J . Phys. Chem. 1988, 92, 2462-2471
2462
of conditions of the current investigation. Future experiments are planned with helium and other bath gases to extend the present knowledge of the role of collisions in this imDortant recombination reaction.
Acknowledgment. We gratefully acknowledge useful discussions with Albert F. Wagner. David M. Wardlaw. and William C. Gardiner. We also tcank'Paul F. Sawyer for developing the
computer codes used in the data analysis and Dariusz Sarzynski for his assistance in performing experiments (both at IIT). The research at IIT was supported by the Office of Basic Energy Sciences. Division of Chemical Sciences. U.S. DeDartment of Energy under Contract DE-AC02-78ER-14593 and tiat at Oxford by S.E.R.C. Registry No. CH3, 2229-07-4; acetone, 67-64-1
Study of the Recomblnation Reaction CH, 4- CH,
-
C,H,.
2. Theory
Albert F. Wagner* Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439
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and David M. Wardlaw+ Department of Chemistry, Queen's University, Kingston, Ontario, K7L 3N6 Canada (Received: August 1 1 , 1987; In Final Form: December 15, 1987)
A microcanonical variational RRKM rate constant calculation, based on the flexible transition-state theory of Wardlaw and Marcus (Chem. Phys. Lett. 1984, 110, 230) and an adjustable semiempirical potential energy surface, is compared to the measurement of part 1 (preceding paper in this issue) and other experimental studies. The calculationscontain two adjustable parameters: a potential parameter a,which influences the tightness of the transition state, and (AE),,,, the total average energy change in metastable C2H6*per collision with the buffer gas. When the parameters are optimized, the resulting calculated rate constants have a 9.9% root mean square error with respect to the measurements in Ar buffer gas. The final parameter values are a = 0.70 i 0.13 k'and (AE)to,= -205 & 65 cm-'. Similar calculations for He buffer gas produce 22.4% error when compared to the less extensive measurements and give an optimized (AE),,, value of about -135 cm-'. The rate calculations for Ar buffer gas have been extended to 2000 K and then fit over the full temperature and pressure range to a modification of the functional form of Gilbert et al. (Eer. Bunsen-Ges.Phys. Chem. 1983,87, 161). The resulting cm3/(molecules), F , = 0.381e-T/73.2 + 0.619e-T/'180, fitted functions required by that form are ko = 8.76 X 10-7T703e-'390/T[Ar] and k , = 1.50 X 10-7T1~18e-329/Tcm3/(molecule s). When used in the Gilbert et al. function, this parametrization produces an excellent representation of the theoretical and experimental rate constants in Ar.
I. Introduction The recombination of methyl radicals in the presence of buffer gas M (reaction 1) has been and continues to be a reaction of CH3 + CH3 + M CZH6 + M (1) considerable interest to both experimentalists and theoreticians in the field of chemical kinetics. This elementary reaction has been the subject of numerous laboratory studies'-' because of its importance as a major termination process in the pyrolysis and oxidation of hydrocarbon~,82~ its frequent use as a reference reaction in kinetic studies of radical-molecule reactions,I0," and its popularity as a model for testing and improving statistical theories of unimolecular r e a ~ t i o n s . ~ J ~ - ' ~ Most of the experimental studies of reaction 1 to date have been conducted at or near the high-pressure limit.1~2~4 The best available measurement of this limit is in ref 4, where Hippler, Luther, Ravishankara, and Troe (HLRT) have measured k l at ambient temperature in Ar from 1 to 200 atm of total pressure. There have been three experimental studies of reaction 1 in the falloff region. The two earliest studies both used argon as the collision partner. Glanzer, Quack, and Troe3 (GQT) measured the apparent bimolecular rate constant k , at temperatures close to 1350 K and pressures from 2 to 20 atm. More recently, Macpherson, Pilling, and (MPS) studied reaction 1 as a function both of temperature (296-577 K) and of pressure (5-500 Torr). In the MPS study, the rate constants obtained at each temperature were fit to a pressure-dependent functional form for recombination rate constants introduced by TroeI7 to obtain as comprehensive
-
'Natural Science and Engineering Research Council of Canada, University Research Fellow.
0022-3654/88/2092-2462$01 S O / O
a picture as possible of the high-pressure limit and the falloff behavior. At room temperature, this high-pressure limit agrees closely with that directly measured by HLRT. The most recent study of the falloff behavior is described in the collaborative preceding paper,7 hereafter designated part 1,
( I ) Baulch, D. L.; Duxbury, J. Combust. Flame 1980, 37, 313. (2) Skinner, G. B.; Rogers, D.; Patel, K. B. In?. J . Chem. Kine?. 1981, 13, 481. (3) Glanzer, K.; Quack, M.; Troe, J. Chem. Phys. Lert. 1976, 39, 304. (4) Hippler, H.; Luther, K.; Ravishankara,A. R.; Troe, J. Z . Phys. Chem. Neue Folge 1984, 142, 1. (5) Macpherson, J. T.; Pilling, J. J.; Smith, M. J. C. Chem. Phys. Lett. 1983, 94, 430. (6) Macpherson, J. T.; Pilling, J. J.; Smith, M. J. C.J . Phys. Chem. 1985, 89, 2268. (7) Slagle, I. R.; Gutman, D.; Davies, J. W.; Pilling, M. J., preceding paper in this issue. (8) Warnatz, J. In Combustion Chemisfry; Gardiner, W. C., Ed.; Springer-Verlag: New York, 1984; Chapter 5 . (9) Westbrook, C. K.; Dryer, E. L. Prog. Energy Combust. Sci. 1984, 10,
1.
(10) Kerr, J. A. In Free Radicals; Kochi, J. K., Ed.: Wiley-Interscience: New York, 1973; Vol. 1, Chapter 1. (11) Kerr, J. A.; Moss, S. J. CRC Handbook of Bimolecular and Termolecular Gas Reactions; CRC Press: Boca Raton, FL, 198 1. (12) Waage, E. V.; Rabinovitch, B. S. Inr. J . Chem. Kiner. 1971, 3, 105. See also references in the following: Olson, D. B.; Gardiner, W. C. J . Phys. Chem. 1979,83, 922. (13) Quack, M.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 240. (14) Hase, W. L. J . Chem. Phys. 1976, 64, 2442. ( 1 5 ) Wardlaw, D. M.; Marcus, R. A. J . Phys. Chem. 1986, 90, 5383. (16) Wardlaw, D. M.; Marcus, R. A. J . Chem. Phys. 1985, 83, 3462. (17) Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 161.
0 1988 American Chemical Society
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Theoretical Study of C H 3 + CH3
-
C2H6
in which Slagle et al. measured kl over a wide temperature range (296-906 K) and pressure range (1-600 Torr). Both Ar and H e buffer gases were used. For the Ar bath gas, some of the lower temperatures of the study were chosen to be the same as those employed by MPS. This overlap of conditions was planned in order to create an extended view of the falloff behavior of reaction 1 for a particular bath gas over a wide pressure range. Part 1 lists 130 new and MPS-measured rate constants spanning nine different temperatures. There have been several previous theoretical studies of CH3 self-recombination or of the reverse process of the thermal dissociation of ethane. These include conventional RRKM calculations3J2where the transition state is fixed at some point along the reaction path, a statistical adiabatic channel model calculation,I3 and a variational RRKM c a l ~ u l a t i o nwhere ~ ~ the location of the transition state is temperature dependent.I8 All of these studies can be ~ r i t i c i z e d 'for ~ both an incomplete treatment of hindered rotor effects and an inability to quantitatively, and often qualitatively, reproduce experimental information on the highpressure limit. In this study, values of k , were calculated by using a microcanonical variational version of RRKM theory, here denoted flexible transition-state theory (FTST). The method was developed by Wardlaw and M a r c ~ s ' ~and ~'~ applied ~ ' ~ by themI5 to calculate k l at the high-pressure limit. The theoretical model yielded limiting values of k , that were in good agreement with experiment, both in magnitude and temperature dependence. In the current study, this same model, including the potential energy surface of ref 15, is extended to include the role of collisions in the falloff region. In section I1 of this paper, the potential energy surface and dynamics theory used in this study are described. In section 111, the calculated results are compared to experiment. Section IV discusses a fit of the final calculated rate constants to an analytical function for k , as a function of temperature T and buffer gas concentration [MI (equivalent to pressure). This fitted function is suitable for modeling studies. The conclusions are reviewed in section V.
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988 2463
11. Theoretical Model for Reaction 1 A . Potential Energy Surface. The parameterized potential energy surface used here is identical with that used in previous calculation^^^^^^ and will be described only briefly. The surface, while physically reasonable, is not based on any ab initio electronic energy calculations. A limited comparison to available ab initio calculations will be made later. The potential energy for CH3-CH3* is approximated as the sum of two terms: V, for the transitional degrees of freedom (those vibrational degrees of freedom in C2H6 that arise from rotational and translational degrees of freedom in C H 3 CH3) and V, for the conserved vibrational degrees of freedom (vibrations in both C2H6 and CH3 CH3). The potential V, is the sum of two terms: V N B for the nonbonded interactions and VB for the bonded interactions. VNBis the sum of pairwise Lennard-Jones potentials between all nonbonded H-H and C-H pairs. VB is the product of a Morse oscillator potential for the C-C coordinate r times an orientation factor measuring the alignment of each methyl radical symmetry axis (coincident with the axis of the radical bonding orbital) with the r vector. In order that the structure of each CH3 group evolves smoothly when the reaction coordinate is varied, the equilibrium C-H distance and equilibrium H-C-H bond angle in each radical is assumed
to change from methyl radical to ethane values according to the interpolation function where the reaction coordinate R is the separation of the centers of mass of the CH, fragments and R, is the value of R at the C2H6 equilibrium. (This functional form appears in the statistical adiabatic channel model of Quack and Troe.I3) The CH3 structure and the relative orientation and relative separation of the two CH3 groups together determine all the interfragment coordinates necessary to specify V,. In addition, the CH3 structure determines three principal moments of inertia that are used to assign a (rigid rotor) rotational energy to each fragment in the sum of states calculation described in section IIB. In VB the Morse PM0 parameter is determined in standard fashion from the dissociation energy and the harmonic force constant of the C-C bond in C2H6. The potential V, for the conserved vibrational degrees of freedom is assumed to be separable and harmonic. The switch from CH3 vibrational frequencies to C2H6 vibrational frequencies is again controlled by the interpolation function e-a(R-Re). The full specification of V, and V, can be found in various tables of ref 16. Several parameters control how the C2H6 and C H 3 regions of the surface are connected: the Lennard-Jones parameters for the nonbonded interfragment pairwise interactions, pMo, which controls the steepness of the energy change along the reaction path, and a,which here controls the tightness of certain degrees of freedom along the reaction path. These last two parameters are similar to but not identical with the a and PM0 parameters in the statistical adiabatic channel m 0 d e 1 . I ~ The ~ ~ ~most important difference is that the a parameter here influences the transitional degrees of freedom only indirectly by controlling small geometry changes within each CH3 fragment (bond lengths and bond angles). The pairwise representation of the nonbonded interactions provides a fully anharmonic description of this part of the potential with no imposition of rigid rotor or harmonic oscillator models. In the statistical adiabatic channel model, a controls directly the evolution of rigid rotor rotational constants to harmonic oscillator vibrational frequencies for the transitional degrees of freedom. Without reference to ab initio calculations, there is no way to select a value for a short of comparison to dynamics studies. Of the three published ab initio c a l c ~ l a t i o n s on ~ ~ the - ~ ~potential energy surface, none examines this aspect of the surface. Furthermore, two of the calculation^^^^^^ have a barrier to recombination that probably is incorrect and is due to t h e - M I N D 0 approximations used. Consequently, in this study, cy will be considered an adjustable parameter. While PM0 can be determined as described above, a very recent ab initio calculation21(the only one that has no recombination barrier) suggests that a large value of PMo (about 2.4 A-' rather than the 1.8 A-' used here) is a more accurate representation of the change in energy along the reaction path. This conclusion is similar to that derived from semiempirical modeling20 and ab H recombination. However, the actual initio s t ~ d i e s ~forl , CH3 ~~ calculations for CH3 CH3 involved a small basis set expansion and a novel but insufficiently tested reduced potential representation. Further ab initio studies along these lines would be very valuable. For this study, PMo was not varied from the value of 1.8 A-' used in ref 15 and 16. The high-pressure rate constants derived from this work are in excellent agreement with the results in ref 15. Furthermore, as the results will show, the agreement with measured recombination rate constants could hardly improve with variation in PMo. Future theoretical studies are planned that will include both comparisons of calculated and measured dis-
(18) Two reviews discussing variational RRKM theory are contained in the following: Truhlar, D. G.; Hase, W. L.; Hynes, C. J. Phys. Chem. 1983, 87, 2264. Hase, W. L. Arc. Chem. Res. 1983, 16, 258. Several recent applications of variational RRKM theory are given in the following: (a) Rai, S.N.; Truhlar, D. G. J . Chem. Phys. 1983, 79, 6046. (b) LeBlanc, J. F.; Pacey, P. D. J . Chem. Phys. 1985, 83, 4511. (c) Duchovic, R. J.; Hase, W. L. J. Chem. Phys. 1985,83, 3448. (d) Hase, W. L.; Mondro, S. L.; Duchovic, R. J.; Hirst, D. M. J . A m . Chem. SOC.1987, 109, 2916. (e) Mondro, S. L.; Hase, W. L.; Vande Linde, S. J. Chem. Phys. 1986, 84, 3783. (0 Vande Linde, S. R.; Mondro, S. L.; Hase, W. L. J . Chem. Phys. 1987, 86, 1348. (19) Wardlaw, D. M.; Marcus, R. A. Chem. Phys. Letr. 1984, 110, 230.
(20) Cobos, C. J.; Troe, J. Chem. Phys. Letf. 1985, 113, 419. (21) Evleth, E. M.; Kassab, E. Chem. Phys. Lett. 1986, 131, 475. (22) Dannenberg, J. J.; Tanaka, K. J . A m . Chem. SOC.1985, 107, 671. (23) Dannenberg, J. J.; Rayez, J. C.; Rayez-Meaume, M. T.; Halvich, P. THEOCHEM. 1985, 123, 343. (24) Duchovic, R. J.; Hase, W. L.; Schlegel, B.; Frisch, M. J.; Raghavachari, K. Chem. Phys. Lett. 1982,89, 120. Peyerimhoff, S.; Lewerenz, M.; Quack, M. Chem. Phys. Lett. 1984,109, 563. Hirst, D. M. Chem. Phys. Left. 1985, 122, 225. Brown, F. B.; Truhlar, D. G. Chem. Phys. Letf. 1985, 113, 419. Schlegel, H. B. J . Chem. Phys. 1986, 84, 4530.
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The Journal of Physical Chemistry, Vol. 92, No. 9, 1988
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sociation rate constants and the effect of variations in PM0. Construction of the potential energy surface involves specification of the well depth for the c-C bond in C2H6. Because of small differences in the AH,-(O) for CH, and C2H6,25326 the value of AHc-c(0) has been listed at either 87.6 kcal/m01~~ or 88.4 kcal/moL2’ The uncertainty in either value is probably in excess of 0.5 kcal/mol. The lower value is used here. The small uncertainty in this well depth adds a negligible uncertainty to the calculated values of k , . The influence on calculations of k-, will be greater. B. Dynamics Theory. Theoretical rate constants for reaction 1 were obtained by using variational RRKM theory in which conservation of both total energy E and total angular momentum J is explicit. The expression to be evaluated is
where Q ( r ) is the partition function for an isolated methyl radical at temperature T, k(E,J) = N ( E , J ) / h p ( E , J )is the RRKM expression for the unimolecular dissociation at E and J , N(E,J) is the sum of states at the variationally determined transition state, p(E,J) is the density of states of and k, is an effective rate constant for stabilization of C2H6* by buffer gas collisions. The factor of I/., in eq 2 arises from the usual approximation that only systems initially on the singlet potential surface lead to recombination products. Each of the quantities N ( E , J ) ,p ( E , J ) , and k, is described in turn; Q(7‘) is evaluated in standard fashion and is not further discussed. N ( E , J ) . Since C H 3 self-recombination has no barrier on the potential energy surface, N(E,J) must be determined variationally, Le., the desired value of N ( E , J ) is the minimum in N ( E , J , R )as a function of R . The calculation of N ( E , J ) is via FTST,15,16,’9 and only a brief description is given here. In FTST the conserved vibrational modes and the transitional modes are assumed uncoupled from each other (apart from an indirect coupling via a dependence on the reaction coordinate). N(E,J,R) is then expressed, for a given J value, as a convolution of a quantum sum of states for the conserved modes with a classical density of states for the transitional modes subject to conservation of the total energy E . The resulting expression for N ( E , J , R )is a 12-dimensional integral multiplied by a symmetry number u. The 12 integration variables are 6 angular momentum action coordinates and their 6 conjugate angle coordinates that describe all motion in transitional coordinates except motion along the reaction path. The range of these variables is restricted by certain angular momentum triangular inequalities. The integrand is the sum of states for all conserved vibrational degrees of freedom consistent with total vibrational energy less than or equal to the total energy reduced by the transitional mode energy (Le., rotational, orbital, and potential energy); the latter energy is dependent on the values of the integration variables, as well as R and J. The integral is evaluated by using the Monte Carlo method described in ref 15. The value of the symmetry number u is approximated** (25) Baulch, D. L.; Cox, R. A.; Hampson, Jr. R. F.; Kerr, J. A.; Troe, J.; Watson, R. T. J . Phys. Chem. Ref.Data 1984, 13, 1259. The quoted value for AHAO) for C H 3 is 35.6 k 0.2 kcal/mol and basically derives from ion threshold measurements by Chupka ( J . Chem. Phys. 1968,48,2337) that have been confirmed by others. The quoted value of M A O ) for C2H6is - 16.3 f 0.1 kcal/mol and basically derives from the most recent flame calorimetry study by Pittam et al. ( J . Chem. SOC.,Faraday Trans. I 1972, 2224). (26) For CHI, a frequently quoted measurement of AHA273 K) is 35.1 i 0.15 kcal/mol determined by Benson et al. (In!. J. Chem. Kine!. 1979, 1 1 , 147) from halogenation kinetics and equilibria measurements. If the change from 273 to 0 K is as given in ref 25, then AHXO) would be 35.9 kcal/mol. For C2Hb,a frequently referenced value of AHd273 K) is 20.2 & 0.1 kcal/mol from the flame calorimetry study of Rossini (J. Res. Natl. Bur. Stand (US.) 1934, 12, 735). Again if the change from 273 to 0 K is as given in ref 25, then AHf(0) is -16.5 kcal/mol. (27) McMillian, D. F.; Golden, D. M. Ann. Reu. Phys. Chem. 1982, 33, 493. The actual value quoted (in Table 2) is the enthalpy change during dissociation at 273 K as derived from ref 26. Reduction to the 0 K value in the text is as described in ref 26.
Wagner and Wardlaw by its “loose” value of 2(2 X 3)2 = 72. The minimization of N(E,J,R) with respect to R is done numerically on a 0.1-A grid over an appropriate range of R values. For convenience, minimized values of N(E,J) were obtained only over a fixed grid of values for E and J . The grid of values for J was evenly spaced from 0 to 200 in units of 25. Sixteen values of E (in kcal/mol) were selected for the energy grid: 0.13, 0.22, 0.44, 0.71, 0.89, 1.18, 1.78, 2.36, 3.44, 4.73, 5.86, 9.53, 11.89, 22.95, 39.10, and 63.52. This selection is based on previous c a l c ~ l a t i o n sof~ the ~ high-pressure rate constant in which a numerical quadrature was employed to evaluate the thermal average of k , ( E , J ) . The value of N(E,J) is also dependent on the parameters of the potential energy surface, in particular on the cy parameter, which is adjusted to obtain best agreement with experiment. In ref 15, results for two different values of cy were reported, 1.0 and 0.8 Here a set of N(E,J) was also calculated for a = 0.55 k ’ .So that N(E,J) at intermediate values of E, J, and a could be obtained, an interpolation scheme was developed and is described in the Appendix. The estimated errors in the interpolation procedure result in uncertainties in the calculated values of k l that are much smaller than those in the experimental values. It is to be emphasized that the FTST approach used here to evaluate N(E,J) does not require either the approximate form of the potential energy surface or the numerical interpolation described above. In subsequent studies of the CH3 + CH3 reaction, more detailed descriptions of the potential energy surface will be incorporated into the dynamics study. Unlike RRKM or statistical adiabatic channel models, the FTST approach is designed to fully account for the anharmonicity in the transitional degrees of freedom. Very recently, an estimate, in the high-pressure limit, of the error introduced by the classical treatment of the transitional modes has been provided by Klippenstein and Marcus.29 These authors apply a quantum Monte Carlo path integration method within the FTST framework to obtain the limiting high-pressure value of k , for selected temperatures. For a = 1.0 A-’ and a canonical variational determination of the transition state, the discrepancies between the quantum and classical results are found to be I500 K). While expected to be minor, these channels impose further uncertainty on the representation of the recombination rate constant at the highest temperatures. The rate constant for loss of CH, to all products might substantially increase over just the recombination rate constant at higher temperatures because of this channel. V. Conclusions The major conclusions of this study are as follows: 1. A weak-coupling corrected RRKM theory calculation of the rate constant reproduces the measurements of part 1 with an average error of about 10%. The calculations are selected to reproduce the single directly measured high-pressure limit by Hippler et al. at room temperature and are in satisfactory agreement with the measured rate constant by Glanzer et al. at 1350 K. The calculations employ a variationally determined transition state for each energy and angular momentum and include effects of the loose degrees of freedom via the flexible transition-state theory proposed by Wardlaw and Marcus. The
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Theoretical Study of CH3
+ CH,
-
C2H6
intermediate- to long-range portion of the potential energy surface used to deal with the transition state was also proposed by Wardlaw and Marcus. The calculation of the rate constant here involves two adjustable parameters, a potential parameter a used to adjust the evolution of some of the degrees of freedom perpendicular to the reaction path and (AE),,,, which is the average energy change in metastable C2H6by collision with buffer gas. These two parameters are optimized to produce the best agreement with experiment. 2. A simple functional form, similar to but not identical with that proposed by Gilbert et al., fits the RRKM calculations over an extended range of pressure and from room temperature to 2000 K. This form is suitable for modeling. Because of the accuracy of the RRKM calculations in reproducing the available measurements, this function is expected to be accurate to within about 10-20% below 1000 K and to within about 40-50% by 2000 K. 3. The final value of (AE),,, is -205 f 6 5 cm-I. The experiments are most sensitive to this parameter at 800-900 K. At lower temperatures the above error bars on (AE),,, become considerably larger. Measurements over extended ranges of pressure (especially over the lower ranges of temperature) are needed to reduce these error bars. and is lower 4. The final range of values of a is 0.57-0.83 than those found in other recombination studies with related semiempirical otentials having a similar a parameter (typical values are 1.0 1 - l ) . Possible reasons for this low value are suggested. 5. Weak-coupling corrected RRKM calculations can partially reproduce new measurements of the recombination rate constant of C H 3 C H 3 in He. These measurements are over a narrow pressure range for three temperatures and are very similar to the Ar measurements at the two higher temperatures but are 30-40% lower at the lowest temperature. The calculated rate constants can fall within all the error bars of these measurements only if ( AE),,, has a significant temperature dependence. This is the first in a series of studies that will explore many features of the surface and the sensitivity of these features to both recombination and dissociation. New experiments, incorporation of C2H6dissociation, and ab initio work on the potential surface2’ will be used in future studies.
+
Acknowledgment. We gratefully acknowledge discussions with David Gutman, Irene Slagle, and Michael Pilling. This research was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, U S . Department of Energy under contracts W-31-109-Eng-38 and DE-AC02-78ER-14593 and by the Natural Sciences and Engineering Research Council of Canada. Appendix: Interpolation of N ( E , J ) with Respect to E , J , and a The order of the interpolation is first with respect to J , then a,and then E. The interpolation with respect to J has three basic steps that are carried out for all three values of a and for all values of E for which calculations were explicitly performed. Let the desired value of N(E,J)be N(E’,J’) where E’is one of the energies for which explicit calculations have been made and where J’ is not one of the total angular momenta for which explicit calcu-
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988 2471 lations have been made. In the first step, determine the threshold energy, denoted E?, at which the N(E,J’)goes to 1 as a function of E . (For E < Ej,, N(E,J’) is set to 0.) This is obtained from a spline fit of threshold energies E j at the values of J for which explicit calculations were performed. In the second step, for each J for which explicit calculations were performed, find N(E”,J) at a value E” that is shifted from E’by the difference EJ, - E,. This is obtained from a spline of the calculated values of N(E,J) with respect to E (see below). Upon completion of this step, a series of values for the sum of states has been constructed with the same displacement above threshold but with different total angular momentum. In the third step, spline these values of N(E”,J) together, and from that spline obtain N(E’,J’). This procedure builds in a realistic threshold behavior for N(E,J’)and allows for a more stable interpolation with respect to E (as described below). This J interpolation scheme can be tested. In ref 15, at room temperature for a = 1.O A-’, explicit calculations are made for Jvalues separated by 12.5 instead of 25 used for the other values of a. By explicit calculation, a 4.2% drop in the high-pressure limit is produced by halving the grid of Jvalues. The interpolation scheme results in about a 10% drop in the high-pressure limit, both for a = 1.O A-1 and for a wider range of a values. At 2000 K, the interpolation scheme predicts a
