J . Phys. Chem. 1994, 98, 7994-7999
7994
New Reduced Dimensionality Calculations of Cumulative Reaction Probabilities and Rate Constants for the H H2 and D + H2 Reactions
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Desheng Wang and Joel M. Bowman' Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322 Received: February 14, 1994; In Final Form: April 27, 1994'
Several refinements to our reduced dimensionality/adiabatic bend theory of reactive scattering are presented and applied to the H H2 and D H2 reactions. First, the harmonic treatment of the three-atom bending energy is replaced by an accurate numerical calculation of the adiabatic bending energies. Second, the standard J-shifting approximation for linear transition states is replaced by an adiabatic calculation of the overall rotational energy. In addition, the J-shift approximation is reanalyzed for anharmonic bending motion of the transition state, and an obvious improvement is suggested. Cumulative reaction probabilities and thermal rate constants are calculated for H H2 and D H2 and compared to accurate quantum ones for selected values of the total angular momentum. Based on these comparisons and previous calculations it is concluded that the new treatment of the adiabatic bending energy is a more significant refinement of the reduced dimensionality theory than is the adiabatic treatment of overall rotation. The thermal rate constant is calculated for D H2 using the new theory and compared with other previous calculations, a recent full-dimensional accurate one up to 900 K and with experiment up to 2000 K.
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I. Introduction The exchange reaction of the isotopic systems H + HZ has been the subject of intensive experimental and theoretical studies for many years. It is one of the simplest systems which can be described theoretically in essentially an exact fashion, beginning with the potential energy surface to the quantum dynamics of the nuclear motion. Exact results for this reaction play the role of benchmarks against which simple and more general approximation methods can be tested. Over the past decade, several approximate theories of reactive scattering have been developed in which more than one but less than all the degrees of freedom are fully coupled. Thus, for an atom-diatom reaction the problem reduces to a two-mathematical dimensional one with effective potentials. We refer to these theories as reduced dimensionality theories to emphasize the reduction in treating all degrees of freedom exactly. Therefore, in principle the possible sources of error associated with the theories are the approximations introduced by the theories. One of these reduced dimensionality approaches is based on the adiabatic bending approximation,in which the bending degrees of freedom are treated adiabaticaly.1-8 There are several variants of this particular approximation that have used efficient and reasonably accurate approximations to obtain the bending energy. In the simplest version of the theory,lq2 the reaction probability is calculated for ground bend state, and the reaction probability for the nth excited bend state at the total energy E is obtained from the ground bend probability evaluated at the energy, E E;, where E: is the energy of nth bend state of the transition state referenced to the ground-state bending energy. A more sophisticated version of the theory has been given and applied to the D + H2 and H + D2 reactions,9JOin which the reaction probabilities are explicitly calculated for each bending state including all excited states, for zero angular momentum, J. In both these approaches, however, the bending energies are evaluated in the harmonic approximation over the grid of internuclear bond lengths for the which the potential is a minimum with respect to the internal angular degrees of freedom. The other major approximation which is used often in reduced dimensionality theories is the J-shift approximation.l-4JJ,l1-13In 0
Abstract published in Advance ACS Abstracts, June 1, 1994.
this approximation the reaction probability for J > 0 at the total energy is obtained from J = 0 probability calculated at the energy E - EjK, where EJK is the overall rotational energy at the transition state. As has been pointed out by several authors, including ourselves, the J-shift approximation can be regarded as the molecular version of the modified wave number approximation of inelastic scattering.I4 (That approximation was introduced within thecontext of the distorted wave approximation of inelastic scattering in which the centrifugal potential is replaced by a constant.) For transition states that are nonlinear the rotational energy depends on J , and for symmetric tops K,the projection of J on the top symmetry axis. For linear transition states the rotational energy depends only on J , as is very wellknown. As shown by Sun et al.," for linear transition states the J = 0 probability contains contributions from the even bending states only, whereas for J > 0 both even and odd bending states contribute to the reaction probability. Sun et al. suggested a simple, approximate method to include the contribution of odd bending states when J-shifting is done with J = 0. This reduced dimensionality theory was applied recently to a calculation of the rate constant for the D + H2 and H + D2 reactions and compared to experimentlo over a very wide temperature range, 167-1979 K for D + Hz and 256-206 1 K for H Dz. Very good agreement with experiment was found over the entire temperature range for the two reactions. The very good agreement found at the higher temperatures was brought into question by a subsequent full dimensional calculation of the rate constant for D + Hz by Park and Light (on a slightly different potential energy surface).*S Their results are in good agreement with experiment and reduced dimensionality calculations up to roughly 900 K, but at higher temperatures their calculations are significantly below both experiment and the reduced dimensionality calculations. At 1500 K, the highest temperature they considered, their rate constant is roughly 50% of the measured and reduced dimensionality ones. Partly as a result of this situation, we decided to extend the reduced dimensionality theory in two ways and to recalculate the rate constant for D + H2. First, we treat the bending degrees of freedom numerically, using the full bending potential at a given value of the internuclear coordinates. Second, we treat the overall rotational energy adiabatically at each nuclear configura-
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0022-3654/94/2098-7994%04.50/0 0 1994 American Chemical Society
Calculations of Reaction Probabilities and Rate Constants tion. For a linear minimum configuration, which pertains to the H3 system, the adiabatic rotational energy is given simply by BJ(J+Z), where B is the usual linear-molecule rotation constant evaluated over the minimum energy space of the three-atom system. In addition, we reanalyze the J-shift approximation for anharmonic bending motion and suggest improvements to the approximation. It is of course necessary to test new approaches whenever possible, and that is one of the main points of this paper. Specifically, the cumulative reaction probability (CRP) is calculated for J = 0 and 4 for the H H2 reaction on the DMBE potential16 and compared to exact quantumcalculations of Truhlar and CO-workers.17-18The thermal rate constant for J = 0 is also calculated and compared with exact quantum calculations for H H2.18J9 CRPs are also calculated for D H2 for J = 6 and 9, using the DMBE surface, and compared to very recent exact calculations.20 Thermal rate constants are also calculated for D + H2 for J = 0 and 6 and compared with previous full dimensionality quantum calculations of Park and LightIs (which were done on the LSTH p ~ t e n t i a l ~ l -and ~ ~ )also to very recent exact calculations done on the DMBE potential.20 In addition to testing the above refinements to the reduced dimensionality theory, the thermal rate constant for the D H2 reaction is calculated and compared to the previous one of Park and Light and very recent full dimensional ones of Mielke et aL20up to 1500 and 900 K, respectively. Comparisons with experiment are also made up to 2000 K. This comparison with experiment was made earlier using the J-shift approximation and the harmonic treatment of the bend.10 Finally, a test of an extension to the J-shift approximation is also made. This extension is similar in one respect to an extension of the J-shift approximation that has been recently suggested by Truhlar and co-w0rkers.2~ [While this paper was in review, a test of the J-shift approximation was reported by Auerbach and Miller,zs for rate constants for the D + H2 (u = 1) reaction using the LSTH surface. They reported that the J-shift approximation was within 12% of their exact calculation for rotationally state-selected rate constants except for the state j = 1 where the approximation was poor. For the fully averaged rate constant the J-shift approximation was within 4% of their exact results at 310 K and "nearly exact" (to quote them) up to 700 K and "semiquantitative" (again quoting) up to 1000 K.] The paper is organized as follows. Section I1 presents the theoretical formulation and the details of the calculations. Section I11 presents results and discusses the calculations and comparisons with previous calculations and experiment. Section IV gives a summary and conclusions.
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11. Theory and Calculations
In the reduced dimensionality theory of reactions with a linear transition state, the bending quantum number n = 0, 1 , 2... and the projection quantum number of J o n the body-fixed z-axis, 52, are assumed to be conserved. For linear transition states the quantum numbers Q and K are identical, and we will use 52 throughout in this paper. Thus the effective Hamiltonian for an A BC reaction in the center of mass, mass-scaled coordinate system can be written ads2
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yy=--h2( -+a2 2P
aR2
);;
+ V(R,r,y = 0) +)',@,ne
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where
is the reduced mass, R is the mass-scaled distance from A to the center of mass of BC, r is the mass-scaled BC internuclear distance,
The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 7995 and y is the angle between the two vectors R and r. EJ(R,r) is the rotational energy, which is evaluated over the space of the two internuclear distances rAB, rBC, and transformed to R and r. For the H3 system the minimum energy configuration is a linear one, so the transformation to R and r is trivial, and the rotational energy is given by
E,(R,r) = B(R,r)J(J + 1)
(3)
whereB(R,r) is therotationalconstantfor Randr. Inour previous work132 the rotational energy Ej(R,r) was given in the so-called centrifugal sudden (CS) approximation, which on simple theoretical grounds is not expected to be as accurate as eq 3. This is because the adiabatic approximation is made in the principal axis system, whereas the CS approximation is not.26 This was verified recently in numerical calculations on HC0,Z7where the adiabatic treatment of rotation was shown to be more accurate than the CS treatment of rotation. Note that the adiabatic treatment of the rotational energy is quite similar in spirit to earlier semiclassical work by McCurdy and Millerz8 and also to the treatment of rotation in the "reaction Hamiltonian" of Carrington and Miller.29 Also, note that the J-shift approximation is obtained by replacing the rotational energy Ej(R,r) by the constant rotational energy a t the transition state, B*J(J+Z). In the present work, the adiabatic bending energies are obtained by solving the following equation
subject to the boundary condition
where Gee is the Wilson G-matrix element, 0 is the ABC bond angle, tS(rAB,rBC,e)is the adiabatic bending wave function, and E$(rAB,r.BC) is the bending eigenvalue. For a linear minimum energy configuration, as pertains to H3, R and r are given trivially in terms of and rBC. In addition the bending energy is doublydegenerate, and thus the total bending energy is the sum eS + E$ 0. Vibrational state-to-state reaction probabilities are calculated with this approach for a given bend state (n,IQl) and total angular momentum J at the total energy E. These state-to-state probabilities are summed over initial and final vibrational states to yield partially cumulative reaction probabilities denoted q$E;n), which are then used to obtain the full cumulative reabtion probability as described below. Thermal Rate Constant. The thermal rate constant for an A BC reaction can be expressed as the thermal trace of the cumulative reaction probability, N ( E ) (which has also been denoted as P ( E ) ) , as follows
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The Journal of Physical Chemistry, Vol. 98, No. 33, I994
where E is the total energy, h is Planck's constant, k~ the Boltzmann constant, and Z ( is the total atom-diatom electronictranslational-rotational-vibrational partition function. The CRP, can be written as
TABLE 1: Thermal Rate Constants H(T)for H + Hz, for J = 0, on the DMBE Surface this work exact quantuma T (K) 200 300 400 600 700 1000
(II
N ( E ) = C(2J+ l)NJ(E)
(7)
J=O
where the partial wave cumulative probability " ( E ) is defined by
where ejedYn*(E) is the exact state-to-state reaction probability between the initial and final vibrational/rotational states ( u j , i l ) and (u'j',il'), respectively. The bending quantum numbers n and 1ilI of the triatomic complex are assumed to be conserved in the adiabatic-bend theory, the relationship ]ill I n continues to hold, and thus, the reduced dimensionality approximation to the partial wave CRP is given by (9)
where
with the restriction that Iill 5 n, in steps of two. This latter restriction is not an approximation, although it holds also for the harmonic model for the bending motion. This will be made clear by an example which we give in section 111. To summarize, the present calculation requires reaction probabilities for each J and I!2l and for a corresponding set of bending states (n,Iill). This is fairly compute-intensive, however, considerably easier than an exact calculation. One approximation that we have used in the past which greatly simplifies the calculation is the J-shift approximation.l4JI-l3 We review that next and suggest an obvious extension to it, somewhat along the lines also suggested recently by Truhlar and c o - ~ o r k e r s . ~ ~ J-Shift Approximation. The J-shift approximation has been described in detail el~ewhere.'.~Jl-l3The approximation gives the reaction probability for J greater than zero and a t the total energy E by the reaction probability for J = 0 a t the energy E minus the rotational energy of the transition state. For linear configurations this energy is B*J(J+ I), where B* is the rotation constant of the transition state. Obviously the approximation applies to any reaction probability; however, we focus here on the CRP. In our previous applications of the adiabatic treatment of the bend together with J-shifting, the harmonic approximation was assumed for the bending energies. This approximation led to significant simplifications in the theory for linear transition states, because the bending energy does not depend on In1in the harmonic approximation. Even within that approximation it was recognized that J = 0 probabilities contain contributions only for even bend states, i.e., n = even, whereas for J > 0 both even and odd bend states contribute. Thus, an additional bend energy-shift approximation was introduced to account for the odd bend states as we1l.l' Thus, in eq 12, t $ E ) was approximated by11
where is the bending energy of the transition state measured relative to the energy of the ground bend of the transition state. A better approximation is to shift from J = 1 since this probability contains contribution from all bend states. Thus, the
6.8(-20) 9.3(-18) 1.4(-16) 2.1(-15) 4.5(-15) 1.6(-14)
6.43(-20) 8.51(-18) 1.29(-16) 1.99(-15) 4.25(-15) 1.56(-14)
Chatfield et al., ref 18 and Day and Truhlar, ref 19.
TABLE 2 Numerical and Harmonic (HO) Bending Energies (eV) for D + Hz at the Transition State numerical 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.1127 0.2298 0.3469
0.3554 0.4725
0.5980
0.4887 0.6058
0.7313 0.8646
0.6288 0.7459
0.8714 1.0047
1.1448
1.0177 1.1510
1.2911 1.4374
1.3031
1.5895
1.3275 1.4608
1.6009 1.7472
1.8993 2.0570
1.0442 1.1698
1.4432
1.7416
0.7751 0.8921
1.6241 1.7642
1.9105 2.0626
1.9329 2.0792
0.1084 0.2167 0.3251 0.4335 0.5418 0.6502 0.7586 0.8669 0.9753 1.0837 1.1920 1.3004 1.4088 1.5172 1.6255
The harmonic energies are n + 1-folddegenerate, and the numerical ones are 2-fold degenerate for Q > 0. new approximation is
where AEJ is the rotational energy difference relative to J = 1, i.e., B*[J(J 1) - 21. Both the previous and this new J-shift approximation will be tested below. This approach is a special case of an extension of J-shifting that Truhlar and co-workers have recently proposedOz4They suggest using a higher value of J from which to do J-shifting and noted quite correctly the advantage in including more bending states contained in the higher J-state reaction probabilities. They also use a rotation constant which is determined by a best-fit procedure to relate a C R P at a higher value of J from one at a lower J using J-shifting. It should be emphasized, however, that in those instances where a J = 0 calculation is the only feasible one, J-shifting from J = 0 will continue to be quite useful. Calculations. The potential energy surface used in the calculations is the DMBE fitI6 to ab initio calculations.21.22.16 The bending energies were obtained over a two-dimensional grid in the space of the two internuclear distances using the ColbertMiller DVR representation of the angular kinetic energy operator.30 The linear triatomic rotation constant, B, was calculated over the same grid. For each value of J and for each bending state, reaction probabilities were calculated using a modification of the collinear hyperspherical code of Schatz and c o - w o r k e r ~for , ~ ~J u p to 30. In principle, bend states with Iill up to J should be included in the summation in eq 12. However, the energy of the bend states a t and in the vicinity of the transition state increase rapidly with Iill (cf. Table 2), and the following scheme for selecting bend states gave well coverged probabilities. For J up to 15 all [ill-valueswere included, and for J greater than 15, ICtl ranged from 0 to 15. For each Iill the quantum number n varied from to nmx, where nmx = 30 for even n-states and nmax= 29 for odd n-states. This set of bend states was sufficient to obtain converged reaction probabilities up 2.0 eV. Reaction
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Calculations of Reaction Probabilities and Rate Constants
The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 7997 70.0
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E(eW Figure 1. Comparison of present (open circles) and previous reduced dimensionality(open squares)cumulativereaction probabilitieswith exact quantum results (solid curve) of Chatfield et al. (ref 18) for H + H2 as a function of the total energy E. The previous reduced dimensionality results were done with the harmonic approximation for the bend and for J = 4 with the J-shift approximation.
probabilities were calculated for 88 energies in the total energy range 0.3-2.0 eV. Thermal rate constants were evaluated by fitting the calculated values of the cumulative reaction probabilities with cubic splines and integrating eq 6 numerically. We tested other methods of integration, including fitting the logarithm of the CRP to splines. Exact diatomic vibration/rotation energies were used to evaluate the partition function, Z(7'). 111. Results and Discussion
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H H2. We calculated cumulative reaction probabilities for H + Hz on the DMBE surface for J = 0 and 4 in order to compare to the corresponding exact quantum calculations of Chatfield et ~ 1 . 1 8We also calculated reduced dimensionality CRPs using the harmonic approximation for the bend energy. For J = 4 the harmonic results were obtained using the J-shift approximation based on the harmonic-bend J = 0 CRP. The comparisons are shown in Figure 1. For J = 0 the harmonic and the numericalbend CRPs are both in very good agreement with the exact C R P for total energies below 1.OeV. Above 1.OeV both approximations are somewhat out of phase with the exact CRP; however, both remain within roughly 10%of it. For J = 4 the harmonic-bendJ-shifted CRP exceeds the exact one above 1.0 eV by 20% or more; however, the numerical-bend C R P remains in very good agreement with the exact result over the entire energy range shown. In order to determine whether the harmonic treatment of the bend or the J-shift approximation (or both) is responsible for this disagreement with the exact result at higher energies we did a J-shift calculation for t h e J = 0 reaction probability obtained with the numerical bend. The resulting CRP was very similar to the CRP shown in Figure 1 using the adiabatic rotational
0.4
0.6
0.8
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E(eW Figure 2. Linear an semilog comparison of present (open circles) and exact quantum (solid curve, from ref 20) cumulative reaction probabilities as a function of total energy E for D + Hz for J = 6.
energy. Thus, for this example, the harmonic treatment of bending motion is mainly responsible for the decreased accuracy of the harmonic-bend CRP at higher energies. A comparison of the present and exact18-19 thermal rate constants for J = 0 is given in Table 1. The agreement is very good, with the present results roughly 5-10% higher than the exact ones. D + H2. Before presenting the CRPs and rate constants for D + Hz, we show the numerical and harmonic bend energies for the DH2 transition state in Table 2. As seen, the numerical bend energies are uniformly larger than the corresponding harmonic ones. The n+l-fold degeneracy of the harmonic energies splits into two 2-fold degenerate energies (for J > 0) in the numerical case. For example, for n = 3 the 4-fold degenerate harmonic energy splits into 2-fold degenerate numerical energies with lsll = 1 and 3. As seen, the higher 2-fold degenerate numerical energies correspond to the higher Exact quantum cumulative reaction probabilities for D Hz have recently been calculated by Mielke et al.,20924 using the DMBE potential. Comparisons of the present CRPs with these exact ones for J = 6 and 9 are shown in Figures 2 and 3, respectively. As seen there is very good agreement with the exact calculations, with the approximate calculations generally slightly larger than the exact ones in the tunneling region. The thermal rate for constant J = 6 has also been calculated by Mielke et al.32 using their CRP, and, in earlier work, Park and Light15 also reported thermal rate constants for a large range of Jvalues (but on the LSTH potential). A comparison of the present results for J = 0 and 6 is made with these other calculations in Table 3. First, note the very good agreement among all the calculations for J = 0. For J = 6 the present results are within 20% of the exact ones at the lower temperatures and within 10% or less a t
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799% The Journal of Physical Chemistry. Vol. 98, No. 33, 199’4
Wang and Bowman 10’1
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TABLE 3: Thermal Rate Constants kJ( = 0 and 6
1.2
1.4
1.6
for D
+ HZfor J
~
9.0(-18) 5.7(-16) 3.2(-15) 7.9(-15) 1.4(-14) 2.0(-14) 2.5(-14)
0.80
1.o
1.2
1000 K I T
E(eW Figure 3. Same as Figure 2 but for J = 9.
300 500 700 900 1100 1300 1500
0.60
9.2(-18) 5.6(-16) 3.2(-15) 8,1(-15) 1.4(-14) 2.1(-14) 2.7(-14)
2.8(-18) 3.2(-16) 2.6(-15) 8.7(-15) 1.9(-14) 3.3(-14) 5.0(-14)
2.32(-18) 2.76(-16) 2.32(-15) 7.81(-15) 1.73(-14) 3.06(-14) 4.71(-14)
2.7(-18) 2.8(-16) 2.1(-15) 6.7(-15) 1.4(-14) 2.4(-14) 3.4(-14)
This work on the DMBE potential. b Park and Light, ref 15, on the LSTH potential. c Mielke, S.L. et al., ref 32,on the DMBE potential. Q
the higher temperatures. The earlier results of Park and Light are in good agreement with the present and exact ones at lower temperatures, but are symmetrically lower than the exact ones and the present ones at the higher temperatures, for J = 6. A comparison of the present thermal rate constant with other calculations is given in Table 4. The SRA calculations are those of Mielke et al.20 based on their separable rotation approximation,24whichwasbrieflydescribedabove. TheSRAresults shown are based on exact calculations for J = 6 and 9. A detailed comparison of the Park-Light results with SRA and accurate ones on the LSTH potential is given in the preceding paper. (The accurate rate constants on the LSTH and DMBE potentials differ by 3% or less for T between 300 and 900 K.20) At higher temperatures, up to 900 K the present results are roughly 20% higher than theSRA andaccurateones. Thecomparison between the present calculations and the SRA ones between 900 and 1500 K continue to show differences of roughly only 17-20%. This comparison and the more detailed one in ref 20 indicate that the Park-Light results underestimate the rate constant at the higher temperatures. The present rate constant is compared with experiment35 in Figure 4, where very agreement is seen over the very wide range
Figure 4. Comparison of present thermal rate constant for D
experiment.
1.4
+ H2 with
TABLE 4 Thermal Rate Constants for the D + H2 Reaction this work Park and Light0 SRb Mielke et aleb T(K) DMBE LSTH DMBE DMBE ~~
200 300 400 500 600 700 800 900 lo00 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 0
1.9(-18)‘ 3.3(-16) 6.1(-15) 3.8(-14) 1.4(-13) 3.5(-13) 7.3(-13) 1.3(-12) 2.1(-12) 3.2(-12) 4.6(-12) 6.3(-12) 8.3(-12) 1.1(-11) 1.34(-11) 1.64(-11) 1.97(-11) 2.32(-11) 2.72(-11)
3.2(-16) 3.3(-14) 2.8(-13) 9.7(-13) 2.2(-12) 4.1(-12) 6.6(-12)
1.64(-18) 2.77(-16) 5.12(-15) 3.24(-14) 1.16(-13) 2.98(-13) 6.20(-13) 1.12(-12) 1.82(-12) 2.76(-12) 3.94(-12) 5.40(-12) 7.13(-12) 9.15(-12)
1.62(-18) 2.78(-16) 5.13(-15) 3.24(-14) 1.16(-13) 2.98(-13) 6.22(-13) 1.13(-12)
Reference 15. Reference 20.
of temperatures from 200 to 2000 K. At the highest temperatures there is some deterioration in the agreement with experiment. Finally, we test the J-shift approximation from J = 0 and from J = 1 CRPs. This is done by comparing the present CRPs for J = 4 and 10 (with the adiabatic treatment of overall rotation) and the two J-shift approximations. The comparisons are shown in Figure 5 , where the J-shift approximation from J = 1 is found to be somewhat more accurate than from J = 0, as expected. Both approximations are quite accurate, Le., within 10% or less of the adiabatic results up to total energies of 1.6 eV for J = 4; however, for J = 10 they become less accurate at energies above roughly 1.3 eV. For J = 10 the rotational energy of the DH2 transition state is approximately 0.18 eV, which is a substantial fraction of theelectronic barrier height of 0.42 eV. Thus, somedeterioration
Calculations of Reaction Probabilities and Rate Constants
An obvious extension of the J-shift approximation was made in which the J-shift is applied to the J = 1 C R P instead of the J = 0 CRP, as had been done previously. This extension eliminated the need to use a bend-energy shift, which is necessary for J-shifting from J = 0 (for linear transition states). That shift is necessary because the J = 0 C R P contains contributions from even bend states only. This shift is unnecessary for the J = 1 CRP which contains contributions for even and odd bend states, although only for values of the vibrational angular momentum of zero and one.
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Acknowledgment. Support from Department of Energy (DEFG05-86ER13568) is gratefully acknowledged. Also, we are grateful to Professor Don Truhlar for sending preprints of ref 20 and 24 and for helpful comments and discussion. References and Notes
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(1) Bowman, J. M. Ado. Chem. Phys. 1985, 61, 115. (2) Bowman, J. M.; Wagner, A. F. In Theory of Chemical Reaction Dynamics; Clary, D. C., Ed.; Reidel: Dordrecht, 1986; p 47. (3) Bowman, J. M. J. Phys. Chem. 1991,95,4960. (4) Bowman, J. M.; Wang, D. In Advances Molecular Vibrations and Collision Dynamics, Bowman, J. M., Ed.; JAI: Greenwich, 1994; Vol. IIB, Chapter 5. (5) Walker, R. B.; Hayes, E. F. In Theory of Chemical Reaction Dynamics; Clary, D. C., Ed.; Reidel: Dordrecht, 1986; p 105. (6) Hayes, E. F.; Pendergast, P.; Walker, R. B. in Advances in Molecular
J = 10
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v 7
z
Vibrations and Collision Dynamics; Bowman, J. M., Ed.; JAI: Greenwich, 0.6
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1 .o
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E(eW Figure 5. Comparison of present cumulative reaction probabilities (solid line) for D Hz with twoJ-shift approximations;onefromJ=0 (triangles) and one from J = 1 (circles) cumulative reaction probabilities.
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1994; Vol. IIA, Chapter 3. (7) Ohsaki, A,; Nakamura, H. Phys. Rep. 1990, 187, 1. (8) Takada, S.;Tsuda, K.-I.; Ohsaki, A.; Nakamura, H. In Advances in
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of the J-shift approximation is not surprising especially at higher energies where the reactive flux can sample regions of the potential surface that are significantly removed from the transition state. Another related possibility is that variational effects due to the large rotational energy, which we have ignored in the J-shift approximation, may be growing in importance for larger values of J a n d at the higher total energies. It wouldof course be possible to include these effects, by using a rotation constant corresponding to the variational transition state instead of the fixed constant given at the saddle point. IV. Summary and Conclusions The adiabatic-bend theory of reactive scattering was extended in two ways. First, the bending energies were calculated numerically instead of the harmonic approximation. Second, the overall rotational energy was calculated adiabatically instead of using the J-shift approximation. The new theory was tested by comparing cumulative reaction probabilities (CRPs) with exact quantum ones for H H2 for total angular momentum J = 0 and 4, and also by comparison of the thermal rate constant for J = 0. Similar tests were also done for D + Hz for J = 6 and 9 and also for the thermal rate constant for J = 0 and 6. These tests demonstrated a significant imporvement in accuracy with the new methods compared with the previous approximations. Comparisons of t h e t h e r m a l rate constant for D + HZwere made with accurate quantum calculations up to 900 K and with experiment up to 2000 K. Based on the good agreement with the former, the good agreement between theory and experiment throughout the temperature range 200-2000 K seems secure.
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