5472
J. Phys. Chem. 1987, 91, 5472-5480
A Classical Kinematic Model for Direct Reactions of Oriented Reagents I. Schechter, M. G. Prisant,+ and R. D. Levine* The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91 904, Israel (Receiued: February 12, 1987)
A simple kinematic model based on the concept of an orientation-dependent critical configuration for reaction is introduced and applied. The model serves two complementarypurposes. The very concept of a critical configuration of no return provides a practical link between the experimental laboratory-based measurements and the features of the potential energy surface. It thus provides a route from the observations to the orientation dependence of the barrier for reaction. In the predictive mode the model provides an easily implemented procedure for computing the reactivity of oriented reagents (including those actually amenable to measurement) from a given potential energy surface. The predictions of the model are compared against classical trajectory results for the H + D2reaction. By use of realistic potential energy surfaces the model is applied to the Li + HF and 0 + HC1 reactions where the HX molecules are pumped by a polarized laser. A given classical trajectory is deemed reactive or not according to whether it can surmount the barrier at that particular orientation. The essential difference with the model of Levine and Bernstein is that the averaging over initial conditions is performed by using a Monte Carlo integration. One can therefore use the correct orientation-dependent shape (and not only height) of the barrier to reaction and, furthermore, use oriented or aligned reagents. Since the only numerical step is a Monte Carlo sampling of initial conditions, very many trajectories can be "run". This suffices to determine the reaction cross section for different initial conditions. To probe the products, we have employed the kinematic approach of Elsum and Gordon. The result is a model where, under varying initial conditions, examining final-state distributions or screening different potential energy surfaces can be efficiently carried out.
1. Introduction proach. At the present stage when our physical intuition regarding reactivity of oriented reactants is not yet perfectly developed, the Mapping specific features of the potential energy surface to observed dynamics is an importantiw4but difficult task. Recent studies4I7 suggest however that reactions of oriented or of aligned reagent^^,'^-^^ can effectively probe the angular dependence of the (1) Polanyi, J. C. Acc. Chem. Res. 1972, 5 , 161. barrier to reaction. In the limit when the approach motion is rapid (2) Kuntz, P. J. In Dynamics of Molecular Collisions; Miller, W. H., Ed.; Plenum: New York, 1976. on the time scale of molecular rotation, the connection is a very (3) Hirst, D. M. Potential Energy Surfaces; Taylor and Francis: London, direct one: The (classical) differential cross section as a function 1985. of orientation is given in terms of a simple criterion for reaction, (4) Levine, R. D.; Bernstein, R. B. Molecular Reacfion Dynamics and namely, that enough energy is available to surmount the barrier. Chemical Reacfivify;Oxford University Press: New York, 1987. Detailed examination12*22 of potential energy surfaces for atom(5) Smith, I. W. M. J . Chem. Educ. 1982, 59, 9. (6) Jellinek, J.; Pollak, E. J . Chem. Phys. 1983, 78, 3014. diatom exchange reactions shows that chemical forces are indeed (7) Levine, R. D.; Bernstein, R. B. Chem. Phys. Left. 1984, 105, 467. short range and that particularly if the barrier is early, the ori(8) Blais, N. C.; Bernstein, R. B.; Levine, R. D. J . Phys. Chem. 1985, 89, entation at the critical configuration for reaction is likely to closely 10. relate to the initial orientation. Such low velocities for which this (9) Pollak, E. Chem. Phys. Left. 1985, 119, 98. is not the case are often too low to cross the barrier in any case. (IO) Kuppermann, A.; Levine, R. D. J . Chem. Phys. 1985, 83, 1671. The central assumption of the model is thus of a sudden approach (1 1) Bernstein, R. B. J . Chem. Phys. 1985, 82, 3656. (12) Schechter, I.; Levine, R. D. Inf. J . Chem. Kinef. 1986, 18, 1023. to the barrier, with an unperturbed motion. The same assumption (13) Loesch, H. J. Chem. Phys. 1986, 104, 213. is made in other kinematic type model^.*^-^^ It can be removed most readily by integrating two classical equations of m o t i ~ n . ' ~ ~ ~ ~(14) Kornweitz, H.; Persky, A,; Levine, R. D. Chem. Phys. Lett. 1986, 128, 443. Here, however, we retain the sudden-like approach. (15) Jalink, H.; Parker, D. H.; Meiwes-Broer, K. H.; Stolte, S. J . Phys. The essential input to the model in its predictive mode is the Chem. 1986, 90, 552. orientation dependence of the location and height of the barrier (16) Alvarino, J. M.; Basterrechea, F. J.; Lagana, A. Mol. Phys. 1986, 59, 559. to reaction. The second assumption of the model is that such (17) Levine, R. D.; Bernstein, R. B. Chem. Phys. Left. 1986, 132, 11. trajectories with sufficient energy (in the direction perpendicular (18) Brooks, P. Science 1976, 193, 11. to the barrier) to cross the barrier will do so and proceed to form (19) Stolte, S. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 413. products. With this assumption, the computation of the cross (20) Zare, R. N. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 422. section is reduced to averaging over initial conditions. This can (21) Bernstein, R. B. Chemical Dynamics via Molecular Beams and Laser for simple barrier shapes be analytically performed5~7J0J7~34~36,37 Techniques; Oxford University Press: New York, 1984. (22) Schechter, I.; Levine, R. D.; Bernstein, R. B., in this issue. and heights. However, as soon as an accurate barrier height and (23) Cross, R. J.; Wolfgang, R. J . Chem. Phys. 1961, 35, 2002. shape is employed, the analytical results (essentially a series (24) Suplinskas, R. J. J . Chem. Phys. 1968, 49, 5046. expansion) are no longer very t r a n ~ p a r e n t . ~We ~ furthermore (25) Light, J. C.; Chan, S. J . Chem. Phys. 1969, 51, 1008. wish to consider oriented reagents. For a finite impact parameter (26) Kuntz, P. J. Trans. Faraday SOC.1970, 66, 2980. collision, Figure 1, there is a whole range of angles 7, cos y = (27) Marron, M. T. J . Chem. Phys. 1973, 58, 153. bR,corresponding to a particular orientation angle 6 of the dia(28) Case, D. A,: Herschbach, D. R. J . Chem. Phvs. 1978, 69, 150. (29) Mahan, B. H.; Ruska, W. E. W.; Winn, J. S . J . Chem. Phys. 1976, tomic molecules (a circumflex designates a unit vector). Since 65. 3888. the barrier height and location depend on 7, determining the (30) Dixon, R. N.; Truhlar, D. G. In Atom-Molecule Collision Theory; opacity function as a function of 0 is no longer as simple as it is Bernstein, R. B., Ed.; Plenum: New York, 1979. for randomly oriented reagents or for a given y. The proper (31) Elsum, I. R.: Gordon, R. G. J . Chem. Phys. 1982, 76, 3001. averaging over initial conditions (performed numerically by Monte (32) Prisant, M. G.: Rettner, C. T.; Zare, R. N . 1.Chem. Phys. 1984, 81, Carlo integration) is an important ingredient of the present ap2699. 'Present address: Department of Chemistry, University of California, Berkeley, CA 94720.
0022-3654/%7/2091-5472$01 .50/0
(33) Safron, S.A. J . Phys. Chem. 1985, 89, 5713. (34) Evans, G. T.; She, R. S.; Bernstein, R. B. J . Chem. Phys. 1985, 82, 2258. She, R. S. C.; Evans, G. T.; Bernstein, R. B . Ibid. 1986, 84, 2204.
0 1987 American Chemical Society
Direct Reactions of Oriented Reagents
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5473
X
Figure 1. The laboratory (space-fixed) coordinate system. The initial relative velocity is along the 2 axis. The angle s2 is used to sample the impact parameter. 7,the angle between R (directed toward A) and r (directed from B to C), is 180" for a collinear attack. The location of ; the height of the barrier the barrier to reaction is specified as ~ ( 7 )hence is a function only of y . 9 is the orientation angle of the diatomic molecule.
facile ability to generate easily visualized results is quite useful. We have encountered situations where the numerical results were not immediately obvious even for such trivial aspects as the distribution in cos y for all reagents vs. the distribution in k-i as a function of the polarization of the laser used to align the reagents. For computing the reactivity of oriented reagents, the two assumptions made so far (a sudden approach to the barrier; no recrossing) are sufficient. We wish, however, to add a third assumption, namely that the retreat of the products from the barrier is equally sudden. This additional assumption is not required and is not invoked in computing the reactivity. It does, however, enable us to compute the angular momentum disposal in the products and, in particular, their vector correlation^.^-^^^^^ Such computations do show that directional properties of the products are sensitive to the orientation dependence of the barrier shape and height. Work is in progress on whether reported dev i a t i o n from ~ ~ ~kinematic ~ models can be reconciled by allowing for a proper barrier functionality. Studies of the directional or vector aspects of chemical reactivity are intimately related to the disposal and the consumption of angular momentum, both orbital and interna1.39,48The combined assumption of a sudden transition from reagents' to products' configuration establishes a correspondence between the initial and final angular m ~ m e n t a . ~ There , ~ ~ , is ~ ~however a contribution dependent on the location and height of the barrier. The kinematic model, where conservation of total angular momentum is inherent, ~~
Kornweitz, H.; Persky, A.; Levine, R. D. Chem. Phys. Lett., in press. Parker, D. H.; Stolte, S., in this issue. Gardiner, W. H.; Levine, R. D. Chem. Phys. 1987, I l l , 1. Such a series representation is however very useful in going from the measured cross section to the potential. See: Stolte, S.; Chakravorty, K. K.; Bernstein, R. B.; Parker, D. H. Chem. Phys. 1982,68, 1. Van Den Ende, D.; Stolte, S . Ibid. 1984, 89, 121; and Prisant, M. G.; Schechter, I.; Levine, R. D., to be published. (39) Herschbach, D. R. Faraday Discuss. Chem. SOC. 1962, 33, 283. (40) Case, D. A.; Herschbach, D. R. Mol. Phys. 1975,30,1537; J. Chem. Phys. 1976, 64, 4212. (41) Hijazi, N. H.; Polanyi, J. C. Chem. Phys. 1975, 1 1 , 1. (42) Kafri, A.; Shimoni, Y.; Levine, R. D.; Alexander, S . Chem. Pkys. (35) (36) (37) (38)
1976, 13, 323. (43) Prisant, M. G.; Rettner, C. T.; Zare, R. N . J . Chem. Phys. 1981,75, 2222. (44) Simons. J. P.. in this issue. (45j Vernon; M. F.; Schmidt, H.; Weiss, P. S . ; Covinsky, M. H.; Lee, Y. T.J . Chem. Phys. 1986, 84, 5580. (46) Johnson, K.; Simons, J. P.; Smith, P. A.; Washington, C. Mol. Phys. -1986. -57 255 --(47) Schatz, G.C.; Amaee, B.; Connor, J. N. L. Chem. Phys. Lett. 1986, 132, 1. (48) Herschbach, D. R. Adv. Chem. Phys. 1966, I O , 319.
--. .
determines the angular momentum disposal consistent with a realistic barrier functionality. It remains to be seen whether such vector properties can serve as a direct probe of the barrier or that an additional ingredient-the impulse imparted during the s w i t c h o ~ e r ~ ~ ~ ~ ~ -ben eincorporated. ed Work is in progress49on a kinematic model where this aspect is included within the present framework. The orientation dependence of the barrier height and location are determined from a potential energy surface for the reaction. We have used the results from our examination of the energy relief in the reaction plane.1z,50Other numerical procedures can equally well be employed. The present results indicate that not only the overall reactivity but also the final-state distributions (such as rotational angular momentum disposal) are sensitive to the barrier functionality. Hence, good quality surfaces need be employed in the predictive mode. Section 2 deals with the computation of the reactivity. From the potential energy function we extractz2the critical shell, s(y), which is the barrier to reaction as a function of the distance s from the atom A to the center of mass of BC (and may depend on the orientation angle y). Trajectories are assumed to reach s unperturbed. At s a decision is made whether the trajectory can or cannot cross the barrier. The real effort is in the Monte Carlo sampling of initial conditions. Computations are reported for aligned reagents which are selected in the laboratory-fixed system of coordinates. Hence the development in section 2 is concerned primarily with the transformation between a molecule-fixed and a space-fixed system of coordinates. Initial state selection is performed by the trajectory program by mimicking the laboratory procedure, that is, by giving nonuniform weight to initial conditions. This is discussed in section 3 where we reap the benefit of the transformations discussed in section 2. Two examples, using realistic potential energy surface^,^^*^^ that of 0 HC1 and Li + HF, are discussed in section 4. A third model assumption, a sudden, unperturbed, departure of the products from the barrier, is introduced in section 5. With this assumption, one can readily compute the angular momentum disposal along the lines of Elsum and Gordon.27 The difference with their results is that here the reactive trajectories take proper cognizances of the orientation dependence of the barrier height and shape. We find that this does make a difference. Results for angular momentum disposal are presented in section 6.
+
2. Reactivity in the Molecular and the Laboratory Frames A sample of oriented or aligned BC molecules is prepared in the laboratory w$h reference to a fixed Z axis. The directional distribution w(i.Z) of molecular axes r in the laboratory frame will depend on the method of preparation as discussed in the next section. The potential energy is a function of interatomic distances and angles. Hence models of the reactivity are most readily expressed in a molecule fixed system of coordinates. To simplify the transformation between the two frames we take the molecule to be along the x axis in the molecular system (lower case vectors) and the rotational angular momentum to be along the z axis. From the Euler angle definitions of ZareS2or of Brink and Satchlers3 I$ is the azimuthal orientation O f j in the laboratory-fixed system, 0 is the orientation angle of j with respect to the (laboratory system) Z axis, and x is the angle of rotation of the diatomic molecule in its plane of rotation, Le., it is an azimuthal angle with respect to the z axis. The Cartesian components of the three unit vectors in the molecule based system are then given, in the laboratory system, by ~
~~
Gordon, R. G.; Levine, R. D., work in progress. Persky, A.; Broida, M. J . Chem. Phys. 1984, 81, 4352. Chen, M. M. L.; Schaefer, 111, H. F. J . Chem. Phys. 1980, 72, 4376. Zare, R. N. Angular Momentum in Quantum Mechanics; Wiley: New York, 1987; Section 3.3. (53) Brink, D. M.; Satchler, G. R. Angular Momentum; Clarendon: Oxford, U.K., 1968. (49) (50) (51) (52)
5474
(;j=(
cos 4 cos 8 cos
ZX
s=
( ) ( jr
=
\jz/
x
1
4 cos 0 sin
x - sin 4 cos x
-sin 4 cos 0 sin
x + cos 4 cos x
/-cos
/jx\
x - sin 4 sin x
x + cos 4 sin x
sin 4 cos 8 cos
-sir 6 cos
9
Schechter et al.
The Journal of Physical Chemistry, Vol. 91, No. 21, I987
\sin 8 sin
x
(5, 1
H + D2-HD
+D
I
(2.1)
\
1
/
cos 4 sin 6
2
=
= (sin 4 sin 8
(2.3)
ip ip
=
(n 9 i)
(2.4)
is the matrix of direction cosines which transforms a (column) vector written in terms of its components in the molecular frame to the same vector with components in the laboratory frame. Conversely, the matrix 9Trotates from the laboratory to the molecular frame. In practice we have evaluated all scalar products in the laboratory system. For example, with R, the vector to A from the center of mass of BC, being taken to lie in the Z-Y plane
R.i = Rrix+ R&
(2.5)
where iz = i.k. The two components of R can equally be specified by the length d of the vector R and the angle R with the Z axis R = (:sin”) -d
COS w
The impact parameter ( b = Rx)can thus be specified in terms of the angle R
b = d sin D
(2.7)
It follows from (2.5) that aka given b and orientation of r in the laboratory (specified by bk), FORcan have a range of values. Physically, this is because the aziquthal angle q5 of the rotor can vary, and hence, in (2.5) so can i.X. In the model, the approach to the barrier is sudden. The motion of R to the barrier is thus unperturbed, being a straight line, parallel to the Z axis, in the X-Y plane. Hence, at the barrier we use (2.6) with d = s where s is the distance of the barrier to reaction22along R. The decision whether a trajectory is reactive or not is made in terms of the value at the barrier of PR, as follows. The component of the initial kinetic energy, at a distance s, which is along R is ET(1 - b2/s2) = ET(1 - ( d / ~cos2 ) ~0). This energy needs to be sufficient to exceed the height, Vb(P.R), which depends on the orientation angle in the molecular frame i-R. As is discussed in detail elsewhere,22 the critical configuration s for reaction may itself depend on the orientation angle y, cos y = i.R. It is for this reason that we distinguish between the spherical shell of radius d , d Imax(s), which is used to sample b (vide infra), and the reaction shell. Trajectory computations are performed by a Monte Carlo sampling of initial conditions as follows: the R vector is confined to the X-Z_p!ane and the impact parameter is sampled by choosing sin2 R = (RsX)~,cf. (2.6), to be equiprobable on the interval [0,1]. This makes for a uniform sampling of b2 and hence4 means that any cross section of interest is ui = r&N,(i)/N(i)
(r RJ 011 reactnnts
k) all reactmts
i r h i JII rsditant\
+
cos 8
The matrix
(J
Figure 2. Distributions of reagent orientation as generated by the model for the H Dz(u=O,j=1) reaction at ET = 0.9 eV. The notation (a-b) denotes the scalar product of unit vectors (a-b) E (5-b), Le., the cosine of the angle between a and b. The bottom panels are the distributions over all reactants for the three angles of interest. For aligned reagents the three distributions will be. different. For the present case of randomly
(2.8)
where i is the set of initial conditions and N(i) is the total number of trajectories of which N,(i) were reactive. The direction of the molecular axis r is specified in the laboratory system by the three Euler angles, 4, 6, x, cf. (2.1). The
oriented reagents the distributions should be uniform and the slight ripple reflects the finite number of trajectories used in the computation. To resolve details of the reactive reactants we use 40 bins which require some 100000 trajectories for such low noise level. The upper panels are for reactive reactants, Le., the distribution over the trajectories that do cross the barrier. As expected,’J2 the preferred configuration for reaction is collinear. In agreement with the quantal results,1° there is a definite preference for the alignment of j. angles 4 and x are taken equiprobable in the interval [0,2r].The orientation of the molecular angular momentum j (which is along the axis z in the molecular frame) is specified by its angle 0 with the space fixed Z axis, cf. (2.3). Hence cos 8 is taken as equiprobable in the interval [-1,1]. The four angles 9, 4,8, and x specify the orientation variables. They are sampled as indicated for all possible initial-state selections. The cross section for a particular preparation is determined by assigning weights to individual trajectories, as discussed in the next section. When the weighing is uniform we have randomly oriented reagents. The velocity variables are the initial relative velocity along the 2 axis and the vibrational and rotational velocity of the molecule which are along the x and y axes in the molecule-fixed frame. The vibrational phase is selected as usual for a Morse potential in r. The distribution of orientations is shown both for all reagents and for reactive reagents (Le., for those trajectories that do cross the barrier) for H D2 collision in Figure 2. The potential used is the SLTH54fit as shown in detail elsewhere’2,22at an initial translational energy of 0.9 eV. The approach is clearly preferentially from the collinear direction when viewed in either the molecular or the laboratory systems. The quantitative accuracy of the model is shown in Figure 3 where results of exact trajectory computations12for the differential cross section du/d(i+’) are compared with those of the kinematic model (r’ is the AB distance). There is a slight overestimate of the reactivity at higher collision energies since exact trajectories which did cross the barrier have a certain tendency to recross back’* and hence not to react. Otherwise, the results do show that du/d(i+’) is well reproduced by the model assumptions. Of course, du/d(i-i’) is not directly amenable to measurement. We thus proceed to the results for experimentally realistic reagent selections.
+
3. Reagent Selection The trajectory program samples the laboratory distribution of the molecular axis r as if it is randomly oriented. To allow for reagent state selection, each trajectory was assigned a weight reflecting the method of preparation. When no state selection (54) Siegbahn, P.; Liu, B. J . Chem. Phys. 1978,68,2457. Truhlar, D. G.; Horowitz, C. J. J . Chem. Phys. 1978, 68, 2466.
Direct Reactions of Oriented Reagents
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5415
I
E, = 0.55 eV
I
1
I
E T = 0.55 eV
ET= 0.9 eV
n 3 I W
b \
n
-I
1 -I
0
0
1
-1
I
0
-2 E,= 1.3 eV
W
b 1
0
-1
(rsr') Figure 3. Comparison of the predictions of the kinematic model with exact classical trajectories12 for the H + D2reaction (using the S L T H potential energy surface) a t a number of energies. Shown is the dependence of the reaction cross section on the angle between the old and new bond distances a t the barrier. The reaction cross sections are normalized by their value for a collinear collision.
is camed out, the weighting was uniform. The program also allows for a transformation to a kinetic frame where the distribution of reagents is referred to the relative velocity vector k
w(i.2)f(k.2)
w(i&) = J d k
(3.1)
wheref(k.2) is the laboratory directional distribution of the relative velocity vector. In this paper we consider only the ideal limit where the laboratory direction of k is sharply defined,55and we define the Z axis to be in the direction of k so that
= w(i.2) (3.2) Otherwise, it is w(i.2) which is useful for the transformation to the molecular frame, since Fk = rz. The distribution of molecular axes for different methods of state selection has been extensively d i s c u ~ s e d . ' ~ -Here ~ ' ~ we ~ ~consider ~~~ specifically the case ofioptical alignment. ZareZohas shown that the distribution of ?.E for a plane-polarized radiation can be expressed as w(i.2) = [ 1 A ~ P ~ ( c oe)]s / 4 ~ (3.3) W(i.k)
+
where A, is the alignment parameter (which depends on the initial state of the molecutc) and P2(cos O) is the second-order Legendre polynomial. Explicit expressions for the initial-state dependence of the alignment parameter for dipole absorption by a symmetric top and by a rigid rotor have been provided.z0 Here we shall use the classical limiting case
,(?.E) CE (i.E)2 (3.4) The averaging over the initial azimuthal and polar angles of the rotor will be performed within the trajectory program as part of the Monte Carlo sampling. The only change from the standard program for random reagents is that each trajectory is assigned the weight determined by the orientation angles of the rotor, cf. (3.4), and the components of the electric field E in the laboratory system. As an example, for a plane-polarized radiation as discussed by ZareZo(m (of photons) = 0), one readily finds using (2.1) and performing the average over 4 and x that (3.4) leads to
w(i.2)
0:
1
+ cos2 o
(3.5)
which is the classical limitz0 of (3.3). ( 5 5 ) For the required transformations when this is not the case see, e.g.,
ref 43.
00
I
-1 0
05
00
05
in
cos( '{ i
+
Figure 4. The height of the barrier for r e ~ c t i ~ of n ~Li~ ~ HF ~ ' as a function of the angle y,cos y = ( R v ) (Rei).
4. Example I. Reactivity
The first example chosen for detailed study is Li + H F for which there are extensive exact classical trajectory computat i o n ~ . ~Using ~ - ~ ~an accurate fit59,60to the ab initio computed points,51we show the barrier height as a function of y in Figure 4. This figure is central for interpreting the following results. Trajectories will be reactive for that range of y where crossing the barrier is classically possible and the magnitude of the reactivity at a given y is determined by the excess energy which delineates the range of impact parameters which lead to reaction.' All the rest is purely kinematics and proper weighting and averaging. Figure 5 shows three differential cross sections at ET = 0.4 and 0.9 eV. The opening up of the cone of acceptance with increasing translational energy is quite clearly seen, in both the molecular and the laboratory systems. Now provide the same additional energy (beyond 0.4 eV) by pumping HF to L: = 1 using polarized light, Figure 6. Each panel shows not only the distribution of reactive trajectories but also, as a reference, the distribution of ( 5 6 ) Alvarino, J. M.; Basterrechea, F. J.; Hernandez, M. L.; Lagana, A. Mol. Phys. 1986, 59, 559. (57) NoorBatcha, I.; Sathyamurthy, N. Chem. Phys. 1983, 77, 6 7 . (58) Zeiri, Y . ;Shapiro, M.; Pollak, E. Chem. Phys. 1981, 60, 239. (59) Sorbie, K. S.; Murrell, J. N. Mol. Phys. 1975, 29, 1387. (60) Murrell, J. N.; Carter, S.; Farantos, S. C.; Huxley, P.; Varandas, A. J. C. Molecular Potential Energy Functions; Wiley: New York, 1984.
5476 The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 Li + FH
Schechter et al.
- LiF + H
L i t FH'-
LiF t H reactive reactanis 90'
I
all
reactants 90'
n
1
.I
4 . k ) reactive reactants
.
n
.1.
1
(r.R) reactFve reactants
.
-1
n
I
(r.k) reactiie reactants. reactive reactants
0
I
I
(j.k) reactive reactants
0
-1
I
(r,R) reactive reactants
-1
0
60'
1
(r,k) reactive reactants
ieactive reactants 45'
all
reactants 45'
ij A) all reaclants
( r R) a11 ieactantc
reactive reactants
( r I) all reactants
Figure 5. Orientation distributions for Li + HF collisions. Bottom panels: all reactants (but see Figure 6 ) . Middle panels: reactive reactants at ET = 0.4 eV. Top panels: reactive reactants at ET = 0.9 eV. These are to be compared with Figure 6 where the total energy is also 0.9 eV but the HF is laser pumped. The asymmetry of the distribution of reactive reactants in cos y = ( P R )is a direct reflection of the asymmetry of the barrier as shown in Figure 4. Note that the preferred orientation of j is opposite to that of H + DZ,Figure 2. all trajectories. Since the trajectories are weighted according to the preparation this serves to illustrate the changes in the distribution of reagents as the field orientation is changing. The initial relative velocity is along the Z axis. Hence the distribution in r-k is the laboratory polar distribution. The changes for all reactants are as expected from (3.4). The distribution of reactive reactants in r-k is essentially a weighting of the distribution shown in Figure 5 for randomly distributed reagents by the distribution in r-k as imposed by the field. The same is the case for the distribution in j-k. Since a sidewgys approach is preferable, the reaction is preferred for higher Ij-kl. A field at 90' is thus most conductive for re_action, Figure 7 . Somewhat unexpected is the distribution in i.R. Since R is randomly oriented in the X-z plane (with a uniform weight of cos2 Q),the distribution in i.R is 1 cos2y independent of the orientation of the field. Seemingly that should imply the same reactivity. But no. All that is implied is that the range of orientations y that is reactive is the same at all field orientations. What that means is that the opening angle of the cone of reaction in the molecular frame is independent of the orientation of the field. This is, of course, only to be expected. Within the cone, the reactivity does vary, an$ that is the role of the impact parameter. At a given value of R-i, the distribution of possible impact parameters is intertwined, cf. (2.5), with the distribution of r in the X - z plane. Take, for example, E along the Z axis. Collisions for i-R 1 require, in this case, low impact parameters. Hence the higher barrier, Figure 4, for collinear collisions can be surmounted, since the centrifugal barrier is low. For a field at 90°, much higher impact parameters are required to reach near-collinear configurations and the reactivity there is reduced. The potential energy surface50for 0 HCl has a rather narrow cone of acceptance about the collinear direction.22 For y > 4 2 O reaction is practically excluded. Since the larger C1 atom acts as a bulky group, it is tempting to regard the molecule as a "painted sphere", where the barrier is constant up to y = 42' and very high otherwise. Even at fairly high collision energies, Figure 8, this is not a realistic approximation. The variation of barrier height within the cone does strongly influence the range of contributing impact parameters. Not only the reactivity but also the final-state distributions (see section 6) are quite sensitive to the range of impact parameters that can contribute to reaction and
+
-
+
all reaclanrs
(r R)
I 1 I)
30'
30'
ir I)
Figure 6. Orientation distributions of all reactants and of the reactive reactants for Li + aligned (laser-pumped) HF at E = 0.9 eV. The HF is laser pumped to L! = 1. The angle of the electric field E with the laboratory Z axis is shown at the side of each row of panels The absolute reactivity is shown in Figure 7. I I ?
LI + FH--
LiF + H
; h
09
\
00
0.5
!.O
I F.7,
Figure 7. The dependence of the reaction cross section for Li + aligned (laser-pumped)HF on the polarization angle of the laser (with respect to the laboratory Z axis; the Z axis is the axis of the relative velocity). Shown is the cross section normalized by that for randomly oriented reagents (cf. Figure 5 ) at the same total energy. The finite but not large effect is due to the comparatively moderate variation of the potential (Figure 4) with y. This can also be seen from Figures 5 and 6. In particular, note the distribution of reactive reactants (in Figure 5 ) in j.k. The largest effect would thus be manifest for as low initial translational energy as possible. (In Figure 7 ET N 0.4 eV.) Unfortunately, the absolute reaction cross section will then be low. hence to the detailed orientation dependence of the barrier height. The comparatively tight cone of acceptance in 0 + HC1 is clearly reflected when HC1 is aligned by pumping to c = 1, whether at a low or at a higher collision energy, Figure 9. Here too, it is not only the distribution of reactive reagents but also the total
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5411
Direct Reactions of Oriented Reagents OtHC1 -0HtC1 I
I
‘1
I
I
1
R --I
0
I
0.k) reactive reactants
-I
I
(PR)reactive reactants
0
-1
0 + HCI’-
OH
+ CI
/*
1
(r,k) reactive reactants
o ! 00 (J
k ) reactibe reactants
( r R) reactive reactants
Figure 8. Orientation distributions of reactive reactants for 0 + HCI (u=O,j=1) at ET = 1 eV. Bottom panels: using the optimized LEPS potentiaLS0 This has a cone of acceptance with an apex angle of about 2 X 42O. The potential barrier to reaction does however vary within the cone.22 If the barrier within the cone is replaced by a constant barrier, the top panels are generated. Even at a fairly high initial translation, there is a clear difference. Using aligned reagents (note the left column) one should be able to probe the orientation dependence of the barrier. 0 + HClt-
OH + CI
90“
1
reactive reactants 90’
~
I 0.45 eV
90’
0’
I
Figure 10. The reaction cross section for 0 + aligned HCI a t a total energy of 0.45 eV vs. the polarization angle of the laser. (2is the axis of the relative velocity and E is the electric field.) Note that the direction of the effect is opposite to that in Li H F , Figure 7 .
+
+
For both 0 HCl and Li + HF we have used aligned reage n t ~ . Electric ~ , ~ ~ field focusing can be used to prepare orientled reagents4,*’where it is the distribution of ?.E rather than of li-E12 which is selected. For the cases considered (and for many other cases), the additional ability to place the molecule in a favorable or unfavorable direction would clearly make a large difference.
5. Angular Momentum Disposal The computations thus far presented were based on two assumptions: (i) sudden approach of the reagents to the barrier and (ii) all trajectories which can cross the barrier, do so, and do not return. We now add a third assumption: (iii) sudden departure of the products from the barrier. The introduction of this third assumption does not, of course, affect the computed distribution of reactive reactants or the reactivity. It only serves to compute the final-state distribution amongst those trajectories deemed reactive by assumptions (i) and (ii). Moreover, including the probe of angular momentum disposal hardly adds to the execution time. Essentially all the numerical effort is in the Monte Carlo sampling of initial conditions for the trajectories to be run. We therefore include routinely all three assumptions. The model is then a sudden switchover from reagents to products at the barrier configuration for those trajectories that can cross it. The transformation from the reactants’ (unprimed) to products’ (primed) coordinates is2’
(R’) - (-cot a’
reactive reactants 0’
1 0.45 eV d i
1.o
0.5
(E.Z)
(r k) r e a m \ 2 redctant?
r’
-
1
ai2,”) (p)
where @ is the ubiquitous4 skew angle COS’ P = mAmC/(mB + mc)(mA mB)and a and a’are the angles between the -R’ and R axis and between the I‘ and R axis, respectively; cot CY = m C / ( m B + mc). The transformation (5.1) is valid at any configuration. We shall use it specifically at the barrier in order to express the coordinates of the products in terms of those of the reagents. For several purposes it is of interest to work with mass-scaled coordinate~~,~~
+
I
0’
+
Figure 9. Reactants orientation distributions for 0 (laser-pumped) HCI, at two different total energies and two different laser polarizations (as indicated a t the side of each row of panels). Note the differences between this collinearly dominated reaction (with a “bulky” group steric hindrance) and the corresponding results, Figure 6 , for the sideways approach in Li HF.
+
reactivity, Figure 10 (and also the angular momentum disposal) which is affected by the orientation of the field. As is to be expected, reactivity is highest with a field oriented along the relative velocity ( Z ) axis, and the variation with the field orientation spans an order of magnitude, Figure 10. This suggests that an experimental study of this reaction with aligned reagents would be worthwhile, particularly so since even the minimal barrier height suffices to practically rule out reaction with thermal reagents.jO
Q = aR q = ( b sin P)r
Q’= bR’
(5.2)
q’ = ( a sin P)r’
where
a2 = mA(mB+ m c ) / M
(5.3)
b2 = mC(mB4- m A ) / M With b cot CY’ = a cos P and a cot a = b sin P, (5.1) has the symmetric form (61) Smith, F. T. J . Chem. Phys. 1959, 31, 1352. Hirschfelder, J. 0.Int. J . Quantum Chem., Symp. 1969, No. 3, 17.
5478
Schechter et al.
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 (5.4)
contribution to j’sk will sample R h r:
j’ak = mB cos2 Pk.(RAi)
+ cos2 Pj-k
(5.10)
The model assumption of an instantaneous switch from reagents to products means that (5.1) or (5.4) are equally valid for the velocities. Explicitly and using a dot to designate a time derivative
In terms of angular momenta, our argument is that often the range of 1 values that contributes to a reaction is such that 1 >> j . Hence, unless cos2 P and/or x is unusually large, j’ is determined by a large vector sin2PI which is perpendicular to k, plus a much smaller - (t, ff’ sii2!) vector cos2P(j x ) which is oriented randomly or otherwise fr_oq ( 5 . 5 ) i ‘ - 1 the tip of I. Whether the direction of j is or is not selected, j’.k has a limited range of variation about 0, unless (i) cos2 (3 1 The result (5.5) is written down since it will be used in an essential and/or (ii) x is unusally large. way in order to compute the angular momentum disposal. It is On the other hand and as already discussed, the distribution important however to emphasize that (5.5) can be modified without much change in the spirit of the model. A m ~ d i f i c a t i o n ~ ~ of the magnitude of j’does exhibit also the I term contribution and hence spans a wider range of values. Other vector correlaof (5.5) is necessary in order to (i) compute a realistic product tions28.40can also be readily computed. Using vibrationally excited angular distribution and (ii) to conserve energy. Within the model, reagents would be of particular interest since this makes i-i? the the change can be made by allowing for an impulse at the barrier. dilection of r. Hijazi and Polanyi4’ have suggested that j4 and This will decouple (5.5) from (5.1). Here, and as in previous ?.I’ proyide direct understanding of the d,ynamics. The identity we shall use ( 5 . 9 , which has the advantage that the only (R A R)-(r h i) = (R.r)(R+) - (R.r)(R-r) for the quadruple input required from the potential energy surrface is the barrier product shows why. These angles directly explore the correlations location. of orientations of both positions ??d velocities. At low impact Introducing, as usual, the orbital 1 = FR A R and rotational j parameters when RllR we expect j.1 to be preferentially zero for = pABrA i angular momenta and using primes for products, (5.1) sideways attack (R-r i= 0). For the products, however, we find and (5.5) imply2’ that whenever 1 >> j , j’ and I’ (which are both linear combinations j’ = sin2 PI cos2 Pj cos2 Ox of j and )! tend to be both preferentially in the direction of 1 and (5.6) hence j’4’ is peaked near unity. An anticorrelation of j’ and I’ I’ = cos2 PI sin2 Pj - cos2 Px is primarily brought about by x. Hence, we expect to see this in 0 HCI, where x is larger (vide supra) for initial rotational with excitation. In considering the implications of (5.1)-(5.8), one must bear x = mB(Rhi 4- rAR) (5.7) in mind that they refer to individual trajectories. The observed Since the mass-scaled coordinates correspond to reduced masses distributions will reflect averaging over the distribution of reactive of unity, I = Q A Q and j = q A q so that (5.6) is unchanged but trajectories as discussed in sections 3 and 4. As an example consider Li HF where sideways attack (Le., r.k or PR near zero, x = tan P ( Q h 4 + qAQ) (5.8) cf. Figure 5) is favored j’sr will then span a wide range and this can be seen even in angular distributions since R’ will be in the The additional term x in (5.6) is the impulse imparted during direction of r.65 the switchover from reagents to products coordinate. As can be The kinematic model can also be used to explore the systemseen from (5.7)or (5.8) its magnitude is critically dependent on atics1,2,66 of the conversion of reagents to products energies. Here the configuration at the barrier. Thus, a preferentially collinear too, we find that steric considerations also play a role. As an approach to a high barrier (e&, 0 HC1) means that r h R example, AB vibrational velocity, ?’si’, is more effectively pumped 0. (The reason is that to surmount the barrier, R need be prefby reagent vibrational excitation or reagent translation in a erentially along R, which is preferentially along 1.) A sideways preferentially collinear approach (as can be seen by using (5.1) approach (e.g., Li HF) would make this term of dominant for r’ and (5.5) for r’). This is intuitively reasonable but much importance. Similarly, molecular vibration (where r is along r) of our intuition is essentially kinematic in origin. will contribute to x for a sideways crossing of the barrier while molecular rotation (i perpendicular to r) is important for a 6. Example 11. Final-State Distributions preferentially collinear attack. Angular momentum disposal and other final-state distributions Angular momentum disposal can thus provide diagnostics of were computed from (5.5)-(5.7) with r and R evaluated at the the stereoselectivity of the reaction. Even when the exchanged barrier. For 0 + HCl, the high barrier implies that only a limited atom is heavier or the incident or departing atoms are lighter and range of 1 values will contribute to the reaction. The high value 1, so that j ’ = 1. Under such circumstances we are cos2 of cos2 /3 further restricts the contribution of 1 to j’. Starting from probing the opacity f ~ n c t i o n which, ~ * ~ ~in~the ~ ~present ~ model, HCl at j = 0 it follows that essentially all the O H rotational is very much a result of the orientation dependence of the barrier. excitation.reflects the x term in (5.6). Specifically, for i = 0, j’ IJrSucts’ alignment4,64is often characterized by the distribution = mH(rA R). At low energies where near-collinear collisions (Rllr) of j’.k. There are two terms in j’ predominate, the O H is computed to be rotationally very cold. j’ = FABr’A (R cot ai) (5.9) Laser pumping of the HC1 reagent is of limited use, since the vibrational velocity is along r and r is predominantly along R, and For many.reactions, the relative motion is faster than the BC hence the R A i term in x does not help. Only rotation of HC1 rotation, R > i.. Under such circumstances, a large component is effectively channeled into O H rotation. At higher collision of j’ is perpendicular to k. Hence the angle between j’ and k is energies, some translation is also channeled into OH rotation since preferentially 90’ as is also found in exact classical trajectory off-collinear collisions can now be reactive, Figure 11. That it computation^.^^ In the kinematic model it is only the reagent is the orientation dependence of the potential barrier that is reinternal velocity which can contribute to j’sk. Such a contribution sponsible for the ET j ’ conversion can be demonstrated by will be particularly important for vibrationally excited reagents. replacing the true barrier by one that has a constant value for Since vibrational frequencies can be high, j’sk can be large under approach from the H cone of acceptance22and is very high othsuch conditions and since the vibrational velocity is along r, its erwise. (A, so called, painted sphere model.) Much more OH rotational excitation is now possible, Figure 11, since a constant
(e>
(”3
+ +
+
-
+
+
+
+
-
+
-
+
-
(62) Mosch, J . E.; Safron, S. A,; Toennies, J. P. Chem. Phys. Lett. 1974, 29, 7 . (63) Noda, C.; McKilop, J. S . ; Johnson, M. A,; Waldeck, J. R.; Zare, R. N. J . Chem. Phys. 1986,85, 856. (64) Greene, C. H.; Zare, R. N. Annu. Reo. Phys. Chem. 1982, 33, 119.
(65) McDonald, J. D.; LeBreton, P. R.; Lee, Y . T.; Herschbach, D. R. J . Chem. Phys. 1972, 56, 769. (66) Polanyi, J. C. Faraday Discuss. Chem. Sor. 1973, 55, 389.
Direct Reactions of Oriented Reagents
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5479 Li + FH
O+HCI -OH+CI
O+HCI -OH+CI I
0
-1
1
0
- LiF + H
E, = 0.9 eV
E, = 0.76 eV
. -
I
16
(j’.r) j‘ Figure 11. Products’ distribution in 0 + HC1 at E T = 1 eV. j’ and 1’ are the OH rotational angular momentum and orbital angular momentum in units of h. Bottom panels: using the optimized LEPS potential.50 Top panels: using the painted sphere potential. As discussed in the text in connection with eq 5.9 and 5.10, ( j ’ e k ) is confined about 0 for low initial j . The reactants distribution for both potentials is shown in Figure 8. The results for the products further support the conclusion that the orientation dependence of the barrier can be probed by dynamical experiments.
Li + FH
- LiF + H (j.1) reactive reactants
Figure 13. The vector correlations of angular momenta for 0 + HC1 ( ~ 0j=1) , (left column, E, = 0.76 eV) and Li HF ( ~ 0j=1) , (right column, ET = 0.9 eV). Each column shows GI) the reactive reagents and (j’4’) for the products at three initial values of j . Note the difference in GI) for a collinear vs. a sideways attack, the kinematic positive correlation between j’ and I’ and the role of x, cf. eq 5.6, in giving rise also to an anticorrelation of j’ and I’ in 0 HC1.
+
I
I
+
I
O+HC1 -OH+Cl j=4
j=l I
0
i
o
18
54
36
fj’,r)
72
j’
0
8
n
I’
Figure 12. Products’ distribution for Li + HF at two different translational energies. Note the very definite contribution of the initial translation to the j’ magnitude and orientation distribution.
barrier allows a wider range of angles y to react, Figure 8. As is only to be expected from the spectator model4 (which is the limiting case of the kinematic model), HCl vibration is effectively channeled into OH vibration. One expects4*LiF rotational excitation for the Li HF reaction with the departing light H atom. What is less expected, Figure 12, is the considerable enhancement of LiF rotation with initial translation and the sensitivity to the detailed orientation dependence of the barrier and to reagent alignment. All these effects are due to the me(r A R) contribution in the x term of (5.6). A heavy transferred atom compensates for the low value of cosz p, and a sideways attack makes r h R contribute significantly. HF vibrational excitatioClwill also _enhance the LiF rotation. The distribution of j4 and of j’4’ is shown in Figure 13 for several initial values o f j . The results are a?-expected from the general considerations. The differnce in j4 between a collinear and sideways approach is quite evident. For 0 HCl, the R h i term in x is the important one a s j increases. For this mass combination, j = pAB(rh i) and x = mB(Rh r) are very comparable. (The r h R term in x does not contribute much for this collinearly dominated reaction. Hence j’4’ = 2 j 4 - j) and an anticorrelation of j’ with 1’ is favored at high fs. For Li HF, the correlation is quite different. At low initial j both j’ and I’ tend to be in the direction of I (since I > j and cos2 p j , j’ remains in the direction of 1.) Hence, as j increases, j’J’ approaches j-I. The intermediate structure at j = 3 results from the j contribution to I’, which makes I’ not quite preferentially along 1. For the sideways
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 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.
p-k
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 RbI + CH3, and H for barriers to reaction Vo(cos yr) resembling the cases Ba + N 2 0 BaO* N2, Rb + CHJ 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
