J. Phys. Chem. 1984,88,4064-4068
4064
Spectroscopy of the Transition State (Theory). 3. Absorption by H,S in the H, Three-Dimensional Reaction H
+
H. R. Mayne,+ J. C. Polanyi,* N. Sathyamurthy,t Department of Chemistry, University of Toronto, Toronto, Canada MSS 1 A1
and S. Raynor Department of Chemistry, Rutgers University, Newark, New Jersey 071 02 (Received: October 28, 1983: In Final Form: February 16, 1984)
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An earlier calculation of the absorption spectra for transition-state configurations in the reaction H H2 H,* H2 + H has been extended to three dimensions (3D). The density of H3*in configuration space was computed by means of a 3D classical trajectory study on the Siegbahn-Liu-Truhlar-Horowitz (SLTH) potential energy surface (pes). Optically induced transitions were assumed to occur, with constant transition moment, to a 3D upper pes fitted to points obtained from a diatomics-in-molecules(DIM) calculation. The effect of increasing collision energy was examined. The computed spectra showed a singie peak with wavelength related to the distance of closest approach of the collision partners and breadth related to the range of angles present in Ha at closest approach. With increasing collision energy, ET = 2.3-23 kcal/mol, the location of the transition-state spectral peak shifted to lower u by -27000 cm-’.
I. Introduction In the previous paper to this one (part 11’) we calculated absorption spectra for the transition-state configurations in the reaction H + H2 H3* Hz + H (the ”transition states”, H3*, denote all configurations intermediate betwen reagents and products). The calculations involved computing batches of trajectories for various initial-state energies, across the one-dimensional (1D) Siegbahn-Liu-Truhlar-Horowitz2 (SLTH) ab initio ground-state potential energy surface (pes). For the optically linked excited-state pes we used four different approximations to a small number of ab initio SCF-MO points that we had generated on the pes that correlates with H*(2p) + H, (term symbol 211 in the collinear configuration or 2pA” in the more general C, geometry; see Figure 1). The transition-state spectra generated in this fashion exhibited a number of features which could be related to the dynamics of reaction on the lower pes. In the light of our 3D computations of transition-state spectra (in part 13) for some model exothermic pes, we expected that spectral features would also be observed for H,* in the 3D reaction H H,. We have now performed this 3D computation for H + H2and once again observe informative features in the transition-state spectrum. These are reported and discussed here. The reaction proceeded across the 3D SLTH pes. The form of the 3D electronically excited pes was obtained by fitting a smooth function to points calculated in the diatomics-in-molecules (DIM) appr~ximation.~ In Figure 1 we show the four lowest energy surfaces for the H3 system. The figure is not to scale. It shows qualitatively how the potential energy of these surfaces varies with nuclear geometry. The ground-state surface is substantially lower in energy than any of the excited states, except in D3h,where it rises to a conical intersection with the first excited state. In D3hthese two lowest energy states both have 2pE’ symmetry (the 2p refers to the united atom limit). At higher energy than these degenerate E’ states lie the 2sAl’ and 2pA2” states. These are stable and have been observed as the lower states in spectroscopic work by Herzberg and coworkems These higher states correlate with H*(2s) + H2(’Zg+) and H*( 2p) + H2(’Zg+),respectively. Diatomics-in-molecules4 (DIM) calculations, performed as part of the present study, have shown that these two states, 2sA’ and 2pA”, lie very close in energy for all geometries, differing by at most 1000 cm-’. We have
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‘Present address: Department of Chemistry, Eastern Michigan University, Ypsilanti, MI 48197. *Permanent address: Department of Chemistry, Indian Institute of Technology, Kanpur, India 208016.
0022-3654/84/2088-4064$01.50/0
used the nomenclature 2sA’ in order to describe in its most general symmetry (c,)the state that in D3his 2sA,’; similarly, we have used 2pA” to describe the state that in D3his 2pA,”, The DIM calculations also give approximate values for the transition moments involved. The atomic transition from 1s to 2p (the Lyman a line) is strongly allowed ( A = 6.2 X lo8 s-I). In C, (and hence in D3*)symmetry, however, the transition from the ground state to the 2pA” state is forbidden. For configurations between these limiting cases, the A factor is nonzero ( 1 X lo8 s-l) for less symmetric configurations than Czv. The 1s 2s atomic transition is forbidden, but the approach of H*(2s) to H2 gives the 2sA’ state to which transitions are optically allowed. An M O calculation6 gave the A factor for 2pE’ 2sA’ of A = 3.2 X lo7s-I. Thus, since there is always an optically allowed pathway to one or the other of these upper states, and since they are nearly degenerate, we have computed the transition-state spectra using a (3D) upper pes fitted to points for the 2pA” state. We have taken the optical transition moment to be constant. In fact, as H + H, approach, in various geometries, there will be configurations in which the increasing A factor for transitions to the 2sA’ state do not fully compensate for the decreasing transition probability to the 2pA“ state. As we note in section IV, this may increase the information content by making certain configurations more prominent than others in the transition-state spectrum. The assumption of constant transition probability was made as a matter of convenience at the present stage of our calculations. The other low-lying state, which is degenerate with the ground state in D3h,correlates with the asymptotic state of H(1s) + N
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(1) H. R. Mayne, R. A. Poirier, and J. C. Polanyi, J . Chem. Phys., 80, 4025 (1984). (2) B. Liu, J. Chem. Phys., 58, 1925 (1973); P. Siegbahn and B. Liu, ibid., 68, 2457 (1978); D. G. Truhlar and C. J. Horowitz, ibid., 68, 2466 (1978); 71. (1979). -, 1514 -(3) J. C:-Pofanyi and R. J. Wolf, J . Chem. Phys., 75, 5951 (1981). (4) F. 0. Ellison, J . Am. Chem. Soc., 85, 3540, 3544 (1963); J. C . Tully
in “Semiempirical Methods of Electronic Structure Calculations”, Part A, G. A. Segal, Ed., Plenum Press, New York, 1977; P. J. Kuntz in “Atom-Molecule Collision Theory: A Guide for the Experimentalist”, R. B. Bernstein, Ed., Plenum Press, New York, 1979, p 79; A. D. Isaacson and J. T. Muckerman, J . Chem. Phys., 73, 1729 (1980); M. B. Faist and J. T Muckerman, ibid., 71, 233 (1979). (5) G. Herzberg, J . Chem. Phys., 70,4806 (1979); I. Dabrowski and G. Herzberg, Can. J . Phys., 58, 1238 (1980); G. Herzberg and J. K. G. Watson, ibid., 58, 1250 (1980); G. Herzberg, H. Lew, J. J. Sloan, and J. K. G. Watson, ibid., 59,428 (1981); G. Herzberg, Faraday Discuss. Chem. SOC.,No.71, 165 (1981). (6) S. Raynor, unpublished data.
0 1984 American Chemical Society
Absorption by H3t in the 3D Reaction H
+ H2
The Journal of Physical Chemistry, Vol. 88. No. 18. 1984 4065
T,i
L
AB
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Figure 2. Coordinate system used in the rotated-Morse-curve spline (RMCS) interpolation. H+H2
O
w
“3h
Figure 1. Correlation energy diagram for the four lowest potential energy surfaces for H3*, together with the ground-state surface H,, as a function of nuclear geometry. At each side, the separated atom-molecule limit is given. In the center, the triatomic species has equilateral triangle (Qh) geometry. Note that the lowest excited-state and ground-state surfaces meet here in a conical intersection. On the right-hand side, the curves are appropriate to those obtained by deforming the isosceles triangle (C2”) geometry in such a way that the apex angle is reduced to 60°. The symmetry species in square brackets refer to the general C, symmetry. In the inset we show that the lowest excited-state surface correlates diabatically with H H,*(’Z,+). The resultant adiabatic curves are given here. Also shown is the potential energy for the ionic species H,+.
+
H2*(3Zu+).Transitions from the ground state to this surface are highly allowed for near collinear geometries ( A N 1 X los s - ~ ) ~ and are much weaker in noncollinear geometries. This excited state will predissociate strongly to yield ground-state H + H2. This in no way precludes observation of upward transitions to this lowest excited state (2pA’), since the absorption of a photon can be made to evidence itself as laser-assisted reaction-for example D
+ H2
-+
DH2*-
hui
DH2*
pre
DH
+H
where “pre” indicates predissociation and the products D H and H have an energy enhanced by hv,--or can be detected by twophoton ionization into the H3+continuum by using a sufficiently intense laser to generate the ionizing radiation (hv2,with v2 2 vl; Figure 1) so that the rate of ionization competes with the rate of predissociation. This transition-state spectrum will be shifted as compared with those computed in this study. We propose to report on these spectra in a later paper in this series (part IV). 11. Potential Energy Surfaces For the ground-state potential energy surface we used the accurate a b initio potential due to Siegbahn, Liu, Truhlar, and Horowitz* (SLTH), as in our previous study in 1D.I The excited (H3*) pes corresponded to the 2pA“ state which correlates with H*(2p) H2(lZg+). Values for the potential energy were calculated at various geometries by using the DIM4 procedure and were then interpolated and fitted. The details of the DIM calculations have been reported elsewhere.’ Briefly, the potential energy curves for 22 electronic states of the H2 diatom were spline fitted and then used in the usual DIM approach. Further approximations appropriate to the treatment of Rydberg states7 (since this is the nature of the H3* states) enabled straightforward generation of potential energies for arbitrary nuclear geometries. The nuclear geometries at which the potential energy was calculated were chosen to facilitate a rotated-Morse-curve spline RMCS) interpolation.* For each value of the bond angle y (where
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(7) S. Raynor and D. Herschbach, J . Phys. Chem., 86, 1214 (1982). (8) J. N. L. Connor, W. Jakubetz, and J. Manz, Mol. Phys., 29, 347 (1975); J. M. Bowman and A. Kuppermann, Chem. Phys. Letr., 34, 523 (1975); S. K. Gray and J. S. Wright, J. Chem. Phys., 66, 2867 (1977); J. S. Wright and S. K. Gray, ibid., 69, 67 (1978).
iJ’
“9 i-zoc
, ’o g
Figure 3. Electronically excited-state (H,*) and ground-state (H,) potential energy surfaces along the minimum-energy path of the groundstate surface, SLTH, with y = 37, 66, and 90°.
y is the deviation from collinearity), progress from large rAB to large rBCwas mapped by using the variable 4 = arctan [(rsr A B ) / ( rs r B C ) ]where , rs is some large distance, here taken to be 5.3 A. The coordinates used in generating the pes are shown in Figure 2. The surface was formed by assuming a Morse potential at each 4 extending along 1. The variable I was defined by 1’ = (rs - rAB)2 (r, - rBC)2.For fixed 4 the position of the minimum energy is I,. The potential was calculated at I, and 1, f 0.053 A. These values were then used to calculate the second derivative of the potential and hence the Morse curvature parameter, @. Thus, for a given y, we can fit the potential along any “ray” (fixed $) to a Morse curve
+
V(L4,7) =
W M y ) ( X ( l , 4 , 7 )- 2)
where D is the Morse well depth and X is given by
WL4,y) = exP[@(67)(K4>y)- lc(4,r))l in which p is the Morse curvature parameter and I, is the location of the potential minimum. (Note that this expression differs from the traditional Morse function in having a + sign in the exponent, so that V 0 approximately as 1 0.) Values of the potential energy were computed for y = 0, 60, 90, and 120’. For each value of y,the DIM energy was computed at 4 values of 0, 10,20, 30, 35, 40, and 45’ (hence, by symmetry, also at 50, 5 5 , 60, 70, 80, and 90’). The Morse parameters for each y and 4 were determined as described above, and the dependence of the Morse parameters on y and 4 was fitted by using a 2D cubic spline interpolation? The Morse parameters obtained by fitting the DIM points are listed in Table I. +
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(9) D. R. McLaughlin and D. L. Thompson, J . Chem. Phys., 59, 4393 (1973); N. Sathyamurthy and L. M. Raff, ibid., 63, 464 (1975).
4066
The Journal of Physical Chemistry, Vol. 88, No. 18, 1984
Mayne et al.
TABLE I: Morse Parameters Obtained from Diatomics-in-Molecules Data Points and Used in the Rotated-Morse-CurveSpline (RMCS) Interoolation
6 deg Y,
den
0
10
20
30
35
40
45
(a) Morse Well Depth, D, kcal/mol 0 60 90 120
109.53 109.53 109.53 109.53
109.53 109.59 109.59 109.59
109.72 109.71 109.77 109.97
111.59 111.91 112.79 116.18
121.95 123.83 129.30 143.04
159.80 169.91 183.03 198.35
199.10 209.77 221.57 231.36
(b) Location of Potential Minimum, I,, 8, 0 60 90 120
4.547 4.547 4.547 4.547
4.617 4.616 4.617 4.617
0 60 90 120
1.9466 1.9466 1.9466 1.9466
1.9198 1.9451 1.9192 1.9192
4.839 4.840 4.840 4.839
5.256 5.251 5.257 5.256
5.582 5.590 5.599 5.591
6.040 6.045 6.026 5.990
6.349 6.327 6.278 6.173
1.5749 1.5632 1.6127 1.5074
1.4258 1.5415 1.4001 1.2351
1.1864 1.1788 1.1898 1.3024
(c) Morse Curvature Parameter, 6, 8,-’ 1.8633 1.8357 1.8351 1.8612
The energy profiles of this surface as progress is made along the SLTH reaction coordinate, for values of y = 37,66, and 90°, are shown in Figure 3. 111. Classical Trajectory Calculations Full 3D classical trajectory calculations were carried out on the lower (SLTH) potential surface. The absorption intensity, Z(v), was taken to be proportional to the amount of time the trajectory spent at the geometry where the difference potential (AV = v* - pLTH) was v. The transition moment was assumed to be a constant. The configuration space of H3 was partitioned in the following manner. The range of cos y (-1 to 1) was split into five regions. The limits of these regions were varied as a check on the results but typically would be as follows: -1.0 Icos y < -0.2; -0.2 I cos y < 0.2; 0.2 ICOS y < 0.6; 0.6 5 cos y C 0.8; 0.8 ICOS y i 1. Within these angular regions the bond lengths were “boxed”. To reduce memory requirements, the symmetry of the H3system was utilized, and only the smaller of the TAB and rAcbond lengths was boxed. This was labeled rl (r2 was invariably rBC). Typically 150 intervals in rl and 60 in r, were used, yielding 9000 boxes within an angular region. Thus, for example, in the saddle point region the box side lengths were dr, = 0.0056 8, and dr, = 0.016 8,. After each integration step (4.2 X lO-”s), the value of (rl,r,) was noted, and the “counter” for the box containing (r,,r2), in the appropriate angular region, was incremented by one. For each set of initial conditions, the various trajectory variables (vibrational phase, diatom orientation, and impact parameter) were selected by the appropriate Monte Carlo techniques,1° and 500 trajectories were run. The rotational quantum number, J , was always zero. After all the trajectories had been run, a cumulative record of box “visits” was available. This was converted into a probability density function by normalizing to unit box size. To assign the probability density to a frequency, the following procedure was adopted. The difference potential for each angular region was assumed to be that for some average angle. We arbitrarily chose that angle, p, whose cosine was the mean of the cos y values at the limits of each region. The sum of all densities contributing to an absorption frequency was considered to be the intensity of the absorption spectrum for that v.
IV. Results and Discussion In Figure 4 we show the transition-state absorption spectra, for 3D transition states, calculated at three translational energies: ET = 2.3, 5.8, and 23 kcal/mol. In each panel, the total spectrum is given by the heavy line. Beneath this we give the contributions (10) R. N. Porter and L. M. Raff in “Dynamics of Molecular Collisions”, Part B, W. H. Miller, Ed., Plenum, New York, 1976, Chapter 1; D. G. Truhlar and J. T. Muckerman in “Atom-Molecule Collision Theory: A Guide for the Experimentalist”, R. B. Bernstein, Ed., Plenum Press, New York, 1979, Chapter 16.
1.7064 1.6745 1.6072 1.6138
from various transition-state geometries: “near collinear” (1 1 cos y 1 0.6), “off collinear” (0.6 I cos y 1 0.2), and “perpendicular” (normally 0.2 Icos y I-0.2, but 0.2 1 cos y I-0.3 for E T = 23 kcal/mol). As was observed in the collinear case,’ the integral wing intensity decreased as collision energy increased. This simply reflects the shorter lifetime of the higher energy collisions. We also note that, for all three collision energies, there is observable structure in the wings. All three display a marked peak. This prominent peak is located at about 76 000, 68 000, and 49 000 cm-’ for E T = 2.3, 5.8, and 23 kcal/mol, respectively. In order to understand these features, we consider the trajectory probability density superimposed on the difference potential. In the three-dimensional case the density plot is created by recording the density originating from all trajectory configurations (r1,r2) within a certain range of angles. This is then superimposed on the difference potential (AV(rl,r2) corresponding to the angle whose cosine is in the center of the angular distribution interval. For example, in Figure 5 we show the trajectory density for ET = 2.3 kcal/mol. The same three angular intervals were employed in obtaining the three components of the 3D transition-state spectrum (i.e,, the set of component curves in each panel of Figure 4). The trajectory probability densities correspond to instantaneous values of rl, r2, and cos y. A given trajectory may contribute probability to all three panels of Figure 5 as it moves through configuration space. The AV contours shown in Figure 5 are for cos y = 0.8, 0.4, and 0, respectively, for the near-collinear, off-collinear, and perpendicular cases (bottom, middle, and top panels of Figure 5 , respectively). No trajectories were found in the cos y < -0.2 angular region for this energy; consequently, neither the AV contours nor the trajectory density plot is recorded for this region of y. Comparison of the three panels of Figure 5 shows that the value of rl, at the translational turning point (the point at which these nonreactive trajectories stop before returning along the entrance valley) becomes larger as y increases. This is a consequence of the behavior of the SLTH surface as y is varied, as can be seen from Figure 3. The potential barrier is lowest (little more than the collinear value of 9.8 kcal/mol) for y = 37’ (Le., cos-] 0.8). The barrier has increased to 17 kcal/mol at 66’ (cos-’ 0.4) and has reached 30 kcal/mol by 90’. It follows that, for a given collision energy, the nearest approach that H makes to H2 will become progressively larger as the transition state becomes more bent. Inspection of Figure 5 reveals, however, that the buildup of probability at the translational turning point occurs near to the same value of the AVdifference potential, since the contours on the AV difference potential move to larger rl as y increases. This has the effect that turning points with all y fall approximately under the same peak in the transition-state spectrum, contributing to its intensity and its breadth.
Absorption by H,* in the 3D Reaction H
+ H2
The Journal of Physical Chemistry, Vol. 88, No. 18, 1984 4067
E T = 23
I
2500
40000 I
I
I
2000
30000
50 000
I
1
2 500 40000
2000
I500 20000
I
I
40000
1
I
I750
70000
'
I
1750
Id6
'Ii1
'1
60000
I
1500
J
1250 1216 10000
20000
30000
125;
10000 1 -
I
24
J,'. ET=23
,[81 AY[G rn-I]
I
E T =2 3
Y
P
05 05
20
IO
rl
30
[il-
Figure 5. Difference potential (AV = P - pLTH) contour plots for y = 37" (lower), 66" (middle), and 90° (top) showing the trajectory density distribution for the angular regions centered on these three angles (ET= 2.3 kcal/ol). The contours are at the following frequencies: v(1) = 35000, 4 2 ) = 41 000, 4 3 ) = 46000, 4 4 ) = 50000, 4 5 ) = 53000, 4 6 ) = 56000, 4 7 ) = 62000, v(8) = 68000, v ( 9 ) = 71 000, and v(10) = 74000 (all in cm-').
L
2500
1
2000
I
1750
I
1500
I 1
1250
1216
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Figure 4. Absorption spectra for the 3D transition states, H3* H3*, at collision energies ET = 2.3 kcal/mol (bottom), 5.8 kcal/mol (middle), and 23 kcal/mol (top). Contributions to the spectra originating in the angular intervals centered on y = 37" 66" (- - -), and 90" (-) are shown together with their sum (-). (e-),
In order to test whether this result was an artifact of the angular intervals used, calculations for ET = 2.3 kcal/mol were also carried out with a different partitioning of cos y: 1.0 to 0.94,0.94 to 0.77; 0.77to 0.50, 0.50 to 0,O to -0.20. This did not noticeably affect the summed spectrum. It is interesting to note (from the curves at differing y's in Figure 4) that the contribution from each of the angular intervals was comparable in magnitude; Le., there is as much signal arising from perpendicular geometries of the transition state as from nearcollinear ones. This observation involves the assumption that all configurations have identical transition moments. In Figure 4,for ET = 23 kcal/mol, we note once again that there is a large contribution from perpendicular geometries. In fact, at this high energy, the perpendicular contribution dominates. This is to be expected if one considers that, a t low energies, the
triatomic system has time to "align" toward collinearity,' whereas at higher collision energies this is no longer possible. The peak in the transition-state spectrum, for ET = 23 kcal/mol (Figure 4), is due to the "translational turning point" evident in the trajectory probability density plots shown in Figure 6. The peak in the spectrum (Figure 4)has shifted to a very much longer wavelength than the peak observed at lower ET. This is due to the fact that the translational turning point occurs at much smaller rl (compare Figure 6 with Figure 5); at ET = 2.3 kcal/mol the 2 A whereas at ET = 23 kcal/mol it is turning point is &t r , at r l 1 A. At the higher collision energy, in addition to trajectories that return to re-form reagents, one sees in Figure 6-particularly for the near-collinear configurations-probability density in the product valley, indicative of reaction. Low rotation in the product has the result that trajectories in the exit valley tend to be registered in the lowest panel of Figure 6; Le., they are near collinear at the barrier crest and remain so in the exit valley. Comparing the density plot for the reactive trajectories in this 3D study with that obtained at the same collision energy in 1D in our earlier work (e.g., for the "CG" surface which has a dif-
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(11) H. R. Mayne and J. C. Polanyi, J . Chem. Phys., 76, 938 (1982).
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Mayne et al.
The Journal of Physical Chemistry, Vol. 88, No. 18, 1984
exhibit a correspondingly reduced line-of-centers collision energy. The long-wavelength tail of the transition-state s ectrum is due (Figure 4) at ET = 23 kcal/mol, extending to -2.500 to penetration of the trajectories to smaller rl (0.7-1.2 A). The A V contours for bent configurations (top panel, Figure 5) are shifted out to considerably larger r , ; this has the unexpected consequence that the more bent configurations, despite the fact that they penetrate only to moderate ri ( 1.2 A), make the largest contribution to the long-wavelength tail of the spectrum.
1,
N
V. Conclusions
(u
*
-
/I
I
0.5
ET=23
-
I
I
20
IO
-
The 3D transition-state spectrum for the reaction H + H2 H, + H computed here exhibits a peak which shifts markedly to longer wavelength as the collision energy increases. The peak moves from 1300 A at ET = 2.3 kcal/mol to 2000 A at ET = 23 kcal/mol, a shift of approximately 27 000 cm-’. The existence of this spectral peak can be ascribed to the accumulation of trajectories at a translational turning point,’ a region of configurations in which the momentum along the coordinate of approach, r l , has been largely converted into potential energy. The functional dependence of the spectral peak location on the reagent collision energy embodies information regarding the difference potential map, AV(rl,r2,y). If one supposes that the excited pes (a Rydberg state, that should be accessible to ab initio calculation) is known, then the spectral shift with collision energy-and ultimately with other types of reagent energy changes (see part II1)-will assist substantially in the task of mapping out the ground-state potential, across which the reaction proceeds. In the present computation, linear and bent transition-state configurations were found to contribute to a comparable extent to the total absorption. The relative contributions evidence themselves as a broadening in the major spectral peak. The indistinct differentiation between collinear and bent configurations arises in the first place from the form of the AV(rl,r2,y) contour map which gives rise to much the same AVvalues at the collinear and the bent translational turning points. It is also, however, a consequence of certain simplifying assumptions made in this work. We have assumed a constant optical transition moment everywhere in configuration space, and we have omitted from consideration one of the three low-lying excited states of H3 optically linked to the ground state. We intend to include these additional features in the transition-state spectrum, in a subsequent study.
3.0
r,
Figure 6. Transition-state spectra in in Figure 5 but at ET = 23 kcal/mol. Contours v(l)-v(9) as in Figure 5 .
ference map, AV, that qualitatively resembles that used here), one notes a smaller density buildup at rl less than the barrier crest (Le., smaller density in the “bobsledding” region of our earlier study). This is likely to stem from the fact that, in 3D, most reactive collisions occur at an impact parameter b > 0 and hence
Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and in part by the U S . Air Force Office of Scientific Research. Registry No. H,12385-13-6;Hz, 1333-74-0; H3, 12184-91-7.
