and rotation cannot be rigorously separated, so that the values quoted for the rotational and vibrational energies are based on an approximate separation.

Translational enerees, Etcans (Cal mole-Lj

J

” 0

0

0 0

8 10 12 14 16 18 32 34 0 6

0 0 0 0

0 0 1

Using the methods outlined above, quasi-classical trajectories were calculated for the initial states and energies listed in table 1. Nomdly, 1000 trajectories were run for each initial state at each energy. The work reported here included a total of about 150 000 trajectory calculations. In selecting certain initial states for detailed study, the prirnary criterion was the choice of states which . would be representative of various regions of the quantum number ol energy space, e.g. unexcited molecules, vibrationally excited but rotationally unexcited molecules, and so on. Trajectory calculations were made for a wide range of initial translational energies,

.- 177

1 1 1

2 2 2 2 2 3 3 4 4 4 5 5 5 5 6 6 6 6 6 7 9 9 9 11 11 12 13

18 20 0 8 16 30 32 0 26 24 26 28 0

14 22 24 0 14 20 22 24 20 0 12 14 0 8 0 4

30,50,80,100,150.175,200, 250,300

30,40,50 30 30 30 5,12,20,3O,iO 30 IO,20, 30,50,75, 100 3,6,10, 21.5,31.5,40,50,75,loo 5, 10,20,30,40,50,60,70.80,90,100 10,20,30,50, 100 75,100,125,150,175,200 50 50 30 50 21 3,6,6.5,10,20,30,50,75,100 30 25 24.5 20 6 75.100,125,150,200 50.75 10,20,24,30,50,75,100 3,6, 10,20,30,35,40,50, 75,100 30 36.5 23.5 19.5 3,6, 10, 16.5,30,50,75,100 9,19, 29 6.10,20,35,50,75,100.150 23.5 10.5,20.5,30.5 10,15,24,30,50,75 3,6,10,20 3,6,10,20,30,35,40,50,75,100 3,6,10, 20, 30,50,75

4. Discussion

4.1. Comparikon with other trgiectoly cakulatiins There are no other trajectory calculations on the potential surface used by us, with which direct com-

J.E. 3ove et al_ / Molecular energy transfer iE HZ-He

178

obtained cross sections for inelastic scattering into a number of rotational states of u = 0 and u = l_ Cross sections from his work are compared with some of our results in table 3. The agreement is good, particularly for the lower energy states. The less close agreement for the higher J states may be due to differences in the potential, though in many cases the differences in the cross sections are still within the statistical uncertainties. Note that the differences between our surface and that of Gordon and Secrest are small for the low energy states but become substantial at higher energies. Ekrgeron and Chapuisat [40] have made a Iimited number of trajectory calculations of collisions of He with Ha and Da, using a potential surface of an analytic form matched to angle-averaged results of Gordon and Secrest [ 14]_ All of their trajectories were cahxrlated for zero impact parameter_ Their surface and methods, and the presentation of their results, are sufficiently different from our own that it is diff-

Table 2

Vibmtiomd, rotational and total energies (kczd mole-‘) of initkd states of pcm-Ha which arc considered in detail in the text 9) .I

U

‘%b

0

0

1 1

0

0 0

1;

E tot

E rot

6.23 18.17

0.00 0.00

6.23 18.17

6 16

18.17 6.23

6.60 38.80

24.78 45.03

34 240

6.23 104.48 59.28

112.36 51.21 0.00

118.60 104.48 110.50

a) The dissociation cncr_~ of the non-rotating molecule, from the minimum of the Potential ctnve = 109.50 kcal mole-‘. parisons can be made. However, Pattengill [39] has made trajectory calculations on the Gordon-Secrest [ 141 Hz-He potentiaI surface, for the (0,O) state at tmnsIational energies up to 224 kcal mole-’ _He

Table 3 IneIastic cross sections (9’) for scattering of Ha from (0.0) into (v’, J’), by coJJision with Hc. as a function of translationrtl energy. Comp;lrisun ofrcsults of this work with tmjcctory c&titions of PattengiJJ [39] Fiinsl stnte v: J-

Ref.

lnitil trmslationcd 28-4

1391

0.2

2) 0.4

0,6 0.

s

0.

10

0. I2 I. tJ 1.2 1,4 176 I.8 3) This work.

30.0

37.6

50.0

100

4.14

4.64

3.44

1.80

200 2.86

0.79

1.31

0.88

1.26

0.53

OS6 0.40

0.20

0.26

0.19

0.15

0.02

0.08

0.09

0.17

0.1 0.20

0.13

0.32

0.33

0.22

0.20

0.11

0.11

0.03

0.02

0.17

0.20

0.18

0.22

0.09

0.15

0.13

0.05

0.2 0.17

0.14 0.1

-co.05 0.04

0.47

0.2

0.1 0.07

0.49

0.3

0.1 0.04

0.75

0.1


1.17


co.05

0.02

2.69

0.1

co.05 0.10 -

2.55

0.2 0.40

0.48

300

0.5

o-1 0.06

250

1.1

0.7


224 3.2

3.09

1.65

<0.05 0.23

150

1.5

1.4 I.74

109 3.4

4.6

4.9

1391 0.9 ;r) 1391 <0.05 3 1391 ;I) 1391 n) 1391 ZJ 1391 3) 1391 a) 1391 a) [391 a) 1391 3)

energy, Et=&_ (kcaJ mole-‘)

0.17

0.04

LE. Dove et al. /koIecuIar energy transfer

cult to make a quantitative comparison. However, their fmding that molecular rotation enhances vibrational energy transfer agrees with our own results. It would be interesting to compare our calculations of energy transfer in H2 -He with other quasiclassical trajectory calculations using different inert collision partners. &is and Truhlar [34] have made quasiclassical trajectory computations for Hz-Ar on a potential surface obtained by using statisticai electron gas calculations together with potential energy data from other sources. They calculated 1059 trajectories on this Ha-Arpotentialfor theinitial state u=6,J=O = 64.8 kcal mole-‘, corresponding to a at Ebtotal collision energy of 132.5 kcal mole-’ _At this energy, they found scattering into a wide range of fmal states, with large changes in u and J occurring with signScant probability_ In our work on Hz-He, we did not study collisions involving this initial state (6,0) at this particular total energy. However, we have made quasiclassical trajectory calculations for the similar neighbouring state (5,O) in He at E= 75.0 kcal mole-’ , giving a total collision energy of 134.1 kcal mole-r, which is almost the same as in Bll and Trublar’s calculation. Our results in table 4 may be compared with table VII of Blais and Truhlar. Our total inelastic cross section (10.72 AZ) is substantially

-

in &-He

179.. _

smaller than theirs (33.1 AZ), presumably mainly because of the difference in size of the collision party ner. Allowing for this difference, the similarities between the results of the two sets of calculations are very striking_ Expressed as a percentage of the total inelastic cross section, very many of our cross sections are within a factor of two of those of Blais and Truhlar. The main qualitative difference between our calculations and theirs is that we fiid a somewhat greater propensity for large AJ changes than they do. This considerable general similarity of the energy transfer behavior of two different collision partners occurs for collisions of high total energy which populate a large number of fmal states. It would be very interesting to see whether this general similarity between Ha-He and Ha-Ar extends to other initial states and energies, including cases where the collisions are more selective, i.e. where a much smaller range of final states is populated. Unfortunately, the data for such a comparison are not yet available. 4.2. Comparisonwith quantum mechanical calculations Because of the difficulty and prohibitive expense of making accurate quantum calculations at high energies, only a limited range of data is available for comparison with our results. (The difficulty and expense of the quantum calculations is, of course, an impor-

-2-

0

20

40

E,,,,,

&:I rno&

100

6ig. l_ Logarithmic plot of the computed cross section (A*) for the transition (u. J) = (LO) -* (0, O), as a function of reIative translational energy in Hz-He collisions. A, quasidassic~ trajectory wkulations (this work). 0, quantum mechanical coupled states calculations by Alemnder [42]. 0, distorted wave calculations [25]. The line is a Le Roy type II function [43] fitted to Alexander’s results.

D I 0

20

I

I 40

r

Ia

Iv

III 100

Et,,., 6% mole-‘;1 Fig_ 2. Logarithmic plot of the computid cxoss &Aion (A2) for the transition (1, 0) + (0, 2), as a function of relative translational energy. (Other data as fig. 1.)

Table 4 Ir&astic cross sections for formation of vibration-rotation states (u’.J’) of HZ, by collision with He, calculated by quasiclassi& trajectories. InitiaI state u = 5,J = 0. Etrans = 75.0 kcal mole-‘, corresponding to a total collision energy of 134.1 kui mole-’ (including the zero point energy of Hz)-=)

v'=3

or= 4

u’= 5

u’=6

u’ = 7

0.006 0.05 0.02 0.01 0.04 0.02 0.04 0.05 0.01 0.08 0.03 0 0 0 0 0 0

0.03 0.07 0.05 0.09 0.04 0.12 0.07 0.10 0.07 0.05 0.01 0 0 0 0 0

0.05 0.47 0.26 0.56 0.17 0.30 0.25 0.05 0.01 0.003 0 0 0 0 0

2.9 1.4 0.62 0.44 0.15 0.03 0.03 0.006 0 0 0 0 0

0.07 0.29 0.28 0.19 0.04 0.08 0.03 0.006 0 0.003 0 0 0

0.003 0.05 0.05 0.10 0.12 0.11 0.01 0.01 0 0 0 0

0.41

0.70

2.07

5.59

0.97

0.45

J’

u’=O

u’= 1

“‘C2

0 2 4 6 8 10 12 I4 16 18 20 22 24 26 28 30 32 34 36

0 0 0 0 0 0.003 0.01 0.01 0 0.003 0.006 0 0 0 0 0 0 0 0

0 0 0.01 0.01 0.01 0.003 0.01 0.03 0.02 0.04 0.05 0.01 0 0 0 0 0 0

UJ’)

0.03

0.21

a) The ri& hand column is the sum over u’ of the cross sections for transitions into the givenJ’. The bottom row is the mm over J’ of the cross sections for transitions into the given u’_Because of rounding. the rows and cohzmns do not aUappear to sum CC&&. tant reason for using classical trajectories

to study tbis problem_) No full quantum calculationshave been made on the actual surface which we used. However, Alexander [41] has made quantum mechanical coupled states calctdations for the transitions (1 ,O) + (O,O), (I ,0) + (0,2), and (1,O) + (0,4) on the modified TsaplineKutzelnigg surface of Raczkowski and Lester [18] which resembles our surface in the features which are important for such calculations. AIexander’s calculations are compared with our results in figs. I,2 and 3. The translational energy range of our results does not overlap that treated by Alexander_ However, a Le Roy [42] type II function, fitted to Alexander’s results, in each case extrapolates satisfactorily to our data. Recently, Raczkowski et al. 1431, using a smaller basis set than Alexander; have made coupled charmel and effective potential calculations for these transitions. Their calculated cross sections into (0,O) and (0,2) are substantially larger than Alexander’s and,

if extrapolated in the same way, would agree less well with our results. For scattering into (0,4), the agree0

I

,

1

(

I

D

0

20

40

,

*

60

r

D

,

,

,

1

D

60

100

EIrans (kcol moC’1 Fig. 3. Logarithmic plot of the computed cross section (A*) for the transition (1,O) + (0,4), as a function of reIative translational energy. (Other data as fig_ 1.)

v’=8

u’ = 9

u’= 10

v'=ll

u'=12

v'=13

6 =-14

hllu')

0 0.05 0.02 0.01 0.04 0.06 0.01 0 0 0 0

0.003 0.02 0.03 0.006 ,0.003 0.01 0 0 0 0

0 0.01 0.003 0 0.006 0 0 0 0

0 0 0 0 0.003 0 0 0

0 0 0.003 0 0.003 0

0 0.003 0 0

0 '0 0

0.16 3.89 2.15 1.54 0.90 0.85 0.46 0.29 0.18 0.19 0.09 0.01 0 0 0 0 0 0 0

0.18

0.08

0.02

0.003

0.006

0.003

0

rnent appears satisfactory. We conclude that the very limited comparisons that can be made show reasonable concordance between our classical trajectory results and accurate quantum calculations. We *o note that Patter@ found that for the initial state (0, 0), using the Gordon-Secrest surface, classical calculations agree with the essential features of the quantum calculations. However, the classical method tends to underestimate the cross section at low energies. Fig. 1 may indicate such an effect in our calculations. Cross sections calculated by the distorted-wave method [44,45,25] incorporating the Mes anbarmonic factor [46] are also plotted in figs. 1-3. In all of the cases shown, the distorted wave calculations give smaller cross sections. The differences are particularly great iu cases where u and J change simultaneously. In order to solve the equations of the distorted wave method, approximations were made [25] ‘whichbad the effect of partially decoupling rotation from vibrationT It is evident that these approximations have caused the cat-

culated probabilities bf simultaneous changes in u and J to be substantially too small, as we have previously suggested [25]. 4.3. Collibim dynamia In our calcuhtions, we fmd that there is a substantial qualitative difference between the energy transfer behavior of low-lying initial states and that of highly excited molecules. The trajectories for the relatively low-lying initial states (0, 0), (1,O) and (1,6) show that the lower rotational states are very readily excited, but that vibration is very hard to excite in a molecule with little or no rotational energy. (Cf. tables 5,6 and 7.) From fig.~4 and table 6, we see tbat the cross section for the (1,O) to (1,2) transition has a low threshold, increases rapidly with increasing translational energy, and maximizes at a relatively large value (M A*) at about Ee = 20 kcai mole-l _It then declines at ’ higher translational energies; as transitions from (1 , 0)

J-E_Dove et al_/MolecuIar energy transfer in H2LHe

182

Table 5 Inemic cross sections (6.2) for transitions from (u = 0,

v* J’

M%

J= 0) to (v’. S)

in H2-He collisions, calculated by quasiclassical mjectofies

0(A2) b, 30 cl

50

80

100

150

17.5

200

250

300

0

0

0.00

-

-

-

-

-

0 0 0

2 4 6

1.02 3.37 6.95

4.1 1.7 0.23

4.6 1.8 0.79

3.7 1.8 0.94

3.4 1.6 0.88

3.1 1.3 0.53

e3 1.1 0.56

2.9 1.3 0.56

2.6 1.2 0.49

2.7 0.75 0.47

0 0 0 I I I 1 1 1 2 2 2 2 2 3 3 4

8 10 12 0 2 4 6 8 10 0 2 4 6 8 6 8 6

11.67 17.36 23.89 11.94 12.91 IS.14 18.55 23.02 28.41 23.21 24.13 26.24 29-47 33.70 39-73 43.73 49.34

O-06

0.42 0.07

0.48 0.10

0.01 0.02 0.02 0.003

0.02 0.04 0.04 0.07 0.04 0.04

0.40 0.20 0.19 0.02 0.09 0.20 0.17 0.17 0.08 0.02 0.01 0.02 0.02 0.01 0.003 0.006

0.40 0.21 0.22 0.05 0.17 0.13 0.14 0.12 0.07 O-006 0.03 0.02 0.04 0.02 0.01 0.006 0.003

0.40 0.26 0.15 0.08 0.17 0.13 0.14 0.04 0.09 0.003 0.05 0.03 0.05 0.03 0.02 0.02 0.003

0.32 0.22 0.11 0.03 0.17 0.18 0.09 0.13 D.05

0.33 0.20 0.11 0.02 0.20 0.22 0.15 0.05 0.08 O.Ob3 0.05 0.02 0.07 0.05 0.02 0.01 0.02

0.01

-

-

0.01 0.05 0.07 0.03 0.01 0.01 0.02

a’ nEi,t = E(u’. J’) - E(O.0) is the chan$e in internal cne~g (kcal moIe-‘1. b, Where no cross section is _tien. no transitions were observed (cross section iess tl% about 0.003 A*). From Etrans = 100 kcal molt -’ upwards, additional transitions not listed above wcrc observed. all with cross-sections less than 0.1 X2. Dissociation was &st obscrvcd at Et,,s = 300 kcal mole-‘. c) Column headings are initiaI translational energies (kcal mole-‘).

to hi$er J 1eveIs increase in importance. Transitions to J = 4 and higher levels appear to maximize in a similar way, but the maximum cross section for each successive J value is smalier and occurs at a higher Ethan the preceding one. Vibration is very much harder to excite for these initial states. Fig. 5 shows that the thresho!d for direct excitation of u = 1 from the (0,O) state is at a very Trajectory calculations using a high v&e of EW limited range of impact parameters (0.0 - OS A) at E,,= 65 kcal moIe_ ’ yieided a classical cross section of only 0.001 A* for transitions into u = I. It is clear that direct excitation ofu = 1 from (0,O) will be very slow, and that this process is most unlikely to play any significant role in vibrational relaxation in the temperature range below 3000 K in which relaxation measurements are usually made.

The most striking feature of these trajectory calculations is the Information which they give about the behavior of highly excited molecules. IT is found that such moIecules have a very strong tendency to interconvert rotational and vibrational energy on collision, rather than to exchange internal energy with the translational degrees of freedom. This is illustrated by figs. 6 and 7 which show the behavior of Hz in the states (1,6), (12, O), (0,34), (0,161 and (5,241, on collision with He at E= 50 kcal mole-‘. (Cf. tables 7-l 1.) Figs. 6 and 7 are contour diagrams showing how the vibrational energy of H2 molecules, initially in certain states, is distributed after a collision. The initial state before the collision is marked by a cross. The pattern of contbur lines near that cross represents the final distribution over vibrational and rotational ener-

:..

J.E.Doveetal./iKdlefulmenergyhan;sjerinH2-HeTable 6 ‘loebtic cross sections (~2)

_. for aao&ions

from (U = 1, J = 0) to (0’. S) in Hi -He collisions,

calculated by quasiclassid

183 ..-,I

tr&cio-

-..

ries u’

J’

nEiF$

,(A*, SC)

0

0

0 0

2

0

0 0 0 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3

4 6 8 10 12 0

2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 t 8

-11.34 -10.92 - 8.58 - 4.99 - 0.28 5.42 11.94 -0.00 0.97 3.20 6.60 11.07 16.47 22.65 11.27 12.18 14.30 1752 21.76 26.86 32.70 21.87 22.74 24.73 27.79 31.78

b) 10

20

30

50

40

60

0.01 0.02

-

-

0.29

3.9

4.8 1.7

4.6 2.2

0.004 0.04 0.03

0.04

0.05

0.02 0.004

0.07 0.03

3.9 2.1 1.04 0.26 0.01

3.9 1.9 1.02 0.35 0.05

0.01 0.02 0.01

0.03 0.04 0.04 0.006

-

4.3 1.8 0.84 0.12

70

80

0.02 a.03 0.04 0.10

0103 0.04 0.07 0.10

0.02 0.12 0.07 0.12

0.08 0.07

0.09 0.05 0.01 -

0.10 0.06 0.01 -

0.10 0.12 0.04

0.11 0.09 0.07

3-9 1.8 0.96 0.42 0.09 0.002 0.05 0.07 0.06 0.06 0.02

3.7 1.8 0.93 0.40 0.13 c.004 0.06 0.10 0.09 0.09 0.06

35 1.7 0.83 0.39 0.16 0.002 0.04 0.11 0.12 0.09 0.06

3.4 1.6 0.82 0.42 0.19 0.01 0.07 OJ8 0.15 0.10 0.06

0.002

0.02

0.03

0.03 0.01 0.002 0.005 0.003 0.005 0.01

90

0.003 0.01

100

0.09 0.13

a) &Tint

= E(u’, J’!- E&O) is the change in internal energy (kcal mob-‘). b, Where no cross section is given, no transitions were observed (cross section Iess than about 0.003 A2). Cl Column headings are initial translational energies (kcal mole-’

1.

gies. These diagrams were generated, for a given initial state, at a given initial translational ener,v Et_, by the following procedure. For each trajectory which yielded a bound product molecule, a final (non-integer) vibrational quantum number ur was calculated by the WIG3 method, as describcd above. The vibrational energy was then defined as .&,=E(uf,J=O), using spiine functions to interpolate EGb between integer vibrational quantum numbers. The rotational energy of the product molecule was calculated as the difference between the total inter& energy and .t?Ab_

’

A grid-pattern of “boxes” 10 kcal by 10 kcal in size was then set up in the energy space (E,,, E&, with the box sides parallel to the axes. One of these boxes was centered on the initial energy. The product molecules of the trajectories for a given initial state and energy were then allocated to boxes according to their fmal.energies (E&r, I&,), and a cross section was calculated for transitions to each box. However, before calculating the cross sections, elastic trajec-, tories were eliminated. (For this purpose, “elastic” trajectories were defied as those which would be elastic using the qu&&ation procedure described in’section 2, i.e. f&r which neither u nor J changed by more than 0.5 or 1.0 uni‘t respectively. However, the__

.

184

J.E. Dove et al. /MoIeculnr energy transfer in Hz-He

1

11.21

F!

Lt.7

I

.

__:___,__.__A-_,_,_, 1 1 . . s . I

0

is not substantially

ted by reasonable variations in the criterion sifying coIlSons

as elastic.) We omitted

40

_

L

80

120

Fig. 6. Contours of inelastic cross sections for Hz-He collisions, from trajectory calculations (this work), plotted as a function of final rotational and viirarional energy in kcal mole-‘. Units of the cross sections are A* per 10 kcal square- Contours are plotted at intervals of 0.2 A* in the ranze_Djl-1.0 A*. Initial translational energy = 50 kd mole - The three initial states shown are (1,6) (lower left), (0,34) (lower right), and (12.0) (upper left). In each case, the energy of the initial state is marked by a cross_ Elastic collisions were eliminated from the cross sections before plotting (see text). The strong diagonal elongation of the con-

Fig- 4. Inel;u;tic cross sections for the collisional scattering of Hz (v= 1, J = 0) by He into different rotational states (1, J), xs il function of relative translationti energy, calculated by quasi-chssia trajectories (this work).

form of the final contours

.

___. __ I\

tours for transitions from the initial states (12,O) and (0.34) shows the strong tendency of these states to interconvert

qffec-

rotational and vibrational energy on collision, rather than to exchange internal and translational energy.

for clas-

the elastic

n

trajectories because otherwise the Iarge peak in the

s___. u

80

_;_

.

_:

__;

L

.

I

‘

~

9

’

I

4.

---_--;--

___l-_.r____

n 5 w

: ; ---r-_~_-~--_~--

.

;

_a__

.

,_--i-_~,:_-j-_~-_l

40~_,__~___~--;--,~$&

I

I

-I t

t

---. 1

I

1

i

t rot

Fii_ 5. Logarithmic plot of cross sections for the vibrational excitation of ground state Hz (u = 0. J = 0) + (u’. _pJ’) by collision with He, as a function of reIative translational energy, calculated by quasiclassicai trajectories (this work)_ The lines arc Iabelled with u’.

120

fig. 7. Contour plots of inelastic cross sections for Hz-He collisions, from trajectory CalcuIations (this work), for the initial states (0.16) (lower) and (5,24)_ Other data as fig_6. Like (0.34) and (12.0), these two states also show astrong tendency to interconvert rotational and vibrational energy on collision.

LE.--Doveet aL/bfoiec&r en&

transfer inI& -He :

-185.’

Table 7

:. Inelastic cross sections (A’) for transitions fro& (v = 1, j= 6) to (u’, J’) g Hz--He collisions. calculated by qoasiclassicai trajeci& lies

U’

J’

ei:1

oui* ) b) 1oc)

0 0 0 0 0 0 0 0 1 1 1

1

1 1 1 1

0 2 4 6 8 10 12 14

0 2 4 6 a 10 12

20

50

100

0.05 0.33

0.01 0.06 0.11 0.08 0.09 0.13 0.03 0.06 0.62

0.04 0.04 0.17 0.33 0.07 0.20 0.20 0.12 0.09 0.53

1.9

1.9

1.5

1.5

-

1.7 0.10

1.9 0.53

1.6 0.62

0.13 0.006 0.02 0.03 0.08

0.21 0.08 0.08 0.04 0.10

0.11 0.03 0.10 0.006

0.35 0.10 0.09 0.02

-18.55 -17.53 -15.18 -11.59 -6.88 -1.18 5.34

30

0.01 0.02

11.54

-6.60 -5.63 -3.41 0.00 4.41 9.87

2 2 2 2

14 0 2 4 6

16.05 22.85 4.66 5.58 7.69 10.92

2 2 2

8 10 12

15.L5 20.26 26.10

0.006 0.53

0.98

'1tiint= E(u', J')- &I, 6)isthechange in internal enegy (knl mole-‘). b) Where no cross section is @en, no transitions were observed (cross section less than about 0.006 AZ)_ At 100 kcd translational energy, some additional transitions not listed above, alI withcross sections less than 0.06A*,were observed. c, Column headings are initial ‘Zanslational energies (kcal mole-‘).

centsal box was found to dominate tire subsequent fitting procedure unduly, tending to cause the generation of false minima elsewhere. Contour lines of equal cross section were then calculated using cubic splines, to give the results shown in figs. 6 and 7. These diagrams can be used to visuahse the dynamic behavior of the inelastic collisions as follows. If changes in Evib and Erot are essentiaby random (uncorrelated) and occur with equal probability, circtdar contours about the initial state would be expected. If rotation-translation (RT) interconversion is favoured, then the contours will be elongated horizontally, i.e. along the E,, axis, while if vibration-translation (VT) transfer is favoured, the contours wiIi be elongated verticabyl If vibrational and rotational energy tend to decrease or to Increase together, with accompanying opposed increase or decrease in the transla-

tional energy, then tire contours wiU tend to form an elongated sloping pattern which is directed towards or away from the origin. Finally, there is the case of vibration-rotation (VR) interconversion at approximately constant translational energy, i.e. vibrational energy increasing as rotational energy decreases and vice versa. This also wi.Ulead to a diagram which is elongated in a direction at about 45” to each axis, but this time tire elongation wiU be approxirnateIy parallel to the solid line representing the edge of the continuum. From fig. 6 we’see that, for the low energy state (1,6) at the left of the dir&am, collisions cause both En% and Emt to change in an apparently nncorrelated manner;with changes in E,,, being somewhat favoured over those in &+,. When these results are quantized, the tendency to favour changes in E,, is strongly

LE. Dove et al. /Molecular

186 Table 8 Inelastic cross sxuons

energy

transferin Hz-He

(A*) for transitions from (v = 0, J= 16) to (u’. J’) in Ha-He collisions, c&ulated

by quasiclassical trajec-

to&s

v*

I’

%S

o(A*)b) 5

0 0 0 0 0

8 IO 12 14 16

-27.14 -21.44 -14.92 -7.72 0.00

0 0 1 1 1 1 1 1 2 2 2 2 2 3 3

I8 20 8 IO 12 14 16 18 8 IO 13 14 16 10 12

8.10 16-45 -15.79 -10.39 -4.21 2.60 9.89 17.53 -5.10 0.00 5.84 12.26 19.13 9.74 15.24

9

20

12

0.01

0.21 -

-

0.04 0.01

” fin* = E(u’. J’) - E(O.16) is the change in internal energy (kcal mole-‘). b) Where ~ro cross section isgiven, no transitions were_observed (cress section observed, at these ener$s, are listed above. c) Column Iw~ding arc inititlI tmnsIation31 energies (kzd mole-‘).

accentuated by the fact that at low IJ and J, the energy spacing of theJ states is much closer than that of the v states. Therefore very many trajectories lead to a change of at least one unit in J, but very few of them cause a change in E”ib which is large enough ro correspond to a change in u of 0.5 unit or more. Thus this diagram essenrizdly co&is the picture derived from fig. l-5. However, it does also show that, in a purely classical calculation, the tendency to favour rotational as opposed to vibrational energy transfer is not as grear as might be supposed from an examination of the quamized results. For the states of high internal energy, (0,34) and (12,O) in fig_ 6, and (0,16) and (5,24) in fig. 7, the energy transfer behavior is quite different. VR transitions, with refatively little change of translational energy. are very strongly favoured. This is shown by

30

0.005 0.44

0.15

0.21

0.54 0.08

0.05 1.2 0.26

less than

50 0.01 0.01 0.10 1.1 0.78 0.01 0.02 0.04 0.27 0.88 0.44 0.01 0.005 0.05 0.07 0.09 0.005 0.01 0.01

about 0.005 A*). Ail of ffie transitions

the way in which the contours are elongated along the diagonal, approximately parallel to the boundary of the continuum. The dynamical behavior was also investigated numerically by looking for correlations between the change in rotauonal energy, AE,,, = EL, - Efot, and the change in vibrational energy AEtib =E,& - E&, where E$,, and Esb are the energies of the initial state. If ti,, is strongly correlated with ~“ib, then points representing (Ekt, E&J will tend to fall along a line in the (I?,,. Etib) plane. A particularly simple case -willbe where thk line is straight. Accordingly, for a number of cases, we fitted the results of our trajectory c&ulations by linear least-squares to the expression

(5)

LE. Dove?etai./ hfokc&&~

.-~

mun_vf&inH2_ffe

:

Table9 Inelastic cross sections (A~) for -uamitions from (U = 5, J = 24) to (u’; I’) in Hz& tories

U’ J’

AE;’

30 26 28 30 24 26 28 20 22 24 26 20

6 6 7 7 7 8 8 9 9

22 24 18 20 22 16 18 14 16

4.05 -3.84 2.42 8.12 -4.77 1.08 6.37 --ll.b7 -5.40 0.00

4.88 -5.73 -0.80 3.63 -S-76 -1.34 2.63 -5.50 -1.63 -4.98 -1.68

.: _,-::

..

_:_A-.: ’f

collisions, calculated by ~uas&xs&af i;aje&: ..

u(A2) b, 38

2 3 3 3 4 4 4 5 5 5 5 6

1, :~ .- :IS;-Y,

6

10

1.67

0.02 3.0 0.006

0.23 0.11 1.1 0.01 0.35 2.3 0.006

0.09 -

0.006 -

0.05 -

0.09

0.10

0.11 0.23 3.7 0.20 0.17 1.2 0.23 0.19 0.76 0.21 0.46

O-19

2.6 0.003 0.75 0.006 0.11 0.003

0.23 3.2 0.03 0.17 0.96 0.04 0.46 0.04 0.26

20

30 0.26 I 0.77 0.55 0.17 0.84 1.6 0.16 0.10 0.65

0.27 0.07 0.71 0.18 0.38 2.1 0.42 0.003 0.27 0.42

0.29

0.27 2.8 0.46 0.31 o-43 0.32 0.27 0.24 0.19 0.09

0.41 1.8 0.68 0.13 0.54 0.23 0.05 0.23 0.15 0.19

35 0.30 0.84 0.37 0.30 0.98 1.2 0.35 0.03 0.33

0.39 1.7 0.57 0.32 0.45 0.19 0.18 0.32 0.19 0.17

40

SO

75

.- 100

0.21 0.62 0.41 0.43 0.50 1.6 0.50 0.07 0.37

0.07 0.43 0.47 0.45 0.32 1.7 0.49 0.08 0.63

0.13 0.26 0.25 0.23 0.92 1.7 0.31 0.06 0.65 -

0.10 0.31 0.20 0.10 1.0 1.0 0.29 0.13 0.50

0.43 0.35

0.59 0.26 1.3 0.71 0.17 0.23 0.18 0.15 0.07 0.07 0.13

0.57 0.2s

0.77 0.36

1.1

0.76

0.53 0.39 0.19 0.24 0.08 0.12 0.01 0.08

0.44 0.12 0.26 0.24 0.12 0.12 0.08 0.08

1.3 o-49 0.30 0.25 0.02 0.09 0.23 0.10 0.25

a) hEirIt = E(u’, J’) - E(5,24) is the change in internal ener_g (Ioralmole-‘). b, Where no value of the cross section is given, no transitions were observed (cross section less than about 0.01 A’). The above table lists only the main transitions observed. Many other transitions also occurred, especially at the higher translationalenergies. Co&ion induced dissociation began at Etrsns = 6.0 kcal mole-‘, and at higher energies a substantial proportion of trajectories let to dissociation. At 35 kcal, only 500 trajectories were run, giving larger standard deviations than normal. c) Column headings are initial translational energies &xl mole-‘)-

This

is also equivalent to fitting the values of the foal energies E&, and J!$,_,,to

E&

=aE&,

+(E&

-aE&,+c)=~E&,,_+c’.

(6)

For these fits, trajectories were grouped Into batches according to the initial impact parameter_ This was done in order to see whether there were important differences between the patterns of dymunic behavior of small and large impact parameter trajectories. For the initial state (1,6), at relatively low (10 kcal) and high (50 kcal) translationrd energies, essentiahy no correlation was found between A&,, and A&b at either low or high impact parameters. The coefficient of correlation of the fit (table 12) was small and fluctuated about zero. For the initial states (5,14), (4,20), and (5,24),

a strong negative correlation was found (table 12) between AE,, and AEvrr,, as would be expected if VR transfer is dominant. In most cases, a is close to -I .O, implying a fairly pure VR process with relatively little interconversion of internal and translational energy_ However, for (5,24) at E& = SO kcal, a = -0.7 to -0.9, impIying that VR interconversion is accompanied by substantial RT transfer. The coefficient of correlation, r, for these fits is in nearly all caSf!S between -09 and -1.0, and is often very close to -1.0, indicating a very strong correlation. (The coefr used here is the square root of the correla-f rcrent . tion coefficient 3 which is often quoted, and is negative for a negative correlation.) Large impact parameter collisions are less efficient at transferring energy; which is of course entirely expected. However, there

.- -. .-, ._

2.

Table LO inelastic cross sections (A*> for tramitions

from [u = 12, J = 0) to (u’, .I’) in Hz-He

from quasiclassical tra-

collisions, calculated

jectories v’ .I

o(A’) b)

nEian)t

3 c) 9 12 10 10 11 0 11 2 11 4 11 6 11 8

-2.33 -0.92 -4.02 -3.59 -2.61 -1.16 0.66

12 20 12 4 12 6 12 8 sz 10 L3 2 13 4

0.35 0.00 1.14 2.29 3.68 5.10 3.21 3.78

6.8 1.8 0.03

6

IO

20

30

3s

40

so

75

100

0.006 0.20 0.x1 0.33 0.48 1.2 0.43

0.13 0.45 0.14 0.48 0.71 0.90 0.81

O-IS 0.25 0.45 0.45 0.45 0.72 O-71

0.22 0.34 0.22 0.54 0.97 0.53 0.74

0.17 0.23 0.22 0.81 0.28 0.30

0.25 0.35 0.08 0.70 0.60 0.28 0.45

0.33 0.16 0.33 0.78 O-49 0.49 0.34

0.14 0.21 0.44 0.50 0.43 0.25 0.29

0.09 0.11 0.36 0.54 0.38 0.24 0.41

6.3 A3 0.96 0.04

- 6.4 2.3 1.3 0.16 0.13 O.L6 0.57

4.2 1.5 0.52 O-15 0.13 0.45 OJO

3.8 0.92 0.72 0.14 0.09 0.39 0.16

4.0 _ 1.7 0.57 0.50 0.15 0.67 O-L5

4.0 1.2 0.77 0.13 0.30 0.35 0.07

3.1 1.3 0.39 0.20 0.04 0.24 0.36

3.0 0.65 0.45 0.34 0.11 0.57 0.15

i4 0.74 0.45 0.08

0.09 0.11

0.26 0.14

a’ lint

= ECU’, J’) - E(12,O) is the change in internal energy (Led mole-‘). b, Wi~eherc no value of the cross section is given, no transitions were observed (cross section Iess than about 0.01 A’). The table lists

only the main transitions observed. Many other transitions also occurred, especially at the higher transLational energies; at 100 kc& about three quarters of the bound states were populated by these transitions. Co&ion induced dissociation began at a rranslational energy of 6 kc;tI moIe_‘_ At IO0 kc& host half of the trajectories led to dissociation. The high proportion of dissochrive tmjectories tends fo incwse the seatfer of the cross sections into the boundstates. For translational ener&s of 6, 35 and 40 kcal mole-’ ) only 500 trJjectori% each were run, so that these cases have larger than normal standard deviations. c, Column headings are initial translarional enemes (kc-& mole-‘).

Inelastic cross sections (.A*) for transitions riL% v’

0

J’

0

32 34

0 1 1 1 2 2 2 3 3 3 4 4 4

36 30 32 34 28 30 32 26 28 30 24 26 28

diZi

6.88 -9.17 -2.14 4.33 -10.69 -4.03 2.05 -11.92 -5.65 0.04 -12.85 -7.00 -1.70

collisions, calcuiated

by quasiclassicd

tmjecto-

0(_%3b) SC)

6

10

21.5

31.5

40

so

75

100

-

0.06 -

o-10 -

0.24 -

0.88 -

1.03 -

0.69 -

1.3 -

-

0.03 0.04 2.1 0.09 0.01 0.44 0.05

0.35 0.29 2.2 058 0.15 0.92 0.14 0.11 0.37 0.11

0.72 0.24 2.4 0.68 0.22 0.75 0.22 0.36 020 0.25 0.25 0.09 0.14

0.82 0.24 1.7 0.88 0.20 0.63 0.35 0.22 0.11 0.17 0.12 0.12 O-15

1.3 0.49 1.2 0.80 0.19 0.33 0.31 0.16 O-14 0.17 0.03 0.08 0.05

0.86 0.43 0.79 1.2 0.14 0.14 0.17 0.22 0.11 0.16 0.07 0.03 0.01

1.7 0.25 0.99 0.35 0.15 0.20 0.28 0.03 0.06 0.07 0.04 0.04 0.01

-7.37

0.00

from (u = 0. J = 34) to (v’. J’) in Hz-He

0.16 0.01

0.06 0.02

1D

‘) AEiot = E(u’, J’) - E(0. 3?) is the &USC in internal energy (kc& mole-‘). b, Where no value of the cross section is &WI, no transitions were observed (cross section less than about 0.005 A2). The table lists onLy theprincipal transitions_ Other transitions began to occur above 10 kcal transLational energy; at 75 and 100 kcal, about haLf of the bound states were populated by these transitions. Dissociation began at 21.5 kcal translational energy c, Column headings are initial translational energies (kcaL mole-‘).

,’

10 10

6 6

6 6

1 1

1 1

50

6 6,

50 50 50 SO.

5 14

524 524

5 5 5 5

53.74

23.08

43,26

6,60.

b, Got

59,2a

59028

49,99

18.17

@ib

rC)

-0099 -LOO -0.92 -0,84

O,OO 0,50 OS0 LOO 1.00 2,oo 2BOO3.00

-0077 -0,89 -0,79 -0,70

0.00 0.50 -0.93 2.00 3,oo -0,92

0,OO 0,50 -1.45

0,OO OS0 o,so LOO LOO 2.00 2800 3,oo

O,OO OJO -1.025 2,oo 3‘00 -1,022

-8,3 -4,8 -0,5 0,3

0.19 -0,Ol

-LO

-1,64 OS30 -0,33 -0Jl2

-146 -0,01

-9

-5,s -2.6 *0,5 to,4

-0,94 -0.96 -0,98 -0,96

-16 - 680 - OS -2.8

-0.9993 t7.7 -0,9998 +0,12

-0079

-0.91 -0,92 -0,93 -0,96

(urot)

+lO + 1.4 -0,s +4,4

-8,l -0,14

+7

+3,9 f2.9 -0,9 -a,5

-187 -0,lO

+ I,? 0,oo

+ 0,Ol -OS6 -OS02 -0,03

(&ib)

lo.9997 + 2.0 ,-0.9999 +0,09

+ 0003 -0eO8 0.26 -0,Ol

C

0,oo OS0 2,oo 3,oo

a

-0008 + 0,04

Max

0,oo OS0 2,oo 3,oo

b$[n

27 25 15 4.1

13.5 106

22

27 24 11 1.5

9,9 1.24

5.3 O,O5

0.15 0,03

(luyjbl)

33 29 la 5.6

14s 1.7

la

26 23 12 1,6

i7,8., 205 37 .33’ ;33’ 1.6:

32 30 19 5.2

‘23,j.i

..29 27 .14 2,5

1*43

10.4

.’

,, ,,.’

.I’

::,. ,,’

1,‘: I

“,‘1

2 ,$

6 g ;b”. % p

‘. ,‘I .,

,.

,’ ,,: .“, “Ifs

‘,,

‘,

5 4, m

!

*’

B-

P : ‘. p

‘,’

,I’

6.8 0.14 “’

1.45 ‘, 0.12

(uro&ms

16.6 2.3

26

29 21 13 2.1

1097 1846

9s

1,21

6.1 0,07

0.17 0,03

(uvib)rms

494 OJO

1,07 0405

(Wrotl)

:

‘:, ‘1 The results of the individunltIajCCtOriCswcrc fitted to the cxprcssion&vib = aaErot + c, :’ ’ @Energiesarc in kcalmole” . Superscriptzero indicalesm initial value.rms = root meansquare, .,‘, ‘, ‘1 Irhpnctparamctcrsof trajcctorluswcrcMonteWo scfcctcdovertllc rang b,,in to lrmlrx(,4), d, Tl~c~i~~e;lr~corrclation cocfficicnt,r, is given by I = *Jr?, and is ncgativc for I negative corrclntion (ncgativca),rz is thc,stntisticalcocfficicntof dctcrmination,

24 24 24 24

50 50 50 so

20 20 20 20

4 4 4 4

6’

,6

420 420

50 SO

E&s

u J

Table 12 Test of linear corrclalionof amounts of vibrational and rotationalenergytransfcrrcdon collisionwith Hc, for certaininitial stotcsof H2 a)

is otherwise no strong difference indicated between the dynamic behavior of Iow and high impact paramster collisions. There is some tendency at high impact parameters for rotationa energy transfer to be favoured over vibrational, and for ME,, and A.&,, to be even more cIoseIy correlated, but neither of these :rends is at all strong: Contour lines of constant cross section (%obarns”?) are plotted in fig- 8 as a function of the quantum numbers J and u. (We were able to fit the contours fur this diagram without having to elimiiate rfle elastic peak.) Fig. 8 shows how the cIassical energy changes described above translate into changes of ~Iuantum state. The differences between this diagram

and figs. 6 and 7, in which the contours are plotted against (&,, E,,,), are most significant for the states (1,6) and (12,O). These differences arise mainly from the fact that the energy level spacings increase strongly with J (except at high J) but decrease sIowly with u. In consequence, the tendency for the state (1,6) to undergo RT rather than VT or VR transitions is enhanced, as mentioned above. Also, whereas classically the state (12,O) has a very strong propensity towards VR transitions, on quantization many of these transitions are found to be classified as RT processes because Au < 0.5. The extent to which rotational energy of a molecule enhances the probability of making a vibrational tran-

0

0

36

0

12

J

24

36

24

36

I@- 8. Contour plots of cross sections in A* for scattering into (u’, J’) in Hz-He collisions. from quasiclassical trajectory nlcuktIions (this wurk). Note that, unlil;e fig. 6 and 7, the cross sections are pIottcd as a function of the quantum numbers and not of Ihe intcrnaf encrgicn Initial relative translational energies are (a) 10 kcal mob-‘, (b) 20 kcal mole-‘, (c) 50 kcal mole-t and (d) 100 kcal mole-‘. and the in&i stares (cxh marked with a cross) a;e (1.6), (0,16). (0.34), (5.24) and (12,O). However, the comours for (0. 16) in la) are for a rranslationrrl energ of 12 kc;ll mole -I; also, no trajectories were calculated for (0.16) at 100 kczd mole-L. GencraIly, the contours plotted ;LTC0.5, 1, 7_, 3,4 and 5 A’. However, in some cases, because of crowding. fewer than 6 contours are plotted-Those plotted are then: 2 contours, 0.5 and 5 A*: 3 contours, 05,l and 5 A2;4 contours, 05.1.2. and 5 A’; 5 contours, 0.5,1,2,3, and 5 At_

J.E. Dove et d /Molecular

energ

.191

transfer in &-Se

amount of relative translational energy, in order for energy tratisfec to occur at a sigiSctit rate. Nev&&eless, fig. 9 and the other results discussed.in this settion show that a high cross section for vibrational transitions is most readily attained if a large propo& tion of the collision energy is in the form of rotation. 4.4. Rate constants Rate constants for energy transfer were calculated for a thermal distribution of relative translational energies, E, using the following integration: k(v, J + v’, J’; T) = (SkT/r#‘*

x j=

u(u, J

+u’,J’;E)-

0 20

40

E,

60

80

101 3

(kcol!

Fig_ 9. Graph showing the influence of energy in different degees of freedom, on the cross section for vibrational excitation in Hz-He collisions. The ordinate is the cross section for u - (u + I), and the abscissa is incremental energy. The ori_ti of the energy scale is at (u = 0. J = 0, E,,,,, = 30 kcal mole-‘). The cross sections were calculated by quasiclassical trajectories (this work). The graph shows that adding energy to the rotational degree of freedom (R) is more effective than addition of vibrational (V) or trrmslationai Q energy, in enhancing the cross section for vibrational transitions. Translational energy is relatively ineffective.

sition is shown also by fig. 9. The question which was asked here is the following.‘Consider a collision between a ground state Hz molecule and a He atom. Suppose that we wish to enhance the probability of a transition with Au = +l. Into which degree of freedom should we feed a given amount of additional energy, to get the greatest enhancement? The origin in this graph is the state (u = 0, J = 0) at Et,, = 30 kcal. The results of the trajectory calculations in fig. 9 show that this additional energy should be put into the rotational degree of freedom. This greatly enhances the probability of making a transition with Au = +l . Extra vibrational energy is less effective than rotation, while translational energy is relativkly ineffective in enhancingvibrational transition probabilities. Of course, the choice of origin here was somewhat arbitrary. Moreover, clearly the molecules inust have at least a small

E exp kT

where u is the cross section for vibrational energy transfer at translational energy E. The cross sections used in eq_ (7) where the trajectory values fitted by a least-squares technique to L.e Roy’s class II function 1431.

=

[Q@

=o 3

- Eo)"IE]

exp [-b(E - Eo)] ,

E>Eo. EC.&, (8)

where EO was chosen as IA/& I, and Q, b and II were chosen as least-squares parameters. In each case, cross sections for all the translational energies listed in table 1 were used in the fitting procedure. Results are displayed in table 13 for 1000 K and 10000 K, for four initial states of widely different energy distributions, namely (1,6), (12,0), (5,241 and (0,34). 1000 K and 10000 K are the lowest and highest temperatures considered in our study of the dissociation reaction [26,27,47] - Rate constants are not given for transitions where too few non-zero cross sections were available for the fit in eq. (8), or where the calculated rate constant was less than 10-‘5cm3 mol-’ s-‘. Possible errors in the rate constants were estimated by repeating the above calculations with eq. (8) fitted to u +-da. The rate constants found in this way were typically larger by 60% at 1000 K and by 40% at 10000 K. The& dif-

J-E_ Dove et al. /Molecular

192

energy transfer in Hz-He

Table 13 Rate constants (cm3 molec- 1 s-l j for the inelastic processes (II. Jl + (u’.S)

at 1000 K and 10000 K, calculated from the Hz-He

trajectory cross sections ”

J

I

I

12

5

0

*I The

Type a)

k(1000 K)

k(10000 K)

V V V

+T *RT +RT

l-3(-19) 5.7(-16) l-3(-15)

9.2(-13) l-5(-12) 1.8(-12) -

T T T

+R -+R +R

l-7(-11) l-3(-11) 7.3(-14)

4.9(- 10) 2.2(-10) 6.9(-l 1)

V V V R R

-+T +RT -+RT +T +T

6_4(-17) 3.0(-16) X7(-15) 2.2(-12) 9.3(-12) -

9.7(--12) X0(-12) 6.5(-12) H-11) 2.0(-10)

T +R T -R RT*V

7.3(-13) S-4(-16) 2.0(-17)

l-7(-10) 3.4(--11) 5.5(-12)

IO 11 11 II 12

V V V V

+RT -+T -RT +RT

6.3(-13) l-7(-12) 4.2(-12) S-8(-12) -.

1.8(-11) 2X(-11) 6.7(-11) 7.1(--11) -

12 12 13

T T T

+R +R -V

2.8(-10) 8.3(-l 1) U-12)

5.O(-iO) l-7(-10) 3.8(-H)

u’ 0

6

0

24

34

J’

0 0 0

0 n 0 1 1 1 1 1 2

6 8 10 2 4 6 8 10 4

3 3 4 4 4 5 5 5 5 6 6 6 7

24 28 24 26 28 20 22

V -‘T VT-+R V 4T VT-R VT+R R -tT R -T

24 26 20 22 24 20

1.7(-15) 6.9(-12) 7.3(-13) 6.5(-l 1) X5(-14) 4.2(-16) S-4(-13) -

1.4(-11) 6.6(-11) 7.4(-11) 2.3(-10) 3.5(-11) 6.8(-12) 4.8(-11) -

T R R T R

+R -+VT *VT -V +VT

l.O(-12) l-9(-12) l.O(-10) l-4(-12) 3_2(-11)

5.0(-11) 3J(-11) 2.5(-IO) 6.5(-U) 5.9(-11)

0 0 0 0 I 1 1 2

30 32 34 36 30 32 34 30

R R

+T -+T

l-8(-17) 36-13)

5.8(-12) 7.9(-l 1)

T R R T R

+R -+VT +VT +V *VT

l-41-13) 1.8(-l:) 5.0(-l 1) 4.9(-13) 4.0(--12)

8.7(-11) 3.2(--11) 2.1(-10) 7.7(-11) 6.1(-11)

type of energy conversion process is indicated, with V, R and T denoting vibrational, rotational ad

tra&atio&

energy_

ferences are not large enough to change the trends shown by tabie 13. At 1000 K, the rate constants for each initial level show strong preferences for one type of transition: R*Tfor(i,6)and(lZ,O),andVGRTorVT+R for (5,24) and (12,34). In each case, the V w T rate constantsare several orders of magnitude smaller than those of the most favored transitions. At 10000 K, the Iargest rate constants are about the same as at 1000 K, but the rate constants for all the main transition types are of a similar order of magnitude.

The resuhs of this study indicate stron&y that the

194

J.E. Dove et al. 1Mokcular

transfer. For the higher internal states, vibration-rotation (VR) interconversion becomes increasingly important, and is often dominant. Direct translation-vibration transfer is generally of Iow probability, but the probability of vibrational excitation is strondy enhanced by molecular rotation. These results imply that rotation will have a substantial effect on both vibrational relaxation and dissociation at high temperatures. RT relaxation followed by VR transfer will provide a reIativeIy easy route for vibrational rekation and for access to many upper states from which collisional dissociation can occur_.

Acknowledgement We thank Dr. J.L. Schreiber for discussions about the use of the trajectory programme, Dr. N. Satilyamurthy for making available a contour

pIotting routine,

and hlr. R. Lipson for assistance

with some of the computations_ This work was supported in part by the National Research Council of Canada_

References [I ] J. Tree and H. Gg. Wagner, Ber. Bunsenges. Physik. Chem. 71(1967) 930. [I] R-L- Belford and R.A. StrehIow, Ann. Rev. Phys. Chem. 20 t 1969) 247. [ 31 F. Kautinan, Ann. Rev. Phys. Chem. 20 (1969) 45. [4] 11-S. Johnston and J. Birks, Accounts Chem. Rer 5 (1972) 327. [S] D.L. BauIch, D.D. Drysdate, D.G. Home and A.C. LIoyd, EvaIuared LinetIc data for high-temperature reactions, Vol. 1 (The Chemical Rubber Co., CIeveknd, 1972). [ 6] AA. Westenberg, Ann. Rev. Phys. Chem. 24 (1973) 77. [ 7 j B_ Stevens, CoIIisionaI activation in gases (Pergamon, Oxford. 1967). 181 G.&t- Burnett and A.M. North, Transfer and storage of energy by motcules. VoL 2 OViIex New York, 1969) sects. 2, 3, and 5. [9] J.P. Tocnnies, -Ann. Rev_ Phyr Chem. 27 (1976) 225. [lo] D. Secrest, Ann. Rev_ Phyr Chem. 24 (1973) 379. [ I1 ] %.A. Lester Jr., Advan. Quantum Chem. 9 (1975) 199. [ 121 H-0. Pritchard, Specialist periodical reports, reaction kinetics, Vol. I (The ChemicaI Society, London, 1975) p_ 243. 1131 H-0. Pritchard, Accounts Chem. Res. 9 (1976) 99. [Ia] M-D. Gordon and D. Secrest, J. C&em. Phys. 52 (1970) IZO. [ 151 hl. Krauss and F.H. Mies, J. Chem. Phys. 42 (1965) 2703.

energy transfer in Hz-He 116) T- TsapIine and 1%‘.htzelnigg, Chem. Phys. Letters 23 (1973) 173. [17] C.W. WiIson Jr_, R. Kapral, and G. Bums, Chem. Phys. Letters 24 (1974) 488[IS] A-W. Raakowski and W.A. Lester Jr., Chem. Phys. Letters 47 (1977) 45. [ 191 LE. Dove. D-G. Jones and H. Teitelbaum. Proceedings of 14th Symposium (International) on Combustion (The Combustion Institute. Pittsburgh. 1973) p. 177._ [20] J-E-Dove and 8. Teitelbaum. Chem Phys. 40 (1979) 87. 1211 H. Rabitz and C. 2arur.J. Chem. Phys. 62 (1975) 1425. [22] M-H. Alexander, J. Chem. Phys. 56 (1972) 3030. 1231 P. McGuire and J.P. Tommies. J. Chem. Phys. 62 (1975) 4623. [24] M.H. Alexander and P. McCuire, J. Chem. Phyr 64 (1976) 452. [25] J.E. Dove, D.G. Jones and H. Teitelbaum, I. Phys. Chem. 81 (1977) 2564. [26] J.E. Dove and S. Raynor, Chem. Phys. 28 (1978) 113. [ 27j J.E. Dove and S. Raynor, in preparation. 1281 W. KoTos and L. Wolniewicz, J_ Chem. Phys. 43 (1965) 2429; 49 (1968) 404; them Phys. Letters 24 (1974) 457; J. Mol. Spectry. 54 (1965) 303_ 1291 D.M. Bishop and S. Shi. J. Chem. Phys. 64 (1976) 162; 67 (1977) 4313s [30] R. Gengenbach, Ch. Hahn, and J.P. Toennies, Phys. Rev. A, 7 (1973) . 98.__ - [31] R-J. Le Roy, WlS-TCl-387, Theoretical Chemistry Icstiture. University of Wisconsin, Madison, Wis., 1971. [32] J-1. Schreiber, Ph. D. Thesis, University of Toronto (1973). [33] R.A. LaBudde and RB. Bernstein. J. Chem. Phys. 59 (1973) 3687_ [34] N-C. Blaisand D-G. Truhlar, J. Chem. Phys. 65 (1976) 5335. [35] S. Chapman and S. Green, J. Chem. Phys. 67 (1977) 2317. [36] S.D. Augustin and NH. MiUer, Chem. Phys. Letters 28 (1974) 149. f371 J-W_ Duff and D-G_ TruhIar, Chem. Phyr 9 (1975) 243, and references therein. [38] CS. Lin and D. Secrest, J. Chem. Phys. 67 (1977) 1291. 1391 hl.D. Patter@, Chem. Phys. Letters 36’(197S) 25; J. Chem. phys. 65 (1976) 5033; 66 (1977) 1449. [SO] G. Berge:on and X. Chapuisat. J. them. Phys. 57 (1972) 3436. [41] M.H- Alexander, J. Chem_ Phys. 66 (1977) 4608. [42] R.L. Le Roy, J. Phys. Chem. 73 (1969) 4338. [43] A.\%‘_Raczkowski, W.A. Lester Jr. and W.H. Miller, J.

Chem. Phys. 69 (1978) 2962. [44] J.M. Jackson and N-F. Mott. Proc. Roy. Sot. (London) Al37 (1932) 703. [45] R.E. Roberts, R.B. Bernstein, and CF. Curt&, J. Chem. Phys. 50 (1569) 5163. [46] F.H. Mies. J_ Chem. Phys. 40 (1964) 523. 1471 J-E. Dove andS_ Raynor, 3. Phys_ Chem. 83 (1979) 127.