J . Phys. Chem. 1987, 91, 5480-5486
5480
attack, x does not vary much with rotational excitation and the mass considerations also restrict its contribution. Figure 14 illustrates the two important conditions to p-k as indicated by (5.10). For the 0 + HC1 reaction, cos2 fl = 1 and for this collinearly dominated reaction R is pypendicular to i for rotationally excited reagents. Hence, Lhe_j-k distribution (e.g., Figure 8) is reflected in the results for j’ek at a low initial j and a second contribution, due to the R A r term in (5.10) is evident as j increases. The role of initial translation, which is manifested in the r A R term in x, is limited for this collinearly dominated reaction.
7. Concluding Remarks Using only the location and orientation dependence of the barrier to reaction it is found possible to examine a wealth of dynamic information about direct reactive collisions. In particular, the orientation dependence of the reactive reactants and of the products can be discussed in clear physical terms. The assumption of a sudden switch from reagents to products is an approximation. Examination of realistic potential energy surfacesZZsuggests however that it is not an unrealistic one. Entrance and exit valley
interactions as well as impulses imparted during the switchover will tend to deform the distributions predicted by the simple model. Many of the overall features appear, however, to be quite robust. Experiments on the reactivity of aligned reagents (e.g., Figures 7 and 10) are yet to be performed. It must also be noted that the model makes a number of additional explicit predictions, particularly on the role of reagents vibrational and rotational excitation and its relation to the orientation dependence of the barrier. The model suggests that the orientation dependence of the barrier to reaction will have a marked effect on directly measurable quantities. We have examined here the predictive route, from the potential to observables. Work is in progress on the complementary aspect, that of extracting the barrier from the measurements.
Acknowledgment. We thank Prof. R. G. Gordon for discussions. The Fritz Haber Research Center is supported by the Minerva Gesellschaft fur die Forschung, mbH, Munchen, BRD. Registry No. H, 12385-13-6; Dlr 7782-39-0; Li, 7439-93-2; HF, 7664-39-3; 0, 17778-80-2; HCI, 7647-01-0.
Calculation of Steric Effects in Reactive Collisions Employing the Angle-Dependent Line of Centers Model Maurice H. M. Janssen and Steven Stolte* Molekuul en Laserfysika, Fysisch Laboratorium, Katholieke Universiteit Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands (Received: March 23, 1987)
-
-
The steric dependence of reactivity upon the initial angle of attack yo as well as upon the reaction angle yrhas been examined for barriers to reaction Vo(cos yr) resembling the cases Ba + N 2 0 BaO* N2, Rb + CHJ RbI + CH3, and H D2 HD D. The orientational dependence of the reaction cross section uR upon cos yo turns out to be up to a factor of 2 smaller than the dependence of uR upon cos yr. Large differences between the dependence upon alignment for U,(COS yr) and U ~ ( C O yo) S have been found. Consequently analysis of steric data in terms of the theoretically easier accessible angle yr instead of the experimentally controllable angle yo leads to a considerable underestimation of the anisotropy of the steric barrier Vo(cos yr). For ellipsoidally shaped barriers having an eccentricity parameter X I 1 maximum reactivity has been found at values of cos yo considerably different from the head-on orientation (cos yo = 1). -+
+
1. Introduction The advent of quantitative measurements on the steric dependence of reaction probability, Le., experiments employing reactants for which the extent of the spatial preference of the orientation can be controlled in a well-defined way,’-5 has attracted considerable attention of theorists in analyzing the results in terms of anisotropic potential barriers to reaction. Since elsewhere in this issue an excellent review of the theory of the steric effect is presented by Bernstein, Herschbach, and Levine6 and recent experimental developments are appraised by Parker, Jalink, and Stolte’ a brief introduction to the subject will be adequate. From the steric measurements, mentioned above, experimentalists are capable of gaining information about the reaction (1) Ende, D. v. d.; Stolte, S. Chem. Phys. Lett. 1980, 76, 13; Chem. Phys. 1984, 89, 121.
(2) Parker, D. H.; Chakravorty, K. K.; Bernstein, R. B. J . Phys. Chem. 1981, 85, 466; Chem. Phys. Lett. 1982, 86, 113.
(3) Choi, S . E.; Bernstein, R. B. J . Chem. Phys. 1985, 83, 4463. (4) Jalink, H.; Parker, D. H.; Meiwes-Broer, K. H.; Stolte, S. J . Phys. Chem. 1986, 90, 552. (5) Jalink, H.; Parker, D. H.; Stolte, S. J . Chem. Phys. 1986, 85, 5372. (6) Bernstein, R. B.; Herschbach, D. R.; Levine, R. D., in this issue. (7) Parker, D. H.; Jalink, H.; Stolte, S., in this issue. (8) Stolte, S.; Chakravorty, K. K.; Bernstein, R. B.; Parker, D. H. Chem. Phys. 1982, 71, 353. (9) Bernstein, R. B. J . Chem. Phys. 1985, 82, 3656.
0022-3654/87/2091-5480$01 S O / O
+
+
probability as a function of a single orientation of the molecular reactant axis.’+ Interestingly it turns out that these secalled steric opacity functions exhibit a range of orientation angles of the reactant axis for which the reactivity is essentially zero accompanied by a range for which reaction is found to be substantial, imaging the configuration of the (suggested) transition state. Although ultimately the steric dependence observed should be compared with the theoretical results obtained from extensive dynamical (trajectory) calculations on sophisticated potential surfaces, the unavailability of such surfaces and calculations in practical cases and the dominance of the steric observations attract a search for a direct connection between steric effects and the anisotropy of the entrance barrier to reaction. Recently, preluded independently by Smithlo and Pollak and Wyatt,ll Levine and Bernstein’* developed and generalized an excellent tool for such a (partial) analysis, called the angle-dependent line of centers model (ADLCM), which was shown to be very attractive because of its simple concept; i.e., reaction occurs only when a reactant atom approaches the reactant molecule, surrounded by an imaginary hard shell, with sufficient kinetic energy along the line of centers to surmount the steric barrier, (IO) Smith, I. W. M. J . Chem. Educ. 1982, 59, 9. (11) Pollak, E.; Wyatt, R. E. J . Chem. Phys. 1983, 78, 4464. (12) Levine, R. D.; Bernstein, R. B. Chem. Phys. Lett. 1984, 105, 467.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5481
Steric Effects in Reactive Collisions Vo(cos yr), at this location (see Figure 1). Such hard ellipsoids turn out to be also adequate in describing the essentials of the shape of the anisotropic potentials encountered at rotational inelastic collisions for simple molecules (e.g., ref 13-15). The reliability of ADLCM was tested and confirmed for the reaction H + D2 H D + D by comparing the results with those of classical trajectory calculations on a full potential surface.12J6 The original device of ADLCM for spherical shells was extended later on to a kinetic theory of reactive collisions of hard ellipsoidal surfaces1' incorporating the possibility of nonrigid spherical molecules with angle-dependent activation barriers.18 Recently the effect of loading, i.e., a shift of the center of mass from the geometrical center in the cylindrically symmetric reactant molecule, has been included in the treatment.4~'~In general these extensions of the original spherical model12appear to include minor changes only. ADLCM has been employed to interpret the full classical trajectory calculations for the various isotopic variants of the H H2 H2 H exchange reaction.20,21 Explicit predictions for the (J, K, M) quantum-state dependence of the total reaction cross section for spherical top molecules have been derived from ADLCM also.22 Finally, the steric effects observed experimentally and Ba N20 for the reactions Rb CH31 RbI + CH32*3*8*g,23 BaO* were analyzed with ADLCM. It is important to realize, as noted earlier in ref 16 and 27, that two positions along the collision path are convenient for defining the orientation of the molecular reactant axis, Le., one at (infinite) large distance from the impact region defining the initial angle of attack
-
+
-
-
+
+
+
-
+
X
Figure 1. Artists view of the reactive encounter of an atom with velocity v', impact parameter 6 , and azimuthal orientation & around B following the straight trajectory T which crosses an ellisoid-shaped anisotropically permissive (note the shading) barrier to reaction surrounding the molecule. The barrier is assumed to be cylindrically symmetric and is oriented with its symmetry axis along the z axis. The initial orientation of D (lying in the xz plane) with respect to the symmetry axis is given by yo. The crossing point of T with the barrier defines the reaction angle yr.
N2437324
yo = arccos (i.0,)
(1)
as the angle between the relative velocity 0, of reactant molecule and atom and the reactant molecular axis, i,before the upcoming collision. Since the rotation of the reactant molecule is assumed to be much slower than the collision time (defined as the diameter of the barrier divided by u,), as is typically experimentally enthe direction of i does not change during the approach toward the barrier.10J2,22The alternative definition of the angle of attack, which we call the reaction angle
is conceived a t the position of impact of $he reactant atom onto the shell around the reactant molecule. Roindicates the spatial displacement of the atom with respect to the center of (the ellipsoid around) the molecule at this critical configuration. We want to stress that for some systems our definition of the reaction angle differs from the orientation angle $ as proposed by Smith.lo In our definition the mirror symmetry cenier of the potential is taken as the origin of the position vector Ro. This implies that the reaction angle does not change on substituting isotopic species as one expects the steric barrier not to change from isotopic substitution.21 The effect of the displacement of the center of mass of the molecule with respect to the center of the barrier ellipsoid will be discussed in more detail in section 3. (13) Beck, D.; Ross, U.; Schepper, W. Z . Phys. A 1979, 293, 107. (14) Faubel, M. In Fundamental Processes in Atomic Collision Physics; Kleinpoppen, H., Lutz, H. O., Briggs, J., Eds.; Plenum: London, 1985. (15) Buck, U. Comments A t . Mol. Phys. 1986, 17, 143. (16) Blais, N. C.; Bernstein, R. B.; Levine, R. D. J. Phys. Chem. 1985, 89, 10. (17) Evans, G. T.; She, R. S. C.; Bernstein, R. B. J . Chem. Phys. 1985, 82, 2258. (18) She, R. S.C.; Evans, G. T.; Bernstein, R. B . J. Chem. Phys. 1986, 84, 2204. (19) Evans, G . T. J . Chem. Phys. 1987, 88, 3852. (20) Schechter, I.; Kosloff, R.; Levine, R. D. J . Phys. Chem. 1986, 90, 1006. (21) Schechter, I.; Levine, R. D. Int. J . Chem. Kinet. 1986, 18, 1023. (22) Levine, R. D.; Bernstein, R. B. Chem. Phys. Lett. 1986, 132, 11. (23) Bernstein, R. B. In Recent Advances in Molecular Reaction Dyanics; Vetter, R., Vigut, J., Eds.; Editions de CNRS: Paris, 1986; p 51. (24) Jalink, H.; Janssen, M.; Harren, F.; van den Ende, D.; Parker, D. H.; Stolte, S . In Recent Advances in Molecular Reaction Dynamics; Vetter, R., Vigut, J., Eds.; Editions de CNRS: Paris, 1986; p 41.
In this paper we report the implications of these alternative definitions of orientation for the steric dependence of reactivity. A computational algorithm according to the concept of ADLCM has been developed of which the essentials are outlined in section 2. The calculated results and a discussion about the consequences of expressing the steric dependence in terms of yoand yr on the basis of a few practical cases are presented in section 3. Section 4 summaries the main conclusions. Finally in the Appendix analytical expressions useful in steric calculations are given.
2. Computational Outline of ADLCM The central idea of ADLCM, that reaction can only m u r when there is sufficient collisional energy available to surmount the potential barrier Vo(cos 7,) at the shell surface, is illustrated in Figure 1. To facilitate the calculation, the center of the shell, 0, represented for practical purposes by an ellipsoidal or spherical surface, is supposed to coincide with the origin of the coordinate system, S, in which the collision between the reactants is described. The direction of the molecular axis, i, has been chosen along the z axis of S (initial molecular rotation is assumed to be absent, see also ref 22) and, for reasons of symmetry, is assumed to coincide with the major axis of the cylindrically symmetric shell surface. Putting the length of this axis equal to 2R, and the size of the orthogonal degenerate axes equal to 2Rb, the surface of the ellipsoidal shell can be expressed simply as
(a)'+($+ (E)'=
1
(3)
To account for all possible initial angles of attack, yo, the stream of incoming atoms is assumed to have its velocity v' parallel to the xz plane and to be Gstributed homogeneously over all possible impact parameters b (see Figure 1). Since u' is directed antiparallel to the relative velocity Cr, defined as the velocity of the oriented reactant with respect to the isotropic reactant (5 -CJ, the following parametrization is extracted from eq 1 U^ = -(sin yo2 cos yo i) (4)
+
The introduction of an azimuthal angle, @o, of the impact parameter b around v' completes the characterization of the initial conditions in terms of yo, $o, b, and u . Absence of interaction between the reactants is assumed for distances outside the shell surface of eq 3. Thus the approach of the atom can be described by the straight-line trajectory T d , = bh + si3 (5) where s is a scalar registering the elapsed distance and
5482
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987
do cos yo 2 + sin do P + cos dosin
6 = -cos
yo 2
(6)
According to the extended concept of ADLCM three conditions determining the reaction probability have to be considered. A first requirement for reaction is the occurrence of an intersection of the trajectory T with the steric barrier of Figure 1 . Substitution of eq 5 into eq 3 yields a square equation in s. Its root so can be solved in a straightforward way
Janssen and Stolte
TABLE I
A Rb,A d C M ~A I d W , A2 R,,
Vo,eV V I ,eV cos Yc h
Ba+N20
Rb+CHJ
H + D,
3.43 2.27 0.043 1.21 0.048 -0.048 -0.6 1.283
5.15 4.43 0.728 1.26 0.092 -0.059 -0.644 0.351
1.32 1.32 -0.374 0.699
0
with
a2 = (sin yO/Rb)'+ (cos T ~ / R , ) ~
a , = 26 sin yo cos yo cos do[1/Ra2 - 1 a. = b2(sin2 +o
(7a) (7b)
+ cos2 yo cos2 I#Jo)/Rb2 + b2 sin2 yo cos2 do/R? - 1 (7c)
The contraction of the negative root only for the solution of the square equation in s is induced by our interest in thefirst crossing of the straight-line trajectory T with the steric barrier. Having deterrpined the position of the intersection with the barrier, Ro = b.6 sou^, the second condition for reaction is to have sufficient kinetic energy along the line ofAce!ters to surmount the barrier height Vo(cos 7,)with cos y, = Z.Ro (see eq 2); Le., the component, 2.5, of the incoming velocity along the normal A of the ellipsoidal surface at the crossing position has to be large enough to overcome the b a r ~ i e r , ~ ~ - ' ~ *From ~ ' - ' ~elemen_tay geometry, after defining X o Ro.X, Yo i?o-p,and Zo= Ro.Z, one obtains
+
[X$/Rb2
A=
+ YoP/Rb2+ Z $ / R a 2 ]
[Xo2/R2+ Y o 2 / R 2+ Z02/R,4]1/2
(8)
The third and final condition for reaction to occur in ADLCM arises from the question whether the center of mass of the reactant molecule is positioned on the straight line, resulting from extending A to the interior of the surrounding ellipsoidal barrier. A displacement dCMof the C M of the molecular reactant from the extended A vector occurs for a nonspherical barrier, or when the shell is spherically shaped but the center of the sphere does not coincide with the C M of the molecular reactant itself. In such cases the incoming atom remains incapable of fully utilizing its kinetic energy, directed along A, to surmount Vo(cos 7,).Part of it will be transposed into rotational energy of the frame of the reactant molecule. The molecular C M is assumed to be located on its symmetry axis, i.e., at dCMZin Figure 1. Consequently the incoming kinetic energy along A, E , = 1/2p,2 with Cn = (&?)A and I.L (MatomMmlecule)/(Matm + Mmolecule), is separated into a component toward the molecular CM, Enll,and a component perpendicular to this direction, E,,. For the first on:, corresppnding to the component of 6,directed along RM E Ro - dc&, one obtains E,,, = (0,*kM)2E,
(9)
as kinetic energy fully available for the surmount o,f Vo(cos yr). However, for the second component, E,, = (1 - (u^,.RM)2)E,, only , Ibequalling the moment of the fraction I b / ( I b + p R M 2 ) with inertia of the molecule around the X or Y axis (see Figure l ) , is available for the surmount because of rotational recoil. Thus the total of the kinetic energy E,, avajlable to overcome the barrier Vo(cos yr)at the reaction point Ro,is given by
with E,, = 1 / z ~ u 2 . To yield the reaction cross section as a function of cos yo, U,(COS yo), or as a function of cos y,,C,(COSy,), a computer program has been developed, which calculates the reaction probability (0 or 1) by employing eq 1-10 for the trajectories T , initiated from regularly distributed values of b, do, and cos yo, for all possible
configurations. A straightforward integration scheme, employing the simple trapezoid rule, was used to average with sufficient accuracy over the range of interest, i.e., 50, 50, and 100 individual values were taken for b, do,and cos yo,to cover the ranges 0 R,, 0 2 ~and , -1 1, respectively. The dependence of uR on cos y,was obtained by dividing the total range of -1 Icos y, 5 1 in 100 bins, regularly spaced. Now for each trajectory turning out to be reactive, one adds a 1 to the bin to which the involved value of y,belongs. As a result one extracts U,(COSy,),resolved over 100 evenly distributed values of cos 7,.
-
-
-
-
3. Results and Discussion For the reaction Ba + N 2 0 BaO* + N 2 the orientational dependence of reactivity has been investigated experimentally at the lowest (E,, = 0.075 eV) and at the highest (Elr = 0.13 eV) collision e n e r g i e ~ .For ~ the reaction R b + CHJ RbI + CH, measurements of the steric dependence of the reaction probability have been performed at E,, = 0.13 eV.29399 Finally for the reaction H + D2 H D + H steric effects have previously been studied extensively by means of trajectory calculations and ADLCM at E,, = 1.3 eV.I28l6 Simulations of U,(COSyo) and CJ,(COS 7 , )have been performed for these three reactions. For all cases a considerable decrease of the dependence of uR on cos yoas compared to the dependence of uR on cos y,has been found. This decrease has a purely geometrical origin. Trajectories with large impact parameters departing with unfavorable initial orientation (cos yo Icos yc, with yc representing the cutoff angle at which the steric barrier becomes infinite) can arrive at the barrier with favorable orientation (cos y,-- 1) and consequently yield reaction. The model parameters (R,, Rb,dcM,I b / p ) and the steric barrier Vo(cos yr) used in our calculation resemble reported values from literature and are listed in Table I. The three reactions under investigation turn out to represent different types of steric barriers. Consequently our results and discussion will be presented sequentially for the reactions studied. The results of our calculations, CJ,(COSyo) and U,(COSy,), for the reaction Ba + N 2 0 BaO* + N2 are plotted in Figures 2 and 3. A sphere-shaped barrier (X = 0; remember X = (R,/Rb)' - 1 is the colliding pair anisotropy parameter17) and an ellipsoid ( A = 1.283)-shaped barrier were taken for Figures 2 and 3, respectively. For both figures the same linear dependence of the barrier height upon the approach angle yr was inserted, in conformity with ref 4. For cos y, > cos yc one defines
-
-
-
Vo(c0s 7,)= Eo
+ E,(1 - cos 7,)
(1 1)
for cos Y~Icos ycreaction is supposed to be impossible and V0(m 7,) = a. Note that E l = -VI and Eo = V,, Vl, with Voand Vl listed as parameters in Table I. For the purpose of inspection the actual behavior of Vo(cos 7,) is included in the upper panels of Figures 2 and 3. As a special feature of this reaction one notices the absence of a potential barrier for the most favorable configuration at cos yr = 1, i.e., Eo = 0. Moreover, in contrast to the case of E,, = 0.13 eV in the lower panels of Figures 2 and 3, the presence of the cutoff angle (cosyc = -0.6) remains unrecorded for E, = 0.075 eV in the upper panels of Figure 2 and 3, due to lack of translational energy ( Vo(cosyc) = 0.0768 eV). As expected'* an abrupt switch-off for u,(cos yr) occurs in the lower panels of Figures 2 and 3 when cos y,becomes smaller than cos ye This discontinuity appears not to exist for U,(COSyo). The geometrical averaging
+
Steric Effects in Reactive Collisions
Bar N,O
01 - 10
e c
t
I € t,: 0 13 eVl
n -1 0
0
-
10"
c0sYa
Figure 2. Orientational dependence of the reaction cross section uR for the reaction Ba + N20 BaO* + N2 assuming a sphere-shaped barrier with R, = Rb = 2.8 8, and translational energy E,, = 0.075 eV (upper panel) and E,, = 0.13 eV (lower panel). For both energies the same anisotropic potential barrier Vo(cos7,)has been assumed (for details of parameters see Table I) and is shown in the upper panel with left ordinate. The solid curve displays the dependence of uR upon the initial
-1 0
0
10'
cosYa
Figure 3. All parameters as in Figure 2 but an ellipsoid-shaped barrier with R, = 3.43 ,& and Rb = 2.27 A. f t ' cross sections have been normalized to T R =~37.0 ,&*. For detail. see caption of Figure 2.
expressions for U, with n I 3 can be found in the Appendix. The resulting orientationally averaged cross section, uo, and the loworder Legendre moments ul/uo, u2/uo,and u3/yo are listed in orientation cos yo and the dots upon the reaction orientation cos 7,.The Table I1 for comparison. For the spherical barrier of Figure 1 dashed (---) curve shows the orientation dependence of the cross section there is essentially no difference between the results of eq 12 and upon cos yras predicted by the Evans formula, eq 12. All cross sections our numerically obtained dependence upon cos yr. On the other have been normalized to T R =~24.6 A*. hand the orientational effect ul/uo is found to be about 50% larger in cos yr than in cos yo, and, as was argued earlier,5 appears to for U,(COSy o ) is also responsible for, in comparison with U,(COS decrease with Ew Moreover, in Table I1 the moments ul/uo, u2/u0, yr), the decreased range of (unfavorable) angles of attack (at about and uj/uo exhibit a dependence upon E, which is at least as strong cos y o = -1) for which reaction turns out to be impossible. as that of the orientational averaged reaction cross section uo. To check the accuracy of our numerical treatment and to make Addressing ourselves to the case of the ellipsoidally shaped (A a connection to the work of Evans et al.,17318 we compare our values = 1.283) barrier of Figure 3 the startling maximum of U,(COSyo) of U,(COS 7,) with those following from the Evans approximation at cos yo = 0.3 (and not at the expected head-on orientation (cos O d C O S 7,)= y o = 1)) appears in both panels as an outstanding feature. Inspection of U,(COS 7,)does not offer a clue for this surprising behavior. The peaking of U,(COS7,)at cos yr = 1 turns out to be even more pronounced in Figure 3 than in Figure 2. To explain the maximum of U,(COSyo) at cos yo = 0.3 we first consider the more simple example of the collision probability, uT, for COS y,1 1 - (Eu.- Eo)/E,and COS yr 1 cos yo otherwise U,(COS for a hard ellipsoid and a sphere generated by a barrier with an yr) = 0. For spherically shaped barriers (A = 0) with dcM= 0 identical geometrical shape as the one between Ba and N 2 0 . eq 12 reduces to the original result of Levine and Bemstein forming The total collision cross sectioi j U ~ ( C Oyo) S and u,(cos 7,) an exact ADLCM solution for uR(cos yr). According to eq 12 describing this probability have been obtained by simply calcuwe find indeed that the dashed curves (--- in Figure 2) agree lating the reaction cross section numerically with ADLCM for very well with the points calculated with our numerical procedure. the case of a barrier with vanishing height (Le., with E , = Eo = The relatively small offset of the C M from the center of symmetry 0). The results of these calculations for a spherical and an elwithin the N 2 0 (dcM= 0.043 %I, R, = Rb = 2.8 A!) appears to lipsoidal shell, possessing the same shape as those used in Figure have negligible influence upon the results. 2 and 3, are displayed in Figure 4 and Table 11. Of course the To put the comparison between the calculated U,(COSy o ) and obvious result for a sphere, U~(COS yo) = u T ( m yr) = r R 2 , served U,(COS yr) and the Evans approximation on a quantitative footing only to confirm the reliability of the numerical procedure. In the the steric dependence has been expanded in a series of Legendre case of an ellipsoid, one deduces from simple geometrical conpolynomials in cos y n (a = 0 or r ) , yielding siderations uT(cos yo = f l ) = TRb2 = ~ R 2 / ( 1+ k), U,(COSyo UT(COS yr = f1) = rRa2,and (13) = 0) = rR,Rb = a R 2 / ( 1 4UR(COS 7), = E unPn(COS ya) n=O U ~ ( C O S7r = 0 ) = T R =~rR2/(1 + A). These values are obtained This also allows an easy comparison with experiment v a l ~ e s . ~ - ~ indeed for ~ ~ ( yo), ~ curved 0 s concave down, and UT(COS y,), C U N ~ The Legendre moment, u,, can be obtained by straightforward concave up. It is just this opposite dependence upon orientation for the upper bound cross sections UT(COS 7,)and COS y o ) that integration. For the Evans approximation of eq 12 analytical
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987
5484
Janssen and Stolte
TABLE IIy Ba + N,O (R. = Rk = 2.8 .&) E,, = 0.075 eV E,, = 0.13 eV quantity uo.
yo) UR(COS yr) 9.80 (9.62) 1.06 1.44 (1.44) -0.24 0.24 (0.24) -0.11 -0.19 (-0.19)
UR(COS
A2
uI/uo u2/uo
~j/ro
UR(COS yo) U,(COS yr) 14.1 (13.9) 0.64 0.93 (0.94) -0.32 -0.27 (-0.26) 0.10 0.09 (0.09)
Ba + N 2 0 (R, = 3.43 E,, = 0.075 eV quantity
uR(cOs 700) Yr) 8.65 (9.29) 1.74 (1.70) 0.85 0.92 (0.87) -0.59 -0.17 0.35 (0.29)
go> .A2 U ] / U ~
U Z / U ~
u~/uO
A, Rb = 2.27
UR(c0S Yr) 11.9 ( 1 2.6) 0.58 1.24 (1.21) -0.63 0.33 (0.32) 0.03 0.45 (0.42)
UR(COS 70)
R b + CH31 (R,= 5.15 A,R b = 4.34 A)
+ D, ( R , = R, = 1.32 A)
H
E,, = 0.13 eV vantit)
~ R ( C O SYO)
A2
U,,/U~ U Z / U ~
u~/uO
Ba quantity
~ ( ~ yr) 0 s 1.04 (1.08) 1.70 2.00 (1.93) 0.54 1.21 (1.04) -0.41 -0.14 (-0.25)
OR(COS YO)
0.51
X
(R, = 3.43 A,
(R,=
= 2.8
YO)
A2
u2/uo
~ R ( C O Syi)
+ N,O
Rb
00,
E,, = 1.3 eV
20.0 (20.8) 1.57 (1.47) 1.10 0.50 (0.41) -0.35 -0.30 -0.05 (0.04)
go>
.&)
E,, = 0.13 eV
A)
R, = 2.27 .&) Yr)
24.9 (24.6) 0.13 X lo-' (0.0)
uT(cos YO) OT(COs Yr) 22.1 (21.0) -0.24 0.60 (0.60)
"All values in brackets follow from the Evans approximation of eq 12, except for H Dz where the equation given in the caption of Figure 6 has been used.
+
.-*,..a , o &'..
.'a.
.*...,, .*
:*.-
a*..
.
. ,a=r ...**...**,**'lo I -.*_....._. . . .- ::... \*."-."' *
...e
a,b spherical shell
,/.'-
.?
05
05
VI
I
c , d ellipsoidal shell
I '
e
Figure 4. Influence of the geometrical shape of the barrier upon the total collision cross sections U,(COS yo) and UT(COS yr). The solid curves, b and c, represent the numerical result for UT(COS yo) in the cases of a sphere-shaped (R, = Rb = 2.8 A) and an ellipsoid-shaped (R, = 3.43 A, Rb = 2.27 .&) shell, respectively. The dots labeled a and d show O,(COS yr) for the sphere and the ellipsoid, respectively. The dashed curve (---) displays U,(COSyr) for the ellipsoid as predicted by the Evans formula, eq 12. All cross sections have been normalized to TR;. Notice the yr) (d) reflecting the different extrema of a,(cos yo) (c) and UT(COS effective area of the ellipsoid.
is responsible for the enhanced peaking of O,(COS yr) at cos yr = 1 in Figure 3 compared to the spherical case of Figure 2 and for the "surprising" maximum of C~,(COS yo) in Figure 3 at cos yo = 0.3. The ratio of uo for the total reactive cross section U, and total collision cross section aTwe like to call the average steric opacity, Le., the fraction of reactive encounters out of all collisions between two unoriented reactants.
0 COSY~
-1 0
-
10-
Figure 5. Orientational dependence of the reaction cross section uR for
+
+
the reaction R b CH31 RbI C H 3 assuming an ellipsoid-shaped barrier with R, = 5.15 A and Rb = 4.34 A. The steric barrier Vo(cos yr)used in the simulation is included in the figure with left ordinate (for details of parameters, see Table I). The solid curve shows the dependence of uR upon the initial orientation cos yo, the dots upon the reaction orientation cos y,. The dashed (---)curve displays O,(COSyr) as predicted by the Evans formula, eq 12. All cross sections have been normalized to rRa2= 83.3 A'.
For aT(cosyr)we employed the Evans approximation the more simple variant following from eq 12, recommended by ref 17 and 19 yielding UT(COSyr) = rR;(1
+ X COS'
yr)/(l
+ A)
(14)
Although the Evans approximation is strictly valid only at X
