J. Phys. Chem. 1991, 95, 1621-1625
1621
Kinematic Mass Effect in the Dynamical Stereochemistry of Activated Bimolecular Reactionst H. Kornweitz, A. Persky, Department of Chemistry, Bar Ilan University, Ramat Can 521 00, Israel
and R. D. Levine* The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91 904, Israel, and Department of Chemistry and Biochemistry, The University of California, Los Angeles, California 90024 (Received: June 6, 1990)
The dynamics of elementary A + BC exchange reactions with restricted cones of acceptance can be very much dependent on the location of the BC center of mass. Other things being equal, the heavier B is, the wider is its cone of acceptance for reaction with A, for a nonrotating BC molecule. The reactivity will, however, significantly decline the higher the rotational energy of BC is as compared to the A-BC relative kinetic energy. The opposite conclusions apply for reaction with the C end. The conclusions are motivated by analytical considerations of the variation of the effective potential energy function upon isotopic substitution and by explicit potential maps for the 0 + HCI, CI + HCI, F + H2, CI + H2,and 0 + H2 reactions. These demonstrate that the heavier B is. the more the potential gradient is directed so as to orient reagents into the cone of acceptance.
Introduction Isotope effects in reaction dynamics are not as easily interpretable as one could wish. The reason is that what is invariant under isotopic substitution is, in the Born-Oppenheimer approximation, the potential energy as a function of the physical interatomic distances. Yet the kinetic energy for the motion of the nuclei is not diagonal in these coordinates. To make it diagonal (Le., a sum of squared terms), one needs to transform the coordinates in a mass-dependent fashion.'-4 The most familiar example is the mass-skewed coordinate system for a collinear atom-diatom collision.2-s This transformation, while allowing us to simplify the dynamics, results in a potential energy function that is no longer isotopically invariant. In the studies of vibrational energy requirements and disposal, this leads to the "light (and also heavy) atom a n ~ m a l i e s " . ~These * ~ refer to the qualitative changes in the topography of the potential energy surface (discussed typically for a collinear collision) that may occur for extreme mass combinations. The purpose of this paper is to point out that such anomalies are also possible in the features of the potential energy that govern the steric requirements of atom-exchange reactions. The close correspondence between the stereochemistry and the role of reagent rotation (and also of rotational energy d i ~ p o s a l ) ~provides -'~ another motivation for our considerations. That isotopic substitution can modify the steric requirements was pointed out in the study of the H + H D and H H2 reactions,"-'* resulting in different "cones of acceptance" about the H and D atoms. The implications for the branching ratio in the H H D collision turn out to be quite dramatic." Other studies of mass effects include refs 15 and 16. Hard ellipsoid modeld7J8 can also exhibit the effect, provided the ellipsoid can be loadedIg (i.e., the center of mass need not coincide with the geometric center of the ellipsoid). A general discussion of isotope shifts on intermolecular potentials has also been given.20,21 Our discussion of the collision dynamics for a given potential energy is based on the transition-state-like approximation that trajectories which cross the barrier to reaction will proceed to form products. The reactivity is therefore determined by the flux of trajectories at the barrier. It should be recognized at the outset that, while this assumption is quite common, it can and will break down at sufficiently high collision energies (e.g., Figures 6 and 7 of ref 1 I ) . Moreover, for exoergic reactions with an early barrier,
the trajectories reach the inner hard core with higher energy than their nominal initial collision energy. Hence, the validity of this no-recrossing assumption cannot be taken for granted. For the purpose of visualizing and discussing the proposed mass effect, it is useful to make two additional approximations. Exact trajectory computations show that the effect in question is not dependent on making these approximations. The considerable simplification that is provided by introducing them make it, to us, very worthwhile. The first is that en route to the barrier the j,-conserving a p p r o ~ i m a t i o n ~ J ~is2valid. ~ 2 ~ In this approximation, the relative motion of A as it approaches BC is confined to a fixed plane. We furthermore assume that the BC molecule is not (1) Hirschfelder, J. 0.;Jespen, D. W. Proc. Nutl. Acud. Sci. U.S.A. 1959, 45, 249.
+
(2) Hirschfelder, J. 0. Int. J . Quantum Chem., Symp. 1969, No. 3, 17. (3) Smith, F. T. J. Chem. Phys. 1959, 31, 1352. (4) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics and Chemical Reacfioify;Oxford University Press: New York, 1989. ( 5 ) Mahan. B. H. J . Chem. Educ. 1974. 51. 308. (6) Polanyi; J. C. Faraday Discuss. Chem. Soc. 1973,389. Polanyi, J. C. Science 1987, 236, 680. (7) Kuntz. P. J. In Dynamics of Molecular Collisions; Miller. W . H.. Ed.: Plenum: New York, 1976. (8) Sathyamurthy, N. Chem. Reu. 1983, 83, 601. (9) (a) Loesch, H. J. Chem. Phys. 1986, 86, 213. (b) Grote, M.; Hoffmaster, M.; Schleysing, R.; Zerhau-Dreihofer, H.; Loesch, H. J. Selectivity in Chemicul Reactions; Reidel: Dordrecht, The Netherlands, 1988. (c) Loesch, H. J. Chem. Phys. 1987, 112, 85. (10) (a) Kornweitz, H.; Persky, A,; Levine, R. D. Chem. Phys. Lett. 1986, 128, 443. (b) Kornweitz, H.; Persky, A,; Schechter, I.; Levine, R. D. Ibid.
*Towhom correspondenceshould be addressed at The Hebrew University. 'Dedicated to the memory of the late Professor J. 0. Hirschfelder.
(11) Schechter, I.; Levine, R. D. I n f . J . Chem. Kinet. 1986, 18, 1023. (12) Persky, A.; Kornweitz, H. Chem. Phys. 1989, 130, 129. (13) Persky, A.; Kornweitz, H. Chem. Phys. 1989, 133, 145. (14) Bernstein, R. B.; Herschbach, D. R.; Levine, R. D. J . Phys. Chem. 1987, 91, 5365. (15) Mayne, H. R. J . Phys. Chem. 1988, 92, 6289. (16) Raghavan, L.; Sathyamurthy, N.; Garetz, B. A. Chem. Phys. 1987, 113, 187. (17) Evans, G. T.; She, R. S. C.; Bernstein, R. B. J . Chem. Phys. 1985, 82, 2258. (18) Janssen, M. H. M.; Stolte, S. J . Phys. Chem. 1987, 91, 5480. (19) Evans, G.T. J . Chem. Phys. 1987,88, 3852. (20) van Montfort, J. Th.; Heukels, W. F.; van de Rel. J. Chem. Phys. 1972, 57, 947. (21) Kreek, H.; LeRoy, R. J. J . Chem. Phys. 1975.63, 338. (22) Mulloney, T.; Schatz, G.C. Chem. Phys. 1980, 45, 213. (23) Muga, J. G.;Levine, R. D. J. Chem. Soc., Faraday Trans. 1990,86, 1669.
+
0022-3654191 /2095- 1621$02.50/0
1990,169,489.
0 1991 American Chemical Society
1622 The Journal of Physical Chemistry, Vol. 95, No. 4, 1991
Kornweitz et al. heavier than B’, the shift D in the location of the BC center of mass is positive:
The opening angle of the cone of acceptance, which is the exterior angle, is then larger: T - y > T - y’ (7) D
Figure 1. R and y coordinates and their shift when B’ is replaced by a heavier atom B, resulting in a shift, D, in the location of the center of mass (c.m.) of BC.
vibrating. It must, however, be emphasized that both of these assumptions are only made for the approach to the barrier. With these assumptions, one needs to deal with only two coordinates: R , the A to BC center of mass distance, and y, the R to BC bond angle.9q’0,23The collision can then be traced in the R - y plane as the kinetic energy is diagonal in these coordinates. The potential governing the dynamics is V(R , y ) obtained from the full, threevariable function V(R,r,y) by holding r constant upon isotopic substitution. The real invariant is V(rAB,rBC,rAC) where the r’s are the interatomic distances. We shall point out two aspects of interest. One is the shift in the location of the barrier to reaction upon isotopic substitution. The other is the shift of the equipotential contours of the potential V ( R , y ) . The latter is of importance in understanding the role of the potentials in orienting the reagents during their motion to the barrier. The prime origin of the mass effect is the dependence of the coordinate R, the A to BC center of mass distance, on the masses of B and C. A second mass effect that governs the magnitude of the effect of reagent rotational excitation is the ratio of the BC reduced mass to the A-BC reduced masslo.
Kinematics The A to BC center of mass coordinate R is related to the interatomic coordinates by R =
rAB
+ (MC/(MB + MC))rBC = rAC
-
+ MC))rBC
(1)
y is the angle between R and r rBc. In the j,-conserving approximation, the Hamiltonian for an initially nonrotating BC molecule is93’OJ3
= (rA,BC/2)RZ+ (pBC/2)?2 + v.fdR*Y) VerdR,y) = V ( R , y )+ ETb2/R2
(2)
(3)
Here b is the impact parameter, ET is the initial collision energy, and the dots denote the time derivative. When BC is initially rotating, there is an additional rotational centrifugal barrier in VefP Note that in the Hamiltonian (eq 2 ) y appears as a Cartesian and not as a polar coordinate. It is for this reasong that one plots the potential in a Cartesian R-y plane and not as a polar map. The opening angle of the cone of acceptance at the collision energy ET is the solution of4 Veff(R,y)= ET (4) for R having its value at the barrier. The maximal opening angle is for b = 0. Suppose we replace the exchanged atom by a heavier isotope. We shall use primed variables to refer to the initial, A B’C, system. When B’ is replaced by B, both R’and y’will change to R and y. Consider the triangle whose three vertices are at the old and new positions of the center of mass of BC and at the solution of (4). It follows from the law of sines (cf. Figure 1 ) that sin (T - y) = ( R ’ / R ) , sin (T - y’) (5)
+
where the subscript
* means evaluated at the barrier.
If B is
Note that the conclusion is valid even when the location d of the barrier varies with y, as it will for realistic potentials. Next we examine the “shape” of the BC molecule as seen by the A atom.lob Consider first the “painted-sphere” limit,24where the potential is that of hard spheres. In the R’-y’ plane, the location of the barrier is a y-independent line at R’ = d . Now draw the same contour in the R-y plane. The equation of the curve is that given by the law of cosines dz = 9 + R2 - 2DR cos y. (Recall that y is defined as the interior angle; cf. Figure 1). For a collinear approach, ( R ) , = d - D < d, so that the reactive end of the BC molecule is now recessed into the sphere. The larger the difference in mass terms in eq 6, the larger is the effect. The implication is that the heavier the reactive end, the more recessed is the barrier within the cone of acceptance. Conversely, the lighter the reactive end, the more the barrier protrudes. This implication has key consequences for the reactivity of both initially nonrotating BC molecules and rotating ones. The implications are opposite in character:Iob (i) an oblate nonrotating molecule has significantly higher reactivity than that judged on the basis of a hard-sphere model and (ii) at lower collision energies, increasing the BC rotational energy will lead to a considerable decrease in this reactivity. To avoid confusion, note that in general, the barrier height varies with the approach angle. Hence the location ( R ( y ) ) ,of the barrier is not an equipotential contour. We have been discussing the effect of isotope shift on the location of the barrier. We turn now to the shift in the shape of the potential contours. An oblate potential energy contour is one where a R / a y is negative, assuming the barrier is lowest for a collinear (y = 180°) approach. Now, from the law of cosines, we have for the painted-sphere model - q a R / a COS y) = 2 d s ~ (8) showing that increasing D will result in an even more oblate contour. Consider next the general case. Along an equipotential contour, V = constant. Hence -(av/ay) = (av/aR)(aR/ay) (9) Since dV/dR < 0 en route to the barrier, the torque -aV/dy, and the y motion, is toward larger values of y, i.e. more nearly collinear. A general transformation between the potentials expressed in the old R’,y’ and the new R , y coordinates is conveniently basedZoJ1 on the law of cosines (cf. Figure 1 ) R2 = R l 2 + D2
+ 2RD cos y’
(10)
the law of sines (eq 5 ) , and the Legendre expansion
Retaining only the n = 0 term in (1 1) corresponds to the limit where there is no torque on the A + B’C reagents en route to the barrier. In general, the generic features of the orienting nature of the potential are already well described by the n = 1, 2 terms in (1 1). The n = 1 term is the leading contribution to the orientation of the reagents en route to the barrier, while the n = 2 term induces an alignment. Since V(R,y) U(R’,y’), the explicit expansion for V ( R , y )in a Legendre series I/(R,y) =
Zn Vfl(R)P,,(cos y)
(12)
(24) Beuhler, R. J., Jr.; Bernstein, R. B. J . Chem. Phys. 1969, 51, 5305.
The Journal of Physical Chemistry, Vol. 95, No. 4, 1991 1623
Kinematic Mass Effect in Bimolecular Reactions can be obtained by using the orthogonality property of the Legendre polynomials:21 I
Vn(R)= (2n + 1 ) s d(cos y ) U (R’,y’) Pn(cos y ) (13) -I
If only the Uo(R’) term is used in the expansion, ( 1 1),21 one has, to order D2 Vo(R) = Uo(R) + D2[(1/3R)(dUo/aR) (1/6)(aZUo/aRz)]
v ~ ( R= ) D auo/aR
Vz(R)= D2[-(l /3R)(aUo/aR) + (1 /3)(a2Uo/aRz)]
(14)
0
-s
320
0
n 280
2 40
90
where the derivatives are with respect to R’evaluated at R ’ = R . The explicit results have equivalent implications for our considerations based on eq 9. Both Voand V2(R)have no term linear in D. It is the orienting potential, Vl(R), whose sign is determined by D. For the chemical forces en route to the barrier, -aUo/aR is positive, so that VI and D have the same sign. Including an orienting term ( n = 1 ) in U(R’,y’)will modify eqs 14 by adding terms linear in D.
v,(R) = u o ( ~ + D) Z [ ( ~ / ~ R ) ( ~ U+, / ~ R ) ( 1 /6)(azU0/aR2)1- D [ ( 2 / 3 R ) & ( R ) + (1/3)(aUI/aR)l
+ UI(R) + D 2 [ - ( 3 / 5 R 2 ) U ~ ( R+) (~/sR)(~u,+ / ~( 3R/ )i o ) ( a z u , / a ~ z ) ] 3 ~ ) ( a ~ ~+/ (1a/ 3~ ))( a w o / a ~ z )+]
Vl(R) = DaUo/dR
v,(R) = DZ[-(I/
90
180
180
gommo (deg)
Figure 2. Cartesian plots of V ( R , y ) contours, 2 kcal mol-’ apart, for LEPS type potential functions for the 0 + HBr r e a c t i ~ n . ~ ~The * ~ ~zero *~’ of energy is at the dissociated atoms. The dashed line indicates the collinear, y = 180°, approach for which the barrier is lowest. The solid arrows show the direction of the force, -aV/ay, on the Cartesian y motion (for initially nonrotating BC, Ia2y/at2= -aV/ay, where I is the BC moment of inertia). For the prolate surface I, this force tends to disorient trajectories that could be. otherwise reactive. For the oblate surface 11, this force pulls in trajectories with initial values of y differing from collinear ones. The location of the barrier to reaction is shown as a hemispherical line for lower R values. Note the rapid rise of barrier height with y for surface 11.
-
surface I
O+HC I
-+
OH* C I
surface 1
3Ao.
D [ ( 2 / 3 R W d R ) - ( 2 / 3 ) W 1 / a R ) l (15) The main difference from our point of view is that Ul(R)can now contribute directly to the orienting potential. The explicit conclusion is that, to order D VI(R) = UI(R) - D[(dUo/aR) ( 6 / 5 R ) U z ( R ) ( 2 / 5 ) ( a ~ a ~(16) )1
+
+
A D > 0 mass shift will make V l ( R )more positive than an already positive U I ( R )or less negative than a negative U l ( R ) . (The only limitation is if the aligning potential U z ( R )is more rapidly increasing as a function of R than Uo(R)is decreasing, when the term in square brackets in (16) will be positive.) Since Pl(cos 0’) = 1 and P,(cos 180’) = -1, a positive V l ( R )means that the contour V ( R , y ) = E occurs at a lower R value for y = 180’ than for y = 0’; Le., the potential is oblate. (Recall that Pz(cos y) is an even function of cos y.) To conclude, if the A + B’C potential is already oblate (Vl(R’) positive in the region leading to the barrier), replacing B’ by a heavier isotope B will make the potential even more oblate. a V ( R , y ) / d COS Vz(r) Ul(R) - DdUo/aR (17)
Replacing B’ by a lighter isotope ( D < 0) will make the potential less oblate, or even prolate. The reverse is true if the A + B’C potential is prolate. A heavier isotope will make it less prolate, or even oblate, while a lighter isotope will make it more prolate. Explicit examples of the two alternatives are provided in Figures 7 and 8.
Dynamics on Oblate vs Prolate Potentials An oblate potential energy surface tends to orient the reactants into the cone of a c c e p t a n ~ e . ~The ~ J ~reactivity ~ is higher than for a hard-sphere model (or, in general, when only the n = 0 term is retained in eq 12, since then there is no reorientationz5en route to the barrier) because the asymptotic range in y values that can result in reaction is larger than the range of y values allowed at the barrier. The greater the mass of the reactive end, the more oblate is the potential in the R,y coordinates, and the more important is this reorientation effect, other things being equal. A prolate potential energy surface tends to orient away from the cone of acceptance. The lighter the reactive end of the (25) Bernstein, R. B.; Levine, R. D. J . Phys. Chem. 1989, 93, 1687.
gamma (deg)
Figure 3. Cartesian plots of V ( R , y ) contours, 2 kcal mol-’ apart, for LEPS type potential functions used in dynamical computations for 0 + H C P and CI + HCLZ9 The solid line is the location of the barrier. As in Figure 2, the rise of the barrier height with noncollinearity is steeper for the more oblate type surfaces (right).
molecule, the more detrimental is this reorientation effect. As an aid in visualization, Figure 2 shows two realistic potential energy surfaces for the 0 HBr r e a c t i ~ n . ’ ~ * Surface ~ ~ , ~ ’ I is prolate. The gradient of V, i.e. the force on the trajectory, is shown as an arrow. Its direction is such as to disorientz5 the reactants and prevent their reaching the barrier (whose location is shown as a thick hemispherical line). Surface I1 is oblate and tends to reorient reactants into the cone of acceptance. For an oblate potential, initial BC rotational excitation can fully negate the enhanced reactivity due to the favorable reorientation effect.Iob The reason is the recessed cone of entry. The BC rotation corresponds to motion along the y axis in the R-y plane. S u c h a motion, if fast enough, may make the trajectory miss the
+
(26) Broida, M.; Tamir, M.; Persky, A. Chem. Phys. 1986, 110, 83. (27) McKendrick, K. G.; Rakestraw, D. J.; Zhang, R.; Zare, R. N. J . Phys. Chem. 1988, 92, 5530.
1624 The Journal of Physical Chemistry, Vol. 95, No. 4, 1991
Kornweitz et al.
surfoce I
2 20
-
-108kcol mol-'
IF+HD-.FHtD
lh \
/ A
IIF+DH--
/I
/-----I!-
FDtH
J
gomma (deg)
Figure 4. Same as Figure 3 but for a mass 35 "H atom" exchange. Note how the shift in the location of the HCI center of mass changes (i) the rise of the barrier height with y and (ii) the direction of the force on the y motion (cf. Figure 2).
gamma (deg) Figure 6. Cartesian plots of Muckerman VS3 LEPS type potential functions for F + H2 and isotopic variants as indicated. Other details are as in Figures 2-4. Note the very recessed approach to the barrier. An attack on the lighter end of the molecule occurs at a more outwardreaching barrier. In other words, the lighter end subtends a longer arm.
3'401 -00.3
rotational energy has a beneficial effect, as will be discussed in detail elsewhere.
/
Examples The essence of our argument is contained in the comparison of Figures 3 and 4. These and subsequent figures show V ( R , y ) plotted as a contour map in the RT plane. Figure 3 shows two previously usedZ8sMpotential energy functions, of the LEPS type, for the exchange reactions 0 + HCI CI + HC1
gamma (deg) Figure 5. Mass effect on the Cartesian plot of V ( R , y ) for surface I of the 0 + HCI reaction, for increasing H mass. Other details are as in
Figures 2-4.
entrance. This is in contrast to A-BC motion, which is perpendicular to the y axis. The relevant parameterlo is the ratio of rotational and linear uelocities. For prolate potentials, the BC
--
OH + CI CIH CI
+
Surfaces I are prolate type surfaces (much more so for CI + HCI), tending to disorient the reactants out of the cone of acceptance for reaction. Surface I is possibly more real is ti^,^^,^' but surfaces I1 for 0 + HCI and IV for CI + HCI cannot be ruled out on the basis of the available experimental data. These surfaces are oblate (much more so for 0 + HCI) and tend to orient reagents into the cone of acceptance (quite significantly so for 0 + HC1'0b,25). Figure 4 is identical with Figure 3 except that it is plotted for the "heavy" H isotope of mass 35 amu. The potentials are much more oblate than before. Surface I for CI + 35HC1has the contours nearly y-independent, and so is particularly useful in (28) Broida, M.; Persky, A. J . Chem. Phys. 1984,81,4352. (29) Persky, A,; Kornweitz, H.J . Phys. Cfiem. 1987, 91, 5496. Kornweitz, H.; Persky, A. Chem. Pfiys. 1989, 132, 153. (30) Bondi, D. K.; Connor, J. N. L.; Manz, J.; Romelt, J. Mol. Pfiys. 1983, 50. 461. (31) Schatz, G. C. Private communication. See also: Schatz, G. C. Reu. Mod. Phys. 1989, 61, 669.
3,00r-l
The Journal of Physical Chemistry, Vol. 95, No. 4, 1991 1625
Kinematic Mass Effect in Bimolecular Reactions CI+HH -HCI+H
2.80
]O+HH-+OH+H
.109kcol mol-l
2.401 2.60! 2.20.--'07
2 00
-
2.401 J
2.00
I
-107
mL
2.20
A
1.80
1.40&
..
.
.
.
. . . . . . . . 180
.
. 1
90
180
gamma (deg)
Figure 7. Same as Figure 6 but for the C1+ H2 reaction with an LEPS type GSW potentiaL3' Note that, for this less oblate potential, attack on the lighter end of the molecule already occurs on a prolate surface.
studies where one wants the LEPS potential to neither orient nor disorient the reactants.32 The gradual change in the nature of the surface with the shift, D, in the center of mass of BC is shown for 0 + HCI in Figure 5. Our considerations become particularly relevant for comparing the reactivities on the two ends of the molecule." We illustrate this with three examples. The Muckerman V LEPS type potential33has been extensively applied in studies of the dynamics of the F + H2 reaction. The plot, Figure 6, shows it to be an extremely oblate type potential with a very recessed entry to the barrier region (shown as a solid line giving the location of the saddle point in the R-y plane). Note the clear shift in the location of the barrier upon isotopic substitution. The lighter the attacked atom, the further out is the barrier along the R direction. The potential is so oblate that it remains oblate (but less so) even when the mass ratio is 1/3. Note that the heavy end of the molecule is always more recessed than the lighter one. This is the origin of the effect pointed out by Muckermad3 in explaining the role of HD rotational excitation on the branching ratio. The LEPS type C1 + H2 GSW potentiaV4 is very nearly yindependent, as shown in Figure 7. Hence, isotopic substitution has a clearly noticeable effect. The lighter end of the molecule will extend further in R, will tend to disorient incident trajectories, but will be the more reactive end for rotationally excited reagents at lower collision velocities. A similar result is found for the 0 (32) Benjamin, 1.; Liu, A.; Wilson, K. R.; Levine, R. D. J . Phys. Chem.
1990, 94, 3937.
(33) Muckerman, J. T. In Theoretical Chemistry; Henderson, D., Ed.; Academic Press; New York, 1981, Vol. 6. (34) Stern, M. J.; Persky, A.; Klein, S.F. J . Chem. Phys. 1973.58, 5697. Persky, A. Ibid. 1977, 66, 2932; 1979, 70. 3910.
gamma (deg)
Figure 8. Same as Figure 6 but for the prolate 0 + H2potential?$ Here, attack at the heavier H isotope occurs on an oblate potential. As is also the case in Figure 7, isotopic substitution can change the orienting nature of the potential en route to the barrier.
+ H2 potential,35 plotted in Figure 8. Discussion Trajectory comp~tations'~,'r"2*~~ have clearly established that, for a given mass combination, oblate and prolate type potentials, as illustrated in Figures 2 and 3, can give rise to rather distinct dynamical features. Those aspects that relate to steric requirements have been emphasized here: viz., that an oblate surface will orient the reactants toward the cone of acceptance with the opposite being the case for a prolate surface. Other aspects relating to attributes of products have been discussed in detail elsewhere.'2,13+29The point made in this paper is that the very same potential function can appear prolate in some mass combinations and oblate in others. The form of the Hamiltonian, eq 2, ensures that what is relevant for the dynamics is the shape of the potential when plotted in a Cartesian R--y plane. Of key importance is the direction of the force on the y motion. The ratio of the BC to A-BC reduced masses governs the relative importance of reagent rotational and translational energies but cannot affect the sign of 4 V / h . Hence, the very same potential energy function expressed in terms of interatomic distances may either orient or disorient the reagents, may or may not lead to a detrimental effect of BC rotational excitation, and may have a wider or narrower cone of acceptance, depending on the B/C mass ratio. Acknowledgment. The research at Bar Ilan University was supported by the Fund for Basic Research, administered by the Israel Academy of Sciences and Humanities. The Fritz Haber Research Center is supported by the Minerva Gesellschaft fur die Forschung, mbH, Munich, BRD. (35) Johnson, B. R.; Winter, N. W. J . Chem. Phys. 1977, 66, 4116.
