Ab Initio Chemical Kinetics: Converged Quantal Reaction Rate

Quantum Mechanical Wave Packet and Quasiclassical Trajectory Calculations for the Li + ... A Quantum Wave Packet Dynamics Study of the N(D) + H2 React...
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J. Phys. Chem. 1994, 98, 8000-8008

8000

Ab Initio Chemical Kinetics: Converged Quantal Reaction Rate Constants for the D

+ Hz System

Steven L. Mielke, Cillian C. Lynch, and Donald C. Truhlar. Department of Chemistry, Chemical Physics Program, and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431

David W. Schwenke NASA Ames Research Center, Mail Stop 230-3, Moffett Field, California 94035-1000

Received: February 9, 1994; In Final Form: June 17, 1994'

+

W e present accurate quantal rate constants for the D HZsystem in the 167-900 K temperature range and rate constants calculated in the separable rotation approximation up to 1500 K. W e have calculated rate constants for the three most accurate ab initio potential energy surfaces. The separable-rotation calculations agree to within 2.2% with the present accurate quantal calculations, and the results show substantially better agreement with high-temperature experimental rate constants than do previous quantal calculations. The ab initio rate constants for the LSTH and D M B E surfaces agree well with experiment over a wide temperature range, but the newer BKMP surface gives poor agreement a t low temperatures. From 200 to 900 K, a factor of 4.5 in temperature, the present totally ab initio reaction rate constants agree with experiment within 13% or better a t each temperature, with an average absolute deviation of only 5%.

1. Introduction

contributions from specific total angular momenta

Advances in computational techniques for accurate quantal dynamics have reached the stage where totally ab initio rate calculationsof useful accuracy are now possible for simple enough chemical reactions. In the present article we demonstrate this approach by a careful applicationof accurate quantum rate theory to a simple reaction. We also show how the general applicability of this approach can be broadened by making the route from the potential energy surface or electronic structure calculations to the calculated observables more efficient for a given accuracy or more accurate for a given computational cost. There are several ways to increase the accuracy/effort ratio. These include (i) more efficient quantum dynamics schemes, (ii) shortcuts in the route from quantum mechanical transition amplitudes to rate constants, and (iii) use of accurate results for simple systems to validate semiclassical approximations and separable or reduced-dimentionality approximations that are applicable to larger systems. The accurate results presented here are a consequence of advances in area i, whereas the approximate calculations in the present work are focused on issues ii and iii, and the accompanying paper' by Wang and Bowman is also focused on issue iii. In particular, Wang and Bowman test a reduced-dimensionality approximation, and the present paper presents accurate results and tests a separable approximation. Thermal rate constants for the D H Z reaction have been measured experimentallyby several g r o ~ p s over ~ - ~the temperature range 167-2000 K. Approximate theoretical calculationsl!6 employing statistical, reduced-dimensionality,and semiclassical methods have been presented for the LSTH7and DMBE8surfaces, and full-dimensionality quantal calculationsgon the LSTH surface have been performed by Park and Light. In the present work we present accurate quantal calculations on the LSTH and DMBE surfaces as well as the newer BKMPIO surface for temperatures up to 900 K, and we present separable-rotation-approximation" (SRA) rate constants for temperatures up to 1500 K. We also present rate constants for 900-1500 K based on extrapolating the accurate quantal results to higher energies than those for which they were directly calculated. The thermal rate constant, k(T), may be partitioned into

+

Abstract published in Aduunce ACS Abstrucrs, August 1, 1994.

k(T) = 2 ( 2 J + l)k'(T) J=O

and for a bimolecular reaction A

+ BC we have12

K d E NJ(E)e-E/kT ('

= h@,,(T)

QA(T)

Qec(T)

where re^( T) is the relative translational partition function per unit volume of A with respect to BC, Le.,

arc,(q= ( 2 T , ~ ~ 3 / * p 3 (3) h is Planck's constant, k is Boltzmann's constant, p is the A-BC reduced mass, QA is the internal partition function of A, Qw is the internal partition function of BC, and NJ(E)is thecumulative reaction probability for total angular momentum J. The cumulative reaction probability is defined a d 3

(4) %=og %+og

where n and n'are channel indices, (YO is the arrangement of the reactants (D H2 in the current application), anis the value of LY in channel n, and Si,,,is an element of the scattering matrix for a given J. In terms of the probability, that a collision with a given total energy E, total angular momentum J, initial vibrational quantum number v, and initial rotational quantum numberj will react to produce a product molecule withvibrational quantum number v' and rotational quantum number j', this becomes

+

where e j ( E ) denotes the total reaction probability for selected J , v, and j at total energy E. Let m denote a pair of the quantum

0022-3654/94/2098-8000S04.50/00 1994 American Chemical Societv

Ab Initio Chemical Kinetics of D

+ H2

The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 8001

numbers u and j specifying a particular initial state, and let

Combining this with the equations above gives'"16 Jo"d E K (E,m)e-E/kT

which is equivalent to the form for the exact rate, k( T), used in early discussions of the adiabatic theory of reactions. Instead of computing NJ(E)for all values of E and Jrequired to evaluate eq 2, the rate constants, k( T ) , may be estimated by invoking the separable-rotation approximation'

+

NJ(E + EYm) = NJ'(E E;?')

(8)

where EYTS is the rotational energy of the variational transition state with rotational quantum number J. Then the thermal rate constant may be expressed in terms of information from a single value, J', of the total angular momentum by

k( T ) =

T)eEJYrSlkTP(T )

(9)

which may be considered to be the partition function of the variational transition state. For a linear transition state in the rigid rotor approximation, we can approximate QLY using

E Y =~ B~ LJ(~J +~ 1

tional principle (OWVP)23-28 with complex boundary conditions21J3~24J9and multichannel distortion potentials.20 The GNVP calculations may be interpreted as a special case the OWVP ones,24so we will explain the calculations in the latter context. The distortion potentials include full rotational coupling for all channels with j < &u) for a given vibrational level u of arrangement a,27and they couple the degenerate components of a given vibrational-rotational level of a given arrangement for j L j",u), where j is the rotational quantum number.25.27s28In both the GNVP and the OWVP calculations the wave function is expanded in a multiarrangement basis set,30 where each basis function is either a multichannel half-integrated Green's function, consisting of radial half-integrated Green's functions (HIGFS),~O~ body-frame rotational-orbital functions,3l and vibrational functions, or it is a product of a translational distributed G a ~ s s i a n , 3 ~ a rotational-orbital function, and a vibrational function. The rotational-orbital functions3I are eigenfunctions of the total angular momentum and parity expressed in body-frame coordinates. The vibrational functions are eigenfunctions of the asymptotic diatomic fragment of the given arrangement, expressed as linear combinations of NJHO) harmonic oscillator eigenfunctions. The GNVP and OWVP also require the regular solutions to each of the distortion potential problems. The radial regular functions and the radial components of the HIGFs were computed by finite-difference methods.20bv25-27,28 The GNVP and OWVP reduce the calculation to a set of linear algebraic equations, and thematrix of coefficients and right-hand sides for these equations were obtained by multidimensional q ~ a d r a t u r e s . ~ O - ~ 2 ~ ~ ~ ~ 3 ~ The calculations were decoupled in J , the parity P, and in the symmetry under permutation of the H atoms. Para-H2 has even symmetry under permutation of the spatial coordinates of these two atoms, and it has nuclear spin Z = 0; ortho-Hz has odd permutational symmetry and Z = 1 . Then

and estimate B Z s as

"(E) =

BLT' = h2/2ZVTS where IVTS is the moment of inertia of the variational transition state. More accurate schemes can be implemented if data are available for two or more values of J. In particular, given data for two values, J' and J", of the total angular momentum, we represent EYE as in eq 7 and choose B y such that the separable-rotation approximation for kJ(T), Le., (13) holds exactly for these two values. This yields

from which

2. Computational Methods The H2 partition functions for each surface were obtained using the accurate vibrational-rotational eigenenergies of the asymptotic H2potential and include nuclear spin degeneracies ( 1 and 3), i.e.,

QH2(T)= c ( 2 j jeven V J

+ l)e-ful/kT+ 3 c ( 2 j + l)efoj/x-T

(16)

,%

where tujis the eigenvalue of the state with vibrational quantum number u and rotational quantum number j. Calculations of the cumulative reaction probability (CRP) for the D + para-H2 -, H D H reaction were carried out using the generalized Newton variational principle (GNVP)l7-22 and the outgoing wave varia-

+

(2Z+ l ) N J r ( E ) feO,l

where (21

+ 1 ) is the nuclear-spin degeneracy, and NJ'(E) =

N'"(E) P=+1,-l

Four parameter sets were used for the results presented in the present paper. Threeof these, A, B, and D, take the translational basis functions as HIGFs in open channels and distributed Gaussians in closed channels. Set A was used to calculate CRPs for J in the range 0-20 for 31 evenly spaced points on an energy grid from 0.4 to 1.0 eV and CRPs for J in the range 21-24 for the six energy grid points in the range 0.9-1.0 eV. Set B, which has a substantially larger basis and more stringent numerical parameters, was used to calculate CRPs a t selected values of J over the energy range 0.4-1.6 eV. Full details of these parameter sets are given in Table 1 using notation that has been defined previously.27 The third parameter set, denoted C, that is more stable at low energies was used to calculate CRPs for E = 0.340.40 eV for J = 6 on the DMBE surface. This set is a modified version of set A in which all translational basis functions are HIGFs in both open and closed channels and the vibrational screening parameter, tX, is changed to This translational basis reduces the OWVP to the GNVP. The fourth parameter set, denoted D, is a slimmed down version of set B that was used to calculate CRPs at 1 . 1 and 1.2 eV for J 4 35. A body-frame basis set truncation scheme28-34-36 was employed based on the value, R, of the magnitude of the projection of the total angular momentum on the atom-to-diatom axis in a given arrangement. For channels with R 4 amax we used 6 (set A or C), 7 (set B), or 5-6 (set D) translational basis functions per channel. For open channels with > Q,,, we used one

8002 The Journal of Physical Chemistry, Vol. 98, No. 33, 1994

Mielke et al.

TABLE 1: Parameter Sets set A parameter

explanation

D+H2

a

12 12 10 9

a a a

HD+H 14 14 12 9 6

a a

b b C

d e

f b g

g

i k

I I m m

n 0

P 4

10 4 2.50 0.36 1.26 6 50 30 50b,0' 16 163 13 8 1.4 16.0 1.4 16.0 30 0.9 19 7

m

4 2.80 0.36 1.26 6 50 30 50 16 177 13 8 1.9 17.0 1.9 17.0 30 0.9

r

21

S

7

set B D+H2

set D HD+H

14 14 14 12 10 12 6 2.65 0.30 1.05 7 50 30 50 18 212 13 8 1.o 17.0

14 14 14 12 10 10 12 6 2.45 0.30

1.05 7

50 30 50 18 198 13

8 1.o 16.0

1.o

1.o

17.0 30 0.9 26

16.0 30 0.9 24

7

I

D+Hz

HD+H

13 13

14 14 14 10 6

13 11

9 12 6 2.65 0.30 1.05 6 50 30 50 18 212 13 8 1.o 16.0 1.4 8.0

12 6

2.75 0.30

1.05 5 50 30 50 18 170 13 8 1.o 17.0 1.5 8.O

30

30

0.9 26

0.9 20 7

7

5 100 5 6

5 5 100 11 100 5 5 u 6 W 6 5 5 5 X j,(u) is the maximum value of the rotational quantum number in vibrational level u included in the vibrational-rotational-orbital basis. * See text, R:, is the value of R, at the center of the innermost translational Gaussian (Gaussian source function generatinga half-integratedGreen's function in open channels and Gaussian basis functions in closed channels), where R, is the mass-scaled atom-to-diatom distance defined in ref 27. A is the spacing in R, between successive translational Gaussians. e c is the translational Gaussian overlap parameter, defined in refs 32b and 20b. f m is the number of translational Gaussians per channel for channels with Q 5 Q-. g fld and @$are the number of points in the Gauss-Legendre quadrature used in the single- and multi-arrangement angular quadratures, respectively. (Note that a = 1 is D + H2 and a = 2 and 3 are HD + H.) a = 2 or 3, a' = 1. a = 2, a' = 3. I @'is the number of points in the optimized33vibrational quadrature used with Gauss ground-state nodes33in the evaluation of the vibrational integrals of the interaction potential. N,(F) is the total number of points in the finite-differencegrid used for the calculation of the regular solution of the distortion problems and the half-integrated Green's functions. In the present calculations, no additional points are added to the main part of the finite-differencegrid defined by the quadrature nodes. and are the number of points used in the representation of the second-derivative operator in the main body of the finite-differencegrid and the last grid point, respectively. m R L and R:fl.(o+, are respectively the location of the lower and upper finite-difference boundary condition points. n RPR(")is the lower limit of the radial quadrature grid for &. 0 R Y ( , ) is the upper limit of the radial quadrature grid for R,. p is the number of points appended to the main part of the finite-differencegrid is the number with geometrically decreasing spacing. qfSD is the step size decrease factor for the spacing of the final finitedifference grid points. of repetitionsof WL-point Gaussian quadrature used in the generation of the finite-differencegrid and the integrations over R,. * @is the number is the radial screening parameter.25.z7*a of points in the Gaussianquadrature used in each repetition. ex is the vibrationalscreeningparameter.z5.27.28€4 c, is the translational basis screening parameter.2*~~~.~~ tw is a screening parameter involving the matrix W.25*27*28 is a screening parameter involving the matrix B.2**27928 t

*

f

e,(?

XD

translational basis function per channel, and closed channels with Q > Qmax were not included. One notable efficiency of set A was the omission2* of the calculation of the exchange matrix elements of the variational functional involving both of the identical HD H arrangements (indicated by ne,^ = 0 in Table 1). This reduces the quadrature costs by about a factor of 2 but has a negligible effect on the results for reaction probabilities out of the D H2 arrangement a t moderate energies. Convergence checks of the CRPs obtained from set B were performed at selected energies for J = 6 using larger basis sets that differed simultaneously in a sufficient subset of parameters to demonstrate convergence. For energies greater than 0.5 eV the CRPs from set B were converged to better than 0.296, and

+

+

convergence for the 0.44.5-eV range was 3% or better. Although the calculations performed with set A were substantially more efficient than those using set B, the convergence is still very good. For the 31 CRPs a t J = 6 that were obtained for the DMBE surface using both parameter sets, the mean unsigned difference of the set A and set B CRPs is 0.18%, and the largest difference is 2.35% at 0.40eV. Convergence of the CRPs for set C is better than 0.1% at 0.4 eV and about 2% at 0.34 eV. Convergence of the CRPs for set D was checked in several cases by comparison to results for set C, and the largest difference observed was 0.2%. In two instances CRPs using set B were poorly converged (at 1.14 eV for the -parity block of J = 9 for the BKMP surface and a t 1.46 eV for the -parity block of J = 9 for the DMBE surface) due to instabilities in the finite-difference solution resulting from

Ab Inirio Chemical Kinetics of D TABLE 2

0

+ Hz

The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 8003

Cumulative Reaction Probabilities for J = 0 for the DMBE Surface (Using Parameter Set B)

energy (eV)

D + para-H2

D + ortho-HZ

energy (eV)

D + para-H?

0.5 0.6 0.7 0.8 0.9 1.o

6.99(-4)" 2.00(-1) 9.74(-1) 1.18 1.85 2.71

7.00(-4) 2.00(-1) 9.74(-1) 1.18 1.85 2.70

1.1 1.2 1.3 1.4 1.5 1.6

3.62 4.50 5.58 6.88 8.19 9.59

D

+ 0rtho-H~ 3.63 4.48 5.61 6.90 8.24 9.74

Numbers in parentheses indicate powers of 10.

TABLE 3 Cumulative Reaction Probabilities for J = 6 for the DMBE Surface energy (eV)

parameter set

D + para-HZ

0.34 0.36 0.38 0.4 0.5 0.6 0.7 0.8 0.9 1.o 1.1 1.2 1.3 1.4 1.5 1.6

C C C C

3.2(-11)' 4.38(-10) 4.2 1(-9) 3.13(-8) 9.267(-5) 3.869(-2) 9.157(-1) 2.445 4.725 8.297 13.73 20.66 29.54 40.28 53.64 68.44

0

B B B B B B B B B B B B

D + ortho-HZ 3.1(-11) 4.35(-10) 4.21(-9) 3.14(-8) 9.266(-5) 3.869(-2) 9.157(-1) 2.445 4.725 8.298 13.72 20.69 29.47 40.39 53.66 68.59

Numbers in parentheses indicate powers of 10,

the presence of nearly closed channels in the H D arrangements. These two cases were rerun with the length of the H D finitedifference grid and N p s increased by a factor of 2, and better convergence was obtained. The better converged results were used for the rate constant calculations. The finite-difference grid was similarly extended for calculations with set A a t 0.92 eV for J 1 15 for all three surfaces. Tables 2 and 3 compare cumulative reaction probabilities for J = 0 and 6 on the DMBE surface for the D para-H* and D ortho-Hz reactions; the results are summed over the two identical product arrangements. The CRPs for the two symmetry blocks are not equal by symmetry, but they agree to well within the degree of convergence. Thus, the calculation of only a single permutation symmetry is required to obtain D normal-HZ rate constants. Thus, with the exception of Figures 1 and 2 presented at the beginning of section 3, all further results in this paper were calculated using the approximation NJo(E)= NJ1(E). The integration in eq 2 was performed using Simpson's rule, and results of this procedure agree to better than 0.1% with numerical integration of curves obtained using quadratic fits of the logarithm of the CRPs. The CRPs were extended to the energy region E < 0.4 eV using a third-order polynomial fit obtained with the data at the first four grid points (0.40, 0.42, 0.44, and 0.46 eV). Contributions from this region are negligible except for low J and low temperatures. For the DMBE surface and J = 0, the region E < 0.4 eV contributes 9.6% at 167 K, 2.1% at 200 K, 0.3% at 250 K, and 0.06% a t 300 K. For J = 6 the low-energy extrapolated contributions to the rate constants are 5.5% at 167 K, 1.0% at 200 K, and 0.03% at 250 K. The E < 0.4 eV region contributes 8.1%, 1.5%, 0.2%, and 0.02% to the full rate constants at 167, 200,250, and 300 K, respectively. For J = 6 on the DMBE surface additional CRPs were calculated at 0.34, 0.36, and 0.38 eV, and separable-rotation rate constants obtained with this additional data agreed with those obtained from the extrapolation procedure to within 0.05%. The Boltzmann-weighted CRPs for J = 6 and 9 obtained with set B were extended to the region E > 1.6 eV using an exponential functional form fitted at the last two grid points (1.58 and 1.60 eV). The contribution to the rate constant, k(T), from the region

+

+

+

Q' 10-12

1

u 0.35 0.36 0.37 0.38 0.39 0.40

energy (eV) Figure 1. State-selected reaction probabilitiesfor the reaction D Hz(u=Oj) H HD for the DMBE surface with J = 6. The threshold of u = 0 , j = 3 for H2 is 0.3561 eV.

-

+

+

E > 1.6 eV for the separable rotation rate constants obtained from J = 6 was about 2% at 1500 K-the highest temperature considered. For the accurate calculations using set A, the contributions to the rate constants from the energy region E > 1.O eV were calculated by extrapolating the J-summed CRPs, using a third-order polynomial fit to the data from set A for 0.9 5 E 5 1.0 eV and to the data at 1.1 and h;!eV obtained with set D. If instead, a fourth-order polynomial is used to extrapolate the DMBE results, differences of less than 0.3%are observed in the calculated rate constants. The contributions to the rate constants from the region E > 1.OeV (E > 1.2 eV) for the accurate quantal calculations for the DMBE surface were less than 0.46% (0.013%) for T I 500 K, 1.8% (0.11%) at 600 K, 8.6% (1.3%) at 800 K, and 14% (2.8%) at 900 K. 3. Results and Discussion Figures 1 and 2 show state-selected reaction probabilities for the forward and reverse reaction for the DMBE surface a t J = 6 (summed over the two identical H H D product arrangements for the forward case). Over the selected energy range, j = 1 has the highest reactivity for both reactions. For the D H2 reaction with E < 0.4 eV, the reactivity o f j = 0 is nearly as large a s j = 1 and the reactivity of the higherj is appreciably smaller. Figure 1 does not explain the remarkable fact, mentioned in section 2, that NJO(E) equals NJ'(E) to an excellent approximation, but that finding may be explained by the assumption of global control of reactivity by quantized transition states.37 Both the ortho and the para reactions pass through the same set of quantized transition states, and initial rotational motion is essentially completely converted into bending and overall rotational motions at the transition states in the energy range under study here. Thus, the total flux shows a "symmetry" that is explainable in terms of quantized transition states but not in terms of reactant quantum numbers. W e found that the J-summed cumulative reaction probabilities depend smoothly on energy and their energy derivatives except

+

+

Mielke et al.

8004 The Journal of Physical Chemistry, Vol. 98, No. 33, 1994

TABLE 4 Comparison of kJ( 7)(in Units of cm3 m~lecule-~ s-l) for selected J from tbe Present, Accurate Quantal, Calculations for the LSTH Surface and the Calculations of Park and Light (Ref 9) J 0

1 2

3 5

6 9

source ref 9 present ref 9 present ref 9 present ref 9 present ref 9 present ref 9 present ref 9 present

300 K 9.2(-18) 8.17(-18) 9.5(-18) 7.99(-18) 8.4(-18) 7.06(-18) 7.0(-18) 5.86(-18) 3.8(-18) 3.34(-18) 2.7(-18) 2.3 1(-18) 6.0(-19) 5.23(-19)

d

500 K 5.6(-16) 5.22(-16) 6.1(-16) 5.84(-16) 5.8(-16) 5.46(-16) 5.1(-16) 4.86(-16) 3.4(-16) 3.42(-16) 2.8(-16) 2.7 1(-16) 1.1(-16) 1.06(-16)

700 K 3.2(-15) 3.00(-15) 3.7(-15) 3.86(-15) 3.6(-15) 3.78(-15) 3.3(-15) 3.49(-15) 2.4(-15) 2.70(-15) 2.1(-15) 2.28(-15) 1.1(-15) 1.15(-15)

d

'oli.34 0.35 0.36 0.37 0.38 0.39 0.40 energy (eV)

900 K 8.1(-1 5 ) 7.59(-15) l.O(-14) 1.10(-14) l.O(-14) 1.13(-14) 9.3(-15) 1.07(-14) 7.2(-15) 8.77(-15) 6.7(-15) 7.68(-15) 3.9(-15) 4.47(-15)

1100 K 1.4(-14) 1.33(-14) 1.9(-14) 2.1 3(-14) 1.9(-14) 2.29(-14) 1.8(-14) 2.23(-14) 1.4(-14) 1.90(-14) 1.4(-14) 1.70(-14) 8.9(-15) 1.09(-14)

1300 K 2.1(-14) 1.94(-14) 3.0(-14) 3.35(-14) 3.0(-14) 3.75(-14) 3.0(-14) 3.74(-14) 2.4(-14) 3.30(-14) 2.4(-14) 3.01(-14) 1.6(-14) 2.06(-14)

1500 K 2.7(-14) 2.53(-14) 4.1(-14) 4.66(-14) 4.3(-14) 5.40(-14) 4.2(-14) 5.51(-14) 3.4(-14) 5.02(-14) 3.4(-14) 4.64(-14) 2.5(-14) 3.34(-14)

energy (eV) Fipe3. Polynomialfits of the J-summed cumulativereaction probability as a function of energy for the DMBE surface. The solid line represents a third-order polynomial fit to the CRPs in the energy range 0.9 IE I 1.2 eV, the dashed curve represents a fourth-order polynomial fit to the CRPs in the energy range 0.8 IE I1.2 eV, and the open circles are the accurate quantal values.

Figure 2. State-selectedreaction probabilities for the reaction H + HD(v=Oj)-P D + H2 on the DMBE surface with J = 6. The threshold of v = 0, j = 4 for HD is 0.3425 eV.

TABLE 5: Comparison of k( 7)Calculations for the LSTH Surface and Experiment in Units of cm3 molecule-1s-l

for a small oscillatory structure around 0.9 eV. The well-defined structure in the energy dependence of the individual-J CRPs (resulting from the presence of quantized transition states) is shifted in energy by roughly BJ(J + 1) for each J and thus on summation is mostly washed out. This permits very accurate interpolation and extrapolation in the computationally demanding high-energy region. Figure 3 displays the two different polynomials used to fit the CRPs from basis sets A and D on the DMBE surface. Table 4 compares selected kJ(T) from the flux-flux autocorrelation calculations of Park and Light9 to the results form the present scattering calculations on the LSTH surface. The rate constants for J = 0 and the h i g h e r J results in the temperature range 50&700 K are in fair agreement, but the remaining data differ substantially more than the error estimates of the earlier work,g indicating that the prior resultsg are not fully converged. At the highest temperature (1500 K),the J > 0 results differ by 1 4 4 8 % . Table 5 compares the full rate constants of Park and Light9 to the present accurate quantal and SRA rate constants as well as to experiment. The present results are significantly larger than the rate constants of Park and Light9 a t high temperatures, and they are in substantially better agreement with the experimental rate constants. Table 6 compares the present accuate quantal results to selected SRA results. Tables 5,7, and 8 present accurate and approximate rate constants for the LSTH, DMBE, and BKMP surfaces, respectively, as well as the experimental results. In these tables, results based on the accurate quantum scattering calculations for all J a r e labeled accurate for T I900 K but are labeled "accurate + extrapolated- for T 2

ref 9 SRA J = 6,9 expto 1.63(-19) 1.60(-19) 7.25(-20) 1.76(-18) 1.74(-18) 1.47(-18) 3.19(-17) 3.18(-17) 3.39(-17) 2.76(-16) 3.2(-16) 2.76(-16) 2.96(-16) 5.03(-15) 5.04(-15) 5.11(-15) 3.17(-14) 3.3(-14) 3.17(-14) 3.17(-14) 1.14(-13) 1.14(-13) 1.15(-13) 2.92(-13) 2.8(-13) 2.92(-13) 3.07(-13) 800 6.09(-13) 6.09(-13) 6.67(-13) 900 1.10(-12) 9.7(-13) 1.10(-12) 1.26(-12) 1000 1.80(-12) 1.79(-12) 2.14(-12) 1100 2.72(-12) 2.2(-12) 2.71(-12) 3.39(-12) 1200 3.89(-12) 3.88(-12) 5.04(-12) 1300 5.32(-12) 4.1(-12) 5.32(-12) 7.17(-12) 1400 7.02(-12) 7.03(-12) 9.81(-12) 1500 8.99(-12) 6.6(-12) 9.03(-12) 1.30(-11) a Taken from the experimental fit given in ref 5a. 1000 K. The reason for this is that for T 2 1000 K the region above 1.2 eV, where the results are extrapolated rather than interpolated, begins to make a significant contribution. In particular, for the DMBE surface, the contribution from this region increases from 5.2% at 1000 K to 12% a t 1200 K to 26% at 1500 K. Next we consider the validity of the separable-rotation approximation. This approximation is stated in eq 8 in the form presented in ref 11, which is a generalization of earlier work.38 We stress that the separable-rotation approximation has a long history in both bound-state calculation^^^ and dynamics; in the latter area we especially mention its use in conventional transition-

accurate

accurate +

T (K) quantum extrapolated 167 200 250 300 400 500 600 700

Ab Initio Chemical Kinetics of D

+ Hz

The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 8005

TABLE 6 Separable-Rotation Approximations for the LSTH Surface Compared to Accurate Quantal and Extrapolated Accurate Quantal Calculations T (K) accurate/extrap J=0 J =1 J=2 J =3 J=5 J=6 J=9 J=6,9 1.31(-19) 1.35(-19) 1.41(-19) 1.51(-19) 1.85(-19) 1.63(-19)" 1.50(-18) 1.54(-19) 1.58(-18) 1.65(-18) 1.86(-18) 1.76(-18)' 3.19(-17)" 2.84(-17) 2.93(-17) 2.97(-17) 3.03(-17) 3.23(-17) 2.76(-16)' 2.49(-16) 2.60(-16) 2.62(-16) 2.66(-16) 2.76(-16) 5.03(-15)' 4.41(-15) 4.84(-15) 4.87(-15) 4.90(-15) 4.98(-15) 500 3.17(-14)' 2.64(-14) 3.07(-14) 3.11(-14) 3.12(-14) 3.14(-14) 8.84(-14) 1.10(-13) 1.12(-13) 600 1.14(-13)' 1.13(-13) 1.13(-13) 2.12(-13) 2.80(-13) 2.91(-13) 2.92(-13) 2.92(-13) 700 2.92(-13)' 4.10(-13) 5.76(-13) 6.07(-13) 6.12(-13) 6.11(-13) 800 6.09(-13)' 900 1.10(-12)" 6.88(-13) 1.02(-12) 1.09(-12) 1.11(-12) 1.11(-12) 1.80(-1 2)b 1.05(-12) 1.63(-12) 1000 1.78(-12) 1.81(-12) 1.82(-12) 1.48(-12) 2.40(-12) 2.68(-12) 2.75(-12) 2.76(-12) 1100 2.72(-12)b 1.98(-12) 3.35(-12) 3.80(-12) 3.93(-12) 3.97(-12) 1200 3.89(-12)b 2.54(-12) 4.46(-12) 5.14(-12) 5.36(-12) 5.44(-12) 1300 5.32(-12)b 3.16(-12) 5.72(-12) 6.70(-12) 7.06(-12) 7.20(-12) 1400 7.02(-12)b 3.82(-12) 7.12(-12) 8.48(-12) 9.01(-12) 9.25(-12) 1500 8.99(-12)' Accurate quantal. Accurate quantal but with significant contributions from extrapolated results. 167 200 250 300 400

2.12(-19) 2.02(-18) 3.38(-17) 2.83(-16) 5.04(-15) 3.16(-14) 1.13(-13) 2.92(-13) 6.1 1(-13) 1.11(-12) 1.81(-12) 2.76(-12) 3.96(-12) 5.44(-12) 7.20(-12) 9.26(-12)

3.80(-19) 2.89(-18) 4.07(-17) 3.16(-16) 5.28(-15) 3.23(- 14) 1.14(-13) 2.92(-13) 6.08(-13) 1.10(-12) 1BO(-12) 2.73(-12) 3.91(-12) 5.37(-12) 7.11(-12) 9.15(-12)

1.60(-19) 1.74(-18) 3.18(-17) 2.76(-16) 5.04(-15) 3.17(-14) 1.14(-13) 2.92(-13) 6.09(-13) 1.10(-12) 1.79(-12) 2.71(-12) 3.88(-12) 5.32(-12) 7.03(-12) 9.03(-12)

@

TABLE 7: Comparison of k( 2') Calculations for the DMBE Surface in Units of cm3 molecule-' s-1 SRA accurate accurate + T(KI 167 200 250 300 400 500 600 700

auantum extrapolated J =6 1.43(-19) 1.82(-19) 1.64(-18) 1.86(-18) 3.14(-17) 3.29(-17) 2.78(-16) 2.83(-16) 5.1 3(-15) 5.13(-15) 3.24(-14) 3.23(-14) 1.16(-13) 1.16(-13) 2.99(-13) 2.98(-13) 6.26(-13) 800 6.21(-13) 1.13(-12) 900 1.12(-12) 1.83(-12) 1.86(-12) 1000 2.77(-12) 2.82(-12) 1100 3.95(-12) 4.05(-12) 1200 5.40(-12) 5.56(-12) 1300 1400 7.13(-12) 7.36(-12) 9.12(-12) 9.46(-12) 1500 a

J = 6.9

1.40(-19) 1.62(-18) 3.12(-17) 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)

exW 7.25(-20) 1.47(-18) 3.39(-17) 2.96(-16) 5.11(-15) 3.17(- 14) 1.15(-13) 3.07(-13) 6.67(-13) 1.26(-12) 2.14(-12) 3.39(-12) 5.04(-12) 7.17(-1 2) 9.81(-12) 1.30(-11)

Taken from the experimental fit given in ref 5a.

TABLE 8 Comparison of k( I) Calculations for the BKMP Surface in Units of cm3 molecule-' s-1 T (K)

167 200 250 300 400 500 600 700

SRA accurate accurate + quantum extrapolated J = 6 J = 6,9 experiment' 3.27(-19) 4.23(-19) 3.20(-19) 7.25(-20) 3.68(-18) 3.17(-18) 1.47(- 18) 3.21(-18) 5.54(-17) 5.21(-17) 3.39(-17) 5.24(-17) 4.3 1(-16) 4.20(-16) 2.96(-16) 4.21(-16) 6.93(-15) 6.94(-15) 6.93(-15) 5.11(-15) 4.10(-14) 4.12(-14) 3.17(-14) 4.1 1(-14) 1.41(-13) 1.42(-13) 1.15(-13) 1.42(-13) 3.54(-13) 3.54(-13) 3.54(-13) 3.07(-13) 7.25(-13) 7.23(-13) 6.67(-13) 7.23(-13) 1.29(-12) 1.29(-12) 1.28(-12) 1.26(-12) 2.07(-12) 2.09(-12) 2.07(-12) 2.14(-12) 3.11(-12) 3.15(-12) 3.10(-12) 3.39(-12) 4.40(-12) 4.48(-12) 4.40(-12) 5.04(-12) 5.98(-12) 6.11(-12) 5.98(-12) 7.17(-12) 7.84(-12) 8.05(-12) 7.86(-12) 9.81(-12) 9.97(-12) 1.03(-11) 1.00(-11) 1.30(-11)

TABLE 9 State

Moments of Inertia of the Variational Transition

r" (au)

surface LSTH DMBE BKMP

15 851 16 023 15 884

TABLE 1 0 Moments of Inertia (au) Obtained with the Separable Rotation Approximation (Eq 11) temp 167 200 250 300 400 500 600 700 800

900 1000 1100 1200 1300 1400 1500

LSTH 1.990(+4) 1.862(+4) 1.756(+4) 1.703(+4) 1.650(+4) 1.620(+4) 1.600(+4) 1.584(+4) 1.571(+4) 1.560(+4) 1.551(+4) 1.544(+4) 1.538(+4) 1.534(+4) 1.532(+4) 1.531(+4)

DMBE 1.979(+4) 1.851(+4) 1.748(+4) 1.697(+4) 1.648(+4) 1.620(+4) 1.600(+4) 1.585(+4) 1.572(+4) 1.561(+4) 1.551(+4) 1.543(+4) 1.537(+4) 1.532(+4) 1.529(+4) 1.527(+4)

BKMP 1.983(+4) 1.858(+4) 1.755(+4) 1.703(+4) 1.652(+4) 1.624(+4) 1.604(+4) 1.588(+4) 1.575(+4) 1.564(+4) 1.555(+4) 1.547(+4) 1.541(+4) 1.537(+4) 1.534(+4) 1.534(+4)

discussion of such approximations is provided in the accompanying paper. 1 As employed here and in ref 11, the separable-rotation approximation is based on accurate cumulative reaction probabilities for moderate values of J, whereas in previous work38 and worku carried out approximately contemporaneously with ref 11, it was applied with accurate cumulative reaction probabilities 800 J = 0. W e demonstrated in ref 11 that the approximation for 900 is more accurate if used with higher J , and we explained why. 1000 We now give another quantitative demonstration of this. 1100 1200 The separable-rotation rate constants calculated from CRPs 1300 for J = 6 used a rotational constant obtained from eq 11 using 1400 moments of inertia from a variational transition-state-theory 1500 calculation; the values for Ivm are listed in Table 9. The Taken from the experimental fit given in ref 5a. calculations using two different Jvalues employed a temperaturedependent estimate of the moment of inertia obtained from eq state theory.40 The separable-rotation approximation has not 15 (these values are listed in Table 10). The latter approach only been applied to cumulative reaction probabilities, as in eq yields extremely accurate rate constants as indicated by the close 8 and ref 38, but also to state-to-state reaction p r o b a b i l i t i e ~ . ~ * - ~ ~agreement with the accurate quantal calculations; the largest Bowman and co-workers have fruitfully employed similar but difference is only 2.2% a t 167 K (for the BKMP surface). more sophisticated approximations in the context of adiabatic Separable-rotation calculations using data from only J = 6 are bend reduced-dimensionality theories for atom-diatom and sufficient to obtain the high-temperature rate constants to within diatom4iatom rea~tions,12.~2,~3 and they have denoted such a few percent but deviate from the accurate calculations by up to 30% at the lowest temperatures. As can be seen in Table 6, treatments "rotational energy shifting" or "Jshifting". Further

Mielke et al.

8006 The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 1010

-

10'2

r

In

r

-P) 2 -a

10-l~

6

10-16

z

Y

h

t Y

10-7

10-18

1

'080;

'

'016' ' O i 8 ' '1:O ' '1:2 ' '1j4 ' '1?6

10-200

1

energy (eV)

Figure 4. Cumulative reaction probabilities for the reaction D + paraH2 for total angular momentum J = 6 as a function of energy for the LSTH, DMBE, and BKMP surfaces. The curve for LSTH is not visible because, to within plotting accuracy, it is hidden behind the DMBE curve.

10-lgl 10-20

2.5

3.5

-

4.5

5.5

1ooorr

Figure 5. Accurate quantal thermal reaction rate constants, k(T), for the reaction D + normal-H2 H + HD on the LSTH, DMBE, and BKMP surfaces compared to the low-temperature experimental results of Mitchell and Le Roy.3 the use of J = 0 in the separable rotation approximation leads to large errors (up to 18% at low T and 58% at high 0,but the accuracy of the approximation increases markedly where the reference calculation corresponds to higher J. Figure 4 displays CRPs for J = 6 for the LSTH, DMBE, and BKMP surfaces. Figure 5 displays accuratequantal rate constants for three surfaces compared to the low-temperature data of Mitchell and Le Roy.3 Figure 6 compares separable-rotation rate constants calculated using J = 6 and 9 for the three surfaces to experimental data for the temperature range 167-1500 K. The LSTH and DMBE rate constants are very close to each other and are in excellent agreement with the experimental data. The BKMP rate constants are significantly higher than the measured rate constants and the rate constants for the other surfaces at low temperatures. The calculated rate constant at 1500 K is about 30% below the experimentally observed rate c o n ~ t a n t .A ~ likely explanation for this (barring experimental error) is that some deficienciesremain in the high-energy regions of the potential energy surface. In several instances45 accurate quantal calculations of integral and differential cross sections have been performed at the same

2

3 1ooorr

4

5

6

-

Figure 6. Separable-rotation-approximation thermal rate constants (obtained from J = 6 and 9 ) for the reaction D + normal-H2 H + HD on the LSTH, DMBE, and BKMP surfacescompared to experimental%5 results. energy for more than one surface, but it has been difficult to draw inferences concerning the relative accuracies of the surfaces based on the moderate differences observed. Recently, Partridge et a1.4 reported high-level ab initio calculations for H3 at 484 geometries and calculated the root-mean-square (rms) deviations between their energies and the three analytic surfaces used here. They obtained rms deviations of 0.44, 0.40, and 0.23 kcal/mol for the LSTH, DMBE, and BKMP surfaces, respectively. They also report barrier heights (including corrections for basis set superposition error) obtained with three basis sets as 9.654,9.649, and 9.632 kcal/mol. For comparison, we note that Monte Carlo calculation^^^ place the barrier at 9.61 f 0.01 kcal/mol with the error estimate being one standard deviation and that recent ab initio calculations by Peterson e? yielded a barrier height estimate of 9.60 f 0.02 kcal/mol. The surfaces employed here have barriers of 9.80 (LSTH), 9.65 (DMBE), and 9.54 kcal/mol (BKMP). Partridge et a1.& stated that the BKMP surface "supersedes" the LSTH and DMBE surfaces as it incorporates most of the electronic structure data to which these two surfaces were fit. (Eighteen nonsymmetric points* in the vicinity of D3h symmetry that were used to fit the DMBE surface were not used for the BKMPfit.) Thecurrent results, however, indicatethat the DMBE surface predicts much more accurate low-temperature kinetics than the BKMP surface does. It may, in fact, be the best surface overall since it is the only surface with the correct behavior in the region of the conical intersection, it apparently has the most accurate barrier height, and it yields excellent rate constants. The BKMP surface, despite having the smallest root-mean-square deviation from the most accurate available electronic structure data,& is shown here to yield much worse agreement with lowtemperature rate constants. 4. Concluding Remarks

We have presented accurate quantal and separable-rotation approximation calculations for the rate constants of the D H2 reaction for threedifferent potential energy surfaces. Cumulative reaction probabilities for the D para-Hz and D ortho-H2 reactions were shown to agree quantitatively over the energy range 0.34-1.6 eV, and this agreement permits the calculation of reaction rates from CRPscurresponding to a singlepennutrition symmetry. Comparisons were made between the present results and the prior calculations of Park and Light? and these indicate that the latter were not fully converged, particularly at high

+

+

+

Ab Initio Chemical Kinetics of D

+ Hz

temperature. The rate constants for the LSTH and DMBE surface agree well with experimental rate constants, but the lowtemperature results on the BKMP surface are much higher than the observed rate constants. Separable-rotation rate constants obtained using cumulative reaction probabilities from only two values of Jshow remarkable agreement with the present accurate rate constants-to within better than 2.2%. Studies of the H + H2 reaction and its isotopologs have a long and well-known history.49 Michael and LimS0 recently summarized the comparison of theory and experiment for the rate constant of the reaction D with H2.They noted that the calculated rate constants of Park and Light9 were significantly below the experimental values of Michael et al.s They opined that “Explaining the H H2 reaction is without question the most important problem in chemical dynamics of the twentieth century. A successful theoretical prediction is crucial because the credibility of every other theoretical advancein the field rests with thesolution to the H Ha problem.” In this light it is very comforting to note that Table 7 shows that the present totally ab initiochemical reaction rate calculations agree with experiment4 within 13% from 200 to 900 K, which is a factor of 4.5 in temperature! The average unsigned deviation from experiment .at the nine temperatures tabulated in this range is only 5%. Clearly ab initio kinetics has arrived.

+

+

Acknowledgment. We thank Joel Bowman for a preprint of ref 1. This work was supported in part by the National Science Foundation. Supplementary Material Available: Tables of CRPs for the LSTH, DMBE, and BKMP surfaces for parameter set A for 3 1 evenly spaces energies in the range 0.4-1 .O eV for J I20 (24 for the highest 6 energies), tables of CRPs for parameter set B for J = 0, 1, 2, 3, 5, 6, and 9 on the LSTH surface and for J = 6 and 9 on the DMBE and BKMP surfaces for the 61 evenly spaces energies in the range 0.4-1.6 eV, tables of CRPs for the LSTH, DMBE, and BKMP surfaces at 1.1 and 1.2 eV for parameter set D and J I35, and tables of k6( T ) and k9( T ) for the DMBE and BKMP surfaces (15 pages). Ordering information is given on any current masthead page. References and Notes (1) Wang, D.; Bowman, J. M. J . Phys. Chem., preceding paper in this issue, and references therein. (2) Ridley, B. A.; Schulz, W. R.; Le Roy, D. J. J. Chem. Phys. 1966, 44, 3344. (3) Mitchell, D. N.; Le Roy, D. J. J. Chem. Phys. 1973, 58, 3449. (4) Westenberg, A. A.; De Haas, N. J . Chem. Phys. 1967, 47, 1393. (5) (a) Michael, J. V.;Fisher, J. R. J . Phys. Chem. 1990, 94, 3318. (b) Michael, J. V.;Fisher, J. R.; Bowman, J. M.; Sun,Q.Science 1990,249,269. (6) Garrett, B. C.; Truhlar, D. G.; Varandas, A. J. C.; Blais, N. C. Int. J. Chem. Kinet. 1986, 18, 1065 and references therein. (7) Liu, B. J. Chem. Phys. 1973, 58, 1925. 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). (8) Varandas, A. J. C.; Brown, F. B.; Mead, C. A.; Truhlar, D. G.; Blais, N. C. J . Chem. Phys. 1987,86, 6258. (9) Park, T. J.; Light, J. C. J. Chem. Phys. 1991, 94, 2946. (10) Bmthroyd, A. I.; Keogh, W. J.; Martin, P. G.; Peterson, M. R. J . Chem. Phys. 1991,95,4343. The parameters for this potential, asdistributed by the authors, differ from those as published in that the parameters have the followingvalues: CII= 7.107064647(+3), c12 = -3.7yi421267(+3), ~ 1 = 3 1.757654624(+2), ~ 1 = 4 -9.725998132(+3), CIS = 4.665086074(+3), ~7.1 = -4.435165986(+1), ~ 2 = 2 1.604477309(+2), ~ 2 = 3 5.805142925(+2), ~ 2= 4 6.892349445(+2), cjl = 6.581730442(+2), ~ 3 2 6.078389739(+1), ~ 4 1 2.885182566 ~ 4 2= -1.728169916, ~ 4 3= -1.119535503(+2), CU = 4.052536250(+1), CSI = 2.540505673(+2), ~ 5 = 2 -5.762083627(+2), ~ 5 = 3 1.295901320(+2), C ~ I 2.131706075(+3), ~ 6 = 2 9.084452020(+3), ~ 6 3 = -1.138253963(+4), ~ 7 1= -3.964298833(+1), ~ 7 7= -5.019979693, ~ 7 3= 2.906541488(-l), ~ 7 4= 1.212943686(+3), ~ 7 5= 4.140463415(+1), cgl = 1.752855549(+2), cg2 = -2.496320107(+1), cg3 3.765413052(+2), cgq parameters have the following values: cl1 = -5.480488130(+1). The 1.917166552(+3), C I =~ -6.542563392(+2), ~ 1 = 3 6.793758367(+1), ~ 1 = 4 -1.694968847(+3), CIS = 6.866649703(+2), C Z = ~ -2.137567948(-l), ~ 2 = 2 4.975938228(-2), c23 = 9.364998295(-2), c24 = -2.444320779(-2), Cjl = -2.863126914(+1), c32 = 5.443219625(-2), C ~ = I 9.673956120(-l), c42 =

The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 8007 -1.160159706, ~ 4 3= -2.424199759(+1), CM 8.569424490, CSI -6.517635862(+3), ~ 5 2 1.518098147(+2), ~ 5 3 -2.706514366(+1), c6I e 4.308392956(+1), C62 -1.234851732(+2), c63 2.320626055(+2), c7l = 1.049541418(+1), ~ 7 2= -2.424169341, ~ 7 3= 1.745646946(-l), ~ 7 4= 2.603615561(+1), ~ 7 5= 1.345799970(+1), C ~ I -6.653710513, cg2 5 -2.576854447(-l), cg3 = 6.172608425(+3), cg4 = -1.328142473(+1). In the present calculations, as well as in all our prior quantal calculations (see ref 45 below), we have used the distributed version. (11) Mielke, S.L.; Lynch, G. C.; Truhlar, D. G.;Schwenke, D. W. Chem. Phys. Lett. 1993,216,441. The value listed in this paper for the moment of inertia of the variational transition state on the LSTH surface should read 1.585(+4) insteadof 1.583(+4). Thecorrectvaluewasusedin thecalculations. (12) Bowman, J. M.; Wagner, A. F. In The Theory of Chemical Reaction Dynamics; Clary, D. C., Ed.; Reidel: Dordrecht, 1986; pp 47-76. (13) Miller, W. H. J. Chem. Phys. 1975, 62, 1899. (14) Marcus, R. A. J . Chem. Phys. 1965, 43, 1598. (15) Marcus, R. A. J. Chem. Phys. 1967, 46, 959. (16) Truhlar, D. G. J. Chem. Phys. 1970.53.2041. (17) Newton, R. G. Scattering Theory of Particles and Waues; McGraw-Hill: New York, 1966; Section 11.3. (18) Sloan, I. H.; Brady, T. J. Phys. Reu. C 1972,6, 701. Brady, T. J.; Sloan, I. H. Phys. Rev. C 1974, 9, 4. (19) Staszewska, G.; Truhlar, D. G. Chem. Phys. Lett. 1986, 130, 341; J. Chem. Phys. 1987,86, 2793. (20) (a) Schwenke, D. W.; Haug, K.; Truhlar, D. G.; Sun, Y.; Zhang, J. 2.H.; Kouri, D. J. J. Chem. Phys. 1987, 91,6080. (b) 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. (21) Schwenke, D. W.; MladenoviC, M.; Zhao, M.; Truhlar, D. G.; Sun, Y.; Kouri, D. J. In SupercomputerAlgorithms for Reactivity, Dynamics, and Kinetics of Small Molecules; Lagan& A., Ed.; Kluwer: Dordrecht, 1989; p 131. (22) Schwenke. D. W.: Truhlar. D. G. In Comoutine Methods in Aodied Sciknces and Enkneerink Glowinski, R., Lichkwsiy, A., Eds.; SiAM: Philadelphia, 1990; p 291. (23) Schlessinger,L.Phys.Rev.1968,167,1411;1968,171,1523. Pieper, S. C.; Wright, J.; Schlessinger, L. Phys. Rev. D 1971, 3, 1419. (24) Sun. Y.:Kouri. D. J.: Truhlar, D. G. Nucl. Phvs. A 1990.508.41~. Sun, Y . ;Kouri, D. J.; Truhlar, D. G.; Schwenke, D. W. Phys. Reu. A 1990, 41, 4857. (25) Schwenke, D. W.; Mielke, S.L.; Truhlar, D. G. Theor. Chim. Acta 1991, 79, 241. (26) Sun, Y.; Kouri, D. J. Chem. Phys. Lett. 1991, 179, 142. (27) Mielke,S. L.;Truhlar,D. G.;Schwenke, D. W.J. Chem.Phys. 1991, 95, 5930. (28) 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, 1994; p 45. (29) Kohn, W. Phys. Rev. 1948,74,1763. Percival, I. C. Proc. Phys. SOC. 1960,76,206. Mott,N. F.;Massey,H.S. W. The TheoryofAtomicCollisions, 3rd ed.; Oxford University Press: London, 1965. Crawford, 0. H. J . Chem. Phys. 1971,55,257. Kanimura, M. Prog. Theor.Phys. 1977,Suppl. 62,236. Miller, W. H.; Jansen op de Haar, B. M. D. D. J . Chem. Phys. 1987,86,2061. McCurdy, C. W.; Resigno, T. N.; Schneider, B. I. Phys. Rev. A 1987, 36, 206 1. (30) Seaton, M. J. Philos. Trans.R.Soc. London A 1953,245,469. Micha, D. A. Ark. Fys. 1965,30,411. Smith, K.; Henry, R. J. W.; Burke, P. G. Phys. Rev. 1966, 147, 21. Miller, W. H. J. Chem. Phys. 1969, 50, 407. Truhlar, D. G.; Abdallah, J., Jr.; Smith, R. L. Adv. Chem. Phys. 1974, 25, 211. (31) Launay, J. M. J. Phys. B 1976, 9, 1823. 1982,86,1192. (b)Hamilton, (32) (a) BariC,Z.;Simons,J.J.Phys.Chem. I. P.; Light, J. C. J . Chem. Phys. 1986, 84, 306. (33) Schwenke, D. W.; Truhlar, D. G. Comput. Phys. Commun. 1984,34, 57. (34) Launay, J. M.; Le Dourneuf, M. Chem. Phys. Lett. 1989,163, 178. (35) Zhang, J. 2.H. J. Chem. Phys. 1991,94, 6047. (36) Mielke, S.L.; Lynch, G. C.; Truhlar, D. G.;Schwenke, D. W. Chem. Phys. Lett. 1993, 213, 10; erratum: 1994, 217, 173. (37) Lynch, G. C.; Halvick, P.; Truhlar, D. G.; Garrett, B. C.; Schwenke, D. W.; Kouri, D. J. Z . Naturforsch. 1989, 44A, 427. Chatfield, D. C.; Friedman, R. S.;Truhlar, D. G.; Garrett, B. C.; Schwenke, D. W. J . Am. Chem. SOC.1991,113,486. Chatfield, D. C.; Friedman, R. S.;Schwenke, D. W.; Truhlar, D. G. J. Phys. Chem. 1992, 96,2414. (38) Schatz, G. C.; Sokolovski, D.;Connor, J. N. L. J . Chem. Phys. 1991, 94, 4311. (39) Wilson, E. B., Jr.; Decius, J. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955; p 11. Herzberg, G. Molecular Spectra and Molecular Structure. II. Infrared and Raman Spectra of Polyatomic Molecules; D. Van Nostrand: Princeton, 1945; p 370. (40) Eyring, H. J . Chem. Phys. 1935,3,107. Waage, E. V.; Rabinovitch, B. S. Chem. Reu. 1970,70, 377. Klots, C. E. J. Chem. Phys. 1993,98,206. (41) See for example: (a) Schatz, G. C.; Sokolovski, D.; Connor, J. N. L. Faraday Discuss. Chem. Soc. 1991, 91, 17. (b) Last, I.; Baram, A.; Szichman, H.; Baer, M. J . Phys. Chem. 1993, 97, 7040. (c) Lagan& A.; Pack, R. T; Parker, G. A. J . Chem. Phys. 1993,99,2269. (d) Auerbach, S. M.; Miller, W. H. J . Chem. Phys. 1994, 100, 1103. (42) Bowman, J. M. Adv. Chem. Phys. 1985,61,115. Sun,Q.;Bowman, J. M.; Schatz, G. C.; Sharp, J. R.; Connor, J. N. L. J . Chem. Phys. 1990,92, 1677. Bowman, J. M. J . Phys. Chem. 1991,95,4960. (43) Wang, D.; Bowman, J. M. J. Chem. Phys. 1993, 98, 6235.

8008 The Journal of Physical Chemistry, Vol. 98, No. 33, 1994 (44) Manthe, U.; Seideman, T.; Miller, W. H. J . Chem. Phys. 1993,99, 10078. (45) Comparable cross-section calculationsondifferingsurfacarhavebeen performed for D H2 at several energies and are presented for the LSTH surface in Zhang, J. Z. H.; Miller, W. H. J. Chem. Phys. 1989,91, 1528, on the DMBE surface in Zhao, M.; Truhlar, D. G.; Schwenke, D. W.; Kouri, D. J. J . Phys. Chem. 1990,94, 7074, and on the BKMP surface in Mielke, S. L.; Truhlar, D. G.; Schwenke, D. W., to be published. High-energy D + H2 calculations are available for the DMBE surface in Mielke, S.L.; Friedman, R. S.;Truhlar, D. G.; Schwenke, D. W.Chem. Phys. Lett. 1992,188, 359; Neuhauser, D.; Judson, R. S.;Kouri, D. J.; Adelman, D. E.; Shafer, N. E.; Kliner, D. A. V.; Zare, R. N. Science 1992,257,519; and Kuppermann, A.; Wu, Y.-S. M.; Chem. Phys. Lett. 1993, 205, 577 and can be compared to results on the BKMP surface given in Keogh, W. J.; Boothroyd, A. I.; Martin, P. G.; Mielke, S.L.; Truhlar, D. G.; Schwenke, D. W. Chem. Phys. Lett. 1992, 195, 144. For the H Dz system, calculations at the same initial conditions have been reported for the LSTH surface in D’Mello, M.; Manolopoulos, D. E.; Wyatt, R. E. J. Chem. Phys. 1991,94,5985 and for the LSTH and BKMP surfaces in Mielke, S.L.; Truhlar, D. G.; Schwenke, D. W. J. Phys. Chem. 1994, 98, 1053. Comparable cross-section calculations for H HZon the

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Mielke et al. LSTH surface have been presented in Zhang, J. Z. H.; Miller, W. H.Chem. Phys. Lett. 1988,153,465; Manolopoulos, D. E.; Wyatt, R. E. Chem. Phys. Lett. 1989,153,123; Zhang, J. 2.H.; Miller, W. H. Chem. Phys. Lett. 1989, 159,130; and Launay, J. M.; Le Doumeuf, M. Chem. Phys. Lett. 1989,163, 178 and may be compared to cross sections on the DMBE surface presented in Mladenovib, M.; Zhao, M.; Truhlar, D. G.; Schwenke, D. W.; Sun, Y.; Kouri,D. J. J . Phys. Chem. 1988,92,7035 and Manolopoulos, D.E.; Wyatt, R. E. J. Chem. Phys. 1990, 92, 810. Comparisons between results on the DMBE and LSTH surfaces for H Hz cross sections at selected total angular momenta J are given in Auerbach, S.M.; Zhang, J. Z. H.; Miller, W. H. J . Chem. Soc., Faraday Trans. 1990,86, 1701. (46) Partridge, H.; Bauchlicher, C.W., Jr.; Stallcop, J. R.; Levin, E. J. Chem. Phys. 1993, 99, 5951. (47) Diedrich, D. L.; Anderson, J. B. Science 1992,258,786. Diedrich, D. L.; Anderson, J. B. J. Chem. Phys. 1994, 100, 8089. (48) Peterson, K. A.; Woon, D. E.; Dunning, T. H., Jr., J. Chem. Phys. 1994, 100, 7410. (49) Truhlar, D. G.; Wyatt, R. E. Annu. Rev. Phys. Chem. 1976, 27, 1. (50) Michael, J. V.; Lim, K. P. Annu. Rev. Phys. Chem. 1993,44,429.

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