4112
J. Phys. Chem. 1989, 93, 4772-4779
Recombination of Methyl Radicals: Ab Initio Potential and Transition-State Theory Calculations Katherine Valenta Darvesh, Russell J. Boyd,* and Philip D. Pacey* Department of Chemistry, Dalhousie University, Halifax. Nova Scotia, Canada B3H 4J3 (Received: September 20, 1988; In Final Form: January 3, 1989)
Ab initio calculations employing the configuration interaction method have been applied to the recombination of methyl radicals. The methyl radicals have been restricted to a planar equilateral triangular geometry, but the carbon-carbon distance and five angles defining the mutual orientation of the radicals have been varied. Attractions between the radicals appear to be dominated by u bonding between the singly occupied orbitals on the two radicals, although r bonding also appears to contribute in some orientations. At short distances, repulsive forces between atoms on adjacent radicals are important. All these interactions have been represented by simple expressions, which have been combined to represent the potential energy as a function of the six coordinates varied. Eight parameters in the potential energy expression have been adjusted by least squares, fitting the ab initio points to within 1.4 kJ mol-'. Canonical variation transition-state theory calculations have been performed on the resulting potential energy surfaces. A Monte Carlo method was used to integrate the Boltzmann factor over the five angles. The critical distance or bottleneck was found to decrease from a carbon-carbon distance of 4.0 A at 200 K to 2.7 A at 2000 K. The rate constant is 5 X 1O'O L mol-' s-l at 300 K and decreases with increasing temperature. Recent experimental results parallel the theoretical ones but are 15-3076 slower.
I. Introduction
been followed by mass spectrometry6J4 and by ultraviolet absorption spectr~scopy.~-~*~-" The rate coefficient for the reaction depends on the pressure and nature of the bath molecule^.'^-^^ In low-pressure gases, the rate is limited by energy transfer;I4 at intermediate pressures, it reaches a maximum;I2 in very highpressure gases or liquids, it becomes limited by diffusion.'Zl9 The reverse reaction,2°*2'including the decomposition of photochemically hot has also been carefully studied. There is now a proposal to investigate the reaction in crossed ablated The potential energy surface developed herein will be useful for interpreting all of these types of experiments. In this paper, we shall only calculate the maximum rate constant at intermediate pressures. There is some divergence of opinion regarding the value of this limiting rate constant at room temperature, but recent experimental result^'^*'^-^^ appear to be converging to a value of (3-4) X 1O'O L mol-' s-'. At first, it appeared that the rate constant was independent of temperature, but now, there is evidence of a decline in rate with increasing t e m p e r a t ~ r e . ' ~ J ~ - ' ~ J ~ Theoreticians have had no difficulty in reproducing the roomtemperature rate constant: simple collision theory, with an electronic factor of 0.25 and a reasonable collision diameter, works well. The temperature dependence has been more challenging. G ~ r i considered n~~ that the methyl radicals were drawn together by a force independent of their angle of orientation. The reaction bottleneck, R , , the carbon-carbon distance where the sum of potential and centrifugal energies was a maximum, became shorter as the temperature, T, became hotter. The rate constant increased in proportion to Rice-Ramsperger-Kassel-Marcus (RRKM) theory20 incorporated harmonic orientation-dependent potentials and predicted an even stronger positive temperature dependence. Quack and Troe26and H a ~ noted e ~ ~ that early applications of RRKM had kept the location of the bottleneck fixed and suggested this was partly responsible for the disagreement. They developed theories
The recombination of methyl radicals is of central importance in free-radical chemistry. It is the model for our understanding of all organic radical combination reactions, and it serves as the reference reaction for measurements of hundreds of other reactions of methyl radicals. There have been many experimental and theoretical studies of these reactions, ,but there has remained a critical gap in our knowledge. We have not known the potential energy of interaction of the radicals. The present work is an attempt to begin to fill that gap by providing ab initio calculations of the potential energy and then by using the potential energy to calculate rates. Many experimental techniques have been applied to this reaction, including the rotating sector technique,' pulsed photolysis,2 flash photoly~is,~-'~ molecular modulation spectro~copy,'~*'~ and non-steady-state p y r o l y ~ i s . ' ~The ~ ' ~ decay of methyl radicals has
( I ) Shepp, A. J. Chem. Phys. 1956, 24, 939-943. (2) March, R. E.; Polanyi, J. C. Proc. R. SOC.London, A 1963, 273, 360-371. (3) van den Bergh, H. E.; Callear, A. B.; Norstrom, R. J. Chem. Phys. Left. 1969, 4, 101-102. (4) Braun, W.; Bass, A. M.; Pilling, M. J. Chem. Phys. 1970, 52, 513 1-5143. ( 5 ) Basco, N.; James, D. G. L.; Stuart, R. D. Int. J. Chem. Kinet. 1970, 2, 215-234. (6) Truby, F. K.; Rice, J. K. Int. J . Chem. Kine?. 1973, 5, 721-732. (7) James, F. C.; Simons, J. P. Int. J. Chem. Kine$. 1974, 6, 887-891. (8) van den Bergh, H. E. Chem. Phys. Lett. 1976, 43, 201-204. (9) Hochanadel, C. P.; Ghormley, J. A,; Boyle, J. W.; Ogren, P. J. J. Phys. Chem. 1977, 81, 3-7. (10) Adachi, H.; Basco, N.; James, D. G. L. In?. J. Chem. Kine?. 1980, 12, 949-917. (11) Laguna, G. A.; Baughcum, S. L. Chem. Phys. Lett. 1982, 88, 568-57 1. (12) Hippler, H.; Luther, K.; Ravishankara, A. R.; Troe, J. 2.Phys. Chem. (Munich) 1984, 142, 1-12. (13) Macpherson, M. T.; Pilling, M. J.; Smith, M. J. C. J. Phys. Chem. 1985,89, 2268-2274. ( I 4) Slagle, I. R.; Gutman, D.; Davies, J. W.; Piliing, M. J. J. Phys. Chem. 1988, 92, 2455-2462. ( 1 5 ) Parkes, D. A.; Paul, D. M.; Quinn, C. P. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1935-1951. (16) Arthur, N. L. J. Chem. Soc., Faraday Trans. 2 1986,82, 331-336. (17) Glanzer, K.; Quack, M.; Troe, J. Symp. (Int.) Combust., [Proc.] 1976, 16, 949-960. (18) Pacey, P. D.; Wimalasena, J. H. J. Phys. Chem. 1980,84,2221-2225.
0022-3654,I89,12093-4772$0 1.50 I O I
(19) Sceats, M. G. Chem. Phys. Lett. 1988, 143, 123-126. (20) Waage, E. V.; Rabinovitch, B. S. In?. J. Chem. Kine?. 1971, 3, 105-125, and references cited therein. (21) Quack, M.; Troe, J. In Gas Kinetics nnd Energy Transfer; Ashmore, P. G., Donovan, R. J., Eds.; Chemical Society: London, 1977; Vol. 2, pp 175-238. Quack, M.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 329-337. Quack, M.; Troe, J. Theor. Chem. (N.Y.) 1981, 68, 199-275. (22) Growcock, F. B.; Hase, W. L.; Simons, J. W. In?.J. Chem. Kine?. 1973, 5, 77-92. (23) Polanyi, J. C. Faraday Discuss. Chem. Soc., in press. (24) Harrison, I.; Polanyi, J. C. 2.Phys. D,in press. (25) Gorin, E. J. Chem. Phys. 1939, 7 , 256-264. (26) Quack, M.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 240-25 1. (27) Hase, W. L. J. Chem. Phys. 1976, 64, 2442-2449.
0 1989 American Chemical Societv -
Recombination of Methyl Radicals in which R , varied with temperature or quantum state. Quack and Troe26 predicted a decrease in rate a t high temperatures. Benson2* supposed that hydrogen atoms on different methyl radicals have an infinite repulsive force at short distances; elsewhere, the potential energy of interaction was independent of angle. In effect, certain orientations were allowed and others were disallowed. Wardlaw and Marcus (WM)2e3i developed a more sophisticated potential function, with an angle- and distance-dependent attraction between the carbons and Lennard-Jones attractive and repulsive forces between non-bonded atoms. They obtained a rate constant that declined with increasing temperature, in excellent quantitative agreement with experiment. Such agreement does not prove this is the correct potential function; there may be a family of functions that would work equally well. A way of choosing among this family is to perform ab initio calculations. There have been many a b initio studies of C2H6, but most were for geometries close to that of the stable molecule.32 Historically, many workers directed their attention to the barrier to rotation about the C-C bond.33 Evleth and K a ~ s a have b~~ performed calculations for longer distances between face-to-face methyl radicals but did not vary the orientation. Carlacci et al. have examined many relative orientations of the methyl radicals with the aim of obtaining spin-orbit coupling constants.35 In section I1 of this paper we report quantum chemical calculations for several values of the C-C distance and for different values of five angles describing the mutual orientation of the methyl radicals. Our method is modeled on those successfully used on the CH4system.36 The basis sets are at the double{-level with polarization included for the carbons; correlation is taken into account by the multireference single- and double-excitation configuration interaction (MRD CI) method with Davidson’s extrapolation. W e shall find that the W M potential has a reasonable form but that there are quantitative disagreements between the potentials. Consequently, in section 111, we develop a modified analytical expression for the potential and fit it to the ab initio points. This potential can be understood in terms of simple forces between atoms and between orbitals. Variational transition-state theory (VTST) is used to calculate maximum rate constants for the reaction in section IV. The results are compared with earlier theories and experiments in section V. 11. Electronic Energies The basis set for this study was of double-f quality, with polarization functions added for the carbons only. For carbon, Dunning’s 9s5p/4s2p contraction was used, and for hydrogen, Dunning’s scaled 4 ~ 1 2 contraction s was employed.” A polarization d exponent of 0.75 was given to carbon.38 Sample ghost orbital calculations were performed at carbon-carbon separations (28) Benson, S.W. Thermochemical Kinetics; Wiley: New York, 1976; pp 160-163. (29) Wardlaw, D. M.; Marcus, R. A. Chem. Phys. Lett. 1984, 110, 230-234. (30) Wardlaw, D. M.;Marcus, R. A. J. Phys. Chem. 1985,83,3462-3480. (31) Wardlaw, D. M.; Marcus, R. A. J. Chem. Phys. 1986,90,5383-5393. (32) See,for example: Lathan, W. A.; Hehre, W. J.; Pople, J. A. J . Am. Chem. Soc. 1971,93,808-815. Buenker, R. J.; Peyerimhoff, S. D. Chem. Phys. 1975,8,56-67. Richartz, A,; Buenker, R. J.; Bruna, P. J.; Peyerimhoff, S . D. Mol. Phys. 1977,33,1345-1366.Bemardi, F.;Robb, M. A. Mol. Phys. 1983,48, 1345-1355. Nakatsuji, H.; Hada, M.; Kanda, K.; Yonezawa, T. Int. J . Quanrum Chem. 1983,23,387-397.Musso, G . I.; Magnasco, V. Mol. Phys. 1984,53,615-630. Kello, V.;Urban, M.; Noga, J.; Diercksen, G. H. F. J. Am. Chem.Soc. 1984,106,5864-5871.Gordon, M. S.;Truong,T. N.; Pople, J. A. Chem. Phys. Lett. 1986, 130, 245-248. (33) Mulliken, R. S.;Ermler, W. C. Polyaromic Molecules; Academic Press: New York, 1981. (34)Evleth, E. M.;Kassab, E. Chem. Phys. Lett. 1986, 131,475-482. (35)Carlacci, L.; Doubleday, C., Jr.; Furlani, T. R.; King, H. F.; McIver, J. W., Jr. J. Am. Chem. Soc. 1987, 109,5323-5329. (36) (a) Peyerimhoff, S.D.; Lewerenz, M.;Quack, M.Chem. Phys. Lett. 1984,109,563-569.(b) Hirst, D. M. Chem. Phys. Lett. 1985,122,225-229. (c) Lewerenz, M.;Quack, M. J . Chem. Phys. 1988,88,5408-5432. (37) Dunning, T. H.,Jr. J. Chem. Phys. 1970,53, 2823-2833. (38) Dunning, T. H., Jr.; Hay, P. J. In Modern Theoretical Chemistry; Schaefer, H. F. 111, Ed.; Plenum Press: New York, 1977;Vol. 3, pp 1-27.
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4113
S
Pih
.pi pi
Figure 1. Illustration of the orientations chosen for a b initio calculations. The angles are listed in Table I.
TABLE I: Values of Angles (as a Fraction of Geometries of the ab Initio Calculations geometry
&I*
S
0 0
A
D E F G
I
J K P EP
0 ‘I1
‘I2 ‘I2 ‘I6 ‘I6
‘I6 0
’I1
9 2 1 ~ 0 0
r
XIIT
Radians) at the x2/*
’13
0 0
0 1
‘I1 ‘I2 ‘I1 ‘11
0 0
‘13
‘13
‘I3
0
‘13
‘13
0
‘/a
0
‘I6 ‘I2 ‘I2
0
0
0
0
0
0 ‘/a
0
&I* 0 0 ‘12
0 0 0 0
0 1 1
‘I2
of 3 and 4 A in order to estimate basis set superposition error (BSSE).39.40 The BSSE was found to lower the energy of the system by about 2 and 0.4 kJ mol-’, respectively. This error was deemed to be small enough to justify the omission of corrections due to BSSE. Configuration interaction was performed with use of the multireference single- and double-excitation CI (MRD CI) program of Buenker and Peyerimh~ff.~’Two reference configurations were employed throughout: namely, the HF configuration for the ]A, ground state, la1,21a2>2a *2a2>1e~3ai,21e,4,and PO*^ excitation. This the configuration corresponding to the p> second configuration accounts for the dissociation of ethane into two methyl radicals. All single and double excitations were generated with respect to the reference set, keeping the Is orbitals on both carbons doubly occupied and forbidding occupation of the two highest virtual orbitals. This resulted in the generation
-
(39) Boys, S.F.;Bernardi, F. Mol. Phys. 1970, 19,553-566. (40) Ostlund, N. S.; Merrifield, D. L. Chem. Phys. Lett. 1976, 39, 612-614. (41) (a) Buenker, R. J.; Peyerimhoff, S. D. Theor. Chim. Acta 1974,35, 33-58; (b) Ibid. 1975,39,217-228.(c) Buenker, R. J.; Peyerimhoff, S. D.; Butcher, W. Mol. Phys. 1978,35, 771-791. (d) Buenker, R. J. Molecular Physics and Quantum Chemistry: Into the ’80’s (Seminar/ Workshop); Burton, P. G., Ed.; University of Wollongong, Wollongong, Australia, 1980; pp 1.5.1-1.5.40. (e) Buenker, R. J. In Studies in Physical and Theoretical Chemistry; Carbo, R., Ed.; Current Aspects of Quantum Chemistry 1981; Elsevier: Amsterdam, 1982;Vol. 21,pp 17-34. (f) Buenker, R. J.; Phillips, R. A. J. Mol. Strucr.: THEOCHEM, 1985, 123, 291-300.
4774
Darvesh et al.
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
TABLE 11: Results of ab Initio Calculations geometry 20s 4s 4A 4D 4E 4F 41 4P 3.5s
3s 3A
3E 3F 3G 31
35
3K ~EP 2.5s 2.5F
RIA 20 4 4 4 4 4 4 4 3.5 3 3 3 3 3 3 3 3 3 2.5 2.5
EscFIEh -78.889366 -78.977892 -78.977874 -78.945691 -78.931127 -78.946671 -78.973 687 -78.933482 -79.003967 -79.038870 -79.038 716 -78.904661 -78.975440 -78.903 594 -79.029 035 -79.029 125 -79.016566 -78.939699 -79.083 225 -78.965208
EmdEha -79.359983 -79.360081 -79.360079 -79.358963 -79.360462 -79.360239 -79.359928 -79.359478 -79.362 a41 -79.371 216 -79.371092 -79,340337 -79.352301 -79.28301 1 -79.36745 1 -79.367046 -79.363163 -79.338296 -79.389768 -79.313065
Ehii ab ~ C / k J / m o l -79.385660 0.0 -79.386954 -3.40 -79.386952 -3.39 -1.04 -79.386056 -79.386282 -1.63 -1.59 -79.386267 -79.386701 -2.73 -79.386145 -1.27 -79.389 728 -10.68 -79.397813 -31.90 -31.62 -79.397 705 -79.367831 46.80 -79.378 633 18.45 -79.309998 198.61 -79.394090 -22.13 -79.393692 -21.09 -79.389950 -1 1.26 -79.365933 51.78 -79.416213 -80.20 -79.339560 121.01
' E M R Dis the extrapolated CI energy. b E f u I Iis~the l CI energy in Eh at full CI estimated with Davidson's corre~tion.'~C A Eis the energy difference ER - E 2 s , at full CI level.
of 13 782 configurations for staggered ethane. From these, configurations were selected according to their energy lowerings. From computation of ethane at its equilibrium geometry (tetrahedral angles; C C and CH bond lengths of 1.534 and 1.093 A, r e ~ p e c t i v e l y ~and ~ ) at a C-C separation of 20 A to represent dissociation ( 120° H C H angles; RC-H= 1.079 A43),a C - C bond dissociation energy, De, of 382 kJ mol-' was obtained. Both equilibrium and dissociated geometries were calculated at a selection threshold of 10-5Eh,followed by energy extrapolation and an estimate of the full C I energy with the M R D CI version of Davidson's correcti~n!~.~~The bond dissociation energy obtained in this manner is about 6% less than the experimental value of 406 kJ mol-', calculated from the experimental reaction enthalpy and spectroscopic zero-point energies.44 Various long-range structures were computed. The geometries chosen are illustrated in Figure 1. In all cases, the C-H separation for each methyl radical was held fixed at 1.079 A, with each methyl angle fixed at 120'. Thus, the portion of the potential being explored is the relative orientation of the two methyl radicals at various C-C separations. The geometries are described in terms of a set of angular coordinates in Table I. Here, 8, and O2 are the angles between the C-C axis and the 3-fold symmetry axes of the two methyl groups, xIand x2 are the angles describing rotation of the methyl groups about their 3-fold axes, and 4 is the angle between these 3-fold axes when viewed along the C-C axis. Most energies reported in Table I1 were evaluated with a configuration selection threshold of 0.1 phartree. This led to a selected configuration space of between 7000 and 14000 configurations. Due to excessively large numbers of configurations, several geometries (I, J, K, and EP) had to be computed at somewhat higher thresholds (0.2 or 0.3 phartree). The binding energy, hE, was defined as the energy of a given structure minus the energy at 20 A. 111. Fitted Potential Energy Surface
In the previous section, we presented ab initio energies at 19 geometries. For the calculation of the reaction rate, we need (42) Interatomic Distances, Supplement. Spec. Publ.-Chem. SOC.1965, NO. 18. (43)Herzberg, G. Electronic Spectra of Polyatomic Molecules; Van Nostrand: Princeton, NJ, 1967;p 609. (44) Hehre, W.J.; Radom, L.; Schleyer, P.v. R.;Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986;p 278. (45) (a) Langhoff, S.R.; Davidson, E. R. I n t . J . Quantum Chem. 1973, 7, 999-1019. (b) Davidson, E. R. In The World of Quantum Chemistry; Daudel, R., Pullman, B., Eds.; Reidel: Dordrecht, The Netherlands, 1973; pp 17-30. (c) Langhoff, S.R.; Davidson, E. R. Int. J. Quuntum Chem. 1974, 8,61-72.
1-
on a
I
0
,
i
0
i
L
I
2
R, A
3
L
Figure 2. Relative energies ( E R = -De/AE) for the staggered (S) geometry as a function of C-C distance. Key: -, Morse curve, ref 30; 0,binding energies, AE, and dissociation energy, De, for optimized geometries taken from ref 34;0 , A E for planar CH3 from final column of Table I1 and De for equilibrium geometry from section 11.
energies at 17 000 geometries. Calculation of all these energies by ab initio methods would require too much computer time. Accordingly, in this section, we seek patterns in the CI energies, find semiempirical functions to express these patterns, and adjust the parameters in the functions by least squares to obtain a global potential energy surface. There are two general methods in use for generating a potential surface from ab initio points. The first is an expansion in terms of internal or normal coordinates. Recent examples include surfaces for the combination of hydrogen atoms and methyl radicals, which involved more than 20 parameter^.^^^^' Examples for the C2H6 system are the RRKMZ0surfaces, where most degrees of freedom were treated as independent harmonic oscillators. An examination of Tables I and I1 reveals strong coupling between the degrees of freedom. For points 3 s and 3A, the angle x2 has little influence on the energy, but for points 3E and 3F, a change in x2 makes a big difference. To find a function to accurately couple the 0 and x angles in this approach would require more ab initio calculations. An alternative approach is the many-body expansion favored by Murrell et The aim is to interpret most of the variation in terms of forces between pairs of atoms, interpreting the rest in terms of three-body or many-body interatomic forces. Benson's surface,28 which involves two-body, hard-sphere interactions, is an example of this type. Many of the variations in Table I1 can be explained in terms of atom-atom forces. One that cannot is the 5 kJ mol-' difference between the energies for 3E and 3EP. The only changes in internuclear distances here are at long range, where two-body interactions should be small. A four-body term would be needed to explain this change, indicating that a many-body expansion may converge slowly. Because both types of surface have evident strengths and weaknesses, we have followed Wardlaw and Marcus30 and combined features from both. W M separated the potential energy into a bonding potential, a function of the C-C distance and the angular coordinates, and nonbonding potentials, functions of two-body internuclear distances. We shall present the form of the surface in three stages-first, the distance dependence for the S geometries, then, the angle dependence of the bonding forces, and finally, the distance dependence of the nonbonding forces. We shall also discuss the method of fitting the parameters of the surface to the ab initio points and shall present a simple harmonic surface as an approximation. The Staggered Configuration. The staggered geometry (S) had the lowest energy of those calculated at each distance. This (46)Duchovic, R.J.; Hase, W. L.; Schlegel, H. B. J . Phys. Chem. 1984, 88, 1339-1347. Hase, W.L.;Mondro, S. L.; Duchovic, R. J.; Hirst, D. M. J. Am. Chem. SOC.1987,109,2916-2922. (47) Brown, F.D.; Truhlar, D. G. Chem. Phys. Lett. 1985,113,441-446. (48) Murrell, J. N.; Carter, S.; Farantos, S . C.; Huxley, P.;Varandas, A. J. C. Molecular Potential Energy Functiom; Wiley: Chichester, U.K., 1984; pp 9 and 29.
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4715
Recombination of Methyl Radicals is in agreement with the usual assumption that the minimum energy path or reaction coordinate has this geometry. Evleth and K a ~ s a found b ~ ~ that the energy difference, AE,is often an exponential function of the bond length, R. This is tested in Figure 2, where In ER is plotted against R. Here, ER = -De/&. This plot emphasizes kinetically important values of AE.34 The solid points were calculated with AE from Table I1 and De from section 11. Here, it must be noted that AE values refer to planar methyl radicals and De refers to the equilibrium geometries. The outer three points are on an almost perfect straight line, whereas the inner point is above this line. The open circles in Figure 2 represent the work of Evleth and K a ~ s a b The . ~ ~ calculations agree at 3 A. At other distances, the differences are due to the larger basis set in the present work and the allowance for nonplanarity of C H 3 in ref 34. The solid curve in Figure 2 is a Morse curve, which gives the correct equlibrium properties for ethane."*30 The ab initio points lie above the Morse curve at long distances. A similar result has been obtained in me,^^^*^,^^ but not all,36acstudies of CHI. In such a case, the potential energy has been satisfactorily modeled by allowing the exponential Morse parameter to be a function of R.& We have adopted a similar approach, allowing the exponential parameter in Evleth and Kassab's relation to vary with R . Dependence on Angles. Wardlaw and Marcus,30 in their semiempirical surface for this reaction, assumed that the bonding potential would be proportional to cos2 Ol cosz 02. They did not give a rationale for this form but cite a private communication with W. A. Goddard. Such a form would arise if we follow a suggestion by M ~ l l i k e nthat ~ ~ the bonding energy should be roughly proportional to the square of the overlap integral. BensonM has also suggested that overlap should play a large role in determining the angular potential in this system. If we consider that a singly occupied p orbital on the first methyl radical lies along the axis perpendicular to the CH3 plane, then the component of the wave function along the internuclear axis is proportional to cos 8'. Similarly, the component of the second p orbital is proportional to cos 02. As shown in ref 35, the overlap integral for a-overlap of these orbitals is then proportional to cos Ol cos 0,; the square has the form suggested by WMS3O This form agrees reasonably well with the present ab initio results for small angles 0. For example, geometries 31 and 35 have the same products of cosines and nearly the same energies. The same is true for 4 s and 4A and for 3 s and 3A. More quanti) 0.75, compared to AE41/AE4s,which is 0.80, tatively, cosz ( ~ / 6 = and AE31/AE3s,which is 0.69. Thus, this form is roughly correct, although there are subtleties that this simple form cannot express. Mulliken et al.51have shown that the overlap integrals depend almost exponentially on internuclear distance at large separations, in agreement with our earlier conclusions from Figure 2. We have therefore expressed the overlap as follows:
S, = A , cos Ol
COS
O2 exp[-B,'R
-B,~R~]
(1)
Initial values of the parameters A , and B,, were calculated from the Morse parameters of WM.30 Bs2 was initially set at zero. It is also possible for p orbitals to overlap side by side, forming a ?r bond. There is evidence from the present ab initio calculations of weak A interactions. Thus, the 5 kJ mol-' difference in energy between geometries 3E and 3EP, already noted, can be explained by the possibility of A overlap between p orbitals perpendicular to the CH3 plane in geometry 3E. This possibility is absent in geometry 3EP. Similarly, A overlap can explain the fact that geometries 4E and 4 F are more stable than geometries 4D and
4P. The component of the first p orbital perpendicular to the C - C axis is proportional to sin 0'. The component of the second in the direction of this component of the first is sin Oz cos 4. The overlap integral is then proportional to sin O1 sin O2 cos 4. Mulliken et ~
~
~~~~
(49) Mulliken, R. S. J . Am. Chem. SOC.1950, 72, 4493-4503. (50) Benson, S. W. Can. J . Chem. 1983, 61, 881-887. (51) Mulliken, R. S.;Rieke, C. A.; Orloff, D.; Orloff, H. J . Chem. Phys. 1949, 1 7 , 1248-1 261.
aLS1have shown that A overlap integrals also decrease exponentially with internuclear distance. The form of the A overlap term was therefore taken as
S, = A , sin 0, sin Oz cos 4 exp(-B,R)
(2)
Starting values of the parameters A , and Bp were chosen to fit energies for the geometries above, where evidence of this effect was clearest. The two overlap terms were then combined to give an overall bonding interaction as follows: (3) This expression is empirical; others were tried, but this gave the best fit to the ab initio results, especially for geometry 3K, in which both types of overlap contribute. We also explored the possibility that the interacting electrons have some s character and added s and sp overlap terms but found this did not improve the fit. Nonbonding Interactions. There is evidence in the results of section I1 for both a long-range attraction and a short-range repulsion. The negative energy differences for geometries 4D and 4P, with zero overlap, cannot be explained by bonding forces but can be explained by nonbonding dispersion forces. Similarly, the bonding overlaps are identical for geometries 3E, 3F, and 3G, but the large increases in energy in this series do correspond to shorter H-H distances and greater H-H repulsion. Accordingly, we added nonbonding interactions to the potential as follows: (4) Here, i and j number the atoms on the first and second methyl fragments, respectively. Wardlaw and Marcus30 included nonbonding H-H and H-C interactions. We have included C-C interactions as well, because there was no C-C repulsion included in the bonding forces. It seemed more reasonable that the C-C repulsions would be anisotropic, like the other nonbonding forces, instead of strongly angle dependent, like the expressions used for the bonding forces. For the individual interactions, Vi, we first tried Lennard-Jones 12-6 functions as in WMS3OThese were found to give a very steep repulsion at short range, which was incompatible with the quantum chemical results. This difficulty could not be overcome by allowing the repulsive exponent to vary with distance, as was recommended by Maitland and Smith.52 A number of worker^^'.^^ have found that short-range repulsions vary approximately exponentially with distance. Accordingly, we tried a Buckingham exp-6 potential, which has been successfully applied in molecular mechanics c a l c u l a t i ~ n s . We ~ ~ found this form gives a very good fit, but at very short internuclear distances, the potential energy becomes negative. This is not physically realistic. R ~ d b e r gand ~ ~Morse potentials were also tried. The Morse potential gave the better fit, and it was selected for subsequent work.
Here, Bxy is the exponential parameter and RmXYand Vmxy are the location and the depth of the attractive well for a particular pair of elements (Le., H-H, C-H, or C-C). Rii is the distance between the repulsive centers of the pair of atoms. For carbon, the repulsive center was at the nucleus. For hydrogen, we have adopted the practice from molecular mechanics5s of allowing the repulsive center to be located a fraction,f, of the distance from the carbon to the hydrogen nucleus. This reflects the shift in electron density from the hydrogen atom toward the bond. The exponential factors Bxy were allowed to vary with distance.
(52) Maitland, G. C.; Smith, E. B. Chem. Phys. Lett. 1973, 22, 443-446. (53) Kolos, W.; Ranghino, G.; Clementi, E.; Novaro, 0. Int. J . Quantum Chem. 1980, 17, 429-448. (54) Bohm, H.-J.; Ahlrichs, R. J . Chem. Phys. 1982, 77, 2028-2032. (55) Allinger, N. L. J . Am. Chem. SOC.1977, 99, 8127-8134.
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The Journal of Physical Chemistry, Vol. 93, No. 12, I989
TABLE 111: Parameters of Potential Enerev Surfaces parameter full surface harmonic surface
Fitted Parameters
f
0.883 572 0.196 590 0.173 155 1.7 19 663 0.016 73 1 56.278219 0.844 886 0.054 742
ARIA AB/A-'
BHHI/A-I
Bxy2IA-I A P /(kJ/mol)'i2
Bs1IA-I B,21A-2 Fl
F2/A-'
Here, S, was given by a shortened version of eq 1.
Parameters Fixed at Initial Values kJ/mol 0.0418 VmcdkJ/mol 0.4 RmccIA 3.88 A,/(kJ/mol)'i2 119.79 Bp/A-I 1.177 vmHH/
S, = A, exp[-B,,R - BS2Rz]
B x y 2 was given the same value for all elemental pairs. Values of B x y l and RmXYfor different elements were tied together by the following equations, which represent arithmetic mean rules: B H H-I &HI = & H I RmHH
- Bcci =
- RmCH = RmCH - RmCC
AB
(7)
= -AR
(8)
Initial values of the parameters Vmxyand RmXYwere taken from WM30 and those for Bxvl from ref 55. Final Fitted Surface. The bonding and nonbonding potential functions, just described, were added together. V ( R , 01, 82, XI,~ 2 b) , = Vbonding(R981, 8 2 3 4)
+ Vnonbonding(R9 81, 829
Examining the forms of eq 1 and 3, neglecting the weaker *-bonding terms, and expanding the cosines as infinite series, we obtain the following equation:
This form assumes that both the angle-dependent and angle-independent terms have the same dependence on R . We can see from Table I1 that this is not true. Geometries with large values of 0' or 82 have negative energies at long distances and positive energies at short distances. Therefore, we have separated the angle-dependent and -independent terms as follows:
0.111842 0.148 081 29.886 807 3.9 12 777 0.913086
As/(kJ/mol)l/2
Darvesh et al.
XI, x2, #)
(9)
Initial values of the parameters were chosen as described previously. These parameters were then adjusted by a nonlinear least-squares routines6 to minimize the sum of the squares of the deviations from the results in the final column of Table 11. All results in Table I1 were given equal weight. The number of fitted parameters was restricted to ensure that the function remained realistic and to avoid convergence problems. Several variations of the final function were tested to see if the fit could be improved. The final parameters, both fixed and fitted parameters, are listed in Table 111. VmcHis given by the geometric mean rule. Thus, there are 13 parameters, 8 adjusted and 5 fixed, that have been used to fit 19 ab initio points. All of the parameter values are reasonable. The value off agrees with the bond length reductions of about lo%, common in molecular mechanics calculations.55 The correction parameters AR,AB, BXY2,and Bs2are all small. B s I ,when doubled because of the square in eq 3, agrees well with normal Morse exponential parameter^.^^ The exponential parameter for the repulsive part of the H-H interaction is twice B H H I , or 3.44 A-'; this also agrees with previous estimate^.^^-^^ The largest deviation between the fitted surface and the quantum chemical results was 1.4 kJ mol-', which occurred for geometry 3E. As a benchmark, one can compare with a maximum 2.3 kJ mol-' deviation in Truhlar and Horowitz' 25-parameter fit to the 267 points of Liu and Siegbahn for H plus H2.57 Because of the reasonable form of the function and the closeness of the fit, we will use the surface to obtain estimates of energies for geometries lying between those in Table 11. Harmonic Approximation. Several previous treatmentsz0 of this reaction have assumed that the rocking motions of the methyl groups could be described as harmonic vibrations. Such an assumption considerably simplifies the calculation of the rate constant. For this reason, we have developed an harmonic approximation to the potential energy surface. (56) Ralston, M. In BMDP Srarisfical Softwure; Dixon, W. J., Ed.; University of California Press: Berkeley, 1983; p 305. (57) Truhlar, D. J.; Horowitz, C. J. J . Chem. Phys. 1978,68, 2466-2476.
(12) Equation 11 was fitted to the ab initio results by nonlinear least squares. Only the negative energies in Table I1 were used in the fit, because the harmonic approximation clearly breaks down where there are large interatomic repulsions. The fitted values of the parameters are given in Table 111. The large changes in the parameters B,, and Bs2 are caused by the lack of any C-C repulsive force in this model. The largest difference between this surface and the energies used as input was 0.6 kJ mol-'. IV. Calculation of Rates In this section, we describe the calculation of rate coefficients by canonical variational transition-state t h e ~ r y . ~In~common ,~~ with other versions of transition-state theory, this theory makes two fundamental assumptions. The first is that all states of the reactants and of reacting complexes are in equilibrium. As long as collisions occur frequently enough to equilibrate these states but not so frequently that diffusion impedes reaction, this assumption is believed to be valid. Our conclusions will only apply, however, to the maxiumum rate constant at intermediate pressures. The second assumption, in classical language, is called the nonrecrossing assumption. For canonical variational TST, it states that at each temperature we may define a distance, R , , and that any pair of radicals that approach each other within this distance always forms products, instead of recrossing this dividing sphere to reform reactants. The distance R , is that for which the calculated rate is a minimum. There has been considerable debate about this assumption, and we consider it further in the next section. For combination reactions, Klippenstein and Marcus60 have derived a CVTST expression, which may be paraphrased as follows: NLK
k( T , R ) = g,LaR2u,B,BI'/2NMc-' exp(-V,,/kBT) (13) n=l
Here, g,, the ratio of electronic partition functions, is taken as 1/4, the fraction of methyl radical collisions that occur on the singlet potential surface. L is the ratio of symmetry numbers in the reactants and complex (1/2, with the present geometries), and u, (8kBT/7rh)is the average relative velocity for CH3 collisions. The treatment of symmetry has been checked by comparison with collision theory6I and the maximum free energy method.21 B, is the ratio of vibrational partition functions between complexes and reactants for the conserved modes, the C-H stretches, and in-plane and umbrella bending modes in CH,. BI is the ratio of the products of moments of inertia for CH, groups in the complex and in reactants. With rigid methyl radicals, the moments of inertia do not change and this factor is unity. NMc is the total number of randomly generated geometries in the Monte Carlo (58) Quack, M.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 329-337. (59) Truhlar, D. G.; Garrett, B. C. Annu. Reu. Phys. Chem. 1984, 35, 159-189. (60) Klippenstein, S. J.; Marcus, R. A. J . Chem. Phys. 1987, 87, 3410-3417. (61) LeBlanc, J F.; Pacey, P. D. J . Chem. Phys. 1985, 83, 4511-4515.
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4777
Recombination of Methyl Radicals integration scheme, Vn is the potential energy for one of these geometries, p is the reduced mass, and kBis Boltzmann's constant. Readers will recognize the similarity between this equation and simple collision theory. In an earlier publication,61 if was called modified collision theory, since it is the collision theory rate constant multiplied by ratios of partition functions and an average Boltzmann factor. In applying this equation, B, was assumed to be unity. Our ab initio calculations did not involve variation of these conserved degrees of freedom. We will assume for now that these vibrations are not perturbed by the methyl-methyl interactions, but we will return to this point in the next section. Although it does not appear explicitly in eq 13, Klippenstein and Marcusm included a factor sin O1 sin O2 in the Monte Carlo integration to give a pro,per weighting to the orientations of the methyl fragments. Thus, the factor Nhlc-land the summation in eq 13 may be replaced by the following: NhE
NhE
C s i n 0, sin O2 exp(-V,,/kBT)/
sin 0, sin O2
(14)
n= 1
n-1
Klippenstein and Marcus integrated over six angles, including angles dl and d2for both methyl radicals. We note that the energy only depends on the difference between these two angles, and we have only integrated over the five angles in Table I. The number of Monte Carlo calculations required for convergence may be reduced if an approximate function is available that can be integrated analytically.62 In this case, we have the harmonic approximation, developed at the end of section 111. Replacing V by Vh-onk and sin el and sin O2 by el and 02, we have -
-
exp(-Vharmonic/kB7') d o l dOZ dd dxl dX2 = (r3/9D2)[1 - e x p ( - D ~ ~ / 4 )exp(C) ]~ (15) Here C = S?/kBT
(16)
and The approximation consists of subtracting the argument on the left side of eq 15 from that in eq 14, integrating by the Monte Carlo method, and then adding the integrated form on the right side of eq 15. The final equation for the rate constant becomes NhE
k ( T , R ) = 0 . 1 2 5 r R 2 u r [sin ~ O1 sin O2 exp(-Vn/kBT) n= 1
"
0102 exp(-vharmo,ic,n/kB7'))1/
n- I
sin
OI
sin
O2 -k
kharmonidT, R ,
(18) Here, kharmonic is the harmonic approximation to the rate constant. kharmonic(T, R, = (0.03125rR2ur/D2)[1 - e ~ p ( - D r ~ / 4 )exp(C) ]~ (19) This expression, except for the factor in square brackets, would result from a simple version of TST, treating the rocking motions as classical vibrations and the motion as a free rotor. The factor in square brackets results from our use of r / 2 as the upper limit in the integrations over Ol and 02, instead of the usual limit of m for classical vibrations. In implementing eq 18, random values of the five angles within the integration limits in eq 15 were generated from the program RANIin ref 63. Values of R were set at 0.1-A intervals from 2.4 (62) Davis, P. J.; Rabinowitz, P. Numerical Integration; Blaisdell: Waltham, MA, 1967; pp 142-148. (63) Press, W. H.; Flannery, E. P.; Teukolsky, S. A,; Vetterling, W. T. Numerical Recipes, The Art of Scientific Computing; Cambridge University Press: Cambridge, U.K., 1986; pp 191-197.
-
x
1
25
30
40
35
R , i Figure 3. Rate coefficients as a function of distance for four temperatures calculated for the full potential energy surface from eq 18. TABLE I V Critical Distances and Calculated Rate Constants' TI R,/A k ( r ) / l O l o L mo1-ls-l 200 300 400
500 600
800 1000 1200 1500
2000
3.96 3.70 3.54 3.42 3.32 3.17 3.05 2.96 2.84 2.67
6.1 f 0.2 5.0 0.1 4.3 f 0.1 3.8 f 0.1 3.47 f 0.09 2.98 f 0.08 2.62 f 0.07 2.34 f 0.06 2.01 f 0.06 1.62 f 0.05
*
"Quoted uncertainties are 95% confidence levels for the Monte Carlo integration only; other uncertainties are discussed in section V. to 4.0 A. Cartesian coordinates of all the atoms were calculated, as were values of Vbndlng, v,,o,,bnding, and Vharmonic. Equation 18 was then applied to calculate the rate constant. Some typical results are seen in Figure 3. At each temperature, k ( T , R ) has a minimum at a distance R , corresponding to a dynamical bottleneck or variational transition state. At longer distances, k(T, R ) increases because of the factor R2 in eq 18; here, there is a large inward flux of particles, some of which will be turned back at or before the bottleneck. At shorter distances, k( T, R ) increases because of the Boltzmann factor; products that have passed the bottleneck are accelerated in this region. Values of R , decrease as the temperature increases. The minima are reasonably shallow, allowing reliable determination of the lowest value of k( T, R ) with the 0.1-A grid of R values. The minimum value of k(T, R ) at each temperature was taken as the rate constant, k( 7'). Values of R , and k(7') are listed in Table IV. The rate coefficient declines consistently with increasing temperature, approximately in proportion to Tilz. The quoted uncertainties are 95% confidence levels, calculated by the method of ref 62. These uncertainties of about 3% were obtained with 1000 sets of random angles. The substitution made in eq 18 allowed about a IO-fold reduction in the number of sets of angles required. V. Comparison with Previous Work and Discussion In this section, we compare the results of the preceding section with earlier theoretical and experimental work and we discuss possible sources of error. Figure 4 is a comparison of the present results (the solid line) with other theoretical calculations. The other work will be discussed in approximately chronological order to illustrate the development of our understanding of this reaction. Benson'sso The earliest model shown is the Gorin calculations with this model are represented by a dashed line. The model allows the reaction bottleneck to change position with changing temperature, but it assumes the bending motions are free rotations. This assumption should work well when the barrier to rotation is less than or equal to kBT.64 This is approximately (64) King, S . 3 29-3 3 8.
C.; LeBlanc,
J . F.; Pacey, P. D. Chem. Phys. 1988, 123,
4778
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989
Darvesh et al.
' 4
ob
I
1000
I
2000
T, K
Figure 4. Comparison of theoretical rate constants: --,Gorin model, ref 50; 0 , ref 20; B, ref 21 and 26; 0,ref 27; 0,ref 65; -, recommended value from present work, eq 18; -.-, kharmonlc, eq 19.
true in the present work at 200 K, where the critical distance is about 4 A. In Figure 4, we see that the two models are in good agreement at this temperature. At 1100 K, the critical distance in Table IV shifted to 3 A, where the barrier to internal rotation is 5kBT. For this reason, the free rotational model does not work well at higher temperatures. The first method used to introduce hindrance into the rocking motions was to treat them as harmonic oscillators. This was done in a number of TST and RRKM calculations in which R , was held fixed. An example is the work of Waage and Rabinovitch,20 shown as the filled circles. The rate is slower than for the Gorin model, but the increase with temperature is more dramatic. The reason can be understood by examining Figure 3. For fixed values of R at the right of this figure, an increase in temperature leads to an increase in rate. For the reaction of H with CH3, it has been shown6' that this effect is even stronger when bending is treated as a vibration, instead of the hindered rotational treatment of Figure 3. A forward step was the combination of a variational criterion with hindered rocking potentials. The treatments of Quack and Troe (filled squares),21,26in which a separate bottleneck was identified for each quantum state, and of Hase (open circles),27 in which R , was selected as the point with the minimum density of states, are pictured in Figure 4. The rates are close to those from the present work, although the temperature dependence differs. A closer parallel with the present work is obtained from the semiempirical potential surface of Wardlaw and Marcus30and a hindered rotational treatment of bending. The only such calculations shown are the results of Wagner and Wardlaw (open who adjusted the surface to fit experimental rates and found bottlenecks for reactants with specific energies and rotational quantum numbers. The dashed and dotted curve shows values of khammlcfrom the present work. Here, we see that changes in the form of the surface and in the method of integration make modest changes in the rate. Both the full surface and the harmonic surface are based on the same set of quantum chemical points at 4 A, the bottleneck at 200 K; consequently, the rates are the same at this temperature. This curve and the work of Wagner and Wardlaw indicate the robustness of variational transition-state theory. In Figure 2, it can be seen that there are differences of as much as a factor of 2 between the present surface and the Morse curve of WM. By shifting the bottleneck to another position, VTST to some extent compensates for these changes. A comparison of the present theory with experiment is seen in Figure 5 . Early experiments' indicated that the limiting rate constant was about 2 X loio L mol-' s-' and independent of temperature. More recent values of the room-temperature rate constant are greater, and many4,7*'o,12-'4 exceed 3 X 1 O l o L mol-I s-l. The high-temperature results of Glanzer, Quack, and TroeI7 ( X ) and of this laboratory'* (0)suggested a decline in rate with increasing temperature. The most precise recent results are those of Hippler et a1.I2 (a) and of Pilling and c o - ~ o r k e r s(+, ~ ~ *v). ~~ (65) Wagner, A. F.; Wardlaw, D. M. J . Phys. Chem. 1988.92, 2462-2471
10
20
30
io3/ T,K - I Figure 5. Comparison of present theoretical results, -, with experiment: 0, ref 1; 0 , ref 2; A, ref 3; 0, ref 4; V, ref 5; m, ref 6; A, ref 7; 8, ref 8; m, ref 9; 0 , ref 10; 0 , ref 11; 0 , ref 12; ref 13; v,ref 14; 0,ref 15; c], ref 16; X, ref 17; 0, ref 18.
+,
Here, the downward trend with increasing temperature is clear. The parallel between the recent experimental results and the present theory is striking and encouraging. The experimental results range from 15% to 30% below the present results. We will now consider possible errors in this work to try to understand this difference. The selection threshold (0.1 phartree) employed in section I1 at large C-C separations minimized errors due to inadequate consideration of electron correlation. For example, the sum of the energy lowerings for all configurations not included in the calculation at geometry 4 s was only 74 phartrees, or about 0.2 kJ mol-I. Another source of error is the basis set superposition error, estimated to lower the AE value by 0.4 kJ mo1-I at the same geometry. The sum of these two possible sources of error is comparable to the 0.7 kJ mol-' difference between this ab initio point and the final fitted surface. The calculated dissociation energy was 6% less than the experimental value. To test the influence of errors in the surface on the rate, we have performed calculations using two surfaces scaled to give the experimental dissociation energy. The first of these (surface C) was scaled by multiplying the entire surface (eq 9) by a factor of 1.06. The other (surface D) was obtained by multiplying Vbndingby 1.03 and leaving the atom-atom terms unchanged. Rates calculated with both surfaces were faster than those for the unscaled surface, although the bottlenecks shifted 0.01-0.04 A outward to partially compensate. The increases in rate were between 1%, for surface D at 200 K, and 9%, for the same surface at 2000 K. Because we used a more stringent threshold for the fitted geometries than for the equilibrium geometry, we recommend the use of our unscaled surface instead of surface C or D. We have assumed that the methyl radicals retain a planar geometry as they approach each other. If this assumption is relaxed, there will be two opposing effects on the rate. First, the most stable geometry at each value of R will have a lower electronic energy, increasing the value of k(T, R ) . Secondly, the umbrella mode of the radicals will become stiffer, decreasing the values of the vibrational partition function and of k( T, R ) . In the reaction of H with CH,, both effects are small. H i d 6 has calculated energies for three values of the umbrella angle at a C-H distance of 3 A. Fitting these to a quadratic, we estimate that the optimized geometry is 0.86 kJ mol-I lower in energy than the planar geometry at this value of R. Aubanel and Wardlaw& have calculated anharmonic vibrational levels for the umbrella mode (66) Aubanel, E. A,; Wardlaw, D. M. Submitted for publication in J. Phys.
Chem.
Recombination of Methyl Radicals at this same distance and at infinite separation. We have calculated partition functions from these levels and find that the extra stiffness would reduce the rate by 4% at 1100 K and 3 A. Combining these two effects, we find that freeing the umbrella mode can increase the flux of H toward CH3 by 5% at these values of T and R . For the recombination of two methyls, there are two umbrella modes and steric interactions may increase the effects. We understand that ab initio calculations on the combination of two nonplanar methyl radicals are being perf~rmed.~' These should indicate the magnitude of this effect. In-plane vibrations have been shown6' to have a smaller effect than the umbrella mode for H plus CH3. We have assumed that the dynamics of the transitional modes in this reaction are classical. Replacing the classical partition functions in eq 19 for khmk by quantal partition functions would reduce the calculated rate by 6%. Klippenstein and Marcus60 found that the quantum correction on a similar surface was only 2%. We have also assumed that systems do not recross, in a classical sense, through the canonical variational transition state. These calculations, then, give an upper bound to the rate constant.*' The best ways to check the size of this error would be to perform full quantum dynamical calculations or classical trajectory calculations on this surface. Until such calculations are performed, we must rely on the results of WM,31 who found for their surface that CVTST gave rates 18-22% faster than a theory that made a more stringent assumption that there is no recrossing through critical hypersurfaces that are specific to each energy and rotational quantum state. We note that correction of the dissociation energy and allowance for nonplanar methyl radicals are likely to move the present theoretical results away from the experimental ones, whereas incorporation of classical recrossing effects and quantal rocking motions would move them closer together.
VI. Summary and Conclusions Ab initio calculations have been performed with a double-{ basis set with polarization functions on carbon for 19 orientations of pairs of planar methyl radicals. Electron correlation was included by the MRD C I method with extrapolation, selecting configurations with a threshold of 0.1 Khartree in most cases. For the equilibrium geometry, where the selection threshold was 100 times larger, the dissociation energy was within 6% of the experimental value. The semiempirical potential energy surface of Wardlaw and Marcus30was found to be remarkably prescient. Bonding energies depended on angle approximately in proportion to cos2 O1 cos2 O2 (eq 1 and 3). Evidence was also found for attractive and repulsive forces between atoms not bonded to each other. The ab initio (67) Hirst, D. M.; Wagner, A. F.; Wardlaw, D. M., private communication.
The Journal of Physical Chemistry, Vol. 93, No. 12, 1989 4119 calculations also revealed some new features. There was evidence of attraction between methyl radicals lying side by side, as in geometries E and F. This could be understood in terms of 7r bonding (eq 2). The a-bonding forces occur at shorter distances than in a standard Morse curve (Figure 2). The nonbonding interactions could be modeled best by shallow Morse potentials (eq 5 ) . A potential function (eq 9), incorporating five fixed and eight adjusted parameters, was fitted to the 19 ab initio points. The surface was tested by performing canonical variational transition-state calculations. An analytical harmonic-oscillator/freerotor approximation (eq 19) was developed and was used to reduce the time required for the more exact Monte Carlo integration (q 18) over the full potential energy surface. Recent experimental results are 1 5 3 0 % slower than the canonical variational theory. This difference reflects the fact that canonical VTST gives an upper limit to the rate constant.2' The temperature dependence is reproduced very well. The surface will be useful for a variety of other calculations. Microcanonical variational TST,59 flexible TST,29 statistical adiabatic channel,26 and classical trajectory calculations can provide improvements to canonical VTST. These theories can also be used to interpret hot-molecule experiments and the pressure dependence of the rate. Combining the surface with a model for diff~sion'~ should make it possible to study rates in high-pressure gases and in liquids. Trajectory calculations can also be used to predict the results of experiments in crossed ablated beams.24 VTST is robust; the rate is not greatly affected by changes in the surface. The converse is that it is difficult to learn the details of the surface by studying the limiting rate. It would be interesting to see if this is also true for crossed-beam and hot-molecule experiments. The basic strategy of the present study was to calculate electronic energies for a limited number of geometries at widely spaced angles in the critical range of internuclear distances. It was hoped that this would reveal the principal forces at work in this system. Such a strategy may be of value for larger molecules, where ab initio calculations would become increasingly time consuming. Indeed, semiempirical surfaces, based on that of WM30 and on eq 1-9, may be useful for these systems. However, it is also important to perform quantum chemical calculations at a larger number of geometries for ethane to refine the surface and to ensure that significant features have not been missed. Acknowledgment. We thank the Natural Sciences and Engineering Reseach Council of Canada (NSERC) for financial support and J. W. Davies, D. M. Hirst, J. C. Polanyi, M. J. Pilling, M. Quack, A. J. Thakkar, A. F. Wagner, and D. M. Wardlaw for valuable discussions and for sending results prior to publication. K.V.D. thanks NSERC and the Killam Trust for postdoctoral fellowships. Registry No. Methyl radical, 2229-07-4.
