J. Phys. Chem. 1994, 98, 10794-10801
10794
An Application of Conventional Transition State Theory To Compute High-pressure Limit Thermal Rate Coefficients for the Reaction: H(D) -t 0 2 == H(D)Oz == OH(D) 0
+
Ronald J. Duchovic' and J. David Pettigrew? Indiana University-Purdue University Fort Wayne, 2101 Coliseum Boulevard East, Fort Wayne, Indiana 46805-1499 Received: June 24, 1994; In Final Form: August 18, 1994@
Several ab initio studies have focused on the minimum energy path region of the hydroperoxyl potential energy surface (PES) (J. Chem. Phys. 1988, 88, 6273) and the saddle point region for H-atom exchange via a T-shaped HO2 complex ( J . Chem. Phys. 1989, 91, 2373). Further, the results of additional calculations (J. Chem. Phys. 1991, 94, 7068) have been reported which, when combined with the earlier studies, provide a global description (but not an analytic representation) of the PES for this reaction. In this work, information at the stationary points of the ab initio PES is used within the framework of conventional transition state theory (TST) applied to both unimolecular and bimolecular processes in the high-pressure limit to compute estimates of the thermal rate coefficients for the forward and reverse reactions. Because these reactions proceed via a bound complex, a simple probability model is utilized to interpret the calculated statistical rate coefficients and to compare the present calculations with both the most recent experimental measurements and the results of quasiclassical trajectory calculations completed on the (analytic) DMBE IV PES (J. Chem. Phys. 1992, 96, 5137).
the forward rate coefficient for reaction 1 differ from one another by approximately a factor of 2. This is an unacceptably large Bimolecular reactions of molecules containing hydrogen, uncertainty for the value of such a critical parameter. carbon, oxygen, and nitrogen play an essential role in all The current effort by NASA to develop both a SCRAMjet hydrogen and hydrocarbon combustion processes occurring (a supersonic RAMjet) propulsion system and a turbojet power under atmospheric conditions. In particular, the reaction plant for various air-breathing, transatmospheric vehicles are but two examples of technologies which depend on an accurate knowledge of combustion reactions.lo1 These engineering efforts have produced several prototypes of single-stage-to-orbit plays a crucial role in these combustion processes. The forward (SSTO) and two-stage-to-orbit (TSTO) vehicles utilizing a direction of reaction 1 is prototypical of a radical-diatom hydrogedoxygen combustion propulsion system. Further, the reaction occurring without a potential energy barrier on a single High Speed Civilian Transport project is exploring a variety of potential energy surface (PES) and proceeding via a relatively hydrocarbon fuel systems in which reaction 1 will play a key deep potential energy well. In this direction, reaction 1 has role. been characterized as the single most important step in combusThe present work addresses two related issues. Firstly, the tion' and is the primary step initiating the chain-branching recently completed ab initio calculations of Walch et phenomenon in the oxidation of molecular hydrogen and of represent the most accurate and most extensive quantum hydrocarbon fuels. Secondly, the reverse direction of reaction chemical analysis of reaction 1. A small subset of this calculated 1 is also prototypical of a neutral-neutral reaction with no data has been utilized by Pastrana et aLgOto recalibrate the potential barrier and again proceeding via a relatively deep analytic DMBE IIIS4PES. This new analytic representation, potential energy well. Further, in this reverse direction, longidentified as DMBE IV, was then utilized in several dynamical range forces seem to play a crucial role in the microscopic investigation^^^^^^ of reaction 1. However, it is fundamentally dynamics.2-8 important to investigate the dynamics of reaction 1 on the In recognition of its critical importance, reaction 1 has been unadjusted ab initio PES of Walch et al. This study, utilizing the subject of numerous experimental studies focusing on transition state theory (TST) applied to both spectroscopic,g-39 thermodynami~,4~-~~ and d y n a m i ~ a l ~ ~ - ~conventional ~ bimolecular and unimolecular (decomposition of the H(D)02 measurements. Several of recommended rate intermediate) processes in the high-pressure limit, represents an coefficients for reaction 1 have been compiled on the basis of initial step in such an investigation. Since the experimental data this extensive experimental work. Complementing the intense experimental activity, there has been a number of ab i n i t i ~ ~ l - ~ ~do indicate that reaction 1 proceeds in both the forward and reverse directions with no barriers to reaction, this chemical and theoretical d y n a m i ~ a l ~studies ~ - l ~ of this reaction. In total, system presents a particularly difficult challenge to conventional these investigations now span more than a quarter century of TST. The theory, by its very assumptions, is only capable of time. Despite this concerted effort to understand reaction 1, in providing an estimated upper bound to the rate coefficients particular, to provide an accurate recommendation for the value governing reaction 1. Hence, this work cannot address all the of the forward rate coefficient, significant discrepancies remain. dynamical issues; the subsequent application of more powerful In fact, at high temperatures, the various estimated values of theoretical tools will be necessary in order to understand the detailed dynamics of reaction 1 on the PES of Walch et al. Graduate research assistant. However, this work does establish some preliminary statistical Abstract published in Advance ACS Abstracts, September 15, 1994.
I. Introduction
al.63364$67
@
0022-365419412098-10794$04.50/0
0 1994 American Chemical Society
An Application of Conventional Transition State Theory
J. Phys. Chem., Vol. 98, No. 42, 1994 10795
bench marks for the analysis of the most recent quantum chemical PES describing reaction 1. The reader should also note that reaction 1 possesses a low-lying excited electronic state (A') which correlates with the ground electronic state of the OH(D) 0 products. This state is not included in the analysis presented here, and it is another component which must be considered in a more extensive theoretical treatment of reaction 1. Secondly, there has been some disagreement in the literature about the use of statistical rate theories as an appropriate tool to study reaction 1. While C ~ b o has s ~ argued ~ that statistical rate theories can be used to estimate rate coefficients for reaction 1, others, notably Miller and along with Leforestier and Miller,99 have emphasized the apparent nonstatistical character of reaction 1. Following the proposal of Larson et a1.,'02we will present in this paper a relatively simple Statistical analysis of reaction 1 and compare our computed rate coefficients with the most recent experimental data53*54 and quasiclassical trajectory calculation^^^ performed using the analytic DMBE IV PES. The plan of this paper is as follows: In section 11, we review briefly the characteristics of the stationary points on the ab initio PES of Walch et al. along with the determination of harmonic and anharmonic vibrational frequencies at selected points of this surface. The underlying assumptions of the TST calculations are presented in section III along with a simple statistical model for reaction 1. The calculated high-pressure limit thermal rate coefficients for H(D) 0 2 OH(D) 0 and OH(D) 0 H(D) 0 2 over a temperature range of 1000-6000 K are presented in section IV. Finally, the paper concludes in section V with a discussion of these results and directions for continuing theoretical investigations.
+
+
+
-
+
+
-
11. Global Potential Energy Surface The recent quantum chemistry calculations of Walch et al. have provided the most complete theoretical description currently available of the molecular geometries and energy relationships which control the progress of reaction 1. An analysis of these data along the constrained energy minimum (CEM) path using full two-dimensional cubic polynomials in roo and 8 (the angle defined by the forming OH bond and the 00 bond) for the H 0 2 channel and rOH and 8 (again the angle between the OH bond and the stretching 00 bond) for the OH 0 channel and full three-dimensional cubic polynomials in the variables roo, rOH, and 8 (the angle determined by the OH and 00 bonds) augmented by quartic terms in each variable (but no quartic cross-terms) at the two saddle points on the PES has yielded optimal geometries and predicted energies and harmonic vibrational frequencies at 13 distinct molecular configurations. Additionally, both harmonic and anharmonic analyses of the molecular vibrations have been completed at the 0 2 and OH asymptotes and at the HO; intermediate, using quartic polynomials in the variables roo (at the 0 2 asymptote) and rOH (at the OH asymptote) and a full three-dimensional quartic polynomial in the variables roo, roH, and 8 at the HO; intermediate. The reader is referred to ref 67 for a detailed discussion of the complete active space selfconsistent field/externally contracted configuration interaction (CASSCFKCI) ab initio calculations and the accompanying vibrational analyses of the asymptotic species, the H-02 and OH-0 transition states, and the HO; intermediate. In Table 1, we have summarized these earlier calculations. The table includes information for the H 0 2 * OH 0 reaction as well as a newly completed analysis of the D 0 2 =+ OD 0 reaction. It should be noted that the OH(D) vibrational
+
+
+
+ +
+
TABLE 1: Stationary Points, Frequencies, Zero-Point Energy, and Moments of Inertiaa Hf02 i
~
~50.000
+
CUI
w2 W3
D E VI
v2 v3
I1 I2
13
D 02 1631.9 2.3329 1597.9 1 1.620 11.620 -
D-02 1518.3 259.97 285.26i 2.5421
-
23.150 19.001 4.1491
OH-0
HOz
H-2
4.0102 2.2853 roo 2.2779 0 120.00 117.28 energyb -150.46392 -150.46303 WI 1631.9 1518.4 316.95 w2 395.631 w3 2.6238 ZPE 2.3329 VI 1597.9 v2 v3 I1 11.620 17.602 14.868 12 11.620 2.7342 13 i
1.8382 2.5197 104.20 -150.54576 3707.0 1449.4 1202.8 9.0909 3355.7 1415.3 1205.6 15.888 15.068 0.81986
1.8416 5.6131 47.159 -150.44589 3811.2 145.30 173.41i 5.6561 -
DO2 2701.0 1212.9 1066.3 7.1194 2516.7 1217.7 1046.3 17.456 15.954 1.5016
OD-0 2774.7 121.66 147.79i 4.1406
-
71.721 71.227 0.49424
-
72.811 71.861 0.95004
OH+O
1.8361 9.oooO 90.000 -150.44485 3764.2 5.3812 3584.1 0.89507 0.89507 -
OD+O 2740.5 3.9176 2645.0 1.6888 1.6888 -
Distances are measured in Bohr, angles in degrees; mi are harmonic frequencies and vi are anharmonic frequencies, both measured in cm-I. Zero-point energies are reported in kcal mol-I; moments of inertia are reported in a m u Bohr2. Energies are from polynomial approximations to the ab initio CASSCF/CCI energies at the stationary points of the PES and are reported in hartrees; 1 hartree = 627.50955 kcal mol-'.
TABLE 2: Frequencies of HydrogenlOxygen Isotopic Combinations SURVIB SPECTRO harmonic harmonic anharmonic anharmonic suecies MRCI" CCIb MRCI" CCIb OI6-OI6 OI6-O1'
1575.0 1551.6 016-018 1530.5 017-017 1527.8 0 17-0 18 1506.4 018-018 1484.7 016-H' 3732.5 OI7-H' 3725.9 018-H1 3720.2 016-HZ 2717.3 017-H2 2708.3 018-H2 2700.3
1631.9 1607.6 1585.8 1583.0 1560.8 1539.4 3764.2 3757.6 3751.8 2740.4 2731.3 2723.3
1545.5 1523.0 1502.7 1500.0 1479.4 1458.5 3529.3 3523.4 3518.3 2609.6 2601.4 2594.0
1597.9 1574.6 1553.7 1551.0 1529.7 1508.1 3584.1 3578.1 3572.8 2645.0 2636.5 2629.0
a Frequencies in cm-' computed from MRCI energies in ref 65 and 67. Frequencies in cm-I computed from CASSCF/CCI energies in ref 67.
frequency, in both the H(D)O; intermediate and the OH(D) diatom, exhibits significant anharmonic character. We have recently extended the vibrational calculations to include various isotopic combinations of 0 2 and OH(D). Using the SuRVIBlo3 and SPECTROIM programs, harmonic and anharmonic vibrational frequencies have been computed using the CASSCFKCI ab initio data and the multireference configuration interaction (MRCI) ab initio data of Walch and D u c h ~ v i cas~ ~well as the MRCI data of Bauschlicher and L a n g h ~ f f .These ~ ~ calculations are summarized in Table 2. The data contained in Table 2 illustrate the ability of the CASSCFI CCI methodology to represent accurately the PES of reaction 1. The reader is again referred to ref 67 for a more detailed comparison between the ab initio calculations and the experimentally measured geometries, frequencies, energies, and ther-
Duchovic and Pettigrew
10796 J. Phys. Chem., Vol. 98, No. 42, 1994 mochemistry of reaction 1. The reader should note, however, that the highly anharmonic character of the OH(D) bond is again very evident. 111. Rate Coefficients-Theory
A simple kinetic model can be written to describe reaction 1:
We write k3[M] = pw where p is a collisional efficiency factor, w is the collision frequency, and [MI is the concentration of the bath gas. In general, the experimentally measured rate coefficient will depend on whether the loss of H(D) and/or 02 is monitored or whether the appearance of OH(D) andor 0 or H(D)02 (the stabilized intermediate) is observed. Historically, experimental investigations of reaction 1 have focused on the forward direction by using a variety of techniques (e.g., laserinduced fluorescence, Lyman-a, atomic resonance absorption spectroscopy) to monitor H-atom concentrations or by deducing a rate coefficient for the forward reaction based on the OH concentration of the products. The very recent experiments of Shin and Michael53monitor the loss of H(D). Consequently, the theoretical analysis presented in this paper begins by addressing the forward direction of reaction 1. Using a simple Lindemann mechanism (so that k-2 in eq 2a is assumed to be zero), the steady state approximation applied to the intermediate species H(D)Oi yields a simple differential expression describing the loss of H(D):
where [I indicates the concentration of the respective species and
In the falloff region, the unimolecular rate coefficients for the decomposition of the H(D)O; intermediate can be expressed in terms of the densities of states as:
(4) where e:(,!?) represents the density of states at each transition state, @(E) represents the density of states of the H(D)O; species, the E: are the critical energies (Le., the minimum energies at which reaction can occur) in the forward and reverse directions, h is Planck’s constant, and a = -1 or 2. The microscopic rate coefficient kl is calculable with high accuracyIo5by using k-I(E) and a ratio of partition functions as:
and g ( E ) represents the population of reacting species with energy E. In contrast to the falloff region, however, each high-pressure limit microscopic rate coefficient contributing to eq 3, along with k-2, can be computed directly and simply from the transition state theory expressions for the unimolecular and bimolecular processes:
where Qi and Q’ are the partition functions of the reacting species and the two transition state species H(D)OO and OH(D)O, respectively, the E; are the critical energies for reaction in each direction, kB is Boltzmann’s constant, and h is Planck’s constant. In the high-pressure limit:
To this point in the discussion, it has been assumed that the intermediate species H(D)Oi is always collisionally stabilized. This is the essential characteristic of the high-pressure regime. However, this stabilized radical species may also thermally decompose, and in fact, this thermal decomposition process plays a critical role under conditions which do not approach the high-pressure limit. The use of a conventional TST model which does not include the thermal decomposition process can yield estimates of the rate coefficient of the forward reaction which differ dramatically from experimental measurements completed under low-pressure conditions. Hence, in order to create a model of reaction 1 which is consistent with the large body of low-pressure regime experimental data, the kinetic mechanism in eq 2 must be expanded to include the reverse of the collisional stabilization step. Following the proposal of Larson et al., let p represent the probability of the thermal decomposition of H(D)O2 to the products OH(D) 0. Then, eq 3b can be written as:
+
kloss =
k,(k, + (1 - P ) P W > k - , k, Po
+ +
Similarly, in the high-pressure limit, eq 8 becomes lim k,,,, = (1 - p ) k y
W--
where
(9)
(10)
The probability p can be calculatedIo2from the following ratio of thermal rate coefficients:
An Application of Conventional Transition State Theory
J. Phys. Chem., Vol. 98, No. 42, 1994 10797 TABLE 3: Rate Coefficients:' H 0 2 OD 0
k2t
+
* = k-l,+ k2,
+
-+
temp
Du
Shin
k,,= B
&-W(E)dE q E-' ~ - l ( E )+ k,(E) + Pw
(12)
where B(E) is a Boltzmann distribution function and the E: represent the critical energies for reaction. (Note that this definition of p requires that the quantity (1 - p ) represents the thermal decomposition of H(D)02 to the reactants H(D) 0 2 . ) The reader should further note that in writing eq 11, Larson et al. assume that there is a simple statistical partitioning between the forward and reverse directions and that the thermal decomposition of the intermediate species is controlled by purely statistical processes. The rate coefficients for the reverse processes, OH(D) 0 * H(D) 02, can be computed via the equilibrium constant for reaction 1:
+
+
+
kfonvard
Kequihbnum
=-
(13)
k r w erse
where kfonvard and kreverserepresent overall forward and reverse rate coefficients for reaction 1. The utilization of conventional TST to calculate kfonvard and kreverSeignores two important characteristics of the reaction: variational effects which displace the reaction bottlenecks away from the ab initio saddle point and dynamical recrossing effects. The contributions of nonstatistical effects to reaction 1 have been noted and discussed p r e v i o u ~ l (although y ~ ~ ~ ~ not ~ ~quantified in terms of conventional TST calculations). The conventional TST expression in the high-pressure limit for Kequlllbnum depends on the four rate coefficients k;, El,ky, and kY2 as follows: TST
K equilibrium .. =-
k?ky
,
kZlky2
1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 4000.0 4500.0 5000.0 5500.0 6000.0
-13.04 -11.95 -11.41 -11.09 -10.87 -10.72 -10.60 -10.51 -10.44 -10.38
-12.94 -11.94 -11.44 -11.14
VarandasC
H + 02 -13.00 -13.26 -11.95 -11.43 -11.52 -11.11 -11.24 -10.90 -11.04
D
1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 4000.0 4500.0 5000.0 5500.0 6000.0
-12.98 -11.97 -11.47 -11.17
0 2
MHBd
(K) Hessle$ MichaelC YF
where
+
-
OH
+ 0 and D
model I
model
II
TST
-12.42 -11.58 -11.16 -10.91 -10.74 -10.63 -10.54 -10.47 -10.41 -10.36 -10.33
-11.18 -10.83 -10.66 -10.55 -10.48 -10.43 -10.40 -10.37 -10.35 -10.33 -10.31
-9.939 -9.727 -9.583 -9.475 -9.388 -9.315 -9.253 -9.199 -9.151 -9.107 -9.067
-12.55 -11.71 -11.28 -11.03 -10.86 -10.74 -10.65 -10.58 -10.52 -10.48 -10.44
-11.32 -10.08 -10.97 -9.868 -10.80 -9.725 -10.70 -9.617 -10.63 -9.530 -10.58 -9.457 -10.54 -9.395 -10.51 -9.341 -10.49 -9.293 -10.47 -9.249 -10.45 -9.209
+ 02 -13.05 -11.22 -11.06
Loglo of the rate coefficients in units of cm+ molecule-1 s-l are reported. Reference 54. Reference 53. Reference 57. e Reference (I
92.
and Bird.'@ Following the recommendations of Neufeld et al., a new algorithm for the calculation of has been added to both KAPPA and UNIMOL. The calculated rate coefficients of this work (reported as loglo(k), where the rate Coefficients for the loss of H-atom, k, are in units of cm3 molecule-' s-') for the forward direction of reaction 1 are summarized in Table 3. The computed values are compared with the recent experimental work of Du and H e ~ s l e r Shin , ~ ~ and Michael,53 and Yu et a1.57 For H 02, the experimental results were computed using the recommended expressions of Du and Hessler (eq 15) and Shin and Michael (eq 16): C;2(2s2)
+
Consequently, the physically correct equilibrium constant characterizing reaction 1 can be expressed as y K ~ ~ ~ ~ b l i u mk = (1.62 f 0.12) x 10-10exp((-7474 f 122)/T) (15) where the empirical parameter y represents the effects which are not included in the simple conventional TST calculation. k = (1.15 f 0.16) x 10-'0e~p((-6917 f 193)/T) (16) At this most elementary level of theory, y must remain a factor where eq 15 is recommended for a temperature range of 960 K which is determined empirically. < T < 5300 K and eq 16 is recommended for the temperature range of 1103 K -= T 2055 K. For D 0 2 , only Shin and IV. Rate Coefficients-Calculations Michael recommend an analytic expression for the rate coefA. H 0 2 and D 0 2 . The TST calculations reported in ficient: this work were completed using the KAPPAIM program and the UNIMOL107suite of programs. The information calculated k = (1.09 f 0.20) x lO-'Oexp((-6937 f 247)/T) (17) from the ab initio PES and utilized by these programs is summarized in Table 1. Both programs used the energies from where eq 17 is recommended for temperature range 1085 K < the polynomial approximations as well as the associated T < 2278 K. In Table 3, the expression of Du and Hessler has moments of inertia and harmonic frequencies at each stationary been used over the range 1000 K < T < 5500 K, while the point of the PES. The symmetry characteristics of the reactants, expressions of Shin and Michael were used over the range 1000 transition states, intermediate, and products have been included K < T < 2500 K. In addition, the results of quasiclassical in these calculations. In computing the rotational partition trajectories by Varandas et dg2 on the analytic DMBE IV PES functions, both programs treated the molecular system as having are included for comparison. one adiabatic two-dimensional external rotor and one fully active The results of this work are reported in the three columns one-dimensional external rotor. Finally, it should be noted that labeled model I, model 11, and TST. The TST values are both programs have been modified to improve the calculation conventional transition state theory calculations in the highof the reduced collision integral C;2(2,2) which is based on Qe pressure limit and, hence, do not account for variational effects, Lennard-Jones parameters c and u. Work by Neufeld et a1.1°8 recrossing effects, and the role of the thermal decomposition identified errors in the original tabulation of Hirschfelder, Curtis, of the intermediate. As a result, the TST values are a factor of
+
+
+
Duchovic and Pettigrew
10798 J. Phys. Chem., Vol. 98, No. 42, 1994 nearly 20 too large at 5500 K and more than a factor of 1200 too large at 1000 K when compared to the recommended experimental values of Du and Hessler for H 0 2 . In the case of D 0 2 , the TST calculations are a factor of nearly 36 too large at 2500 K and a factor of nearly 800 too large at 1000 K when compared to the recommended experimental values of Shin and Michael. The values identified as model I represent an application of the probability model of Larson et al. in which the probability for the thermal decomposition of the stabilized intermediate to OH(D) 0 varies from 0.920 to 0.995 for H 0 2 and from 0.925 to 0.995 for D 0 2 . These ranges for the probabilities were determined by treating p as an empirical parameter (increments of 0.005) in eq 10 and choosing those values of p from the range p = 0.900 to p = 0.995 which, when used in a linear least-squares analysis of kloss (from eq 10) as a function of UT, yielded the best agreement with the recommended experimental value of Du and Hessler for H 0 2 and those of Shin and Michael for D 0 2 . For H 0 2 , the agreement is nearly exact at 5500 K, while the calculated values are larger than the experimental measurements by a factor of slightly more than 4 at 1000 K. In the case of D 0 2 , the calculated values are larger than the recommended experimental values of Shin and Michael by 38% at 2500 K and by a factor of nearly 3 at 1000 K. The column identified as model I1 contains the results of calculations in which the value of p is held fixed at 0.930 and the rate coefficients are calculated from eq 10. These rate coefficients are then approximated by a linear least-squares analysis as a function of 1/T. In the case of H 0 2 , the agreement between the calculated and experimental values is again nearly exact at 5500 K, while the calculated values are larger than the recommended experimental values of Du and Hessler by a factor of nearly 73 at 1000 K. For D 0 2 , the model I1 results are larger than the recommended experimental values of Shin and Michael by a factor of 3 at 2500 K and by a factor of nearly 46 at 1000 K. Finally, the results of quasiclassical trajectory calculations by Varandas et al. on the analytic DMBE IV PES over a temperature range of 1000-3000 K are reported in Table 3. The DMBE IV PES does not exhibit a barrier in the H(D) 0 2 entrance channel. The geometry of the H(D)Oi intermediate is very similar to that reported in Table 1 for the PES of Walch et al., with roo = 2.5143 Bohr, rOH(D) = 1.8345 Bohr, and 8 = 104.29'. However, the stationary point in the OH(D) 0 exit channel of the DMBE IV PES occurs at a somewhat different molecular geometry, with roo = 5.083 Bohr, rOH(D) = 1.820 Bohr, and 8 = 40.2". The reader is referred to ref 90 for a more detailed discussion of the DMBE IV PES. For both H 0 2 and D 0 2 , the rate coefficients calculated from the quasiclassical trajectories are consistently smaller than the experimentally measured values. The agreement between theory and experiment is better for D 0 2 than for H 0 2 . The probabilities used in the model I calculations for H 0 2 and D 0 2 along with the probabilities computed from the theoretical model of Larson et al. are included in Table 4. The high-pressure limit conventional TST rate coefficients were utilized in eq 11 to compute the probabilities of the theoretical model. While the model I calculations agree essentially exactly with the reported experimental rate coefficients at 5500 K, the probability factors ( i e . , the probability for the thermal decomposition of the stabilized intermediate to the OH(D) 0 products) used in model I are significantly larger than those calculated from the simple statistical theory of Larson et al. This discrepancy suggests that the forward direction of reaction
+
+
+
+
+
+
+
+
TABLE 4: Model and Theoretical Probabilities: H 0 2 -OH 0 and D 0 2 - OD 0 H +0 2 D + 02 temp(K) theory" model1 theory" model1
+
+
1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 4000.0 4500.0 5000.0 5500.0 6000.0 a
+
0.003 17 0.0285 0.0788 0.138 0.194 0.243 0.284 0.319 0.347 0.373 0.393
0.995 0.990 0.980 0.970 0.965 0.955 0.950 0.945 0.940 0.930 0.920
0.00576 0.0414 0.103 0.170 0.23 1 0.283 0.326 0.362 0.390 0.4 16 0.435
0.995 0.990 0.980 0.965 0.960 0.950 0.945 0.940 0.935 0.930 0.925
Reference 102.
+
t
+
+
+
+
+
+
+
+
+
+
+
+
ais
as
a56
a10
am
as
1.00
1 / T x loo0 /K-'
+
Figure 1. H 0 2 * HOf OH + 0. Solid line with error bars, experimental data; open circles, model I (this work); open squares, model I1 (this work). @
1 cannot be described quantitatively by a theory (either conventional TST or the proposed model of Larson et al.) which is based on simple statistical assumptions. Hence, the quantitative results presented in this work support the conclusions of Miller and along with those of Leforestier and Millerw which call into question the ability of simple statistical theories to represent accurately the rate characteristics of reaction 1. The use of the highly accurate ab initio calculations of Walch et al. suggests that reaction 1, at least in the forward direction, exhibits dynamical properties not well-represented by simple statistical rate theories. In Figures 1 and 2, the rate coefficients for the forward direction of reaction 1 are plotted (log&) as a function of lo3/ r). For H 0 2 (Figure l), the experimental curve has been extrapolated to 6000 K and includes the experimental error bars. Model I values (open circles) and model 11values (open squares) are plotted over the temperature range of 1000-6000 K for comparison with the experimental measurements. For D 0 2 (Figure 2), the experimental curve, including the appropriate error bars, has been extrapolated to 6000 K to facilitate comparison with the model I (open circles) and model I1 (open squares) calculations. B. OH 0 and OD 0. In Table 5 , the recommended experimental rate coefficients of Shin and Michael over a temperature range of 1000-2500 K are compared with TST calculations of the rate coefficients, with a model calculation which utilizes eq 13 and an empirically corrected equilibrium constant, and, finally, with the quasiclassical trajectory calculations of Varandas et al. on the analytic DMBE IV PES. The
+
+
+
+
J. Phys. Chem., Vol. 98, No. 42, 1994 10799
An Application of Conventional Transition State Theory D -L
DO;
e OD t o
OH t
no
1
-'w
,
a5o
aIo
a3o
am
a#
am
am
+
-
+
TABLE 5: Rate Coefficients:" OH 0 H 0 +O-D+Oz temp (K) Shin Michaelb Varandas' model OH+O -10.65 -10.49 -10.57 1000.0 -10.66 -10.80 -10.73 1500.0 - 10.74 -10.80 -10.81 2000.0 - 10.79 -10.83 -10.86 2500.0 (-10.82) -10.83 3000.0 (-10.85) 3500.0 (-10.87) 4000.0 (-10.88) 4500.0 (-10.89) 5000.0 (-10.90) 5500.0 (-10.91) 6000.0 OD+O -10.81 -10.71 -10.78 1000.0 -10.78 -10.86 1500.0 -10.82 - 10.90 2000.0 - 10.92 -10.84 - 10.92 2500.0 (-10.85) -10.93 3000.0 (-10.87) 3500.0 (-10.87) 4000.0 (-10.88) 4500.0 (-10.88) 5000.0 (-10.89) 5500.0 (-10.89) 6000.0
2
and OD TST
-9.758 -9.938 - 10.05 - 10.07 -10.08 -10.09 -10.10 -10.10 -10.10 -10.10 -10.11 -10.09 -10.16 -10.20 -10.21 -10.22 -10.22 -10.22 -10.22 -10.23 -10.23 -10.23
L1 The loglo of the rate coefficients in units of cm-3 molecule-' are reported. Reference 53. Reference 92.
s-l
recommended expressions of Shin and Michael were used to calculate the rate coefficients for OH 0 * H 0 2 (eq 18) and for OD 0 ==D 0 2 (eq 19):
+
+
k = (8.75 f 1.24) x
+
exp((ll21
+
f 193)/7') (18)
k = (9.73 f 1.79) x 10-12exp((526 f 247)/7')
(19)
where eq 18 is recommended for a temperature range of 1103 K < T < 2055 K and eq 19 is recommended for the temperature range of 1085 K < T < 2278 K. In Table 5 , the expressions of Shin and Michael were used over the range 1000 K -= T < 2500 K. The results of this work are reported in the columns labeled model and TST of Table 5 . The TST values are conventional transition state theory calculations in the high-pressure limit. The TST values are a factor of 6.2 too large at 2500 K and a factor of 6.5 too large at 1000 K when compared to the recommended experimental values of Shin and Michael for OH
H tO2
I
am 1 / T x loo0
I
am
I
am
1.m
/K"
+
Figure 3. OH + 0 HO; H 0 2 . Solid line with error bars, experimental data; open squares, TST calculations (this work); open circles, model (this work); open triangles, ref 45. @
OD 0. Solid line with error bars, Figure 2. D + 0 2 * DO; experimental data; open circles, model I (this work); open squares, model I1 (this work).
+
1
ado
1.a
1 / T x loo0 /K"
G H0;
a
-1i.m
ais
O
+
+
0. In the case of OD 0, the TST calculations are a factor of 5.1 too large at 2500 K and a factor of 4.9 too large at 1000 K when compared to the recommended experimental values of Shin and Michael. The values identified as model were calculated from eq 13 using model I values for kfonvard and an empirically corrected equilibrium constant. The factor y was determined by requiring the calculated reverse rate to be 20% greater than the experimental rate. (This range represents a commonly achieved level of agreement between theoretical calculations and experimental measurements.) For the temperature range 1000 K < T < 2500 K, the actual experimental data were used to determine the value of y , while for temperatures > 2500 K, a simple linear extrapolation using a least-squares analysis of the model calculations between 1000 and 2500 K as a function of 1/T yielded the rate coefficients to 6000 K. The values above 2500 K are parenthesized in Table 5 to indicate that they represent an extrapolation based on the experimental data at lower temperatures. Finally, the results of quasiclassical trajectory calculations by Varandas et al. on the analytic DMBE IV PES over a temperature range of 1000-3000 K are reported in Table 5. Unlike the behavior observed in the case of the forward reaction, the rate coefficients calculated from the quasiclassical trajectories are not consistently smaller than the experimentally 0, the measured values. Beginning at 2000 K for OH quasiclassical trajectory estimates of the rate coefficients are slightly larger than the reported experimental values. In the 0, the trajectory estimate equals the rate case of OD coefficient reported experimentally at 2500 K. In Figures 3 and 4, the rate coefficients for the reverse direction of reaction 1 are plotted (loglo(k) as a function of lo3/ r). For OH 0 (Figure 3), the experimental curve includes the recommended error bars. TST values (open squares) and model values (open circles) are plotted over the temperature range of 1000-2500 K for comparison with the experimental measurements. In addition, the values of the rate coefficient recommended by Cobos et aZ.45(open triangles) are included for comparison. Note that the recommendation of Cobos et al. is temperature independent over the range 1000 K < T < 2500 K. For OD 0 (Figure 4), the experimental curve, including the appropriate error bars, is compared with the TST calculations (open squares) and the model (open circles) calculations. It is interesting to note that the unadjusted TST calculations for both OH 0 and OD 0 exhibit nearly the same temperature
+
+
+
+
+
+
-~~,
,
10800 J. Phys. Chem., Vol. 98, No. 42, 1994 OD
t o e
DO;
e D t02
-tam
I
I
Duchovic and Pettigrew to achieve a quantitative understanding of reaction 1 (at least in the forward direction) and that there are significant nonstatistical factors which control the course of reaction 1. Subsequent investigations may permit an analysis and quantification of these nonstatistical factors. The direction for future work lies in the more complete utilization of the extensive ab initio data of Walch et al. In particular, work in progress is directed toward the construction of a new analytic representation of the PES for reaction 1 using a switching function formalism and including long-range electrostatic forces in the OH(D) 0 channel. Such an analytic representation of the PES will permit the quantification of variational effects, dynamical recrossing effects, and, finally, the apparent nonstatistical factors involved in reaction 1 through the application of more sophisticated theoretical tools (variational TST, statistical adiabatic channel, master equation, quasiclassical trajectory, and quantum scattering calculations). Further, these additional tools may allow a more detailed analysis of the very simple, one-parameter model of Larson et al. (which yields excellent agreement between calculated and experimental rate coefficients at the highest temperatures) and may suggest a nonstatistical interpretation of that model. Finally, while previous work63,64,67 indicates that the ab initio PES used in this study is highly accurate, the PES may also play a role in the failure of the statistical model proposed by Larson et al. The ab initio PES does possess a very small barrier in the H(D) 0 2 channel of reaction 1. The experimental data suggests that reaction 1 proceeds without baniers in both directions. Hence, the small barrier on the ab initio PES reflects a limitation in those calculations. Also, the role of long-range forces participating in reaction 1 and not represented in the ab initio PES has not been considered here. Lastly, as noted earlier, reaction 1 possesses a low-lying excited electronic state (A') which correlates with the ground electronic state of the OH(D) 0 products. It has been argued by Troello that this state must be included in any analysis of the rate coefficients of reaction 1. Again, this issue requires continued investigation.
+
l / T x loo0
/K-l
+ 0 = DO; = D + 0 2 . Solid line with error bars, experimental data; open squares, TST calculations (this work); open circles, model (this work). Figure 4. OD
dependence (Le., nearly the same slope) as the experimental values over the temperature range 1000 K < T < 2500 K.
V. Discussion and Conclusions Estimates of the rate coefficients in both the forward and reverse directions of reaction 1 have been computed in the highpressure limit over the temperature range 1000 K < T < 6000 K using conventional TST. The TST calculations have been compared with previously reported experimental measurements and quasiclassical trajectory results. The rate calculations completed in this work have utilized the highly accurate data of the most extensive ab initio study of reaction 1 completed to date. Further, the computed rate coefficients have been compared to the predictions of a simple statistical model which had been proposed to represent combustion reactions proceeding via an intermediate complex. The conventional TST calculations exhibit poor agreement with the recommended experimental rate coefficients for both the forward and reverse directions of reaction 1. However, the TST calculations for the reverse process (for both H and D) show much better agreement with the experimental values than do the TST calculations for the forward direction of reaction 1. In particular, over the temperature range of 1000 K < T < 2500 K, the TST calculations almost parallel the recommended experimental values (see Figures 3 and 4). The fact that the conventional TST calculations do not include variational effects and dynamical recrossing effects makes this lack of agreement unremarkable. However, the conventional TST calculations were completed using the unmodified ab initio data of Walch et al. They will serve as a quantitative point of comparison for the application of more sophisticated rate theories to this same ab initio data in the study of reaction 1. Of greater interest is the failure of the simple modification of conventional TST proposed by Larson et al. when it is applied to high-pressure limit rate coefficients. While quantitative agreement between the estimates of the rate coefficients calculated from this model and the recommended experimental values of the rate coefficients for reaction 1 in the forward direction can be achieved at high temperatures, this agreement occurs only by assuming that the probability for the decomposition of the stabilized H(D)02 intermediate to the OH(D) 0 products is greater than 0.900. This value is inconsistent with the predictions of the simple statistical model applied to highpressure limit rate coefficients. As noted earlier, this result suggests that the simple statistical rate theories are inappropriate
+
+
+
Acknowledgment. R.J.D. wishes to acknowledge helpful conversations with a number of colleagues: Professor R. S. Friedman, Dr. G. Lendvay, Dr. C. E. Dateo, and Professor S. J. Klippenstein. Professor Friedman's careful and critical reading of this manuscript is greatly appreciated. Dr. Dateo of the NASA Ames Research Center is currently working to develop a new analytic representation for the PES of reaction 1. In addition, R.J.D. thanks undergraduate research assistants Jolene Elett and Gregory Reed for their participation in this project. Finally, acknowledgment is made to the Donors of The Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. R.J.D. also thanks Indiana University-Purdue University Fort Wayne (IPFW)for its partial support of this research in the form of a Faculty Summer Research Grant and the Chemistry Department of IPFW for providing the computational resources for this work. References and Notes (1) Miller, J. A. J . Chem. Phys. 1986, 84, 6170. (2) Claw, D. C.: Werner. H. J. Chem. Phvs. Lett. 1984, 112. 346. (3) (a) Clary, D. C. Mol. Phys. 1984, 3,-53. (b) Clary, D. C. Mol. Phys. 1985, 54, 605. (4) Clarv, D. C.: Smith. D.: Adam. N. G.Chem. Phvs. Leu. 1985, 119, 320. ( 5 ) Varandas, A. J. C. Faraday Discuss. Chem. Sac. 1987, 84, 353. (6) Graff, M. M.; Wagner, A. F. J. Chem. Phys. 1990, 92, 2423. (7) Philips, L. F. J. Phys. Chem. 1990, 94, 7482. (8) Varandas, A. J. C.; Marques, J. M. C. J. Chem. Phys. 1992, 97, 4050. (9) Troe, J. J . Phys. Chem. 1986, 90, 3485.
An Application of Conventional Transition State Theory (10) Dieke, G. H.; Crosswhite, H. M. J . Quant. Spectrosc. Radiat. Transfer 1962, 2, 97. (11) Foner, S. N.; Hudson, R. L. J . Chem. Phys. 1962, 36, 2681. (12) Milligan, D. E.; Jacox, M. E. J . Chem. Phys. 1963, 38, 2627. (13) Getzinger, R. W.; Blair, L. S. Combust. Flame 1969, 13, 271. (14) Krupenie, P. H. J . Phys. Chem. Ref. Data 1972, 1, 423. (15) Friswell, N. S.; Sutton, M. M. Chem. Phys. Lett. 1972, 15, 108. (16) Jacox, M. E.; Milligan, D. E. J . Mol. Spectrosc. 1972, 42, 495. (17) Paukert, T. E.; Johnston, H. S. J. Chem. Phys. 1972, 56, 2824. (18) Kurylo, M. J. J . Phys. Chem. 1972, 76, 3518. (19) Ahumada, J. J.; Michael, J. V.; Osbome, D. T. J. Chem. Phys. 1972, 57, 3736. (20) Smith, D. W.; Andrews, L. J . Chem. Phys. 1974, 60, 81. (21) Wong, W.; Davis, D. D. Int. J . Chem. Kinet. 1974, 6, 401. (22) Beers, Y.; Howard, C. J. J . Chem. Phys. 1976, 64, 1541. (23) Slack, M. W. Combust. Flame 1977, 28, 241. (24) Johns, J. W. C.; McKellar, A. R. W.; Riggin, M. J. J. Chem. Phys. 1978, 68, 3957. (25) Barnes, C. E.; Brown, J. M.; Canington, A,; Pinkstone, J.; Sears, T. J.; Thistlewaite, R. J. J . Mol. Spectrosc. 1978, 72, 86. (26) Coxon, J. A.; Sastry, K. V. L. N.; Austin, J. A,; Levy, D. H. Can. J . Phys. 1979, 57, 619. (27) Coxon, J. A. Can. J . Phys. 1980, 58, 933. (28) Howard, C. J. J . Am. Chem. Soc. 1980, 102, 6937. (29) Nagai, K.; Endo, Y.; Hirota, E. J . Mol. Spectrosc. 1981, 89, 520. (30) Charo, A.; De Lucia, F. C. J . Mol. Spectrosc. 1982, 94, 426. (31) Yamada, C.; Endo, Y.; Hirota, E. J . Chem. Phys. 1983, 78,4379. (32) Lubic, K. G.; Amano, T.; Uehara, H.; Kawaguchi, K.; Hirota, E. J . Chem. Phys. 1984, 81, 4826. (33) Uehara, H.; Kawaguchi, K.; Hirota, E. J . Chem. Phys. 1985, 83, 5479. (34) Jacox, M. E. J . Phys. Chem. Ref. Data 1988, 17, 269. (35) Christensen, H.; Sehested, K. J . Phys. Chem. 1988, 92, 3007. (36) Schwab, J. J.; Brune, W. H.; Anderson, J. G. J . Phys. Chem. 1989, 93, 1030. (37) Nelson, D. D.; Zahniser, M. S. J . Mol. Spectrosc. 1991, 150, 527. (38) Adhikari, N.; Hamilton, I. J . Phys. Chem. 1991, 95, 6470. (39) Burkholder, J. B.; Hammer, P. D.; Howard, C. J.; Towle, J. P.; Brown, J. M. J . Mol. Spectrosc. 1992, 151, 493. (40) Sawyer, D. T. J . Phys. Chem. 1989, 93, 7977. (41) Fisher, E. R.; Armentrout, P. B. J . Phys. Chem. 1990, 94, 4396. (42) Holmes, J. L.; Lossing, F. P.; Mayer, P. M. J . Am. Chem. Soc. 1991, 113, 9723. (43) Schott, G. L. Combust. Flame 1973, 21, 357. (44) Frank, P.; Just, Th. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 181. (45) Cobos, C. J.; Hippler, H.; Troe, J. J . Phys. Chem. 1986, 89, 342. (46) Hsu, K.-J.; Durant, J. L.; Kaufman, F. J. Phys. Chem. 1987, 91, 1895. (47) Fujii, N.; Shin, K. S. Chem. Phys. Lett. 1988, 151, 461. (48) Pirraglia, A. N.; Michael, J. V.; Sutherland, J. W.; Klemm, R. B. J . Phys. Chem. 1989, 93, 282. (49) Hsu, K.-J.; Anderson, S. M.; Durant, J. L.; Kaufman, F. J . Phys. Chem. 1989, 93, 1018. (50) Masten, D. A.; Hanson, R. K.; Bowman, C. T. J. Phys. Chem. 1990, 94, 71 19. (51) Yuan, T.; Wang, C.; Yu, C.-L.; Frenklach, M.; Rabinowitz, M. J. J . Phys. Chem. 1991, 95, 1258. (52) Rubahn, H.-G.; van der Zande, W. J.; Zhang, R.; Bronikowski, M. I.; Zare, R. N. Chem. Phys. Lett. 1991, 186, 154. (53) Shin, K. S.; Michael, J. V. J . Chem. Phys. 1991, 95, 262. (54) Du, H.; Hessler, J. P. J . Chem. Phys. 1992, 96, 1077. (55) Hughes, K. J.; Lightfoot, P. D.; Pilling, M. J. Chem. Phys. Lett. 1992, 191, 581. (56) Carleton, K. L.; Kessler, W. J.; Martinelli, W. J. J . Phys. Chem. 1993, 97, 6412. (57) Yu, C.-L.; Frenklach, M.; Masten, D. A,; Hanson, R. K.; Bowmam, C. T. J . Phys. Chem. 1994, 98, 4770. (58) Baulch, D. L.; Cox, R. A.; Cutzen, P. J.; Hampson, R. F., Jr.; Kerr, J. A.; Troe, J.; Watson, R. T. J . Phys. Chem. Ref. Data 1982, 11, 327. (59) Cohen, N.; Westberg, R. Chemical Kinetic Data Sheets for HighTemperature Chemical Reactions. Aerospace Report No. ATR-82(7888)3; Aerospace Corporation: El Segundo, CA, 1982. (60) Cohen, N.; Westberg, K. R. J . Phys. Chem. Re5 Data 1983, 12, 531. (61) Melius, C. F.; Blint, R. L. Chem. Phys. Lett. 1979, 64, 183. (62) Komomicki, A.; Jaffe, R. L. J . Chem. Phys. 1979, 71, 2150. (63) Walch, S. P.; Rohlfing, C. M.; Melius, C. F.; Bauschlicher, C. W. J . Chem. Phys. 1988, 88, 6273. (64) Walch, S. P.; Rohlfing, C. M. J . Chem. Phys. 1989, 91, 2373. (65) Bauschlicher, C. W.; Langhoff, S. R. J . Chem. Phys. 1989, 90, 3230. (66) Francisco, J. S.; Zhao, Y. Mol. Phys. 1991, 72, 1207. (67) Walch, S. P.; Duchovic, R. J. J . Chem. Phys. 1991, 94, 7068. (68) Sicilia, E.; Russo, N. Mol. Phys. 1992, 76, 1025.
J. Phys. Chem., Vol. 98, No. 42, 1994 10801 (69) Bunker, P. R.; Hamilton, I. P.; Jensen, P. J . Mol. Spectrosc. 1992, 155, 44. (70) Bauschlicher, C. W.; Partridge, H. Chem. Phys. Lett. 1993, 208, 241. (71) Dobbs, K. D.; Dixon, D. A. J . Phys. Chem. 1994, 98, 4498. (72) Lauderdale, W. J.; Cheng, V. G.; Wierschke, S. G. J. Phys. Chem. 1994, 98, 4502. (73) Gauss, A., Jr. J . Chem. Phys. 1978, 68, 1689. (74) Blint, R. L. J . Chem. Phys. 1980, 73, 764. (75) (a) Miller, J. A. J . Chem. Phys. 1981, 74, 5120. (b) Miller, J. A. J . Chem. Phys. 1981, 75, 5349. (76) Bottomley, M.; Bradley, J. N.; Gilbert, J. R. Int. J . Chem. Kinet. 1981, 13, 957. (77) Miller, J. A.; Brown, N. J. J . Phys. Chem. 1982, 86, 772. (78) Rai, S. N.; Truhlar, D. G. J . Chem. Phys. 1983, 79, 6046. (79) Kleinermanns, K.; Schinke, R. J . Chem. Phys. 1984, 80, 1440. (80) Lemon, W. J.; Hase, W. L. J. Phys. Chem. 1987, 91, 1595. (81) Ju, G.; Feng, D.; Cai, Z.; Dong, C. Theoret. Chim. Acta 1988, 74, 403. (82) Davidsson, J.; Nyman, G. Chem. Phys. 1988, 125, 171. (83) Quintales, L. A. M.; Varandas, A. J. C.; Alvarino, J. M. J . Phys. Chem. 1988, 92, 4552. (84) Varandas, A. J. C.; Brandao, J.; Quintales, L. A. M. J. Phys. Chem. 1988, 92, 3732. (85) Cobos, C. J. Chem. Phys. Lett. 1988, 152, 371. (86) Markovic, N.; Nyman, G.; Nordholm, S. Chem. Phys. Lett. 1989, 159, 435. (87) Schneider, F.; Zuelicke, L.; Chapuisat, X. Mol. Phys. 1990, 71, 17. (88) Davidsson, J.; Nyman, G. J . Chem. Phys. 1990, 92, 2407. (89) Myman, G.; Davidsson, J. J . Chem. Phys. 1990, 92, 2415. (90) Pastrana, M. R.; Quintales, L. A. M.; Brandao, J.; Varandas, A. J. C. J . Phys. Chem. 1990, 94, 8073. (91) Jacobs, A.; Volpp, H. R.; Wolfrum, J. Chem. Phys. Lett. 1991,177, 200. (92) Varandas, A. J. C.; Brandao, J.; Pastrana, M. R. J. Chem. Phys. 1992, 96, 5137. (93) Zuhrt, C.; Zuelcke, L.; Chapuisat, X. Chem. Phys. 1992, 166, 1. (94) Klimo, V.; Bittererova, M.; Biskupic, S.; Urban, J.; Nicov, M. Collect. Czech. Chem. Commun. 1993, 58, 234. (95) Klimo, V.; Bittererova, M.; Biskupic, S.; Urban, J. Chem. Phys. 1993, 173, 367. (96) Nyman, G. Chem. Phys. 1993, 173, 159. (97) Pack, R. T; Butcher, E. A,; Parker, G. A. J . Chem. Phys. 1993,99, 9310. (98) Lauderdale, W. J.; Cheng, V. G.; Wierschke, S. G. J. Phys. Chem. 1994, 98, 4502. (99) Leforestier, C.; Miller, W. H. J . Chem. Phys. 1994, 100, 733. (100) Varandas, A. J. C. J . Chem. Phys. 1993, 99, 1076. (101) Jachimowski, C. J. An Analytical Study of the Hydrogen-Air Reaction Mechanism With Application to Scramjet Combustion. NASA TP-2791; 1988. (102) Larson, C. W.; Stewart, P. H.; Golden, D. M. Znt. J. Chem. Kinet. 1988, 20, 27. (103) Harding, L. B.; Ermler, W. C. J . Compur. Chem. 1985, 6, 13. SURVIB: ANLIStevens Tech Surface Fitting and Vibrational Analysis FORTRAN Program. (104) Gaw, J. F.; Willetts, A.; Green, W. H.; Handy, N. C. In Advances in Molecular Vibrations and Collision Dynamics; Bowman, J. M., Ed.; JAI Press: Greenwich, CT, 1990. Willetts, A.; Gaw, J. F.; Green, W. H., Jr.; Handy, N. C. SPECTRO: A FORTRAN program for the derivation of spectroscopic constantsfrom provided quarticforce fields and cubic dipole fields, Version 1.2; 1990. (105) Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and Recombination Reactions; Blackwell: Oxford, 1990; p 304. (106) Wagner, A. F. (with modifications by Duchovic, R. J.) KAPPA: A FORTRAN program to calculate rate coeffcients for unimolecular and recombination reactions. Unpublished. (107) Gilbert, R. G.; Jordan, M. J. T.; Smith, S. C. UNZMOL: A suite of FORTRAN programs to calculate rate coefficientsfor unimolecular and recombination reuctions. Unpublished; available from the authors. (108) Neufeld, P. D.; Janzen, A. R.; Aziz, R. A. J . Chem. Phys. 1972, 57, 1100. (109) Hirschfelder, J. 0.;Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. (1 10) Trce, J. Twenty-Second Symposium (Intemational) on Combustion; The Combustion Institute: Pittsburgh, 1988.
