J . Phys. Chem. 1988, 92, 3190-3195
3190
-
+
A Centrifugal Sudden Dlstorted Wave Study of the CI -t HCI CIH CI Reaction: Results for a Scaled and Fitted ab Initio Potential Energy S-urface Having a Noncollinear Reaction Path George C. Schatz,*+ Theoretical Chemistry Group, Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439
B. Amaee, and J. N. L. Connor* Department of Chemistry, University of Manchester. Manchester M13 9PL, England (Received: July 17, 1987; I n Final Form: October 27, 1987)
-
We present the results of a centrifugal sudden distorted wave (CSDW) quantum scattering study of the reaction C1 + HC1 C1H + C1. The potential energy surface used in this calculation (denoted sf-POLCI) has been chosen to fit a scaled ab initio surface for CI-H-C1 angles greater than 150’ (angles for which the latter surface has been determined), and to fit an extended London-Eyring-Polanyi-Sato (LEPS) surface at smaller angles. This sf-POLCI surface has a noncollinear Cl-H-Cl saddle point with a C1-H-Cl angle of 161.4’. We also compare our CSDW results with those from a LEPS surface, which has a collinear geometry saddle point, but is otherwise similar to the sf-POLCI surface. Results presented include partial wave reaction probabilities, integral and differential cross sections, product rotational distributions, and thermal rate coefficients. The sf-POLCI results are generally similar to the LEPS results, although there are a few important differences. In particular, the integral cross sections in the threshold region increase more slowly with energy for the sf-POLCI surface. As a result, the activation energy is smaller for the LEPS surface, even though it has the higher barrier. Both the sf-POLCI and LEPS cross sections exhibit high product rotational excitation, with the sf-POLCI products more excited than the LEPS. Also, the rotational state which contributes most to the thermal rate coefficient is higher for the sf-POLCI surface than for the LEPS. For both surfaces the CSDW rate coefficients agree with experiment within the experimental uncertainties.
I. Introduction The reaction C1+ HC1- ClH + C1 is an important prototype for hydrogen atom transfer reactions between heavy partners (i.e., heavy + light-heavy atom reactions). It has been the focus of both extensive theoretical’” and experimental (thermal rate’) studies. Of particular interest has been the importance of special features associated with the small skew angle (13.6’) between the reagent and product arrangement channels for the collinear (ID) configuration of the atoms. For example, in accurate 1D quantum reactive scattering calculations for C1 + HCl,293it has been found that the small skew angle leads to large tunneling probabilities (due to “corner cutting”) at energies below the quasi-classical threshold for reaction and to oscillatory reaction probabilities at energies above this threshold. The discovery of these reactivity oscillations has led to speculation* that the differential cross sections might also show oscillatory structure, which has inspired further theoretical work aimed at describing the reaction dynamics in three dimensions (3D) by using quasiand q ~ a n t a l ~methods. .~.~ Recently, we presented the first results of a centrifugal sudden (or coupled states) distorted wave (CSDW) calculation for CI + HCl.5 The CSDW method is a 3D quantum reactive scattering approximation, which has been shown to produce nearly exact results in studies of the H + H2 reaction9 at energies where the reaction probability for each partial wave is less than about 10%. Related distorted wave methods have also been applied to H + H 2 by Suck Salk’O and by Bowers et a1.I’ The CSDW results for C1 + HCI that we have presented so far5,6have used an extended London-Eyring-Polanyi-Sato (LEPS) empirical potential energy surface that was introduced by Bondi et al.3a Our CSDW calculationsss6show that oscillating integral cross sections and oscillating angular distributions do not appear in 3D at energies near or below the quasi-classical threshold. Rather, the most striking feature of the 3D dynamics is the high rotational excitation found in the product HCI molecule (40-50% of the available energy) and, along with this, the strong effect of reagent Visiting scientist. Permanent address: Department of Chemistry, Northwestern University, Evanston, IL 60201.
0022-3654/88/2092-3190$01 S O / O
rotation in enhancing the reaction. Subsequent comparison of the CSDW cross sections with the results of quasi-classical trajectory computations shows that the same dynamical effects for reagent and product rotation occur in both calculations.6b One possible limitation of these CSDW calculations is that the LEPS surface used is empirical, with a I D barrier that has been adjusted to a height of about 0.37 eV, as recommended by Kneba and Wolfrum.12 Recently, Garrett et al.4 performed a b initio calculations in order to produce a potential surface which is expected to be more realistic. This surface differs from the LEPS one in that the reaction path is noncollinear, with a saddle-point (1) (a) Thompson, D. L. J . Chem. Phys. 1972,56,3570. (b) Thommarson, R. L.; Berend, G. C. Inr. J . Chem. Kinet. 1973, 5 , 629. (c) Smith, I. W. M.; Wood, P. M. Mol. Phys. 1973, 25,441. (d) Smith, I. W. M. J . Chem. SOC., Faraday 2 1975, 71, 1970. (e) Wilkins, R. L. J . Chem. Phys. 1975,63, 534. (f) Baer, M.; Last, I. Chem. Phys. Lett. 1985, 119, 393. (8)Last, I.; Baer, M. Int. J . Quantum Chem. 1986, 29, 1067. (h) Last, I.; Baer, M. J. Chem. Phys. 1987, 86,5534. (i) Persky, A,; Kornweitz, H . J. Phys. Chem. 1987, 91, 5496. (2) (a) AbuSalbi, N.; Kim,S.-H.;Kouri, D. J.; Baer, M. Chem. Phys. terr. 1984, 112, 502. (b) Pollak, E.; Baer, M.; Abu-Salbi, N.; Kouri, D. J. Chem. Phys. 1985, 99, 15. (3) (a) Bondi, D. K.; Connor, J . N . L.; Manz, J.; Romelt, J. Mol. Phys. 1983, 50, 467. (b) Bondi, D. K.; Connor, J. N. L.; Garrett, B. C.; Truhlar, D. G.J . Chem. Phys. 1983, 78, 5981. In eq 4, for c,,,(n=s) read c,,,(n,s). In the caption to Figure 4, for distances, read distance s. (4) Garrett, 9.C.; Truhlar, D. G.; Wagner, A. F.; Dunning, T. H., Jr. J . Chem. Phys. 1983, 78, 4400. On p 4406, for R, = 2.80677 a,, read R, = 2.80677 A. (5) Schatz, G. C.; Amaee, B.; Connor, J. N . L. Chem. Phys. Leu. 1986, 132, 1 . (6) (a) Schatz, G . C.; Amaee, B.; Connor, J. N. L. Compur. Phys. Comm. 1987, 47, 45. (b) Amaee, B.; Connor, J. N. L.; Whitehead, J. C.; Jakubetz, W.; Schatz, G.C. Faraday Discuss. Chem. SOC.,in press. (7) (a) Klein, F. S.; Persky, A.; Weston, R. E. J . Chem. Phys. 1964, 41, 1799. (b) Kneba, M.; Wolfrum, J. J . Phys. Chem. 1979, 83, 69. (8) Hiller, C.; Manz, J.; Miller, W. H.; Romelt, J . J . Chem. Phys. 1983, 78, 3850. (9) (a) Schatz, G. C.; Hubbard, L. M.; Dardi, P. S.; Miller, W. H. J . Chem. Phys. 1984,8J, 231. (b) Schatz, G . C. J. Chem. Phys. 1985,83, 5677. (10) Suck Salk, S. H. Phys. Reu. A 1985, 32, 2670. ( 1 1 ) Bowers, M. S.; Choi, B. H.;Tang, K. T. Chem. Phys. Left. 1987,136, 145 (12) Kneba, M.; Wolfrum, J. Annu. ReL;.Phys. Chem. 1980, 31, 47.
0 1988 American Chemical Society
CSDW Study of the C1
+ HCl
-
ClH
+ CI Reaction
CI-H-CI angle 0 of 16 1.4O, but other properties of the surface such as the saddle-point HC1 distance and saddle-point energy are quite similar. Unfortunately the Garrett et al. surface is not a global surface in the sense that it was constructed from ab initio calculations done only for geometries near to collinear ClHC1, with a quadraticquartic expansion for the bend angle dependence. However, it is not difficult to modify the Garrett et al. surface so as to be globally acceptable, and in this paper we present CSDW results using a modified surface of this type. This is the first fully rotationally coupled quantum scattering calculation for any reaction involving a noncollinear reaction path. (Earlier studies of nonlinear reaction path systems such as Li HFI3 and H e + Hz+l 4 used reduced dimensionality scattering treatments of rotational motions.) We will therefore pay close attention to differences between results on this new surface and those on the LEPS. Results to be presented include partial wave reaction probabilities, differential and integral cross sections, rotational distributions, and thermal rate coefficients. We also compare the thermal rate coefficients with experimental values so as to assess (in part) the accuracy of the potential surfaces used. A more detailed comparison of rate coefficients with those from other methods, such as variational transition-state theory (VTST), will be reported elsewhere. It should be emphasized that the energy range considered in most of this calculation is at or below the quasi-classical threshold energy for reaction (10.5eV total energy). Since CSDW is a perturbative method, it is unlikely to be reliable at higher energy. Convergence of the method can be tested, however, by varying the vibration/rotation basis set as this basis defines the partitioning of the Hamiltonian into reactive and nonreactive parts.9a Work on the LEPS surface6 has verified convergence of our results at energies below 0.5 eV, and we have verified this for the sf-POLCI surface as well. Because the CSDW calculation allows for coupling between nonreactive states in each arrangement channel, the partial wave transition probabilities are not forced to be monotonic functions of energy (as sometimes happens in simpler distorted wave methods). However, if the reactivity oscillations that have been seen in collinear C1 HCl calculationsz~3 were important in 3D then the CSDW would not converge at energies where the oscillations are significant, since the oscillations involve substantial flux crossing back and forth from reagent to product regions. The fact that our CSDW calculations are well converged at subthreshold energies is, we feel, good evidence for the lack of importance of reactivity oscillations at these energies. However, this does not preclude their possible importance at higher energies. To summarize the rest of this article, in section I1 we present the modified potential surface and compare its properties with the LEPS surface, while in section I11 we describe the CSDW calculations. Section IV presents the results, and section V summarizes our conclusions.
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3191 TABLE I: Parameters for the sf-POLCI Potential Surface Defined by Equation 1
auantitv value Switch Parameters fIo/rad 2.6 ylrad-’ 20.0 Parameters in DJHCI)/& r,(HCI)lao P(HCl)/a,-’ S(HC1)
+
+
11. Potential Surface The potential surface we have used is based on the polarization configuration interaction (POLCI) plus dispersion interaction calculations of Garrett et al.4 These calculations indicate that the ClHCl saddle point is symmetrically located (equal HCl distances) but slightly bent. A fit to the ab initio points was made4 by using a rotated-Morse-oscillator-cubic-spline f ~ n c t i o n ’for ~ collinear geometries plus a quadratic-quartic function of the angle 8. This fit was then used by Garrett et aL4 to determine thermal rate coefficients for C1 DCl by using VTST with a large curvature ground-state tunneling correction. This resulted in rate coefficients that are too high by factors of 10-22. The fit was then scaled so as to reproduce the experimental rate coefficient for C1 DC1 at 368.2 K. It was subsequently found that this scaled surface (which we will denote s-POLCI) predicts the rate
+
+
(13) Miller, D. L.; Wyatt, R. E. J . Chem. Phys. 1987, 86, 5557. (14) Baer, M.; Nakamura, H.; Kouri, D. J. In?. J . Quantum Chem., Quantum Chem. Symp. 1986, No. 20,483. (15) Connor, J. N. L.; Jakubetz, W.; Manz, J. Mol. Phys. 1975, 29, 347.
Vm.LEpSQ
De(C12)/Eh
rdCM/a0
P(C12)Id S(C1,)
0.169797 2.40846 0.988185 0.115 0.092423 3.756848 1.059392 0.115
Notation is the same as that in Table I of ref 3a. TABLE II: Comparison of Saddle-Point Properties for the sf-POLCI and LEPS Potential Surfaces
property R*HCI/.O
ClHCl angle/deg P/eV
harmonic frequencies/cm-’ sym stretch bend antisym stretch bend eigenvalue/eV ground state P first excited Vt
sf-POLCI bent collinear 2.784 2.784 161.4 180 0.326 0.391
0.371
326 1617 1606i
336
344
1118i 16271
508,508 13981’
0.075 0.144
LEPS collinear 2.772 180
0.065 0.133
+
coefficients for C1 HCl to within experimental error.4 Unfortunately it is not possible for us to do CSDW calculations directly by using the s-POLCI surface, because the quadraticquartic bend potential is extremely repulsive for 8 much smaller than the saddle-point angle, even at large C1-HCl separations where the interaction potential should be negligible. To circumvent this problem we have constructed a new surface (denoted sfPOLCI, for scaled and fitted POLCI), which uses switching functions to join the s-POLCI surface for 8 near to and greater than the saddle-point angle onto a modified LEPS surface at smaller 8. Parameters for the LEPS part of this surface (which we denote m-LEPS) are mostly taken from the work of Bondi et al.,3abut the Morse parameters were slightly revised so as to agree with the s-POLCI parameters at large separation. The specific form of the sf-POLCI surface is ~ s f - P o L c= I
I/,.PoL,cI{~+ tanh [r(0- 00)11/2 + ~rn-LEPS(1- tanh [r(8- eO)ll/2 (1)
where V,.mLc, is exactly as in ref 4, and the other parameters are given in Table I. Table I1 summarizes the properties of the saddle point and the nearby collinear stationary point on the sf-POLCI surface as well as the properties of the saddle point on the LEPS surface of ref 3a, which we used in our earlier CSDW calculation^.^^^ The s-POLCI saddle-point geometries and frequencies are equal to those for sf-POLCI to all the figures indicated in Table I1 except for the bend frequency, which is 2 cm-I smaller. Note that the LEPS surface has nearly the same HCl distance as sf-POLCI at the saddle point, with an energy that is larger than the sf-POLCI saddle-point energy, but smaller than the energy of the sf-POLCI collinear stationary point. The symmetric and antisymmetric stretch frequencies are also similar, but the bend frequencies differ by more than a factor of 3. However, because the LEPS surface has a collinear saddle point while sf-POLCI is noncollinear, the bend zero-point energies turn out to be quite similar on the two surfaces. This is indicated by the entry labeled “bend eigenvalue” in Table 11, where we give the ground- and first-excited-state bend energies on the two surfaces. Note that each bend energy is measured relative to the saddle point energy vt.
3192
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988
Schatz et al.
6,0p&r
4.0
-4.0
1
-6.01 -6.0
0.36’ 0.56e! ,
, -4.0
,
I
-2.0
,
I
,
0.0
. . 2.0 4.0
,
6.0
X lao
Figure 1. Contours of sf-POLCI pJtential surface as a function of the H atom location for a fixed CI-CI separation of 5.60 ao. The origin of the X,Y coordinate system is located at the midpoint of the line joining the two C1 atoms. The saddle-point locations are indicated by crosscs ( X ) on the plot. Note that the maximum which is located at the origin of the plot is the collinear stationary point which is described in Table 11. Other points which appear to be saddle points on this graph are not true stationary points on the full dimensional potential surface.
The bend energies were computed by a numerical finite difference algorithm to solve the bend Schrijdinger equation for coplanar (2D) bend motions, followed by a two bend mode configuration interaction calculation to determine the 3D bend motion eigenvalues. The G matrix element governing bend motions was assumed to be that for 1D ClHCl in all calculations (as was also done in ref 4). This method of treating the bend motions leads to a 3D bend zero-point energy which is only 3 cm-I less than twice the 2D zero-point energy on the LEPS surface, but the difference for the sf-POLCI surface is much larger (294 cm-I). This result is not unexpected since the equality between the 3D zero-point energy and twice the 2D value only holds for harmonic bend potential surfaces with collinear saddle points. In 3D VTST calculations,3b.16it is found that the ground-state vibrationally adiabatic bamer to reaction occurs at the saddle point on the LEPS surface but slightly away from it on the s-POLCI ~ u r f a c e . An ~ estimate given in ref 4 of the 3D ground-state vibrationally adiabatic barrier for s-POLCI is 0.48 eV, while the corresponding 3D barrier on LEPS is 0.50 eV.I6 Figure 1 presents a contour plot of the sf-POLCI surface. In this plot we have fixed the two Cl’s to be separated by 5.60 a, (close to the saddle-pint separation of 5.57 a,,), and the contours are plotted as a function of the H atom location. When plotted in this way, the highest energy contours form circles around each C1 atom. The reaction path is formed by two shallow troughs which connect between the two circles (one on either side of the collinear maximum). The positions of the saddle points are indicated by x’s in the figure. Note that there are no obvious ripples or other unphysical features in the figure associated with the switch In fact we find that if a plot between Vs.poLc, and V,,,.,,,,. analogous to Figure 1 is drawn using V,., everywhere, the only noticeable difference is that the two nonlinear reaction paths are replaced by one collinear path. 111. Dynamics Calculations
The reactive scattering calculations were performed by using the centrifugal sudden distorted wave (CSDW) method, details of which are given in ref 5, 6, and 9. For the sf-POLCI surface, we found that the numerical parameters and basis functions needed to generate converged results are similar to those used in our earlier studies5s6for the LEPS surface of ref 3a. Convergence with respect to the number of rotational basis functions included was somewhat slower however. so since the same basis was used in both calcu(16) Garrett, B.
C.; Truhlar, D. G., unpublished results.
lations, the overall accuracy of the present calculation is somewhat poorer. All the results that we will present were generated by using a basis of 36 vibration/rotation states in each arrangement channel consisting of 14, 14, 6, and 2 rotational states for vibrational levels u = 0, 1, 2, and 3, respectively. Computation time per partial wave per energy with this basis on a Cray X-MP is about 680 s, which is 13% larger than was noted in ref 6a for the LEPS surface. Only the Q = 0 rotational projection quantum number is used in the CSDW computations as this is adequate to accurately determine the cross sections in the energy region near and below the quasi-classical threshold, where the CSDW approximation is valid. (We did a few J = 1 coupled-channel distorted wave calculations to test this point and found that although IQl > 0 makes a substantially larger contribution to the cross sections for the sf-POLCI than for the LEPS, this contribution is still much smaller than the Q = 0 contribution.) When presenting the rate coefficients, we will use the energy-shifting approach developed in ref 17 to estimate the contribution of jQl > 0 to the overall results. This contribution is less than 40% at the highest temperature considered in the rate coefficient measurements. In the cross section calculations, we did computations for every, fifth partial wave (linearly interpolating probabilities for the others) for enough partial waves J (typically about 100) to fully converge the partial wave sum. Calculations were done at total energies E ranging from 0.30 to 0.70 eV in increments of 0.05 eV, where E is measured from the minimum of the HC1 well. However, the results at 0.50 eV and above include some total reaction probabilities (summed over final states) that are greater than 0.1. This behavior indicates that the CSDW method is breaking down,g and therefore, almost all of our analysis considers low energies (0.30-0.50 eV) where the CSDW approximation is expected to be accurate. It is also important to note that our rotational basis set does not include all open states at energies of 0.50 eV or greater. (See ref 6b for a more detailed discussion of this point.) The explicit calculation of differential cross sections requires substantial computer time because computations for each value of J are needed to avoid ambiguities in interpolating the phases of S matrix elements. However, in ref 6a we presented an approximate semiclassical method (which has also been used by others in the past’*) for the calculation of differential cross sections that avoids this problem. In this method, a hard-sphere formula is used to relate the scattering angle to the impact parameter b, and then the CSDW reaction probability is used to determine the probability for a given impact parameter by using the approximation J = p b / h (wherep is the relative translational momentum) to relate J and b. The resulting expression for the state to state ~) OR is the reactive differential cross section G ~ , , ~ ~ , , ( O(where scattering angle and uj and v y ’ are initial and final vibration/ rotation quantum numbers) is
where P is the reaction probability and r is an effective hard-sphere radius that is approximately half the Cl-Cl separation at the saddle point. The relation between b and OR is given by the hard-sphere formula b = 2r cos (0,/2)
(3)
We have used eq 2 and 3 to calculate differential cross sections with the further assumption that the momentum p is the initial momentum for a given transition. To calculate the thermal rate coefficient k( T ) , we have used the standard expression13 k ( T ) = z-’c(2j+ l)e-flJ/qkvI,$,,(T) 01
(4)
V?’
where Z is the reagent vibration/rotation partition function, cv, (17) Colton, M. C.; Schatz, G. C. Inr. J. Chem. Kinet. 1986, 18, 961. (18) Herschbach, D. Appl. Opt., Suppl. 2, 1965, 128. Herschbach, D. Adu. Chem. Phys. 1966, 10, 319. Wyatt, R. E.; McNutt, J.; Redmon, M. J. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 437.
CSDW Study of the C1
+ HCl
-
C1H
+ Cl Reaction
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988 3193
TABLE III: Energy Partitioning Results for the sf-POLCI Surface (LEPS Results in Square Brackets) EIeV 0.30 0.35 0.40
no. of open states
10 7 [61 0.35 (-7)b [0.11 (-6)] 6.3 [6.2] 5.1 0.061 [0.058] 0.047 0.12 0.52 [0.49] 0.40
3
Qjla: (j')j
Ci')" (E,Oj/eV ( E , , ) /ev Eava,,leV
G')j G,)"
11 8 [81 0.13 (-3) 10.13 (-3)] 7.3 [7.3] 6.1 0.080 [0.080] 0.065 0.17 0.48 10.481 0.39
13 9 [91 0.22 (-2) [0.15 (-l)] 8.7 [8.1] 7.2 0.11 [0.098] 0.087 0.22 0.51 [0.45] 0.40
0.45
0.50
14 11 [ l o ] 0.13 [0.31] 10.4 [8.7] 7.9 0.15 [0.11] 0.11 0.27 0.58 [0.42] 0.39
14" 11 [ l l ] 2.56 [ 1.471 10.4 [9.4] 8.8 0.15 [0.13] 0.13 0.32 0.49 [0.42] 0.40
"This value is determined by basis set limitations rather than energy conservation. bThe number in parentheses is the power of 10 by which the entry must be multiplied.
2 ,... ......._
CI +HCl(v=O,j=8)
h
-2
-8.0'
'
'
2
'
4
'
'
6
'
1
I
0.35
0
Q8,/ versus j ' for a-total energy of 0.40 eV, showing the sf-POLCI (solid) and LEPS (dashed) results.
is the vibration/rotation energy, and k, is the state to state rate coefficient. k,,[ is obtained from the degeneracy-averaged integral cross section Q,,,. by numerically integrating the expression l 3
[ $1
112
( k 7 ' ) - 3 1 2 ~ m ~ Q , j ,exp(-E$/kT) ,y
dE$ (5)
where E$ is the translational energy (@$ = E - e!), and p is the translational reduced mass. By including cross sections up to 0.70 eV when numerically evaluating this integral, we are able to calculate rate coefficients up to 500 K. These rate coefficients are, however, less accurate than the other results we present, as they depend on our higher energy cross sections. Finally we should note that all our rate coefficients (but not cross sections) have been multiplied by the usual spin-orbit degeneracy factor19
f = (2
+ exp(-l268K/T)-'
(6)
The masses used in the calculations are mcl = 34.969 u and mH = 1.008 u.
IV. Results IV.A. Integral Cross Sections and Energy Partitioning. We begin by considering the state to state integral reactive cross sections Q,/, where we have omitted the vibrational quantum number labels as they are always zero for the E 5 0.50 eV. Figure 2 plots log QJ, vs j' for j = 8 at E = 0.40 eV and includes results from both the sf-POLCI and LEPSS surfaces. The choice j = 8 has been made because the cross sections for this state are among the largest at E = 0.40 eV. The figure shows that for both surfaces (19) Parr, C. A,; Truhlar, D. G. J . Phys. Chew. 1971, 75, 1844. Moore, C. Atomic Energy Leuels; US.Government Printing Office: Washington,
DC, 1949.
+ HCI(v=O,j)
1
1
8
Figure 2. log
=
CI
1
j'
k,,,
[
0.40
0.45
EIeV Figure 3. log Qoand log Q9versus E for the sf-POLCI (solid) and LEPS
(dashed) potential surfaces. the product rotational distribution peaks at J' = 9, with a rapid falloff on either side of this peak. The shapes of the distributions differ only slightly on the two surfaces, with the falloff for j ' > 9 being less steep on the sf-POLCI surface. The shapes of the distributions f o r j # 8 are similar to those in Figure 2 (as is shown in Figure 1 of ref 5 for the LEPS surface), which indicates that there is no j z j'propensity in the rotational distributions. Oscillations seen at low j'on the sf-POLCI curve in Figure 2 are probably due to incomplete convergence of the CSDW calculation. A summary of the product-state distributions and eilergypartitioning behavior of E = 0.30-0.50 eV is given in Table 111. In this table we use the symbol j^ to denote the rotational quantum number for which the cross section QJ = XJtQ;Jtis a maximum for a given E . The cross section Q; is included in the table along the average product with the average value of j ' (denoted (j');), rotational energy (denoted (E,)J and the fraction ~ , ) ofj the in product rotation. Also included in Table available energy EaVall I11 are microcanonical values (labeled by a superscript zero) for the three average^.^ Table I11 indicates that 1is always 2-3 below the maximum possiblej, being similar to what was obtained for the LEPS surface (given in square brackets). Product rotational excitation is also quite high, as is evidenced by U;)lvalues that are typically around 0.5. These values are consistently higher than the microcanonical averages, and at energies above 0.35 eV, the sf-POLCI are consistently larger than their LEPS counterparts. Apart from this, the sf-POLCI and LEPS results in Table I11 are similar. Figure 3 is a semilog plot of the cross sections Qo and Q9as a function of E for both the sf-POLCI and LEPS surfaces. This graph shows that the energy dependence of the cross sections is different for the two surfaces, with the LEPS Q9 cross section increasing more steeply with E at low energy but then leveling off at higher energies (such that at just above 0.50 eV the sfPOLCI Q9 crosses the LEPS curve). The cross section Qo has a more gradual dependence on energy for both surfaces, with the result that the ratio Q9/Q0 increases with E at low energy but
V,,);
The Journal of Physical Chemistry, Vol. 92, No. 11, 1988
3194
0.3
I r
'
t
CI
'
I
'
,
'
+ HCI(v=O,j)
3
'
)
'
=
E
Schatz et al. -13
l
CI -t HCI
0.40eV
1
-2. h
\
-4. L
a
Y
(JI
-0
-3.
3;
LEPS
sf-POLCI
.
t
3
I
I
-e. s '
IC
20
30
40
53
-18 -
i J
I
50
J
Figure 4. log P9 and log PgS versus J for the sf-POLCI (solid) and LEPS (dashed) potential surfaces at E = 0.40 eV.
0.310
L
m" .
rl
i
0.006t I I
0.DOL 1
0
E = Q.4QeV CI HCI(v=O,j)
+
3.000
45
90
135
J
e R ldeg Figure 5. Differential cross sections o9 and u9,9 versus BR for the sfPOLCI (solid) and LEPS (dashed) potential surfaces at E = 0.40 eV. Note that the sf-POLCI cross sections have been multiplied by 10. Also, the sf-POLCI probabilities in Figure 4 were smoothed (by fitting to a
quadratic polynomial) prior to calculating the differential cross sections. with the sf-POLCI surfaces showing the faster increase. Another way to study the E dependence of the cross sections is to consider the Q; cross sections in Table 111. The table shows that the sf-POLCI result is below LEPS at low E , but above it at the highest E of 0.50 eV. Often it is convenient to define an effective threshold energy Ethas that E where the cross section first becomes equal to a value such as 0.05 ao2.20 According to Table 111, Ethis slightly below 0.45 eV for both surfaces, with a lower value for the LEPS surface. Figure 3 shows that in the LEPS Ethis also lower for Qo and Q9,and in fact this difference is found for all the (3,'s for j = 0-9. This is surprising, since the LEPS surface has both a higher classical barrier and a higher ground-state vibrationally adiabatic barrier. We will discuss this further in section V. 2V.B. Opacities and Angular Distributions. Figure 4 plots the all transitions (dereactive probabilities for the 9 9 and 9 noted P9,9and P9)versus J for the sf-POLCI and LEPS surfaces at E = 0.40 eV. These curves are typical of the J dependence of all the reaction probabilities, with a nearly parabolic falloff for the logarithm of the probability, implying a Gaussian dependence of the probability on J . This behavior is the same as was noted earlier for the LEPS surface.& The differences between sf-POLCI and LEPS are quite small here, with the sf-POLCI probabilities dropping off a little more slowly than LEPS. The small undulations in the sf-POLCI curves at J = 35 and 45 are probably due to quadrature errors in the CSDW calculation. As might be expected, the strong similarity for the J dependence of the sf-POLCI and LEPS probabilities implies that the angular distributions should also be similar. This is born out in Figure 5 , where we plot a(0,) vs OR for the 9 9 and 9 all transitions
- -
- -
(20) Schatz, G. C.; Kupperman, A. J . Chem. Phys. 1976, 65, 4668
where E L n d is the lowest bending energy level associated with vibrational angular momentum Q at the saddle point and Eoknd is taken to be zero. (Actually it would be slightly more accurate to evaluate E&",, at the vibrationally adiabatic barrier rather than the saddle point.) To evaluate eq 7 , the results from the bend energy calculation discussed in section I1 were used (see Table 11). The resulting values of C are similar on the two surfaces, and range from unity at low temperature to 1.2 at 300 K and to 1.4 at the highest experimental temperature (423.2 K). Figure 6 shows that both the LEPS and sf-POLCI rate coefficients agree with the experimental values within the experimental error bars. There are, however, differences between the activation energies E, and preexponential factors A on the two surfaces, with E , = 0.25 eV, A = 1.6 X lo-" cm3 s-l on LEPS and E , = 0.28 eV, A = 5.7 X lo-" cm3 s-l on sf-POLCI (all parameters being evaluated at 368 K by using the method outlined in ref 17). The difference between the activation energies is consistent with the threshold energies discussed earlier, while the larger sf-POLCI A factor is consistent with the fact noted in Table I11 that the sf-POLCI cross sections are larger at high energies. In ref 4 the experimental activation energy is estimated as 0.24 eV with an uncertainty of between 0.01 and 0.10 eV. From this value and the rate coefficients presented in ref 4, one can derive an A factor of 1.0 X lo-" cm3 6'. As has been noted in ref 4, the uncertainties in these parameters are difficult to estimate so the significance of the comparison of these parameters with the theoretical ones is uncertain. It is also important to note that the rotational state that contributes most to the rate coefficient is much higher than the thermally most probable rotational state. Over the 300-400 K temperature range, the largest contribution to the rate coefficient is from j = 11 for sf-POLCI and from j = 10 for LEPS, while the thermally most probable state is j = 3. The j = 11 state of HC1 is 0.17 eV above the ground state, which is 61% of the
J. Phys. Chem. 1988, 92, 3195-3201
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a “figure-of-eight” trajectory as discussed elsewhere6b). If so, then the effective barrier to reaction on the sf-POLCI surface is better thought of as located at the collinear stationary point of Table 11, which is 0.02 eV higher in energy than the LEPS saddle point. Further work (trajectories) will be needed to establish the validity of this argument. One additional topic, which will be deferred to a future paper, concerns the comparison of our CSDW rate coefficients with values obtained from VTST and from reduced dimensionality quantum scattering theories. The observed differences between the sf-POLCI and LEPS rate coefficients bring to question whether vibrationally adiabatic theories that do not impose angular momentum constraints can properly describe threshold energies. We will examine this point in future work.
activation energy on the sf-POLCI surface.
V. Discussion Perhaps the most important finding in this paper concerns the comparison of the LEPS and sf-POLCI results. This comparison indicates that the biggest difference occurs in the energy dependence of the cross sections, with the LEPS results showing lower threshold energies than sf-POLCI, despite the fact that LEPS has the higher ground-state adiabatic barrier. This leads to a noticeable difference (0.03 eV) in the activation energies, even though the overall rate coefficients both agree with experiment within the experimental uncertainties. Other differences between LEPS and sf-POLCI are more subtle, such as the slightly higher product rotational excitation on sf-POLCI. Some quantities, such as angular distributions, show almost no differences at all. The fact that rotation plays an important role in the C1 HCl reaction is a significant result, and although this is true for both potential surfaces, it may also contribute to the differences in the threshold energies for the sf-POLCI and LEPS surfaces. In particular, we note that for highly excited rotational states (j = 10, say) the rotational period (-0.16 ps), while still long compared to the vibrational period (-0.01 ps), is comparable to the collision duration. As a result, it is best to consider reaction from the perspective of Figure 1 wherein the reacting hydrogen atom follows an angular path over the saddle-point region (executing part of
+
Acknowledgment. This research was supported by the Office of Basic Energy Science, Division of Chemical Sciences, U S . Department of Energy, under Contract W-3 1- 109-ENG-38, by N S F Grant CHE-8416026, and by the UK Science and Engineering Research Council. The computations were carried out on the Cray XMP and Cray 2 computers at the National Magnetic Fusion Energy Computer Center, and on the CDC 7600, CDC Cyber 176, and CDC Cyber 205 computers at the University of Manchester Regional Computer Centre. Registry No. C1, 22537-15-1; HC1, 7647-01-0.
Rotational State Distributions following Direct Photodissociation of Triatomic Molecules: Test of Classical Modek Reinhard Schinke Max-Planck-Institut f u r Stromungsforschung, 0-3400 Gottingen, FRG (Received: April 15, 1987; In Final Form: June 11, 1987)
We test the applicability of classical mechanics to calculate rotational state distributions following the direct photodissociation of triatomic molecules under conditions when rotational excitation is str6ng. The agreement with exact quantal calculations for a model system is generally very good. However, we also find that in some cases the agreement is poor. Discrepancies can be traced back to peculiar, nonmonotonic trajectories; We also examine the reliability of classical calculations employing a classical distribution function for the initial state.
I. Introduction Photodissociation of polyatomic molecules in the UV range is an active field, both e~perimentallyl-~ and t h e ~ r e t i c a l l y . ~ - ~ Traditionally, the total absorption cross section as a function of photon wavelength has been measured for almost every molec ~ l e . ~ -It’ is~ related to the energetical ordering of the excited states involved in the photoexcitation process, and its overall shape might give some clues about the nature of the dissociation process. For example, a broad and structureless spectrum is expected for a fast (direct) dissociation mechanism. In contrast, sharp structures indicate an indirect (predissociating) decay mechanism. (1) Leone, S. R. Adu. Chem. Phys. 1982, 50, 255. (2) Simons, J. P.J . Phys. Chem. 1984>88, 1287. (3) Bersohn, R. J . Phys. Chem. 1984, 88, 5145. (4) Jackson, W. M.;Okabe, H. In Advances in Photochemistry; Volman, D. H., Gollnick, K., Hammond, G. S., Eds.; Wiley: New York, 1986: Vol. 13, p 28. (5) Freed, K. F.;Band, Y. B. In Excited States; Lim, E. C.: Ed.; Academic: New York, 1978; Vol. 3. (6)Shapiro, M.; Bersohn, R. Annu. Rev. Phys. Chem. 1982, 33, 409. (7) Shapiro, M.; Baht-Kurti, G. G. In Photodissociation and Photoionization; Lawley, K. P., Ed.; Wiley: New York, 1985; p 403. (8) Schinke, R. J . Phys. Chem. 1986, 90, 1742. (9) Okabe, H.Photochemistry in Small Molecules; Wiley: New York, 1978. (10) Robin, M.B. Higher Excited States of Polyatomic Molecules; Academic: New York, 1974.
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Obviously, more information about the photodissociation process is obtained if the quantum states of the products are also resolved. Due to the availability of sophisticated probing techniques (LIF; REMPI, etc.), the past 10 years or so have witnessed an enormous growth of final state distributions for many dissociation products (OH, NO, CO, e t ~ . ) . l - ~ J Product l state distributions are determined by the preparation of the parent molecule prior to-the electronic excitation and by the dynamics on the excited state potential energy surface. The so-called “final state interaction” is nothing else than the coupling (forces) between the various degrees of freedom induced by the corresponding multidimensional potential energy surface V X .The product state distributions somehow ”reflect” these forces, and thus certain information about Pxmay be deduced which would be difficult to obtain otherwise. Quantum mechanically, the photoabsorption cross section is usually calculated from Fermi’s Golden Rule6-7~12 where Qtj is the nuclear wave function in the ground electronic state and rotational-vibrational level (i), is the nuclear wave function in the excited electronic state asymptotically dissociating (1 1) Buelow, S.;Noble, M.:Radhakrisiinar, G.;Reisler, H.; Wittig, C.; Hancock, G. J . Phys. Chem. 1986, 90, 1015. (12) Weissbluth, M.Atoms and Molecules; Academic: New York, 1978.
0 1988 American Chemical Society
