1053
J. Phys. Chem. 1994,98, 1053-1057
Comparison of Theoretical and Experimental Differential Cross Sections for the H
+ Dz Reaction
Steven L. Mielke and Donald G. Trublar’ Department of Chemistry, Chemical Physics Program, and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431 David W. Scbwenke N A S A Ames Research Center, Mail Stop 230-3, Moffett Field, California 94035-1000 Received: October 22, 1993”
+
W e present accurate quantum mechanical differential cross sections for the H D2 reaction. Results are given for two potential energy surfaces and three initial states at relative translational energies of Ercl= 0.525-0.562 eV. Comparisons are made to quasiclassical trajectory simulations and experiments. The average final rotational quantum number is 12% less than the quasiclassical one, and the quantal and classical differential cross sections have qualitatively different shapes. The theoretical differential cross sections show less forward scattering than experiment.
1. Introduction
+
In recent years the H D2 reaction has been studied extensively experimentally,l-5 and theoretical calculations of the product rotational distributions have been performed using a variety of approximate quantum mechanical methods610 and quasiclassical trajectory (QCT) simulations.ll~2Approximate quantal calculations focusing on other aspects of this reaction have also been reported,I3 close coupling calculations have been performed at low total angular momentum by several groups,”’-16 and stateto-state integral cross sections have been calculated for one initial state.” Differential cross sections (DCSs) have been obtained in recent QCT simulations,12 and in this paper we present the first DCS calculations by accurate quantum mechanics. These calculations are motivated in part by recent experimental advances4v5 that have permitted measurements of the product angular distributions for this system. We present results for the initial states with v = Oj = 0-2 (where v and j denote initial DZ vibrational and rotational quantum numbers, respectively) at two energies on the LSTH’8 potential energy surface that has been used in most of the previous studies, as well as for three energies on the more recent BKMP19 surface. 2. Calculations The present calculations were carried out using the outgoing wave variational principle (OWVP) which is a form of Schlessinger’s scattered wave variational principle and is identical to the scattered wave variational principle we discussed previously.2s28 The OWVP provides a generalization of the KohnZ9and Newton30 variational principles, as discussed elsewhere.22-24 In the present work it is used in the latter way. In particular, when halfintegrated Green’s functions are used for all basis functions, the OWVP reduces for reactive scattering to the generalized Newton variational principle that we have used successfully in earlier work,31J2 but the OWVP also allows the computational efficiency of using energy-independent 1 2 basis functions for a subset of the basis, if desired. In the present work, the basis is a hybrid. The translational basis consisted of energy-dependent halfintegrated complex distorted-wave Green’s functions32 in the open channels and energy-independent distributed G a ~ s s i a n in s ~the ~ closed channels. The final step of an OWVP calculation is the solution of the linear variational equations Ca = b, where C is a matrix representation of H-E in the basis used to expand the Abstract published in Aduance ACS Abstracts, January 15, 1994.
0022-365419412098-1053$04.50/0
outgoing wave. For the present calculations the open-open submatrix of C is identical to the open-open submatrix of C in a corresponding GNVP calculation, the closed-closed submatrix is identical to the closedxlosed part of a Kohn c a l c ~ l a t i o n , ~ ~ - ~ ~ ~ ~ ~ and the open-closed submatrices contain new hybrid matrix elements. The calculations employed space-frame basis functions,36 and full rotational coupling25~32was included in all distortion blocks. Three total energies (B,,,) were studied (0.731 74, 0.739 15, 0.753 94 eV) for the BKMP surface, and two total energies (0.739 15,0.741 80 eV) were studied for the LSTH surface. The resulting relative translational energies for each surface and initial state are given in Table 1. For each E,,, and surface combination, the open states are u = 0, j = 0-1 2 and u = 1, j = 0-6 for D2 a n d v = 0, j = 0-9 a n d v = 1 , j = 0-2 for HD. Table 2 lists the basis and numerical parameters used in our production runs (set A) as well as those used in the convergence checks presented here (set B). (The notation used in this table has been explained previously.26) The two parameter sets differ in a sufficient subset of parameters to fully test for convergence. The calculations decouple in the total angular momentum, J , as well as the parity, P, and the permutation symmetry, S. Each JPS block was solved separately, and the largest JPS block of the production set involved 543 coupled channels, 845 energydependent basis functions, and 2748 energy-independent basis functions. Wedenotestate-to-statedistinguishable-atomreaction probabilities by
where SZ,,.,,.is a scattering matrix element, a is an arrangement label (a = 1 for H + D2, and a = 2 and 3 for the two identical H D + D arrangements), 1 is the orbital angular momentum quantum number, and primes denote final values. Table 3 gives selected transition probabilities for J = 0 and 10 for both parameter sets for the BKMP surface at Et,, = 0.739 15 eV. Convergence is excellent-the largest deviation of the results for the two parameter sets for 41 probabilities greater than 10-4 is 0.24%, and theaverage unsigned deviation for these probabilities is 0.04%. The integral cross sections are converged to within 0.1% by JmaX = 18, but differences of up to 30% are still seen for the differential cross sections in the forward hemisphere when J,,, = 18 results are compared to results calculated with Jmax = 16. Therefore, all cross section calculations include contributions from 0 1994 American Chemical Society
1054 The Journal of Physical Chemistry, Vol. 98. No. 4, 1994 TABLE 1: Relative Translational Energies for the Initial States Studied
Letters
-
TABLE 3: State-to-State Transition Probabilities for J = 0 and 10 for the reaction H + D2(FO, J D + HD(‘Y = O j ?
E (eV)
surface
Ersl(j=O)
Ercl(j=l)
Er&=2)
J
J
i’
set A
0.739 15 0.741 80 0.731 74 0.739 15 0.753 94
LSTH LSTH BKMP BKMP BKMP
0.547 36 0.55000 0.54000 0.547 41 0.562 20
0.539 94 a a 0.540 00 0.554 79
0.525 15 a a 0.525 21 0.540 00
0
0
0 1 2 3 4 5 6 7 8 sum 0 1 2 3 4 5 6 7 8 sum 0 1 2 3 4 5 6 7 8 sum 0 1 2 3 4 5 6 7 8 sum 0 1 2 3 4 5 6 7 8 sum 0 1 2 3 4 5 6 7
2.18(-2) 6.34(-2) 8.17(-2) 5.95 (-2) 2.26(-2) 3.60(-3) 2.09(4) 1.22(-5) 3.50(-7) 2.53(-1)
2.18(-2) 6.34(-2) 8.18 (-2) 5.96(-2) 2.27(-2) 3.61(-3) 2.09(-4) 1.23(-5) 3.47(-7) 2.53(-1)
0.01 0.02 0.03 0.03 0.04 0.07 0.04 1.06 0.81 0.02
6.24(-2) 1.63(-1) 1.83(-1) 1.18(-1) 4.08(-2) 5.95(-3) 3.21(-4) 2.13(-5) 5.63(-7) 5.74(-1)
6.24(-2) 1.63(-1) 1.83(-1) 1.18(-1) 4.09 (-2) 5.95(-3) 3.22(4) 2.15(-5) 5.59(-7) 5.75(-1)
0.05 0.01 0.01 0.03 0.06 0.04 0.24 0.88 0.72 0.01
9.40(-2) 2.03(-1) 1.69(-1) 7.77(-2) 1.97(-2) 1.99(-3) 7.95(-5) 9.25(-6) 1.62(-7) 5.66(-1)
9.40(-2) 2.03(-1) 1.69(-1) 7.77(-2) 1.97(-2) 2.00(-3) 7.93(-5) 9.34(-6) 1.56(-7) 5.66(-1)
0.02 0.00 0.01 0.01 0.00 0.12 0.23 0.99 4.34 0.00
8.96(-3) 1.9 1(-2) 1.6 l (-2) 8.62(-3) 3.57 (-3) 1.04(-3) 1.85(-4) 1.52(-5) 4.66(-7) 5.76(-2)
8.97(-3) 1.92(-2) 1.61(-2) 8.61(-3) 3.57(-3) 1.04(-3) 1.85(-4) 1.52(-5) 4.72(-7) 5.7 6 (-2)
0.04 0.04 0.00 0.03 0.02 0.05 0.10 0.06 1.38 0.02
8.71(-3) 1.93(-2) 1.73(-2) 9.82(-3) 3.96(-3) 1.04(-3) 1.65(-4) 1.34(-5) 4.21(-7) 6.03(02) 7.9 1(-3) 1.85(-2) 1.82 (-2) 1.13(-2) 4.53(-3) 1.07(-3) 1.40(-4) 1.08(-5) 3.59(-7) 6.17(-2)
8.72(-3) 1.93(-2) 1.73(-2) 9.81(-3) 3.96(-3) 1.04(-3) 1.65(-4) 1.34(-5) 4.26(-7) 6.03(02) 7.91(-3) 1.85(-2) 1.82(-2) 1.13(-2) 4.53(-3) 1.07(-3) 1.40(-4) 1.08(-5) 3.63(-7) 6.17(-2)
0.05 0.03 0.02 0.04 0.01 0.07 0.10 0.16 1.15 0.01
Not calculated
TABLE 2: Parameter Sets set A
set B
H+D2
HD+D
H+D2
HD+D
12 12 12 10 8
12 11 10 9 8
2.35 0.33 1.155 6 50 30 50 18 219 13 8 1.o 17.0 0 0.9 30 27 7 0 6 100
2.45 0.33 1.155 7 50 30 50 18 205 13 8 1.o 16.0 0 0.9 30 25 7 0
14 14 14 12 10 8 2.185 0.33 1.056 7 80 60 80 19 285 13 8 0.5 19.0 0 0.9 40 35 7 0 8 120 7 8 7
13 12 11 10 9 8 2.285 0.33 1.056 8 80 60 80 19 285 13 8 0.5 18.0 0 0.9 40 35 7 0
5
6 5
total angular momenta up to J,,, = 22. The convergence of the differential cross sections with respect to J,,, was checked for one set of initial conditions ( u = j = 0 on the BKMP surface with E,,) = 0.54 eV) by comparison of J,,, = 22 results to results with Jmax = 24. All differential cross sections withj’s 6 were converged to within 1.7%, and all differential cross sections greater than 10“ A2/sr were converged to within 0.4%. For J,,, = 22 the integral cross sections are all converged to better than six significant figures with respect to Jmax. In Table 4 we compare our LSTH integral cross sections for the initial state u = j = 0 and Ercl= 0.55 eV to the results of ref 17. The level of agreement is quite satisfactory-and is especially noteworthy for the smaller cross sections, which are always hard to converge. 3. Results and Discussion
Figure 1 displays state-to-state integral cross sections on the BKMP surface for a given total energy (Etot= 0.739 15 eV) for initial states u = O j = 0-2. Figure 2 displays the same results except at a fixed relative translational energy (&I = 0.54 eV). Raising j from 0 to 2 decreases the integral cross section u(j’=O) by 21% in Figure 1 and by 26% in Figure 2. For j f = 1-4, the initial rotational value is seen to have only a small effect, less
1
2
10
0
1
2
8
sum
set
B
% diff
0.04 0.03 0.01 0.02 0.01 0.01 0.12 0.15 0.97 0.01
than or equal to 1275, on the cross sections with final rotational state specified when calculations are compared at a given total energy, whereas changes in the initial rotational angular momentum j a t fixed Erel are more pronounced, up to 32%. For j’ = 5-9, raisingj from 0 to 2 at fixed Etotdecreases 00”) by 1632%, whereas the same change in initial j at fixed Erclincreases uo”) by 35-47% for j‘ = 5-7, by a factor of 2.3 for j ‘ = 8, and by a factor of 5.0 for j f = 9. Figures 3a, 4, and 5 give quantal state-to-state differential cross sections for initial states u = 0, j = 0-2 on the BKMP surface at Ersl= 0.54 eV. Figure 3b displays the quasiclassical
Letters
The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 1055 0.10~
I
I
I
I
I
I
0.040
I
5 0.030
f
0.025
1
;0.020 2
0.00
..#-
i
-
2 0.010
1
0
0.015
Y!
E 0.005
o.ooooL
I
2
scattering angle (deg)
-% 0.0351
0.040~1
I
I
I
I
I
E
-O-j=1 +j=2
8
-
.c
-
I
I
I
I
I
‘
‘
4
- - -j‘ = 3 =4 -i’=5
0.025:
--Y
-
0.0207
-g
-
I
-.-j’=1
.-5
-O.j=O
’
----i‘ = 2
4, 0.030!
I
I
’
-j’=O
‘N
0.10
I
I
0.0157 0.010’
a,
1
5
0.0051
0.000, L
45
i 0.00
0
I
1
1
2
1
3
4
5
-
6
”
4 . 4 .
7
“
8
9
j,
-+
+
Figure 2. State-to-state integral cross sections for the reaction H D2(u=Oj=O-2) D HD(u’=Oj? at 0.54 eV on the BKMPsurface.
-
+
TABLE 4 Integral Cross Sections for the H Dz( v=j=O) HD( v’=Oj’) + D Reaction on the LSTH Surface if present ref 17 % diff 0 1 2 3 4 5 6 7 8
sum
3.82 (-2) 8.57(-2) 8.17(-2) 5.04(-2) 2.03(-2) 4.65(-3) 5.61(4) 3.09(-5) 5.15(-7) 2.84(-1)
3.85(-2) 8.66(-2) 8.23(-2) 5.06(-2) 2.03(-2) 4.65(-3) 5.61(-4) 3.08(-5) 5.21(-7) 2.82(-1)
-
180
+
Figure 3. State-testate differential cross sections for the reaction H D2(u=Oj=O) D HD(u’=Oj? at Ercl= 0.54 eV: (a, top) present
+
results on the BKMP surface, (b, bottom) QCT resultsI2on the LSTH surface. 0.040 .
2
0.035
0.8 1.1 0.7 0.4 0.0 0.0 0.0 0.3 1.2 0.7
results of Aoiz et a1.I2for state-to-state differential cross sections on the LSTH surface for u = j = 0 at Erel = 0.54 eV. Table 5 gives the positions of the maxima in the DSCs for the quantal and quasiclassical results at E,,I = 0.54 eV. (Note that all scattering angles in this paper are defined in the center-of-mass frame and refer to the angle between the final direction of HD and the initial direction of H.) The quantal DCSs are backward peaked with maxima typically located at or near scattering angles 0 of 180°, and only slight differences are observed as a function of the intial rotational state. For jf > 0, there are marked differences from the quasiclassicall* results where the DCSs show peaks as much as 27 degrees more forward than the quantal results and show considerablevariation as a function of the initial state. The differential cross sections from the QCT simulations show the largest differences from the quantum results for j = 0 (see Figure 3a,b for a comparison) but improve somewhat with increasing j. The referee asked for a discussion of the oscillations in the QCT differential cross sections with regard to the fact that unphysical oscillations may result in some cases from statistical sampling errors. The authors12 employed a moments expansion
90 135 scattering angle (deg)
,
-j,=O -.-Y=1
I
”
I
”
-
scattering angle (deg)
Figure 4. State-testate differential cross sections for the reaction
Dz(v=Oj=l)-D+
H+
HD(u’=Oj’)atEre~=0.54eVontheBKMPsurface.
method3’ to produce continuous differential cross sections from a discrete set of trajectories. The uncertainties in thedistribution (68% confidence interval) were provided1* at l l O o , 150°, and 175O, and they are smaller than most of the oscillations. Tests of the stability of the oscillations with respect to adding more moments were not presented in ref 12, but it would be interesting to test this stability in future work. Figure 6 shows quantum mechanical differential cross sections for the initial state u = 0, j = 1 on the LSTH surface at E,,, = 0.54 eV. By comparing the cross sections in Figure 6 to those given in Figure 4 for the same initial conditions on the BKMP surface, we can see that the two potential surfaces yield very similar results. The average unsigned difference in the locations of the maxima of the DCS is 2 O , and the largest difference is 8 O . The largest deviations between results on the two surfaces for the differential cross sections in the backward hemisphere range from 11 to 14%forj’=&2andfrom21 to42%forjf= 3-6. Maximum deviations in the forward hemisphere increase monotonically with
1056
Letters
The Journal of Physical Chemistry, Vol. 98, No. 4,1994 0.040
I
0.12
"
-
A
& 0.035
2 0.030
5
.6
H*
g 0.020
$
2
4
-0- experimental
0.10
l$N
N '
0.015
..... quasiclassical
0.08 0.06
.-5 0.04 4-
P
8
0.02
'0
0.00
0
45
Figure 5. State-to-state differential cross sections for the reaction H + D2(u=Oj=2) D + HD(u'=Oj? at E,,! = 0.54 eV on the BKMP surface.
-.
0.040
5 '
0.035
N
5 0.030
s Pu?
'= .-"
5
,
I
-i'=O
"
l
-.-I,= 1 - - --i' = 3 --i' = 4
0.020
'.
- -i'=5
-r=s
0.015
BKMP BKMP BKMP
0.731 74 0.739 15 0.753 94
LSTH LSTH
0.739 15 0.741 80 0.731 74 0.739 15 0.753 94
45
135
90
180
scattering angle (degj
Figure 6. State-to-state differential cross sections for the reaction H + Dz(u=Oj=l)-D+HD(u'=Oj? atE,,l= 0.54eVon theLSTHsurface.
TABLE 5: Scattering Angle Locations (in degrees) of the Maxima in the State-to-State Differential Cross Sections for the Reaction H + Dz( Y = 0.11 D HD( v'=O$) for Quantal and Quasiclassicall Calculations at Eel = 0.54 eV
-
0 1 2 3 4 5 6
180 180 180 167 154 145 138
134.7 134.4 133.7
133.9 133.3
133.2 132.5
Rotational Quantum Number j'
0.005
j = O
180
Figure 7. Quantum mechanical and quasiclassical12differential cross sections compared to the experimental results of ref 5.
0.010
j'
135
TABLE 6 First Moments of the Scattering Angle and HD Rotational Ouantum Number surface Eiot j=O j = 1 j = 2 Scattering Angle (deg) 0.739 15 135.0 134.6 133.8 LSTH 0.741 80 134.9 LSTH
"
'-i'=2
0.025
90 scattering angle (deg)
scattering angle (deg)
+
QCT
quantal j=1
j=2
j=O
j=1
j=2
180 180 180 168 154 144 137
180 180 180 174 153 143 137
180 165 153 142 128 128
180 180 156 145 134
180 180 180 155 138
j'and range from 31% for j'= 0 a t 6 = 0' to 59% for j ' = 6 at 6 = 28'. Thus, quantitative experimental state-resolved differential cross-sections in the forward hemisphere, if available, could distinguish between these two very similar potential energy surfaces. Table 6 gives the'first moments of the scattering angle and the product rotational distributions for all the initial conditions studied. Table 7 compares the first moments of the scattering angle and product rotational distributions for the BKMP results a t Ere]= 0.54 eV to QCT calculations and experimentalS measurements. The earlier trajectory simulations reported by Blais and one of the authorsll involved a thermal distribution of initial states, and 19% of the trajectories h a d j 2 3. Despite the differing rotational distributions, their results of (j') = 2.27 and (e) = 135' are quite close to the present results and otherI2QCT calculations. The maxima in the quantal DCSs for a given j ' vary by 2' or less as a function of the initial j except for the very broad peak for j' = 3, which differs from j = 0 to j = 2 by 7'. The maxima in the quasiclassical DCSs vary by up to 27' as a function of the initialj and agree better with the quantal results as j increases.
BKMP BKMP BKMP
1.778 1.805 1.847 1.927 2.094
1.821
1.891
1.949 2.088
1.985 2.089
TABLE 7: First Moments at & I = 0.54 eV for Various Potential Surfaces, Methods, and Initial Rotatio~lStates quantity surface method j = 0 j = 1 j = 2 expmix' (O),deg BKMP quantum 135 134 132 134 LSTH QCT 135 135 135 135 experiment 119 U') BKMP quantum 1.85 1.95 2.09 1.93 LSTH QCT 2.2 2.2 2.3 2.2 a 2:l:l ratio of j = 0, 1, and 2, respectively. Figure 7 compares the reaction product imaging differential cross sections of ref 5 to the QCT results of ref 12 and the quantal results for the BKMP surface. In order to simulate the experimental conditions, the results in this plot are averaged over the initial states with weightings 2:l:l for j = 0, 1, and 2, respectively, and summed over all final states. The experimental results were normalized to give the same integral cross section as the quantal results. We note that the QCT resultsI2 were incorrectly plotted in the comparison made in ref 5, but we obtained the correct values from the authors. The sharp dip in the experimental data near 180' can most likely be attributed to difficulties in the image reconstruction process5 rather than to a physical effect. Overall, the three sets of results are in fair agreement, but the experiment shows measurable signal for forward scattering-a finding that is not supported by either theoretical calculation. Examination of the raw data, however, indicates that the forward scattering seen experimentally for 6 5 45' is within the noise level of the e ~ p e r i m e n t .When ~ ~ only the backward hemisphere (8 2 90') is considered, the experimental (6) becomes 134' and thequantal result is 137'. Thus, the 15' discrepancy of theory from experiment in Table 6 is almost entirely attributable to 6 < 90'. 4. Conclusions
We have presented accurate quantal calculations on two potential energy surfaces for several initial states and relative
Letters translational energies. The agreement between integral cross sections and backward-hemisphere differential cross sections calculated for different surfaces is quite good, and fair agreement is also seen when comparing the present results to integral cross sections and rotationally summed differential cross sections from trajectory simulations. The former good agreement does not persist for the small forward-hemisphere state-to-state differential cross sections, and the latter agreement does not persist even for the larger backward-hemisphere state-to-state differential cross sections, for which the quasiclassical simulation12 shows qualitatively different trends than those observed in the present study. The locations of the maxima in the quasiclassical state-to-state differential cross sections differ by as much as 27’ from the quantal results. There is also a significant discrepancy with experiment in that we calculate a forward differential cross section that is 3 orders of magnitude smaller than the backward peak, whereas experimentally this ratio is only 1 order of magnitude. The average scattering angle in the experiment is 15’ less than the theoretical one, primarily due to differences in the 45-90’ scattering angle range.
Acknowledgment. We thank the authors of ref 12 for providing tabulations of their QCT results and D. W. Chandler for helpful discussions. This work was supported in part by the National Science Foundation and the Minnesota Supercomputer Institute. Supplementary Material Available: Tables of differential cross sections tabulated at 4 O intervals for j ’ I 6 for all sets of initial conditions given in Table 1 and of integral cross sections tabulated for these sets of initial conditions for j ’ l 8 (1 1 pages). Ordering information is given on any current masthead page. References and Notes (1) Gerrity, D. P.; Valentini, J. J. J . Chem. Phys. 1983,83, 1605; 1984, 81, 1298; 1985, 82, 1323; 1985, 83, 2207. Veirs, D. K.; Rosenblatt, G. M.; Valentini, J. J. J . Chem. Phys. 1988,83, 1605. Levine, H. B.; Phillips, D. L.; Nieh, J.-C.; Gerrity, D. P. Chem. Phys. Lett. 1988, 143, 317. Valentini, J. J.; Phillips, D. L. Bimolecular Collisions; Ashford, M. N., Baggott, J. E., Eds.; Royal Society of Chemistry: London, 1989; p 1 . (2) Rettner, C. T.; Marinero, E. E.; Zare, R. N. In Physics of Electronic and Atomic Collisions. Invited Papers from the XVIIIth ICPEAC, Berlin, July 27-Aug 2, 1983; Eichler, J., Hertel, I. V., Stolterfoht, N., Eds., North-Holland: Amsterdam, 1984; p 51. Marinero, E. E.; Rettner, C. T.; Zare, R. N. J . Chem. Phys. 1984, 80, 4141. Blake, R. S.; Rinnen, K.-D.; Kliner, D. A. V.; Zare, R. N. Chem. Phys. Lett. 1988, 153, 365. Rinnen, K.-D.; Kliner, D. A. V.; Blake, R. S.; Zare, R. N. Chem. Phys. Lett. 1988, 153,371. Rinnen, K.-D.; Kliner, D. A. V.; Zare, R. N. J . Chem. Phys. 1989, 91,7514. Adelman, D. E.; Xu,H.; Zare, R. N. Chem. Phys. Lett. 1993,203, 573. (3) Johnson,G. W.; Katz, B.;Tsukiyama,K.;Bersohn,R.J.Phys. Chem. 1987, 91, 5445. (4) Schnieder, L.; Seekamp-Rahn, K.; Liedeker, F.; Steuwe, H.; Welge, K. H. Faraday Discuss. Chem. SOC.1991, 91 259. (5) Kitsopoulos, T. N.; Buntine, M. A,; Zare, R. N.; Chandler, D. W. Science 1993, 260, 1605. (6) Schatz, G. C. Chem. Phys. Lett. 1984, 108, 532. (7) Connor, J. N. L.; Southall, W. J. E. Chem. Phys. Lett. 1984,108, 45. Connor, J. N. L.; Southall, W. J. E. Chem. Phys. Lett. 1986,123, 139. (8) Bowers, M.S.;Choi, B.H.;Poe,R.T.;Tang,K.T. Chem. Phys. Lett. 1985, 116, 239. (9) Suck, S. H.; Klein, C. R.; Lutrus, C. K. Chem. Phys. Lett. 1984,110, 112.
The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 1057 (10) Baram, A,; Last, I.; Baer, M. Chem. Phys. Lett. 1993, 212, 649. (11) Blais, N. C.; Truhlar, D. G. Chem. Phys. Lett. 1983, 102, 120. (12) Aoiz, F. J.; Herrero, V. J.; Puentedura, 0.;Sbez Rdbanos, V. Chem. Phys. Lett. 1992, 198, 321. Aoiz, F. J. Private communication. (13) Baer, M. Chem. Phys. 1988, 123, 365. Last, I.; Ron, S.; Baer, M. Isr. J. Chem. 1989, 29, 451. (14) Webster, F.; Light, J. L. J . Chem. Phys. 1986,85,4744. Webster, F.; Light, J. L. J. Chem. Phys. 1989, 90, 300. (15) Zhao, M.; Truhlar, D. G.; Blais, N. C.; Schwenke, D. W.; Kouri, D. J. J. Phys. Chem. 1990, 14, 6696. (16) Last, I.; Baram, A,; Szighman, H.; Baer, M. J . Phys. Chem. 1993, 97, 7040. (17) DMello, M.; Manolopoulos, D. E.; Wyatt, R. E. J . Chem. Phys. 1991, 94, 5985. (18) Liu, B. J. Chem. Phys. 1973, 58, 1924. Siegbahn, P.; Liu, B. J. Chem. Phys. 1978,68,2457. Truhlar, D. G.; Horowitz, C. J. J . Chem. Phys. 1978, 68, 2466; 1979, 71, 1514 (E). (19) Boothroyd, A. I.; Keogh, W. J.; Martin, P. G.; Peterson, M. R. J . Chem. Phys. 1991, 95,4343. (20) Schlessinger,L. Phys. Rev. 1968,167,1411; 1968,171,1523. Pieper, S. C.; Wright, J.; Schlessinger, L. Phys. Rev. D 1971, 3, 1419. (21) McDonald, F.A.;Nuttall, J. Phys. Reu.Lett. 1969,23,361. Nuttall, J.; Cohen, H. L. Phys. Rev. 1969, 188, 1542. (22) McCurdy, C. W.; Rescigno, T. N.; Schneider, B. I. Phys. Rev. A 1987, 36, 2061. (23) Sun, Y.; Kouri, D. J.; Truhlar, D. G.; Schwenke, D. W. Phys. Rev. A 1990,41, 4857. (24) Sun, Y.; Kouri, D. J.; Truhlar, D. G. Nucl. Phys. A I990,508,41c. (25) Schwenke, D. W.; Mielke, S. L.; Truhlar, D. G. Theor. Chim. Acta 1991, 79, 241. (26) Mielke, S. L.; Truhlar, D. G.; Schwenke, D. W. J . Chem. Phys. 1991, 95, 5930. (27) Tawa, G. J.; Mielke, S. L.; Truhlar, D. G.; Schwenke, D. W. In Advances in Molecular Vibrations and Collision Dynamics-Quantum Reactive Scattering, Vol. 2B; Bowman, J. M., Ed.; JAI: Greenwich, CT, in press. (28) Sun, Y.; Kouri, D. J. Chem. Phys. Lett. 1991, 179, 142. (29) Kohn, W. Phys. Rev. 1948, 74, 1763. (30) Newton, R. G. Scattering Theory of Particles and Waves, 1st ed.; McGraw-Hill: New York, 1966; Section 11.3. (31) Schwenke, D. W.; Haug, K.; Truhlar, D. G.; Sun, Y.; Zhang, J. Z. H.; Kouri,D. J.J. Phys. Chem. 1987,91,6080. Schwenke, D. W.; Mladenovic, M.; Zhao, M.; Truhlar, D. G.; Sun, Y.; Kouri, D. J. In Supercomputer Algorithms for Reactivity, Dynamics, and Kinetics of Small Molecules; Lagan& A., Ed.; NATO AS1 Series C, Vol. 277; Kluwer: Dordrecht, The Netherlands, 1989; pp 131-168. Sun, Y.; Yu, C.-h.; Kouri, D. J.; Schwenke, D. W.; Halvick, P.; Mladenovic, M.; Truhlar, D. G. J . Chem. Phys. 1989,91, 1643. Schwenke, D. W.; Truhlar, D. G. In Computing Methods in Applied Sciences and Engineering; Glowinski, R., Lichnewsky, A., Eds.; Society for Industrial and Applied Mathematics: Philadelphia, 1990; pp 291-310. (32) Schwenke, D. W.; Haug, K.; Zhao, M.; Truhlar, D. G.;Sun, Y.; Zhang, J. Z. H.; Kouri, D. J. J . Phys. Chem. 1988, 92, 3202. (33) BaEiC, Z.; Simons, J. J . Phys. Chem. 1982,86, 1192. Hamilton, I. P.; Light, J. C. J . Chem. Phys. 1986, 84, 306. (34) Truhlar, D. G.; Abdallah, J.; Smith, R. L. Adv. Chem. Phys. 1970, 18, 70. (35) Miller, W. H.; Jansen op de Haar, B. M. D. D. J . Chem. Phys. 1987, 86, 6213. Zhang, J. 2.H.; Chu, %-I.; Miller, W. H. J . Chem. Phys. 1988, 88, 6233. (36) Arthurs, A. M.; Dalgarno, A. Proc. R . SOC.London, A 1960, 256, 540. Lester, W. A., Jr. In Dynamics of Molecular Collisions, Part A ; Miller, W. H., Ed.; Plenum Press: New York, 1976; p 1. (37) Truhlar, D. G.; Blais, N. C. J . Chem. Phys. 1977,67,1532. Truhlar, D. G.; Muckerman, J. T. In Atom-Molecule Collision Theory; Bernstein, R. B., Ed.; Plenum: New York, 1979; p 505. Kosmas, A.; Gislason, E. A.; Jorgensen, A. D. J. Chem. Phys. 1981,75,2884. Budenholzer, F. E.; Hu, S. C.; Jeng, D. C.; Gislason, E. A. J. Chem. Phys. 1988,89, 1958. Aoiz, F. J.; Herrero, V. J.; Rbbanos, V. S. J. Chem. Phys. 1991, 94, 7991. (38) Chandler, D. W. Private communication.
