A lowenergy quasiclassical trajectory study of O(3 P)+OH(2Π) →O2(3Σ− g )+H(2 S). I.Cross sections and reaction dynamicsJan Davidsson and Gunnar Nyman Citation: The Journal of Chemical Physics 92, 2407 (1990); doi: 10.1063/1.457984 View online: http://dx.doi.org/10.1063/1.457984 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/92/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A detailed statetostate lowenergy dynamics study of the reaction O(3 P)+OH(2Π)→O2(X̃3Σ g −)+H(2 S) usinga quasiclassical trajectory–internalenergy quantummechanicalthreshold method J. Chem. Phys. 97, 4050 (1992); 10.1063/1.463934 A lowenergy quasiclassical trajectory study of N++H2 and N++D2. Dynamics, cross sections, and rateconstants J. Chem. Phys. 96, 5198 (1992); 10.1063/1.462760 A lowenergy quasiclassical trajectory study of O(3 P)+OH(2Π)→O2(3Σ− g )+H(2 S). II. Rate constants andrecrossing, zeropoint energy effects J. Chem. Phys. 92, 2415 (1990); 10.1063/1.457985 Quasiclassical trajectory study of the reaction O(3 P)+H2→OH+H. The effects of the location of the potentialenergy barrier, vibrational excitation and isotopic substitution on the dynamics J. Chem. Phys. 80, 3687 (1984); 10.1063/1.447191 A comparative study of the reaction dynamics of several potential energy surfaces for O(3 P)+H2→OH+H. II.Collinear exact quantum and quasiclassical reaction probabilities J. Chem. Phys. 76, 3563 (1982); 10.1063/1.443394

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A low-energy quasiclassical trajectory study of O(3P) + OH(2ll) -+ 02(3l:;) + H(2S). I. Cross sections and reaction dynamics

Jan Davidsson and Gunnar Nyman Department 0/ Physical Chemistry, University o/Goteborg and Chalmers University o/Technology, S-412 96 Goteborg, Sweden

(Received 26 April 1989; accepted 7 November 1989)

Quasiclassical trajectory calculations for the title reaction have been carried out using the recent DMBE III (double many body expansion) potential-energy surface by Varandas et al. ( 1988). The dynamics of complex formation were studied in detail and two different types of complexes, referred to as close and distant complexes, were observed. Corresponding cross sections and reaction cross sections are reported as a function of diatom rotational quantum number and atom-diatom relative translational energy in the range 0.25-2.0 kcal/mol. The cross sections decrease with increasing translational energy. The rotational-state dependence of the reaction cross section is found to be complicated. One interesting observation is that the cross section for the rotational ground state is much smaller than for other rotational states. The features in the cross sections are related to the shape of the potential-energy surface and the reaction dynamics.

I. INTRODUCTION

The importance of the reactions Oep) + OHerr)++02eIg-) + HeS) has been docu­mented in several places. I

-3 These reactions have been ex­

tensively studied both experimentally, e.g., Refs. 4-6, and theoretically. Rate constants have been obtained experimen­tally over a large temperature range and also calculated with various theoretical methods. Troe and co-workers per­formed statistical adiabatic channel model calculations,7

Rai and Truhlar used variational transition-state theory,S Clary and Werner did quantum scattering calculations,9

Varandas applied a Gorin-type model, \0 and Davidsson and Nyman developed a so-called extended Langevin model. II In addition to this, numerous trajectory studies have been performed to find rate constants and investigate various oth­er aspects of these reactions,2.12-2o though only two of these studies2

,20 have explicitly concerned the forward reaction. Not only empirical, but also ab initio potential-energy sur­faces21-25 have been developed, fitted to analytical func­tions,25-28 and used in the mentioned calculations.

The present quasi classical trajectory study has provided us with extensive information on the dynamics, cross sec­tions, recrossing ratio, zero-point energy effects, and rates of the ground-state 0 + OH reaction. The material will for clarity of presentation be divided into two papers, this one and the one immediately following. 29 In this first paper we will be concerned with some intriguing features in the cross sections. These features will be related to characteristics of the potential-energy surface and explained in terms of the reaction dynamics.

We employ a recent ab initio surface by Varandas, Brandiio, and Quintales, DMBE III (double many body ex­pansion),28 which is probably the most-accurate complete potential-energy surface presently available for the 0 + OH reaction. There are more recently calculated ab initio points now available,24 but a surface has not yet been fitted to these.

We focus here on the low-energy aspects of the 0 + OH reaction. At low energies the details of the potential-energy surface have a larger bearing on the outcome of a collision

event than at high energies. This makes the low-energy range particularly interesting to study.

At low energies quantum effects may be important. From a tunneling aspect 0 and OH are relatively heavy par­ticles and the potential is also very flat,24 so tunneling should not be a prominent effect. The zero-point energy is, however, important for the outcome of a collision. We still believe the trajectory technique to be a good method for our purposes, but will return to this point in the sequential paper.

II. COMPUTATIONAL PROCEDURE

A. Integration accuracy

All trajectories were run using the integration routine RADAU,3° which has performed well in previous trajectory calculations. 31 ,32 For all selections of initial conditions the energy conservation lilE IIEp was typically 1 X 10- 6

_

2 X 10 - 6 for the product channel, where Ep represents the total energy at infinite product separation. In the same way > 99% of the trajectories have I ilE II Ep < 1 X 10 - 5 and a vast majority have the total angular momentum conserved to better than one part in 1010.

The criterion for products (reactants) to have formed (reformed) was that the distance between the diatom center of mass and the atom exceeded 10 A. (Sometimes 7 A was used instead, which in test runs was found not to affect the calculated results.) For each set of chosen initial conditions, between 1000 and 2000 trajectories were run and over 99% of these were integrated long enough for products (reac­tants) to form (reform). Typical trajectories are 2--4 ps long.

Several test runs indicate that apart from the very long­lived trajectories, most ofthem are backward integrable. The integrations of the equations of motion are thus considered free of numerical instabilities.

B. Potential-energy surface

We have employed the ground-state ab initio surface DMBE III (Ref. 28) obtained by Varandas, Brandiio, and Quintales using a double many-body expansion method. For

J. Chem. Phys. 92 (4), 15 February 1990 0021-9606/90/042407-08$03.00 @ 1990 American Institute of Physics 2407

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2408 J. Davidsson and G. Nyman: O("P)+OH("fl)--+02(" .2'g)+H("S).1

later ease of discussion this potential is illustrated in Fig. 1, but except for that we refer to the original papers27, 28 for details. For convenience, the coordinate system we used is shown in Fig. 2.

C. Initial conditions

Some of the initial conditions were obtained by sam­pling from the relevant distributions while others were var­ied systematically. As is usual, the reactants were initially sufficiently separated (9-10.5 A) to be treated as non inter­acting. The initial translational energy was selected to be 0.25,0.5, 1.0, or 2.0 kcal/mol. For these energies, values of bmax from 7 A down to 4 A were used.

The rotational quantum number J was selected at will in the range (0,9). The initial rotational energy was found as

€"init = J(J + 1 )fz2/2f.1r min r rna" (1)

where r min and r max are the extrema in the OH internuclear separation, found by integrating over one vibrational period.

The relative orientation between the OH axis and the line connecting the OH center of mass and the 0 atom was always sampled randomly. This was done so that the azi­muthal angle was sampled uniformly on (0, 2IT) while the polar angle () was sin () weighted.

The impact parameter b was always sampled to give a triangular distribution for b. This was done by sampling b from

(2)

where {; is a continuous uniform random number and bmax is the largest value allowed for b. The value of bmax was set so that in a histogram representation of the opacity function the outermost class was always empty.

The total energy was found as the sum of the initial translational energy and the vibrational-rotational energy found from the expression

(3)

which approximately gives the energy of a diatom in its vi­brational ground state. In the first term Be is the rotational equilibrium constant and al2 is a modification due to the

(A/----",--------:V""'O'

0-0.13 eV

+O.07eV

(AI

FIG. 1. Illustration of the DMBE III potential-energy surface for 0 ap­proaching OH with the OH internuclear distance held fixed at its equilibri­um value. To the left: three-dimensional view. The grid represents 5.29 A along each side. To the right: contour plot. The curves are spaced 0.2 eV apart with zero at infinite O-OH separation.

H o

FIG. 2. Illustration of symbols used. cm marks the diatom center of mass. rem is the separation between the atom and crn. e is the angle between rem

and the diatom internuclear axis.

vibrational motion of the diatom. The next term is due to the centrifugal force. The third term is the ground-state energy ofa quantum-mechanical harmonic oscillator and the fourth an anharmonicity correction. Reference 33 gives a complete discussion.

The initial vibrational energy is found as the difference between € v ~ 0,' and € "init· Note that the above definitions, although containing a certain degree of arbitrariness, give initial rotational and vibrational energies that are indepen­dent of the vibrational phase.

III. ANALYSIS OF TRAJECTORY RESULTS

A. Opacity function

The opacity function, reaction probability P, (b) vs b, is plotted in histogram form in such a way that the class width is proportional to b 1I2 . This gave histograms where the sta­tistical fluctuations are reduced compared to histograms where each class has the same width.

B. Trajectory types and complex definitions

We distinguish between five main types of trajectories depending on their behavior in the entrance channel. These types are illustrated schematically in Fig. 3. The following paragraphs give the definitions of the five types. Below we use the term effective potential to mean the sum of the fitted ab initio potential energy and the centrifugal energy for the motion of the approaching oxygen atom around the OH cen­ter of mass,

1. Type 1

A trajectory is type 1 if the effective potential never goes below zero and the trajectory never crosses a potential bar­rier, i.e., it is reflected off the first barrier it encounters, be it a centrifugal barrier or a real one.

2. Type2

Trajectories of type 2 pass over an outer barrier but are reflected in an inner barrier or have the effective potential

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E

E

E

~~==~==~~~Type2 T e1

1.9A

.-+-------------------Type5

-I7~==s;::::::::::::_--Type 4

1.9A

FIG. 3. Schematic illustration of the definitions of trajectory types; see text. The effective potential vS rem (see Fig. 2) is shown. The potential barriers vary in height as the trajectory evolves in phase space, which is not shown here. The barrier heights are not drawn to scale.

become negative before encountering a barrier. They have only one inner turning point and this turning point is found at an rem value (see Fig. 2) of at least 1.9 A. (The choice of this value is explained below.) Type-2 trajectories have no outer turning point, thus being unreactive.

3. Type 3

Type-3 trajectories are unreactive. They have three or more turning points, the number always being odd. The in­ner turning points are always found at rem values (see Fig. 2) of at least 1.9 A.

4. Types 4 and 5

For type-4 and type-S trajectories the rem separation is at some stage less than 1.9 A. Type-4 trajectories have at least two turning points outside the inner barrier, the num­ber always being even. Type-S trajectories have no turning points at rem separations larger than 1.9 A. Type-4 and type­S trajectories can be reactive or unreactive.

5. Distant and close complexes

We now give our definitions of distant and close com­plexes. Trajectories of type 3 and type 4 are said to form

2409

distant complexes. The distant complexes are thus made up of trajectories temporarily trapped in a region where we can distinguish the diatom from the atom. These complexes may be quite long lived ( ~ S ps) and have several turning points.

Trajectories of type 4 and type 5 form close complexes. The close complexes are thus made up of trajectories form­ing strongly coupled complexes. Generally, these complexes exist long enough that several minimum distance exchanges occur, i.e., changes in which atom-atom pair has the short­est internuclear separation.

Type-4 trajectories are part of both distance and close complexes, though not at the same time in their evolution. There was no need to determine when a distant or a close complex was formed, just if it was formed during a trajec­tory.

The value of 1.9 A appearing above was chosen on the basis of Fig. 1, from which it is expected that the 0 atom approaching OH will never turn around at an rem value of about 1.9 A. This was confirmed for various initial condi­tions and is illustrated for J = 0 and E, = 0.25 kcal/mol in Fig. 4. In Fig. 4 is shown at which values of rem an 0 atom approaching OH turns around for type 1, 2, and 3 trajector­ies (for type-3 trajectories the first inner turning point is shown). It can be seen that these trajectories turn around at rem values of at least 2 A. The distinction between close and distant complexes is therefore clear.

C. Cross sections

From the trajectory results we calculate distant and close complex formation and reaction cross sections, symbo­lized Sd' So and S" respectively. These are found from

(4)

where Px = N x / N tot and N tot is the total number of trajec­tories, x stands for d, c, or r, and Nx for the corresponding number of trajectories in that group.

IV. RESULTS

In Fig. 5 we show the opacity function for E, = 0.2S and 2.0 kcal/mol for various values of J. The largest value of the

Fract""'on"---______________ ----,

0.5

0.25

FIG. 4. Histogram of the first inner turning point distances 0 to OH center of mass, for trajectories not forming close complexes. J = 0, E, = 0.25 kcallmol. Contributions from trajectories of type 1 are marked with slanted lines, from type 2 shaded, and the remainder is from type-3 trajectories. Close complex forming trajectories all have inner turning points < 1.9 A.

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2410 J. Davidsson and G. Nyman: O(3p)+OHeJ7)--02e~g)+HeS).1

~,~----------~----------~----------~

FIG. 5. The opacity function for an initial translational energy of 0.25 kca1!mol and 2.0 kca1!mol (shaded), for initial diatom rotational quantum numbers as indicated at the top right in each panel.

impact parameter leading to reaction is seen to be consistent­ly smaller for the higher of the two translational energies, for all values of J. This is expected on an attractive potential­energy surface like the one used here. It is also seen that the P r (b) curve is flatter for the lower of the two translational energies. This could be a result of the larger orbital angular momentum resulting at higher translational energies for a given impact parameter. A larger orbital angular momen­tum gives larger barriers in the effective potential which probably influences the P r (b) curve more than smaller bar­riers would. Note that for J = 0 and small values of b, P r (b) is larger for the higher of the two translational energies. The same relation is seen for J = 4, 7, and 8. We will return to the reasons behind this.

In Fig. 6 reaction cross sections and uncertainties for various translational energies are presented as a function of J. Explicit values are given in Table I. The reaction cross sections are seen to decrease with increasing initial transla­tional energy. This agrees with what is expected for an exo­thermic reaction on a potential-energy surface where a close complex can be formed without crossing an energy barrier.

FIG. 6. Reaction cross sections for 0 + OH for various initial translational energies and OH rotational quantum numbers. The translational energies are given in kca1!mol to the right of each curve. The DMBE III potential­energy surface with OH initially in the vibrational ground state was used. The uncertainties were found as given in Table I.

TABLE I. Reaction cross sections S,(E"J) in ,.\, for 0 + OH on the DMBE III potential-energy surface, with OH initially in the vibrational ground state. The second line gives for each entry the standard errors in"\', which are obtained from S,[(N'ol -N,)/N,oIN,jI/2. Nlo" the total number of trajectories, varies from about 1000 to 2000.

E, (kca1!mol)

J=2

J=3

J=4

J=5

J=8

J=9

0.25

23.7 1.1

62.1 2.4

69.2 2.4

56.0 2.3

51.6 2.3

63.1 2.4

64.1 2.5

63.1 2.5

58.2 2.4

7.0

0.5

21.5 1.1

36.8 1.9

47.0 1.8

32.8 1.8

37.7 2.3

47.5 2.8

48.1 2.8

43.0 2.8

39.8 2.7

30.8 2.5

6.5

1.0 2.0

12.5 7.61 0.65 0.48

21.1 11.3 0.78 0.66

28.4 11.6 0.93 0.67

20.7 11.3 0.87 0.66

22.8 13.5 0.90 0.70

28.2 17.5 0.88 0.76

32.0 19.0 1.9 0.78

30.1 18.3 1.9 0.76

25.8 17.6 1.8 0.76

5.0 4.0

a This line gives the chosen maximum value of the impact parameter that can lead to reaction.

Note also how the reaction cross sections increase from J = 0 to J = 2 where they peak, which is particularly notice­able for the smallest translational energy shown. The reac­tion cross sections then show a dip at J = 3 or 4 and a second peak at J = 5 or 6. We will attempt an explanation for part of this in Sec. V A. It is noteworthy that the structure in the reaction cross section as a function J is pronounced at an initial translational energy of 0.25 kcallmol but disappears quickly with energy, hardly being seen at 2.0 kcallmol.

In Fig. 7 we present reaction and complex formation probabilities as a function of rotational quantum number for an initial translational energy of 0.25 kcallmol. The close complex formation probabilities can be seen to generally be somewhat larger than the reaction probabilities, the differ­ence deriving from the fact that some of the close complexes decompose unreactively. Also shown is the distant complex formation probability Pd , which generally decreases with J and is smaller than the other two except for J = O. However, Pd is far from zero, which indicates that a rather large frac­tion of the total number of trajectories can have a very com­plicated path across the potential-energy surface. The corre­sponding figure for an initial translational energy of 1 kcallmol (not shown) shows similar relations.

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0.5

OLO~--~--~2----~3----~4----~5----~6----~J

FIG. 7. Reaction and complex formation probabilities Px ' Eq. (4), vs dia­tom rotational quantum number J, at an initial translational energy of 0.25 kcallmoL The symbols are as follows: (T) close complex formation probability, (.) distant complex formation probability, (.) reaction prob­ability.

In Fig. 8 the fractions of trajectories belonging to the five different types, defined in Sec. III B, are shown as a function of initial J for two choices of initial translational energies, 1.0 (panel A) and 0.25 kcallmol (panel B). The probability of getting a particular type depends on the im­pact parameter (for instance, a very large impact parameter will often give type-1 trajectories). Since different values for bmax were used at 0.25 and 1.0 kcallmol, comparisons in between the panels in Fig. 8 should be made with this kept in mind. It can be seen that the curve for the type-l trajectories are rather flat. This makes sense since these trajectories are almost exclusively reflected in an outer centrifugal barrier, the height of which depends on the orbital angular momen­tum, but not on J. The fraction of type-1 trajectories is, how­ever, somewhat smaller for J 0 than for other J's. The reason for this may be that in the case of J = 0 it should be easier for a trajectory to trace out the minimum energy path

0.4

0.2

_____ -+-- ______ + _______ .... ______ ""i _______ .... _______ ... _____ _

o 0 2 3 4 5 6 7 J

FIG. 8. The fraction of trajectories belonging to a certain type, defined in Sec. III B, as a function of J. The initial translational energy is given in kcallmol at the top of each panel. For panel A, bmax = 5.0 A and for panel B, bmax 7.0 A.

2411

than for other values of J. This may facilitate a closer ap­proach of the reactants than would otherwise have been the case, resulting in another classification of the trajectory.

The most characteristic feature of the type-2 curves in Fig. 8 is their dominance at J = 0, which is explained below. The fractions of type-3 trajectories are rather small and the curves fairly flat. Most of the type-2, -3, and -4 trajectories pass over an outer barrier as 0 initially approaches OR. This means that most of the type-2 and -3 trajectories, which all decompose unreactively, are not represented in statistical theories taking passage ofthe centrifugal barrier on the long­range part of the potential as the criterion for reaction.

The sum of the curves for type-3 and -4 trajectories gives the fraction distant complexes. It can be seen that at the higher of the two translational energies shown type-3 and type-4 trajectories occur to the same extent, while at the lower energy type-4 trajectories are more common than type-3 trajectories. The type-3 and -4 trajectories can be thought of as temporarily trapped between two dynamic barriers in the effective potentiaL The inner one of those two barriers grows faster with increasing orbital angular mo­mentum than the outer one does, thereby favoring type-3 trajectories at high energies. Note also the resemblance of the curves for type-5 trajectories with the corresponding re­action cross sections in Fig. 6.

V. DISCUSSION

In this section the shape of the reaction cross section Sr (E"J) for 0 + OH calculated by us will be discussed. We will also compare with the reaction cross section for 0 + H 2,

calculated by Schinke and Lester. 34

A. The shape of Sr(€t,J) for O+OH

The reaction cross section decreases with increasing translational energy; see Fig. 6. This is due to the increasing difficulty for the attractive long-range part of the potential to capture a passing atom when its translational energy is in­creased.

The shape of the reaction cross section as a function of J is peculiar and not easily explained. The tendencies with a peak at J = 2 and 5 or 6 are very similar for all translational energies. From Fig. 8 it is clear that the curve for the fraction of type-5 trajectories as a function of J resembles the Sr (J) vs J curve. However, the other component in So the type-4 curve, is much flatter. Insight into the shape of the Sr (J) vs J curve may thus be gained by finding an explanation to why type-5 trajectories occur the way they do.

The case where the impact parameter is zero and the diatom rotates in the plane made up by the three atoms was investigated in detail for an initial translational energy of 0.25 kcallmoL It was found that the outcome of a collision was virtually independent of the vibrational phase of the dia­tom. The diatom rotational state and the angle theta (see Fig. 2) were varied systematically so that at 10 intervals a trajectory was run for each value of J. From Fig. 9 it can be seen that for this reduced set of trajectories, the reaction and close and distant complex formation probabilities as a func­tion of J are similar to those for the complete runs reported in Fig. 7. It thus appears that this reduced set of trajectories

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2412

~'----------------------------.~P~r~·~~-·~P~d

OLO~--~--~~--~--~4----~5----~6-----"J

FIG. 9. As in Fig. 7 but with the following modifications. b = 0, the initial angle is varied systematically and the diatom rotates in the plane of the three atoms.

may contain the information needed to interpret the shape of the cross sections. Since in this reduced set of trajectories the orbital angular momentum is zero, it appears that angular momentum restrictions would not have a major effect on the shape of the cross-section curves under the present condi­tions.

Figure 10 shows which initial values of theta give type-5 trajectories, with the same conditions as in Fig. 9. Changing the vibrational phase of the diatom had virtually no effect on the appearance of Fig. 10. It may marginally move the edge of the groups containing type-5 trajectories, but nothing else. Also shown in Fig. 10 are the angles for which the potential­energy surface never goes above 0.25 kcal/mol in the en­trance channel. Note how neatly grouped the type-5 trajec­tories are.

Details of the reaction cross-section curves will now be explained. Consider first the case J = O. The diatom does not rotate at all initially. However, for most angles of approach the incoming atom will feel the outer attractive well at the hydrogen side of the diatom. A slow rotation is then induced

J

9_ 11

8

7

6

4

2

o o ~ - = ~a

FIG. 10. Illustration of which angles lead to type-5 trajectories (shaded) and which do not. The angle a is the initial angle of approach and a = 0, for 0< a < 180 and thereafter a = 360 - 0; see also Fig. 2. For each value of J, 360 trajectories were run, evenly distributed in a. The initial translational energy was 0.25 kca1!mol. Which particular angles lead to type-5 trajector­ies is of no importance, it depends on the initial separation of the reactants, but rather the grouping and fraction of type-5 trajectories. For J = 0 it is indicated (with slanted lines) for which angles of approach there is no po­tential-energy barrier higher than 0.25 kca1!mol.

such that quite often the approaching atom bumps into the diatom on the hydrogen side and is reflected off the potential barrier just inside the outer well; see Fig. 1.

For some angles of approach close to e = 180° direct reflection off the barrier on the oxygen side will occur. For angles somewhat further from 180° the induced rotation will give the approaching atom a chance to slip in around e = 90°, where there is no barrier. It is thus not trajectories initiated with e = 90° that lead to type-5 behavior. This is illustrated in Fig. 10. Remember, compare Figs. 7 and 8, that the curve for the fraction of type-5 trajectories resemble the probability of the reaction curve.

The induced rotation should be the explanation for the larger reaction probability at E, = 2.0 than 0.25 kcal/mol at small band J = 0, seen in Fig. 5. At the higher translational energy the induced rotation will less efficiently rotate the diatom to prohibit reaction than at the lower translational energy. At sufficiently high translational energies the dia­tom will hardly rotate at all before the actual collision takes place. The effect seen at J = 0 in Fig. 5 also appears, though it is smaller, at J = 4, 7, and 8 and maybe higher J's as well. This we are unable to explain.

The induced rotation is also the reason for the excep­tionally high fraction of type-2 trajectories found for J = 0; see Fig. 8. Further, it is also an important reason for the low cross sections for J = 0; see Fig. 6. We conclude that at the low values of E, used here, the combined effect of a potential barrier inside the attractive well on the hydrogen side ofOH is largely responsible for the small values for Sr (J = 0).

The induced rotation is sensitive to details of the angular dependence of the potential. This dependence may be quite different for other 0 + OH potentials as it is difficult to ob­tain accurate information on the angular dependence of the intermediate to long-range part of the potential. However, the attractive well outside the hydrogen atom should always be there as it is caused by the interaction of the OH dipole moment with the oxygen quadrupole moment.

Next consider the case J = 1. The diatom rotates al­ready initially as the atom approaches. The angles now lead­ing to reaction occur more frequently. Their absolute values have no significance, but depend on the initial separation of the atom and the diatom. It appears that ifthe diatom rotates initially, the probability for the approaching atom to slip by the potential barriers is larger than if the diatom does not rotate initially. This probability seems to vary with J in a complicated way, reaching a peak at J = 2. To get a better understanding of this, a large number of individual trajector­ies were studied in detail. From these it was seen that the type-5 groups in Fig. 10 represent trajectories passing rem = 1.9 A at theta values smoothly changing with initial angle of approach. As should be expected, a small change in the initial conditions leads to a small change in the path of the trajectory, at least up to some point where divergence may set in.

In Fig. 11 a few selected trajectories are shown, super­imposed on a graphically simplified representation of the potential, where J = 2, b = 0, E, = 0.25 kcal/mol, and the diatom rotates in the plane of the three atoms. The trajector­ies illustrated are close to the edges of the type-5 groups. It is

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FIG. 11. The same conditions as in Fig. 10, but J = 2. The motion of the approaching oxygen atom is shown in a frame of reference where the .di~tom does not rotate. The diatom vibration is not shown. Some characteristIcs of the potential-energy surface are also shown. The potential is shown for fixed diatom internuclear separation, while the trajectory is for the complete po­tential. b indicates potential barriers and w potential wells. The angle a (see Fig. 10) is indicated. Trajectories with intermediate values of a show inter­mediate behavior. Also see text.

seen that the two type-5 groups represent trajectories slip­ping past the barriers on either side of OR.

For the case J = 3, b = 0, and EI = 0.25 kcallmol, the combination of the rotational and translational motion makes passage to the inside of the barriers on the right-hand side of OR highly unfavorable or impossible; see Fig. 12. The trajectories corresponding to the right part of Fig. 11, where J 2, are in Fig. 12, where J 3, seen to be deflected off a barrier. Similar figures may be made to explain the type-5 behavior seen in Fig. 10 for the other J 's, but this work would not benefit from also discussing these in detail.

In summary, it is seen that at the low energies employed here, intricate details of the potential surface determine the shape of the reaction cross section. The most interesting po­tential surface effect we observed is perhaps the induced ro­tation discussed above.

B. Comparison with 0 + H2

Schinke and Lester34 performed a trajectory study on OeD) + R2e~t) -->ORCZI1) + RCZS). They calculated reaction cross sections as a function of diatom rotational quantum number and initial translational energy, the latter ranging from 0.25 to 5 kcal/mol. They found that the reac­tion cross sections were nearly independent of the initial ro-

FIG. 12. Same as in Fig. II, but for J = 3. Also see text.

2413

8 (AI

FIG. 13. Illustration of the 0 + H, potential used by Schinke and Lester (Ref. 34).

tational state except at the lower energies where the reaction cross sections for J = ° were smaller than those for higher rotational states. This is similar to the behavior we have ob­served in the present study, though we have a much more pronounced change in reaction cross section for J = ° and additional features.

We propose the same mechanism to explain the results by Schinke and Lester34 as we have proposed to explain our own results. The 0 + R2 potential used by Schinke and Les­ter (their surface I) is illustrated in Fig. 13. Note the wells on each side, extending towards right angle approach. Inside the wells there are barriers. This should lead to the same type of induced rotation as discussed above and explain the lower reaction cross section for J = 0, at low energies, than for other values of J. The depth of the outer wells on the Schinke and Lester surface34 are smaller than the corresponding one

dAl '- 20 on the potential by Quintales, Varandas, an vanno, which may explain the smaller J dependence ofthe reaction cross sections on the former surface.

VI. CONCLUSIONS

We have found a complicated dependence of the reac­tion cross section on diatom rotational quantum number, which can only be explained by the detailed dynamical be­havior of the reactants during the reaction. It is proposed that at the low translational energies used here, the attractive well on the hydrogen side of OR induces a rotation of OR relative to O. This causes the approaching 0 atom to hit the potential barrier inside the well and bounce off. It is this combination attractive well-potential barrier that to a large extent is responsible for the exceptionally low cross sections observed for J 0.

ACKNOWLEDGMENTS

This work has been supported by the Swedish Natural Research Council and in part by the Swedish Board for Technical Development. We are most grateful to Nikola Markovic for supplying us with his trajectory program for atom-diatom reactions. J. D. also acknowledges many help­ful discussions on the use of the program.

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