J. Phys. Chem. 1996, 100, 1761-1766
1761
Comparison of ab Initio and Empirical Potentials for H-Atom Association with Diamond Surfaces Pascal de Sainte Claire, Kihyung Song,† and William L. Hase* Department of Chemistry, Wayne State UniVersity, Detroit, Michigan 48202-3489
Donald W. Brenner Department of Materials Sciences and Engineering, North Carolina State UniVersity, Raleigh, North Carolina 27695 ReceiVed: June 19, 1995; In Final Form: October 13, 1995X
Canonical variational transition-state theory (CVTST) is used to compare H + CH3 and H + diamond {111} association rate constants calculated from the Brenner empirical potential function and molecular anharmonic potentials written with switching (MAPS) functions. Previous work [J. Am. Chem. Soc. 1987, 109, 2916; J. Chem. Phys. 1994, 101, 2476] has shown that the MAPS functions, derived from ab initio calculations, give rate constants in agreement with experiment. For the 300-2000 K temperature range, the Brenner potential function gives CVTST H + CH3 and H + diamond {111} association rate constants which are 159-30 and 49-7 times smaller, respectively, than the values from the MAPS functions. An analysis of the Brenner potential function shows that it inaccurately represents the intermediate and long-range H- - -C association potential, which controls the structure of the variational transition state and the CVTST rate constant. The MAPS functions give H + CH3 and H + diamond {111} variational transition states with similar properties. Angular momentum and external rotation have no effect on the H + diamond {111} association rate constant, which makes it approximately an order-of-magnitude smaller than that for H + CH3 association.
Introduction To model diamond film growth by chemical vapor deposition (CVD), rate constants are needed for elementary surface and gas-surface reactions involved in diamond deposition.1-5 These rate constants can be determined by experiment, but another approach is to calculate them by computational/ theoretical methods. In transition-state theory (TST), properties of the TS are used to calculate the elementary reaction’s rate constant. If the reaction has a well-defined potential energy barrier, the TS can usually be placed at this point on the potential energy surface. However, for reactions without potential energy barriers, such as association reactions, variational transition state theory (VTST) must be used to find the TS from properties of the reaction path.6-9 A molecular dynamics (i.e., classical trajectory) calculation of an elementary rate constant requires the complete potential energy surface for the reaction of interest.10-12 Semiempirical,13-21 density functional theory,22,23 and ab initio24-27 electronic structure calculations have been used to study surface and gas-surface reactions associated with diamond CVD. A large cluster was used to represent the diamond surface, in these calculations. Potential energy barriers and reaction energetics were determined from the ab initio calculations.24-27 These properties, as well as transition-state structures and vibrational frequencies, have also been determined from semiempirical MNDO, AM1, and PM3 calculations13,16-20 and from an empirical potential energy function for carbon materials28 developed by Brenner.29 TST was then used to calculate elementary rate constants from the transition state properties.19-21,28 The Brenner potential has also been used in molecular dynamics simulations to determine ratios of elementary rate constants30 and individual elementary rate constants.31-34 † Current address: Department of Chemistry, Korea National University of Education, Chongwon, Chungbuk 363-792, Korea. X Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-1761$12.00/0
An analytic potential energy function for H-atom addition to C-atom radical sites on diamond surfaces, and defined here as MAPS/HDIAM,35 has been determined from high-level ab initio calculations.26 VTST and molecular dynamics calculations,26,36 based on this potential energy function, have been used to calculate the elementary rate constant for association of a gasphase H-atom with a C-atom radical site on the diamond {111} surface. These two theoretical methods give the same, nearly temperature-independent value for this rate constant, which agrees with the value estimated by comparing with the gasphase H + t-C4H9 f C4H10 association rate constant.37,38 The tert-butyl radical has two equally probable sides for reaction, whereas the surface radical has only one side. Thus, the H + diamond {111} association rate constant has been approximated as one-half of that for H + t-C4H9.3 Model VTST calculations for H-atom association with alkyl radicals and the diamond {111} surface support this model for relating gas phase and interfacial rate constants.37 The analytic form of the MAPS/HDIAM potential is similar to the MAPS/CH4 analytic potential developed previously39-41 for the H + CH3 T CH4 system from ab initio calculations and experimental parameters. VTST and molecular dynamics calculations give the same rate constant for H + CH3 association, when using MAPS/CH4.41 The calculations give a D + CH3 association rate constant in excellent agreement with experiment.42 The calculated H + CH3 association rate constant agrees with one experimental measurement43 but is approximately 2 times smaller than another measured value.42 In the work presented here, detailed comparisons are made between the Brenner and MAPS/HDIAM potentials for H + diamond {111} association and between the Brenner and MAPS/ CH4 potentials for H + CH3 association. Reaction path properties, variational transition states, and CVTST rate constants are calculated for the different potentials to facilitate the comparisons. © 1996 American Chemical Society
1762 J. Phys. Chem., Vol. 100, No. 5, 1996
de Sainte Claire et al.
Computational Procedure Potential Energy Surfaces. The expressions and parameters for the Brenner general hydrocarbon potential have been given previously,29,44 from which it is straightforward to construct the Brenner potentials for H + CH3 and H + diamond {111}. The MAPS/CH4 potential was originally developed by Duchovic and co-workers39 and then modified by Hase et al.40 Its next and last modification was by Hu and Hase,41 which is the form used here. Details of the MAPS/HDIAM potential have been described in recent work.26 Reaction Path and Canonical Variational Transition State Theory (CVTST). The potential energy functions studied here are incorporated in the general chemical dynamics computer program VENUS.45 Determining the reaction path and canonical variational transition state, for an analytic potential, is a standard option in VENUS. The reaction path is determined by following the path of steepest descent in mass-weighted Cartesian coordinates.46 This gives the molecular geometry and potential energy as the system moves along the reaction path. Harmonic vibrational frequencies, for the 3N - 7 modes orthogonal to the reaction coordinate, are also determined as the system moves along the reaction path, by diagonalizing a projected 3N × 3N mass-weighted Cartesian force constant matrix.47 Projected out of the standard force constant matrix48 are the reaction coordinate motion and six infinitesimal displacements for overall translation and rotation. The canonical variational transition state is placed at the free energy maximum along the reaction path.6,7,49-51 The free energy, as a function of the reaction path, is written as 3N-7
Gq(s) ) V(s) +
∑ i)1
q q hνi(s)/2 + Gvib (s) + Grot (s)
(1)
where V(s) is the classical potential potential energy, the νi(s) are the harmonic frequencies for the vibrational modes orthogonal to the reaction coordinate, Gqvib(s) is the free energy for q (s) is the free energy for the external these modes, and Grot rotational degrees of freedom. For an association reaction with a surface, there are no external rotational degrees of freedom q (s) is removed from eq 1.37 The harmonic apand Grot proximation is used to represent the zero-point energy in eq 1. This is a good approximation for an association reaction with a loose variational transition state, since there is negligible change in the vibrational frequencies of the reactants as the system moves toward the transition state.26,39d,40 The transitional bending modes, which are formed during the reaction, have very low frequencies (see below), and thus an anharmonic correction would make a small contribution to their zero-point energy. For H + CH3 association at 1000 K, treating the transitional modes as separable harmonic oscillators, as is done here, results in a rate constant only 15% smaller than that found by treating these modes as anharmonic coupled hindered rotors.9 The CVTST rate constant can be written as either the ratio of transition state to reactant partition functions or as the free energy difference ∆Gq between the transition state and reactants;9 i.e.
k)
kBT Qq -Eq0/RT kBT -∆Gq/RT e ) e h QRQS h
(2)
In this equation Qq is the transition state partition function and QR and QS are the partition functions for the reactant radical and surface, respectively. For a radical-radical reaction like H + CH3, QS is replaced by a radical partition function.
Figure 1. Model used for the diamond {111} surface, with an H atom attached to the C-atom radical site (shaded C atom).
Figure 2. Comparison of C-atom motion vibrational frequencies for the model in Figure 1 (without a H atom attached to the radical site) determined from the Brenner and MAPS/HDIAM potentials.
Computational Results The cluster model used for the diamond lattice is shown in Figure 1. It is called a (6+7)-ring model.52 There are six rings of C atoms in the top two layers and seven rings of C atoms in the bottom two layers. This size model gives the same CVTST and molecular dynamics rate constants for H-atom association with the diamond {111} surface as do larger models.26,36 The MAPS/HDIAM potential has force constants for the diamond lattice which reproduce the diamond phonon spectrum.26 Thus, this potential is expected to accurately represent the low-energy, short-range properties of the lattice. Harmonic vibrational frequencies for C-atom vibrational motions of the (6+7)-ring model, as determined from the Brenner and MAPS/ HDIAM potentials, are compared in Figure 2. The H-C stretching and bending frequencies for the (6+7)-ring model are not included in this figure. Though there are some shifts between the two sets of frequencies in Figure 2, overall they are similar, which indicates the Brenner potential is an accurate model for short-range properties of the diamond lattice. Such a result has been found in other studies,53 where it has been shown that the Brenner potential gives accurate bulk properties for the diamond lattice. Finally, the equilibrium C-C and H-C bond lengths are 1.55 and 1.079 Å for the Brenner potential and 1.54 and 1.095 Å for the MAPS/HDIAM potential. The Brenner potential is expected to accurately represent short-range properties of a diamond lattice.29 It is based on a near-neighbor bond-order formalism which takes advantage of the Pauling relationship. This model allows one to fit a large data base of bonding energetics and structures for solid-state and molecular systems. Reaction Path Properties. For both H + CH3 and H + diamond {111} association on the MAPS and Brenner poten-
H-Atom Association with Diamond Surfaces
Figure 3. Reduced H- - -C association potential curves, for the MAPS/ HDIAM, MAPS/CH4, and Brenner potentials. The points are the results of ab initio calculations; see refs 26 and 39.
tials, motion along the reaction path is primarily shortening of the H- - -C separation. Thus, instead of plotting reaction path properties versus the position along the reaction path, here they are plotted versus the H- - -C separation along the reaction path. Reduced reaction path potential energy curves given by the Brenner and MAPS/HDIAM potentials for H + diamond {111} and by the MAPS/CH4 potential for H + CH3 are compared in Figure 3. The reaction path potential is divided by the classical H-C bond energy De54 and plotted versus the H- - -C separation along the reaction path divided by the H-C equilibrium length Re. The MAPS/HDIAM and MAPS/CH4 reduced potential energy curves are nearly identical. However, the Brenner potential energy curve is shorter range and only becomes attractive at R(H- - -C)/Re of about 1.66. The Brenner reaction path potential energy curve is similar to ab initio potentials for H + CtO f HsCdO,55 H + OdO f HsOsO,56 and H + H2CdCH2 f H2CsCH3,57 for which one of the π-bonds of the molecular reactant is broken as the H-atom is associating. The reaction paths for H + CH3 association and H-atom addition to the diamond {111} surface model used here, have C3V symmetry. Thus, as the H atom adds to CH3 or a C-atom radical site on the diamond {111} surface, two degenerate transitional bending modes are formed.9 When following the reaction path, harmonic vibrational frequencies may be determined for the transitional modes.46 These frequencies are compared in Figure 4 for the Brenner and MAPS/HDIAM potentials for H + diamond {111}. It is seen that the attenuation of the frequency occurs at a much shorter range for the Brenner potential than for the MAPS/HDIAM potential. The attenuation of the transitional bending mode frequency for the MAPS/ HDIAM potential is similar to that for the MAPS/CH4 potential, which has been published previously.40 The H•- - -H nonbonded van der Waals potentials of the Brenner and MAPS/HDIAM potential functions are compared in Figure 5. The Brenner potential is less repulsive at longrange but becomes more repulsive at short-range. The shortest H•- - -H separation along the minimum energy path for H-atom association with the diamond {111} surface is 2.56 Å, at which the H•- - -H interaction is zero for the Brenner potential and slightly repulsive (i.e., 0.18 kcal/mol) for the MAPS/HDIAM potential. As a C-H bond ruptures on the diamond surface a geometric relaxation occurs around the resulting C-atom radical and this atom slightly “sinks” into the lattice. The Brenner potential has a near planar geometry between the C-atom radical and the
J. Phys. Chem., Vol. 100, No. 5, 1996 1763
Figure 4. Transitional mode bending frequency versus the H- - -C separation along the reaction path for the MAPS/HDIAM and Brenner potentials.
Figure 5. H•- - -H nonbonded interaction for the MAPS/HDIAM and Brenner potentials.
three carbon atoms to which it is bonded, and the C atom sinks 0.28 Å when the H atom is removed. In contrast, the C atom sinks only 0.13 Å on the MAPS/HDIAM potential. Potential energy contour diagrams of Rp versus the H- - -C distance are given in Figure 6 for the Brenner and MAPS/HDIAM potentials. Rp is the distance between the carbon atom, which loses a H-atom, and a plane formed by the three carbon atoms to which it is bonded. When the H-atom is attached to the carbon atom, there is a near-tetrahedral arrangement at the radical site for the Brenner potential with Rp of 0.5004 and a tetrahedral Rp of 0.5148 Å for the MAPS/HDIAM potential. The relaxation given by the MAPS/HDIAM potential is similar to the results of ab initio24-26 and density functional theory22 calculations for diamond cluster models. The contour diagrams in Figure 6 show that the relaxation about the radical site is more abrupt and extensive for the Brenner potential than for the MAPS/HDIAM potential. This is because the Brenner potential assumes there is a sudden transition at R(H---C) ) 1.80 Å from no interaction between the H-atom and C-atom radical site to a strong covalent bonding interaction (see the reduced potential, transitional mode frequencies, and intermolecular potential in Figures 3-5 and the constant potential energy curves versus R(H- - -C) in Figure 6 for R(H- - -C) > 1.80 Å). The Brenner potential correctly describes short-range properties near the potential energy minimum. This is indicated by the potential minimum’s
1764 J. Phys. Chem., Vol. 100, No. 5, 1996
de Sainte Claire et al. TABLE 1: Variational Transition-State Properties and Rate Constant for H-Atom Association with the Diamond {111} Surface T (K) 300
1000
1500
2000
rqa νqb Eq0c kCVTSTd,e
Brenner Potential 1.77 1.75 424 596 -0.5 -1.2 0.036 0.14
1.74 658 -1.8 0.22
1.73 704 -2.3 0.28
rqa νqb Eq0c kCVTSTd
MAPS/HDIAM Potential 3.32 2.92 2.79 118 227 289 -1.1 -2.6 -3.8 1.8 1.8 1.9
2.69 336 -5.1 2.1
a H- - -C bond length in angstroms at the variational transition state. Degenerate transitional bending mode frequency at the transition state in cm-1. c Difference in classical potential energy between the variational transition state and reactants in kcal/mol. d CVTST rate constant in 1013 cm3 mol-1 s-1. e These rate constants were calculated using the modified SCH parameter in the Brenner potential; see ref 44. If the original SCH parameter of ref 29 is used, the H + diamond {111} association rate constant is 0.034, 0.14, and 0.26 × 1013 cm3 mol-1 s-1 at 300, 1000, and 1500 K, respectively. b
TABLE 2: Variational Transition-State Properties and Rate Constant for H + CH3 Associationa T (K) Figure 6. Potential energy contours in the Rp and R(H- - -C) plane, for H + diamond {111} association on the MAPS/HDIAM and Brenner potentials. The dashed lines identify the reaction paths. Energies on the contours are in units of kcal/mol.
geometry, energy, and vibrational frequencies. However, to attain these correct short-range properties, the transition from long-range to short-range properties is too sharp on the Brenner potential, which results in a potential which is inaccurate for both intermediate and long-range H + diamond {111} interactions. The most abrupt change in Rp, along the reaction path for the Brenner potential, occurs near the minimum energy value for R(H- - -C). This is what gives rise to the dramatic decrease in V/De of 0.05 near R(H- - -C)/Re ) 1 in Figure 3, for the Brenner potential. Association Rate Constants. Reaction path properties were used to calculate CVTST association rate constants for H + diamond {111} on the Brenner and MAPS/HDIAM potentials and for H + CH3 on the Brenner and MAPS/CH4 potentials. The results for H + diamond {111} are in Table 1 and those for H + CH3 are in Table 2. A comparison of the results for the Brenner and MAPS potentials shows that the Brenner potential gives much tighter transition-state structures and much smaller association rate constants. The former could be predicted58 from the reaction path properties described in the previous section, but that the Brenner potential gives much smaller rate constants could not be foretold without actually performing the CVTST calculations. For H + diamond {111} association, the rate constant from the Brenner potential varies from 49 to 7 times smaller than that for the MAPS potential as the temperature is increased from 300 to 2000 K. The rate constant for H + CH3 association ranges from 159 to 30 times smaller for the 300-2000 K temperature range. The long-range H- - -C radial potential is more attractive for the MAPS/HDIAM potential than the Brenner potential (see Figure 3). In previous work,58 it has been shown that, as the attractiveness of the radial potential decreases, the temperature dependence of the rate constant becomes more positive. This is the relationship found here between the radial potential and the rate constants for the Brenner and MAPS/HDIAM potentials.
300
1000 Brenner Potential 1.65 387 -1.2 0.32
1500
2000
rq νq Eq0 kCVTST
1.66 284 -0.4 0.091
1.64 436 -1.8 0.47
1.63 476 -2.4 0.62
rq νq Eq0 kCVTST
MAPS/HDIAM Potential 3.41 3.00 2.84 119 234 298 -0.8 -2.9 -4.4 14.3 18.2 18.7
2.72 354 -6.1 18.8
a See the footnotes in Table 1 for the identification and units of the properties and rate constants.
The MAPS/HDIAM and MAPS/CH4 potentials give variational transition states with very similar transitional mode vibrational frequencies. The primary reason the H + CH3 association rate constant is appreciably smaller than that for H + diamond {111} is because reactant and transition state rotational partition functions affect the H + CH3 association rate constant but not that for H + diamond {111}. For H + CH3 association at 300 and 2000 K, the transition-state rotational partition function is 14 and 10 times larger, respectively, than that for CH3. A secondary effect is that the potential energy difference between the transition state and reactants is not the same for H + CH3 and diamond {111}. This effect suppresses the H + CH3 rate, with respect to diamond {111}, by a factor of 0.55 at 300 K, but enhances the H + CH3 rate by a factor of 1.22 at 2000 K. By combining these two effects, i.e., the rotational partition functions and potential energy differences, the H + CH3 rate is predicted to be larger than that for H + diamond {111} by a factor of 7.7 and 12.2 at 300 and 2000 K, respectively. The factor at 300 K is the same as the ratio of the kCVTST rate constants, while the factor at 2000 K is ∼20% too large. This latter difference results from an increasing difference in the transitional mode vibrational frequencies for H + CH3 and H + diamond {111} as the temperature is increased (see Tables 1 and 2). Free energies, enthalpies, and entropies of activation, for H + CH3 and H + diamond {111} association on the Brenner
H-Atom Association with Diamond Surfaces
J. Phys. Chem., Vol. 100, No. 5, 1996 1765
TABLE 3: Free Energies, Enthalpies, and Entropies of Activationa,b Brenner
MAPS
∆Hq
∆Sq
∆Gq
∆Hq
∆Sq
H + CH3 -25.6 -28.2 -29.2 -29.8
T (K)
∆G
300 1000 1500 2000
7.2 26.2 40.6 55.3
-0.49 -2.04 -3.18 -4.34
4.2 18.2 29.6 41.7
-1.03 -3.80 -5.86 -8.04
-17.3 -22.0 -23.6 -24.9
7.7 27.8 42.9 58.4
H + diamond {111} -0.36 -27.0 5.4 -1.96 -29.8 22.8 -3.06 -30.7 36.4 -4.16 -31.3 50.5
-1.36 -3.60 -5.28 -7.01
-22.6 -26.4 -27.8 -28.8
300 1000 1500 2000
q
a Free energies and enthalpies are in kcal/mol, and entropies are in cal/mol K. ∆Hq includes the potential energy difference between the transition state and reactants. b These values of ∆Gq, ∆Hq, and ∆Sq are for a standard state of 1 atm. To use this ∆Gq in eq 2 to calculate a rate constant in units of L M-1 s-1, eq 2 must be multiplied by 0.08206T.
and MAPS potentials, are listed in Table 3. The positive ∆Gq values arise from the large negative entropies of activation. The ∆Gq values are larger and have a more positive temperature dependence for the Brenner potential. Both the more negative ∆Sq and less negative ∆Hq values, for the Brenner potential, contribute to the larger ∆Gq values for the Brenner potential. It is of interest that the ∆Gq values, for the MAPS/HDIAM and MAPS/CH4 potentials, vary nearly linearly with temperature. Conclusions In this work the empirical Brenner potential29 and MAPS potentials,26,35,39-41 developed from ab initio calculations, are compared for H + CH3 and H + diamond {111} association. The Brenner potential is significantly less attractive and gives tighter variational transition states and smaller CVTST rate constants than the MAPS potentials. The MAPS/CH4 potential gives a D + CH3 association rate constant in excellent agreement with experiment and a H + CH3 association rate constant in agreement with one experimental result and a factor of 2 smaller than another.41 The MAPS/HDIAM potential gives a H + diamond {111} association rate constant which agrees with the value estimated from the H + tert-butyl association rate constant.3,26,36 The CVTST rate constants from the Brenner potential are 1-2 orders of magnitude smaller than those calculated from the MAPS potentials. The Brenner potential does not seem to accurately describe the long-range interactions which influence the association of H-atoms with alkyl radicals and diamond surfaces. However, other studies have shown that the Brenner potential accurately represents short-range properties of diamond materials.53 The MAPS potentials give similar reaction path properties and variational transition states for H + CH3 and H + diamond {111} association. The H + CH3 association rate constant is larger than that for H + diamond {111}, primarily because the former is affected by its external rotational partition functions. External rotation does not participate in the latter reaction. Acknowledgment. This research was supported by the National Science Foundation. The authors wish to thank Dr. Ken Hass of Ford Scientific Research Laboratories in Dearborn, MI for many helpful conversations. Note Added in Proof: In recent work we have used a quasiclassical trajectory simulation to calculate the 1000 K H + diamond {111} association rate constant for the Brenner potential.59 The resulting value, (0.64 ( 0.04) × 1013 cm3 mol-1
s-1, is 6 times smaller than a previous quasiclassical trajectory value reported for this rate constant,34 but 4-5 times larger than the CVTST value reported here. We are currently investigating the possibility that including anharmonic effects when calculating the CVTST rate constant, for the Brenner potential, will bring it into better agreement with the quasiclassical value. References and Notes (1) Frenklach, M.; Wang, H. Phys. ReV. B 1991, 43, 1520. (2) Frenklach, M. J. Chem. Phys. 1992, 97, 5794. (3) Harris, S. J.; Goodwin, D. G. J. Phys. Chem. 1993, 97, 23. (4) Coltrin, M. E.; Dandy, D. S. J. Appl. Phys. 1993, 74, 5803. (5) Dandy, D. S.; Coltrin, M. E. J. Appl. Phys. 1994, 76, 3102. (6) Hase, W. L. J. Chem. Phys. 1976, 64, 2442. (7) Truhlar, D. G.; Garrett, B. C. Acc. Chem. Res. 1980, 13, 440. (8) Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983, 87, 2664. (9) Hase, W. L.; Wardlaw, D. M. In Bimolecular Collisions; Ashfold, M. N. R., Bagott, J. E., Eds.; Royal Society of Chemistry: London, 1989; p 171. (10) Bunker, D. L. Methods Comput. Phys. 1971, 10, 287. (11) Porter, R. N.; Raff, L. M. In Modern Theoretical Chemistry, Miller, W. H., Ed.; Plenum Press: New York, 1976; Vol. 2, Dynamics of Molecular Collisions, Part B, p 1. (12) Truhlar, D. G.; Muckerman, J. T. In Atom-Molecule Collision Theory; Bernstein, R. B., Ed.; Plenum Press: New York, 1979; p 505. (13) Huang, D.; Frenklach, M.; Maroncelli, M. J. Phys. Chem. 1988, 92, 6379. (14) Mehandru, S. P.; Anderson, A. B. J. Mater. Res. 1990, 5, 2286. (15) Valone, S. M.; Trkulo, M.; Laia, J. R. J. Mater. Res. 1990, 5, 2296. (16) Huang, D.; Frenklach, M. J. Phys. Chem. 1991, 95, 3692. (17) Huang, D.; Frenklach, M. J. Phys. Chem. 1992, 96, 1868. (18) Besler, B. H.; Hase, W. L.; Hass, K. C. J. Phys. Chem. 1992, 96, 9369. (19) Skokov, S.; Weiner, B.; Frenklach, M. J. Phys. Chem. 1994, 98, 8. (20) Frenklach, M.; Skokov, S.; Weiner, B. Nature 1994, 372, 535. (21) Skokov, S.; Weiner, B.; Frenklach, M. J. Phys. Chem. 1994, 98, 7073. (22) Pederson, M. R.; Jackson, K. A.; Pickett, W. E. Phys. ReV. B 1991, 44, 3891. (23) Alfonso, D. R.; Yang, S. H.; Drabold, D. A. Phys. ReV. B 1994, 50, 15369. (24) Page, M.; Brenner, D. W. J. Am. Chem. Soc. 1991, 113, 3270. (25) Larsson, K.; Lunell, S.; Carlson, J.-O. Phys. ReV. B 1993, 48, 2666. (26) de Sainte Claire, P.; Barbarat, P.; Hase, W. L. J. Chem. Phys. 1994, 101, 2476. (27) Hukka, T. I.; Pakkanen, T. A.; D'Evelyn, M. P. J. Phys. Chem. 1994, 98, 12420. (28) Chang, X. Y.; Thompson, D. L.; Raff, L. M. J. Phys. Chem. 1993, 97, 10112; J. Chem. Phys. 1994, 100, 1765. (29) Brenner, D. W. Phys. ReV. B 1990, 42, 9458. (30) Brenner, D. W.; Robertson, D. H.; Carty, R. J.; Srivastava, D.; Garrison, B. J. Mater. Res. Soc. Symp. Proc. 1992, 278, 255. (31) Garrison, B. J.; Dawnkaski, E. J.; Srivastava, D.; Brenner, D. W. Science 1992, 255, 835. (32) Peploski, J.; Thompson, D. L.; Raff, L. M. J. Phys. Chem. 1992, 96, 8538. (33) Chang, X. Y.; Perry, M.; Peploski, J.; Thompson, D. L.; Raff, L. M. J. Chem. Phys. 1993, 99, 4748. (34) Perry, M. D.; Raff, L. M. J. Phys. Chem. 1994, 98, 4375. (35) The acronym MAPS denotes a “molecular anharmonic potential written with switching functions”; see: Hase, W. L.; Buckowski, D. G.; Swamy, K. N. J. Phys. Chem. 1983, 87, 2754. (36) Song, K.; de Sainte Claire, P.; Hase, W. L.; Hass, K. C. Phys. ReV. B (37) Barbarat, P.; Accary, C.; Hase, W. L. J. Phys. Chem. 1993, 97, 11706. (38) Tsang, W. J. Phys. Chem. Ref. Data 1990, 19, 31. (39) (a) Duchovic, R. J.; Hase, W. L.; Schlegel, H. B. J. Phys. Chem. 1984, 88, 1339. (b) Duchovic, R. J.; Hase, W. L. Chem. Phys. Lett. 1984, 110, 474. (c) Duchovic, R. J.; Hase, W. L. J. Chem. Phys. 1985, 82, 3599. (d) Hase, W. L.; Duchovic, R. J. J. Chem. Phys. 1985, 83, 3448. (40) Hase, W. L.; Mondro, S. L.; Duchovic, R. J.; Hirst, D. M. J. Am. Chem. Soc. 1987, 109, 2916. (41) Hu, X.; Hase, W. L. J. Chem. Phys. 1991, 95, 8073. (42) Brouard, M.; Macpherson, M. T.; Pilling, M. J. J. Phys. Chem. 1989, 93, 4047. (43) Cobos, C. J.; Troe, J. Z. Phys. Chem. (NF) 1990, 167, 129. (44) In our calculation, the SCH parameter in the Brenner potential is set to 1.92. This value for SCH was chosen to fit an H• + H-C abstraction barrier for diamond surfaces; see ref 30. For the original Brenner potential,29
1766 J. Phys. Chem., Vol. 100, No. 5, 1996 with SCH ) 1.7386, the classical CH bond dissociation energy is 94.6 kcal/ mol, while with SCH ) 1.92 it is 92.7 kcal/mol. (45) Hase, W. L.; Duchovic, R J.; Hu, X.; Lim, K. F.; Lu, D.-L.; Peslherbe, G. H.; Swamy, K. N.; Vande Linde, S. R.; Wang, H.; Wolf, R. J. Quantum Chem. Program Exchange, submitted. (46) Fukui, K. J. Phys. Chem. 1970, 74, 4161. (47) Miller, W. H.; Handy, N. C.; Adams, J. E. J. Chem. Phys. 1980, 72, 99. (48) Califano, S. Vibrational States; John Wiley and Sons: New York, 1976. (49) Horiuti, J. Bull. Chem. Soc. Jpn. 1938, 13, 210. (50) Laidler, K. J. Theories of Chemical Reaction Rates; McGraw-Hill: New York, 1969. (51) Hase, W. L. Acc. Chem. Res. 1983, 16, 258.
de Sainte Claire et al. (52) Accary, C.; Barbarat, P.; Hase, W. L.; Hass, K. C. J. Phys. Chem. 1983, 97, 9934. (53) Drabold, D. A., private communication. (54) The classical dissociation energy De is 92.7, 104.9, and 109.5 kcal/ mol for the Brenner, MAPS/HDIAM, and MAPS/CH4 potentials, respectively. (55) Cho, S.-W.; Hase, W. L.; Swamy, K. N. J. Phys. Chem. 1990, 94, 7371. (56) Lemon, W. J.; Hase, W. L. J. Phys. Chem. 1987, 91 1596. (57) Hase, W. L.; Mrowka, G.; Brudzynski, R. J.; Sloane, C. S. J. Chem. Phys. 1978, 69, 3548. (58) Hu, X.; Hase, W. L. J. Phys. Chem. 1989, 93, 6029. (59) de Sainte Claire, P.; Sheill, P. H.; Hase, W. L. To be published.
JP951693M
