Flexible transition-state theory rate constants for the recombination

Paul W. Seakins, Struan H. Robertson, and Michael J. Pilling , David M. Wardlaw , Fred L. Nesbitt, R. Peyton Thorn, Walter A. Payne, and Louis J. Stie...
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J. Phys. Chem. 1989, 93, 31 17-3124

diazabicyclo[2.2.2]cctane)and studies with other cyano aromatics. Such studies could also confirm the assumption that the intrinsic lifetime of the amine exciplex is determined by the properties of the amine, more precisely the chain length of the diamine, or whether the apparent chain length dependence of the exciplex lifetime is also a function of the geometrical size or substituents of the electron acceptor.

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Acknowledgment. Financial support by Deutsche Forschungsgemeinschaft and the National Science Foundation (CHE-8618994) and compensation of travel expenses by NATO Research Grant 191 1 are gratefully acknowledged. Registry No. 9-CP, 2510-55-6; TEA, 121-44-8; DPEA, 7087-68-5; TEDA2, 150-77-6; TEDA3, 60558-96-5; TEDA4,69704-44-5; TEDAS, 60832-68-0; TEDA6, 7136-51-8; TEDAS, 13989-58-7.

Flexible Transition-State Theory Rate Constants for the Recombination Reaction CH, 4H --* CH4 Eric E. Aubanelt and David M. Wardlaw*yf Department of Chemistry, Queen's University, Kingston, Ontario K7L 3N6, Canada (Received: July 5, 1988; In Final Form: November 15, 1988)

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High-pressure thermal rate constants for CH, H CH4 on the 300-2500 K temperature range are obtained by using flexible transition-state theory with the Hirst potential energy function. The canonically determined rate constants exceed the microcanonically determined ones by 6-10%. The effects on the microcanonical rate constant of an approximate transition-state symmetry number correction and of an ad hoc semiclassical correction to the classical treatment of the transitional rocking modes are found to be minor at all temperatures considered. The inclusion of anharmonicity of the CH3 umbrella vibration produces microcanonical rate constants that are larger than and have a stronger positive temperature dependence than correspondingrate constants based on the usual harmonic treatment of all conserved modes. A limited set of rate constants were computed for CH3 + D CH,D. The ratios of the CH, + D rate constants to their CH3 + H counterparts are (mD/mH)1/2 = 1.4, in disagreement with an experimentally derived ratio of -2.5.

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I. Introduction The relative simplicity of the title reaction provides the rare opportunity of comparing detailed theoretical treatments of the recombination kinetics with experiment. Since realistic potential energy surfaces exist for the CH3-.H system,I4 it is worthwhile undertaking studies aimed at elucidating the reaction dynamics underlying the kinetics. A direct approach using classical trajectories has been taken by Hase and D u c h o ~ i c . ~A- ~commonly chosen alternative is an indirect approach in which a statistical model is invoked to obtain dynamics-free predictions of the association (or dissociation) rate constants. The interpretation of observed rate constants is then attempted by comparison to statistical model results. Two general classes of statistical model that have been applied to the CH3-.H system are variational transition-state theory (VTST)',*-'' and the statistical adiabatic channel model.12 This work describes the implementation for and application to CH, + H recombination of a particular version of VTST recently developed by one of usl3-I6that we call flexible transition-state theory (FTST). FTST is designed to minimize the approximations, beyond the intrinsic statistical assumption itself, involved in the implementation of transition-state theory for those bond-fission and bond-formation processes in which there is no potential energy barrier on the minimum energy path (MEP) from reactants to products. To this end FTST has the following attributes: (i) The transition state is determined variationally for each total energy E and total angular momentum J. Thermal rate constants are then determined by appropriate averaging over E and J . (ii) The full potential energy surface can be explicitly incorporated; here we use the potential based on Hirst's ab initio calculation^'^ as fit by Hase et a].' This so-called Hirst surface is based on over 100 geometries spanning a wide range of orientations and separations of CH3 with respect to H. The Hirst calculations are expected to overcome deficiencies associated with the surface of Duchovic et aL2 by using configuration interaction, instead of 'NSERC of Canada Postgraduate Fellow.

* NSERC of Canada University Research Fellow. 0022-3654/89/2093-3117$01.50/0

Mraller-Plesset perturbation theory, to account for electron correlation. Brown and Truhlar3 have performed a large configuration interaction calculation with a bigger basis set than Hirst to obtain the C-H stretching potential. However, use of their method to calculate the large number of p i n t s needed to generate a potential function comparable to the Hirst surface would require considerably more computer time than the Hirst method and has not been undertaken. Furthermore, the Brown and Truhlar CH3-H dissociation energy of 104.3 kcal mol-' is in poorer agreement with the experimental value of 112.4 kcal mol-' than Hirst's value of 109.5 kcal mol-'. (iii) The transitional modes are not approximated as oscillators, free rotors, or hindered rotors but are given an exact classical treatment via a Monte Carlo evaluation of the associated phase space. In the application of a canonical version of FTST to CH, CH, recombination, Klippenstein and Marcusls have found that quantum corrections

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(1) Hase, W. L.; Mondro, S. L.; Duchovic, R. J.; Hirst, D. M. J . Am. Chem. Sor. 1987, 109, 2916. (2) Duchovic, R. J.; Hase, W. L.; Schlegel, H. B. J. Phys. Chem. 1984, 88, 1339. (3) Brown, F. B.; Truhlar, D. G. Chem. Phys. Lett. 1985, 113, 441. (4) Schlegel, H. B. J . Chem. Phys. 1986,84,4530. ( 5 ) Duchovic, R. J.; Hase, W. L.; Schlegel, H. B.; Frisch, M. J.; Raghavachari, K. Chem. Phys. Lett. 1982,89, 120. (6) Duchovic, R. J.; Hase, W. L. Chem. Phys. Lett. 1984, 110, 474. (7) Duchovic, R. J.; Hase, W. L. J . Chem. Phys. 1985, 82, 3599. (8) Hase, W. L.; Duchovic, R. J. J . Chem. Phys. 1985,83, 3448. (9) Viswanathan, R.; Raff, L. M.; Thompson, D. L. J. Chem. Phys. 1985, 82, 3083. Viswanathan, R.; Raff, L. M.; Thompson, D. L. J . Chem. Phys. 1984, 81, 31 18. Viswanathan, R.; Raff, L. M.; Thompson, D. L. J . Chem. Phys. 1984, 81, 828. (10) LeBlanc, J. F.; Pacey, P. D. J . Chem. Phys. 1985,83, 451 1 . (1 1 ) King,S.C.; LeBlanc, J. F.; Pacey, P. D. Chem. Phys. 1988,123,329. (12) Cobos, C. J.; Troe, J. Chem. Phys. Lett. 1985, 113, 419. Cobos, C. J.; Troe, J. J . Chem. Phys. 1985, 83, 1010. (13) Wardlaw, D. M.; Marcus, R. A. Chem. Phys. Let?. 1984,110,230. (14) Wardlaw, D. M.; Marcus, R. A. J. Chem. Phys. 1985, 83, 3462. (15) Wardlaw, D. M.; Marcus, R. A. J . Phys. Chem. 1986, 90, 5383. (16) Wardlaw, D. M.; Marcus, R. A. Ado. Chem. Phys. 1988,70,231, part 1. (17) Hirst, D. M. Chem. Phys. Lett. 1985, 122, 225

0 1989 American Chemical Society

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The Journal of Physical Chemistry, Vol. 93, No. 8, 1989

to the thermal rate constants are small, being less than 3% of the classical values in the 300-2000 K temperature range. The implementation of FTST for CH3 H is briefly described in section 11. As an improvement to the original applications of FTST14*15we have incorporated the MEP analysis proposed by Miller et aI.I9 and subsequently applied to a variety of systems,*O including CH, H.’y8 This allows us, in the spirit of FTST, to more fully exploit available potential energy surfaces by providing normal-mode frequencies and equilibrium structures along the reaction path, thereby avoiding the previously employed approximate interpolation^'^-^^ for these quantities. Computational aspects of the FTST treatment are briefly described in section 111. Numerical results for FTST thermal rate constants for CH3 H are presented in section IV for the temperature range 300-2500 K and compared to several other canonical VTST rate constants and to several experimentally derived high-pressure rate constants. Assessed are the influences on the FTST rate constants of transitional mode zero-point energy, of conserved mode anharmonicity, and of the transition-state symmetry number; the relevant aspects of these features are outlined in section 111. Limited FTST calculations for CH, D are also presented in section IV, in view of the apparent isotope anomaly for C H 3 H / D recently reported by Pilling and co-workers.21,22

Aubanel and Wardlaw

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Configuration A

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11. Theory

The high-pressure recombination rate constant k is given as a function of temperature by15

where NEj(R) is the sum of (quantum) states in all degrees of freedom except the reaction coordinate R for the given J and having energy less than or equal to E , Rt is the value of R that minimizes NEj for a given E-J pair, and Q, is the partition function for the reactant fragments at infinite separation. The ratio g, of the electronic partition function at the transition state to that for infinitely separated reactants is taken to be corresponding to the usual assumption that only reactants on the singlet electronic potential energy surface lead to recombination products. A primary feature in the FTST calculation of NEj, as described in ref 13, is that the degrees of freedom are subdivided into transitional ones and conserved ones. In the CH4 product the two transitional modes correspond to the doubly degenerate CH,-H rock; in the C H 3 H reactants these transitional modes correspond to free rotation. The six vibrational modes of the CH, reactant are treated as conserved modes and are correlated, via the MEP analysis, with six vibrational modes of CH,. There are thus eight “internal” degrees of freedom to be considered in the evaluation of NEj. The remaining internal degree of freedom of the CH3-.H system, which describes the relative translation of CH, and H and corresponds to the reaction coordinate R, has been explicitly excluded from NE) With this subdivision of coordinates NEj can be written as the conv~lution’~

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where p J ( e ) de is the number of transitional mode states for the given J when their total energy lies in the interval (t,e+dt) and N,,((E’-c) is the number of conserved mode states having energy less than or equal to E’- 6. The quantity E’in eq 2 is the available energy at a given value of R, i.e. E’ = E’(R) = E - v,~p(R) - E,,(R) (3) where VMEp is the potential energy function evaluated on the (18) Klippenstein, S. J.; Marcus, R. A. J . Chem. Phys. 1987,87, 3410. (19) Miller, W. H.; Handy, N. C.; Adams, J. E. J . Chem. Phys. 1980,72, 99. (20) For example: (a) Vande Linde, S. R.; Mondro, S . L.; Hase, W. L. J . Chem. Phys. 1987, 86, 1348. Mondro, S. L.; Vande Linde, S . R.; Hase, W. L. J. Chem. Phys. 1986,84,3783.(b) Rai, S. N.; Truhlar, D. G. J . Chem. Phys. 1983,79, 6046. (21) Brouard, M.; Pilling, M. J. Chem. Phys. Lett. 1986,129, 439. (22) Brouard, M.; Macpherson, M.; Pilling, M. J., submitted.

Configuration B

Figure 1. Various geometrical variables describing the CH3-.H system. C-H4 is the bond being formed and the corresponding “bond” distance. The angles 4i4(i = 1-3) a?d TC-H* determine the transitional mode potential; only 434is shown._Ris the vector from the CH, center of mass to the atom and R = IRI is the reaction coordinate. x is the angle between R and the symmetry axis 2’’ of the CH3 group.

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minimum energy path, E is the total energy (in the center-of-mass frame) available to infinitely separated reactants and measured with respect to VMEp(m)= 0, and E, is the R-dependent zero-point energy of the conserved modes. The value of NYin eq 2 is obtained by the usual quantum count, whereas p J ( t ) is treated classically, resulting in a phase space integral for NEj that is then evaluated by a Monte Carlo method. Salient features of the evaluation of eq 1 and 2 are described in section 111. The quantity p J ( t )is specified below in terms of action-angle variables, as in ref 13-15. This,choice is by no means necessary for the implementation of FTST for any particular system; for example, Klippenstein and Marcus23have demonstrated the viability of using conventional coordinates to evaluate NEj for 2CH3 C2H6. In the present application, the reaction coordinate R is the distance from the H atom to the center of mass of the CH3 fragment. The angular momentum action variables (in units of h = 1 throughout) are (J,J,,IJ’,K), and their respective conjugate angles are (a,&a,,aI,2).J , is the z projection of the total angular momentum vector J o n a set of Cartesian coordinate axes (x,y,z) fixed in the CH,.-H-system; 1 is the magnitude of the orbital angular momentum 1 of the relative motion of the H and CH, fragments; j is the magnitude of the rotational angulai momentum J of the CH3 fragment; K is the z” projection of j on a set of Cartesian coordinate axes (x”,y”,z’? fixed in the CH3 fragment and chosen to diagonalize the CH, inertial tensor. The expression for pJ(c) in terms of these action-angle variables is

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PJ(t)

=

(2a,‘o-’f

...f dJI d l d j dK d a dp da, da, d y A(J,lj) 6(t-Ht)

(4) Implicit in eq 4 is a dependence on R arising from the classical Hamiltonian H, which is defined in eq 5. The limits on the angle variables are 0-2r, the J, integral is over the interval (-JJ), the K integral is restricted by IKJ5 j , the angular momentum actions are restricted by the indicated triangle inequality A and by energy conservation (A equals 1 when the triangle inequality is fulfilled and is zero otherwise), and o is a symmetry number to be discussed in section 111. For CH,--H, H , is written as 12

Ht = E, -I- --k Vt(rC-H*,~14,~24,~34)

(5)

~ F R ~ where is the reduced mass for relative motion of CH, and H, E, is the rotational energy of the CH, fragment, and V, is the potential energy function for the transitional modes. The arguments of V, in eq 5 are given in the notation of Hase et a1.I in which the four hydrogen atoms are labeled H i ( i = l-4), rC+,* is the distance from the C atom to the approaching H4atom, and 4i4 (i = 1-3) is the H4-C-Hi angle; see Figure I . One of the assumptions implicit in FTST is that the transitional modes and conserved modes are uncoupled from each other for the purpose of state counting at the transition state (cf. eq 2).24 ~~

(23) Klippenstein. S J , Marcus, R A J Phys Chem 1988,9.7, 3105

Rate Constants for C H 3 + H

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The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3119

CH4

TABLE I: Barriers to Internal Rotation for the Transitional Mode Potential V,O 1.53 2.02 2.52 3.01 3.51 4.01 4.5 1

-70.272 -31.422 -10.952 -3.002 -0.632 -0.102 -0.012 0.0

m

-1 1.742 -2.352 0.188 0.418 0.198 0.058 0.008 0.0

58.53 29.07 11.14 3.42 0.83 0.16 0.02 0.0

59.5 28.0 10.5 3.23 0.79 0.15 0.02 0.0

”Energies in kcal mol-’. * C = 164.0 kcal mol-’ and A = 0.43 A-2; these values were determined graphically and differ from those of ref 11 because slightly different barrier heights are used here and because R is taken as the reaction coordinate whereas rC-H* is used in ref 11. A physical basis for this assumption is provided in ref 25 in terms of a reaction path decomposition of the Hamiltonian. Such uncoupling implies that Ht in eq 4 must not depend on the internal motion of the CH, group. We therefore invoke the usual FTST approximation that for all orientations of CH, with respect to H, the CH, fragment has its R-dependent equilibrium geometry, as determined by the MEP analysis.26 This geometry has C,, symmetry for all R values, and the R-dependent principal moments of inertia are related by I, = IB < IC so that CH3 is an oblate symmetrical top with symmetry axis C coinciding with the z”axis. The energy E, is thus determined by j and K , and we note that an additional restriction on IKI arises from the requirement that E , must be positive. Values of V, are obtained by evaluating the but with the CH, group Hirst potential for given rc-H*,414,424,434 in its R-dependent equilibrium geometry. With these approximations Ht becomes a transitional mode Hamiltonian since it now depends only on R and all integration variables in eq 4 except for (Jz,a,/3)which specify the orientation of the body-fixed (x,y,z) system with respect to a space-fixed system. Performing the integrals over (Jz,a,/3)and substituting eq 4 in eq 2 yield a sixdimensional integral for NEj: NEj = ( 2 J 1) X (hr)-,u-’ d l d j dK dal d a j d r NdE’-H,) A ( J , l j ) (6)

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l...l

The approximate transitional potential described above admits, for sufficiently small R and e, two disjoint regions of energetically accessible classical phase space. These regions can be characterized by the relative orientat@ of CH, with respect to H. We shall use the angle x between R and the CH, symmetry ax@(z” or C axis) to specify this orientation;28the z”axis, x,and R are depicted in Figure 1. For a fixed value of R and with CH3 in its (R-dependent and nonplanar) equilibrium configuration, a maximum in V, is encountered as x is varied from 0 to a (the orientation x = a specifies the MEP configuration of CH3-.H). The value of this maximum naturally depends on the orientation, in the plane perpendicular to the z”axis, of the axis about which the H atom is rotated, Le., on the azimuthal angle associated with X. All such maxima are found to occur at x a / 2 , Le., in the vicinity of the plane defined by the three methyl H atoms, and

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(24) In existing applications of transition-state theory, some or all of the modes orthogonal to the reaction coordinate are assumed separable so as to obtain more tractable expressions for the sum of states of these modes. Introduction of separability for this purpose is not to be confused with the intermode coupling assumed, in RRKM theory, to occur inside the transition-state hypersurface. (25) Hase, W. L.; Wardlaw, D. M. In Bimolecular Collisions; Ashfold, M . N. R., Baggott, J. E., Eds.; Burlington House: London, in press. (26) We performed the same analysis as in ref 1 and 8 using their VENUS program. The reaction path was obtained by solving the appropriate differential equations using a fourth-order Runge-Kutta algorithm with a step size of 0.000 125 amu’i2 A. The projected harmonic frequencies for motion orthogonal to the reaction path were obtained for 1.8 A -< R 5 4.5 A and agreed with values obtained by Hase.*’ (27) Hase, W. L., private communication. (28) Our angle x is essentially the same as that specified in ref 10 and used there as a coordinate describing the two-dimensional hindered rotational motion of the approaching H’ about CHI. In ref 10, x is the angle between and the 2” axis; here the reference axis a is used instead of

the smallest such maximum, denoted V,,,, occurs when the orbiting H atom bisects the gap between two methyl H atoms. Configurations of CH3-H with x < a / 2 and x > */2 are designated B and A, respectively. Values of V ,and the R-dependent absolute potential minimum VMEpare listed in Table I for R values ranging from 1.53 to 4.51 A. Also given in Table I is a barrier height Vbar, = V,,, - VMEP,measured with respect to I‘M,,. Following King et al.,” we examined the suitability of the functions Ce-AR*and Ce-AR for describing the decrease in Vb,,, with increasing R by plotting In V,, versus R2 and R, respectively. The former plot is essentially linear and yielded the C and A values quoted in Table I; the latter plot was decidedly nonlinear. The V,, values predicted by the C exp(-AR2) model are seen, in Table I, to agree well with those determined from our analysis of the Hirst surface. In the FTST treatment of C H 3 CH, C2H6,I5the substantially larger barriers to rotation of one CH3 group about the other (see Table I1 of ref 15) resulted in the total available energy E’(R) being less than Vb,,(R) for a wide range of energy and R values. This made it necessary to introduce an ad hoc restriction in the calculation of NE, (eq 11.4 of ref 15) so as to eliminate those transitional mode phase space points on the fixed R hypersurface that are unable to evolve, via classical dynamics, to C2H6 products without recrossing the hypersurface. For CH3 H, however, none of the barrier heights V,,, on R = (1.53 A, a) [see Table I] exceeds the minimum energy available to reactants, Emin= 18.6 kcal mo1-I (the zero-point energy for isolated CH, in a normalmode description on the Hirst surface). There are thus no grounds for adopting an analogous rejection procedure here. The implications of the above for the temperature dependence of the rate constants for CH3 H and CH, CH, are discussed in section IV.

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111. Calculation of k In this section a number of related approaches to the calculation of the recombination rate constant are described. In each case the reactant partition function Q,(7‘) appearing in the denominator of eq 1 is obtained in standard fashion by modeling the methyl radical as a collection of separable (quantum) vibrators and a rigid (classical) rotor with vibrational frequencies and moments of inertia determined from the Hirst surface. Except as noted below, the reactant vibrational degrees of freedom are treated as harmonic oscillators. The sum over J in eq 1 is replaced, in each case, by an integral whose lower limit is zero and whose upper limit is taken to be J,,,(E) where J,,, is the maximum value of J for a rigid symmetric top model of equilibrium CH4 when all available product energy (as determined by the total energy E ) is in CH4 overall rotation. Unless otherwise stated, all rate constants are obtained with the symmetry number u in eq 6 set equal to its “loose” value of ul = 6. The normal-mode frequencies of the eight internal vibrational degrees of freedom orthogonal to the reaction coordinate were determined, as a function of R, by the MEP analysis.26 The frequencies of the six conserved modes were used in a direct-count evaluation29of NV in eq 6. The two remaining frequencies were used to estimate a transitional mode zero-point energy czp employed in one of the rate constant calculations described below. In the first approach to the calculation of k , a transition state is determined variationally for each E and J , and the results are then averaged over E and J by using a Boltzmann weighting factor. In the evaluation of NEj via eq 6 the transitional modes are treated in purely classical fashion by ignoring their zero-point energy and including all the allowed phase space with e = H, > 0. The resulting rate constant is of the microcanonical VTST variety and is accordingly denoted k:; it is analogous to the rate constant k’, for 2CH3 C2H6in ref 15. The second approach is a widely used approximation to the first in that a transition state is determined at each temperature, yielding a rate constant of the canonical VTST variety and denoted k$. The remaining ap-

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(29) A grid size of 1 cm-’ was used to obtain an approximate direct count via the method described in ref 14.

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TABLE III: Comparison of NE, NEA,and W, as a Function of R at E'( m ) = 2.363 kcal mol-' and J = 20

TABLE II: Deviation of the CH3 Fragment from Planarity'

R, A

4dR) - *I2

R, A

h ( R )- *I2

1.53 2.02 2.52 3.01

13.5 10.4 7.9 5.7

3.51 4.0 1 4.5 1

3.9 2.5 1.6 0.0

m

R, A

[NE, f uMc]/104 2683 f 156

1.83

[ N t , f u,c]/104

P

N'&/104 4857

0.10

f 0.114 f 0.077 f 0.054 f 0.038 f 0.029 f 0.023 f 0.020 f 0.020 f 0.019

7.68 4.96 3.48 2.59 1.96 1.66 1.49 1.44 1.52 1.64

0.66 0.77 0.74 0.80 0.8 1 0.89 0.94 0.96 0.92 0.96

1.417 f 0.024

2.83

1.oo

2549 f 156

...

"Angles in degrees. proaches to the rate constant calculation are variations on k: and are designed to assess the effect of transitional mode zero-point energy (k",", of an R-dependent symmetry number (k"), and of the anharmonicity of selected conserved modes ( k r J , kFh,''). Rate constants k$, k$,', and kt'jj" for CH3 D CH3D were P ' also calculated and are isotopic analogues of k,: krh,', and kanh," respectively. ( i ) k:. The integrals over E and J in eq 1 are approximated by N-point Laguerre and 2M-point extended Simpson's rule quadratures, respectively, yielding

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Aubanel and Wardlaw

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2.71 2.81 2.9 1 3.01 3.1 1 3.21 3.31 3.41" 3.51 3.61

6.883 f 4.702 f 3.258 f 2.484 f 1.889 f 1.638 f 1.486 f 1.431 f 1.508 f 1.635 f

0.216 0.137 0.092 0.066 0.047 0.037 0.030 0.026 0.024 0.023

4.613 2.902 2.052 1.493 1.128 0.906 0.789 0.748 0.813 0.849

f 0.191

... 4.51

2.829 f 0.028

"Value of RE,' for the specified E and J . TABLE IV: Harmonic and Anharmonic Frequencies of the CH3 Umbrella Motion on the Hint Surfacea

where the quadrature weights wi, wk and the quadrature points Ei, Jkare defined in ref 15. A value of N = 6 and values of A J ranging from 5 to 15 were used in the evaluation of eq 7, with specific values of AJ depending on the temperature T; 2 M was determined by A J and JmaX(E).These values of N and A J were found to provide microcanonical rate constants that are converged within the associated Monte Carlo error bounds. The transition-state location RE> is determined independently for each (Ei,Jk) pair in eq 7 by minimizing NEljkwith respect to R on a -0.1-8, grid over an appropriate range of R values. The integral for NEj in eq 6 is evaluated by adaptation of the Monte Carlo method described in ref 14. The Monte Carlo approximation to NEj is denoted NEj f uMCwhere uMC is 1 standard deviation. Estimates of upper and lower bounds on k t are obtained by using uMC) and (NEIjk- uMC), respectively, in eq 7. (NE!!k ( I I ) kz. The canonical rate constant corresponding to k: is

+

1 /A

N

where the transition-state location RTt is determined by minimizing ~ ~ l ~ with i N respect E l to R on a -0.1-8, grid; this value of Rt is thereby independent of E,. The sum of states NE(R) is defined by

R, A

w

2.02 3.01 4.51

930.5 538.2 400.4 392.0

m

Emu 486.5 270.6 227.6 227.2

WOI

w12

w23

w34

955.3 594.3 564.6 556.7

942.2 674.4 660.0 672.0

940.6 739.0 728.9 731.0

947.4 792.4 784.5 786.4

"Al frequencies and zero-point energies in cm-I; EZP,"and w's are defined in the text.

transition state with smaller RE>, the C H 3 group no longer has a C2, axis and ut = ul/2 = 3. We therefore consider microcanonical rate constants, denoted k;, estimated on the basis of an approximate interpolation that is analogous to that suggested for 2CH3 C2H6in ref 15, namely

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NZJ(R) = p ( R ) NEJ(R) +

i1 -p(R)12NiJ(R)

(I1)

where N;, includes only states corresponding to configuration A , and both NE- and NiJ are obtained by using u = ul in eq 6. The weighting function p should be zero at R = Re (Re is the value of R in equilibrium CH4 product) and unity at R = a. Since NtJ = NEj at R = Re and NtJ = N E J / ~ at R = m, a suitable p is p(R) = 2[NEJ(R) - N$J(R)l/NEJ(R)

(12)

in as much a s p = 0 at R = Re and p = 1 at R = m. At R = Re and a, Nzj therefore equals NCJ and NEj, respectively, as required. Substituting eq 12 in eq 11 yields

(9) N%J(R) = 2[[N$(R)12 + 2[NEJ(R) - N$J(R)121/NEJ(R) (l3) and is evaluated by modifying the Monte Carlo method used for the NEj evaluation to include the additional integration over J . (iii) k.: To account, in approximate fashion, for the influence of transitional mode zero-point energy on the microcanonical rate constant kGf, an ad hoc semiclassical modification of NEj, designed to make it more closely approximate the energy-averaged behavior of its quantum counterpart at small E ' v a l ~ e s ,is~introduced: ~ f i j= 1

+ J:de

NdE"e)

pj(t)

(10)

where the classical treatment of PAC)is retained. The rate constant kp was obtained by substituting PiJfor NEj in the procedure described in (i). (iu) k:. Use of the symmetry number u = u1 = 6 in the NEj expression is, strict1 speaking, valid only for a loose transition state (Le., large REj ) where the methyl fragment is planar. For all E and J values considered in this study, 2.2 8, REJt < 4.1 8, and the equilibrium geometry of C H 3 at the transition state is not planar but umbrella-shaped. Deviations from planar CH,, as measured by the angle @o - r / 2 (@,4 = @24 = @34 = @,, on the MEP) are listed in Table I1 for a range of R values. For a tight

Y

(30) Troe, J. J . Phys. Chem. 1979, 83, 114.

Replacing NE, with Nij in (i) yields k,". The effect of the potential V, on the energetically allowed configuration types A and B as a function of R and the suitability of the loose symmetry number for this recombination system are illustrated in Table 111. The energy under consideration is E'(..) = E - EZp(m)= 2.363 kcal mol-', and the total angular momentum is J = 20. Given are the Monte Carlo estimates for NEj (=N$, @,) and NiJ along with their associated standard deviations uMC, NEj (eq 13), and the interpolation parameter p (eq 12). At 1.83 A, the smallest R value considered for this E and J , p = 0 and as R increases p tends to 1, as expected. Note that at the transition state location of 3.4-3.5 & P = = 1 so that eq 13 yields Ngj ;= NEj, with NEj obtained by using u = ci. A value of p = 1 at R = RE$ is typical of most E-J pairs considered in this study. We emphasize that the interpolation provided by eq 11 and 12 for Ngj is not unique but is certainly reasonable. The advantage of the present approach is that no inspection of the R dependence of NEj and N i J is required to arrive at a suitable interpolation, as would be the case if p(R) were to be modeled as an explicit function of R. ( u ) kyh. The CH3 umbrella motion (which correlates with the out-of-plane CH, bend of isolated reactants), although essentially harmonic at R 2 A, becomes increasingly anharmonic as R increases. This can be seen by comparing, at the R values selected

+

-

Rate Constants for CH3

+H

-

CH,

The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3121

TABLE V Experimental and Theoretical Anharmonic Frequencies for the Out-of-Plane Bending of Isolated CH3"

trans freq WOI

a12 w23

exptl (ref 32)

Hirst surface

ref 33

606.5 681.6 731.1

556.7 672.0 731.0

584.9

Frequencies in cm-I. TABLE VI: Harmonic Frequencies ( w ) , Associated Zero-Point Energies (ZPE), and Moments of Inertia ( I ) for Isolated CH3

w,(A'), cm-' w2(A''), cm-'

w3(E), cm-' w4(E), cm-' ZPE, kcal mol-' I ~ amu , A2 12, amu A2 I,, amu A2

3270 495-545 3285, 3297 1436, 1447 18.9 1.76 1.76 3.55

theoret Hirst surface ref 34 3123 392 3299 1471 18.6 1.78 1.78 3.57

3220 459 3423 1513 19.4 1.75 1.75 3.50

for Table IV, the harmonic bending frequency w to the first four anharmonic frequencies w , , ~ +( ~i = 0-3) for the transitions from level i to level i + 1. Also to be compared in Table IV a r e the anharmonic zero-point energy Ezp,"a n d the corresponding harmonic quantity w/2. The w values are obtained directly from the MEP analysis a n d the anharmonic vibrational levels from a semiclassical treatment of a one degree-of-freedom model for the umbrella motion along t h e MEP.31 Evidence for the reliability of t h e Hirst surface in t h e separated reactant limit is provided by comparing experimental a n d theoretical values of various spectroscopic quantities. T h e experimentally measured snharmonic frequencies for the 0-1, 1-+2,2+3 out-of-plane bending transitions a n d the associated "negative" anharmonicity of this mode a r e seen to be well described by t h e Hirst surface in Table V. T h e largest discrepancy is t h e relatively low Hirst value of mol; the larger theoretical value of wol cited in Table V is the result of a different potential energy calculation a n d indicates t h e possibility of improving this aspect of the Hirst surface. In Table VI experimentally derived harmonic vibrational frequencies, the harmonic zero-point energy, and the moments of inertia for isolated CH3 are compared to two sets of theoretical results: those derived from our analysis of the Hirst surface and those of ref 34, where a different potential energy surface was used. These properties of CH3 are seen to be well described by the Hirst surface, except perhaps for the relatively low value of the out-of-plane harmonic bending frequency w2. Anharmonicity of t h e conserved modes will affect both the reactant partition function Q,a n d the sum of states NEJin eq 7, the effect on the latter explicitly arising from Nd(E'-H,) in eq 6. Treating the umbrella motion anharmonically (31) The umbrella motion was treated as a one-dimensional vibration and WKB semiclassical quantization was used, with the mass term m = 3mHmc/(3mH+ mc), to obtain the anharmonic energy levels for each value of R used in the variational minimization of NE> Only the symmetric umbrella vibration of CH, about its MEP equilibrium structure was considered.2 In this case the angular coordinate A measures the deviation of the CH, group from its R-dependent equilibrium structure and completely characterizes the motion. The umbrella potential V,,,(A) is obtained by evaluating the full angularptential whose form is given by eq 15 of ref 2,, with $,4 = A + $p4, where @,4 IS obtained from the R-dependent MEP geometry of H-CH,, and 8, = cos-' ['/2(3 cos2mi4 - 11 ( i = 1-3). For T = 1000 K and with a six-point Laguerre quadrature for the energy integral (see eq 7), the maximum number of levels needed for any of the R values considered was 24. Higher temperatures were not considered since the large number of energy level calculations required was deemed to be computationally excessive. Furthermore, the temperature range 300-1000 K in which anharmonically corrected rate constants were computed spans the temperatures at which experimental estimates of the high-pressure rate constants are available. (32) Yamada, C.; Hirota, E.; Kawaguchi, K. J . Chem. Phys. 1981, 75, 5256. (33) Surratt, G. T.; Gcddard, W. A. Chem. Phys. 1977, 23, 39. (34) Schatz, G. C.; Wagner, A. F.; Dunning, T. H. J . Phys. Chem. 1984, 88, 221.

temp,

K

theoret

experimentally derived (from ref 34)

TABLE VII: R S T Rate Constants for CH, kz

300 400 500 600 800 1000 1250 1500 1750 2000 2500

+ 6,b

1.42 f 0.02 1.59 f 0.02 1.73 f 0.03 1.82 0.03 1.93 f 0.03 2.06 f 0.03 2.12 f 0.03 2.15 f 0.04 2.16 f 0.04 2.15 0.04 2.20 f 0.04

* *

+H

-

CHIa

ace

k$

1.55 f 0.05 1.71 0.06 1.87 f 0.05 1.98 f 0.06 2.06 f 0.06 2.17 0.06 2.26 f 0.06 2.30 f 0.07 2.34 f 0.07 2.34 f 0.06 2.35 0.07

1.48 1.64 1.75 1.84 1.94 2.06 2.12 2.15 2.16 2.15 2.20

*

* *

1.42 1.59 1.73 1.86 1.94 2.07 2.13 2.18 2.19 2.19 2.24

1.60 1.83 2.00 2.16 2.54

"All rate constants are in 10" L mol-' S-'. ' k ; = 1/2[(kc1)max + ( k $ J and 6, = 1 / 2 [ ( k 3 m a-x(k$,,inl, where (k:Imaxand ( k f ) , , are estimated upper and lower bounds obtained by using NE,,$ uMc and NE/Jk- uMC,respectively, in eq 7. k$ = ' / 2 [ ( k c 1 ) m+ a x(k&,in] and 8, = / 2 [ ( k $ ) m a-x ( k $ ) m i n ]where , (k$)maxand (k&,,in are estimated upper and lower bounds obtained by using NE, + UMC and N!, - UM? respectively, in eq 8. duncertainty in these rate constants IS within *0.01 of the uncertainty 6, in k:.

+

TABLE VIII: ComDarison of Canonical VTST Rate Constants"

temp, K200 300 400 500 600 800 1000 1250 1500 1750 2000 2500

k$

khob 1.24 1.43 1.56 1.64 1.70 1.78 1.83

*

1.27 0.05 1.55 f 0.06 1.71 f 0.05 1.87 f 0.06 1.98 0.06 2.06 f 0.06 2.17 f 0.06 2.26 f 0.06 2.30 i 0.07 2.34 f 0.07 2.34 f 0.06 2.35 f 0.07

kMow

khrC

1.37 1.59 1.73 1.84 1.92 2.05 2.12

d

kLipt

2.4

2.3

2.6

2.6

2.9

2.9

ORate constants in 10" L mol-' s-l. bFrom Table I of ref 1; determined by treating the transitional modes as quantum harmonic oscillators. 'From Table I of ref 1; determined by treating the transitional modes as a classical hindered rotor. dFrom Table 2 of ref 11; determined by modeling the CH,. .H bond dissociation potential with a Morse function. eFrom Table 2 of ref 11; determined by modeling the CH,. -H bond dissociation potential with a Lippincott function.

-

1'! 1 1

0.8

200

1

I

1000

1800

T(K)

2600

+

Figure 2. Plot of high-pressure rate constant for CH3 H versus temperature. Solid symbols are experimentally derived values: ( 0 )ref 22; (m) ref 36; (A)ref 37; (+) ref 38. Open s mbols are FTST calculations are described in the text: (0)k;; (A) k,; (0)kyh,'.

3

a n d all other conserved modes harmonically and following t h e procedure described in (i) yields kFh,'. A second anharmonic rate constant kFh*"was defined by taking into account, in approximate fashion, t h e anharmonicity of t h e three high-frequency stretching modes, in addition t o t h e anharmonicity of the umbrella motion. T h e two remaining conserved modes (which correlate with the doubly degenerate in-plane CH3 deformation of isolated reactants) were treated harmonically. An anharmonic treatment of t h e latter modes would involve calcu-

3122

Aubanel and Wardlaw

The Journal of Physical Chemistry, Vol. 93, No. 8, 1989

TABLE IX: Comparison of Canonical VTST Transition-State Locationsa temp, K RTi hotb RhJC R M ~ ~ R~ LJ~ ~~D ~ ' 200 300 400 500 600 800 1000 1250 1500 1750 2000 2500

3.7 3.7 3.5 3.5 3.3 3.2 3.1 3.0 2.9 2.8 2.8 2.6

3.54 3.42 3.33 3.25 3.19 3.09 3.00

3.67 3.55 3.45 3.37 3.29 3.17 3.07

3.8

3.8

3.5

3.5

3.1

3.2

"Transition state locations in angstroms. bFrom Table I of ref 1; determined by treating the transitional modes as quantum harmonic oscillators. cFrom Table I of ref I ; determined by treating the transitional modes as a,classical hindered rotor. *From Table 2 of ref 11; determined by modeling the CH3.. .H bond dissociation potential with a Morse function. CFromTable 2 of ref 1 I ; determined by modeling the CH3. .H bond dissociation potential with a Lippincott function.

-

TABLE X: WST Rate Constants for CH, + D Isotopic Rate Constant Ratios pr = k,/k,,o temp, K 300 400 500 600 800 1000 1250 1500 I750 2000

-

CH3Daand

kz',D

ki$'

Pz'

pYh9'

1.03 1.14 1.24 1.27 I .38 1.43 I .48 1.49 I .49 1.49

1.14 1.31 1.43 1.54

1.4 1.4 1.4 1.4 1.4 1.4

1.4 1.4 1.4 1.4

1.81

1.4

1.4

1.4 1.4 1.5

"All rate constants in IO" L mol-' s-I.

lation of approximate quantum or semiclassical vibrational eigenvalues for a nonseparable two-degrees-of-freedomsystem. This was not done as the anticipated computational effort was deemed excessive for the present purpose. The stretches are modeled as independent Morse oscillators with energy levels E, = (n I / & - ( n ' / 2 ) z w 2 / 4 Dwhere , w is an R-dependent harmonic stretching frequency determined by the MEP analysis and D is the classical H-C bond dissociation energy of the CH3 fragment, which is chosen to have an R-independent value5 of 110.6 kcal mol-' in the H i n t potential function.]

+

+

IV. Assessment of Results Results for approaches (i)-(v) to the calculation of k( T ) for CH3 + H in the temperature range 300-2500 K are given in Table VII. Selected data from Table VI1 and several experimentally based rate constant predictions are also displayed in Figure 2 as a plot of k versus T. A comparison of k$ to other canonical rate constant calculations on the same potential surface and on a related potential surface is provided in Tables VI11 and IX. Microcanonical rate constants k,3Dfor CH3 D resulting from approaches (i) and (v) are given in Table X, along with ratios p, of the CH3 + D rate constants to the corresponding CH3 + H rate constants. Theoretical Part. The microcanonical rate constant kz (column 2 of Table VI1 and open squares in Figure 2) is seen to increase with temperature in the 300-1000 K range and then to be essentially independent of temperature in the 1250-2500 K range, Le., to agree within the error bound 6, in Table VII. The canonical rate constants kg (column 3 of Table VI1 and open triangles in Figure 2) display the same temperature dependence as kC'but are 5-10% larger. This is expected since NEAR) 1 NEJ(REJ6 at each R, in which case k$ is an upper bound to k.: The positive temperature dependence exhibited by kz and k$ is to be contrasted to the negative temperature dependence of the corresponding rate constants for the related recombination CH, + CH, C2H,.I5

+

-+

This is most likely attributable to larger regions of phase space being excluded at higher energies (where Rt values are smaller) for the interaction between two C H 3 groups rather than an atom and a CH3 group. We are currently attempting to construct simplified models that reproduce detailed FTST calculations to obtain insight into factors affecting the temperature dependence of rate constants for barrierless association reactions. In Table VIII, k$ is compared to canonical VTST rate constants obtained in other studies. khoand khr were computed by Hase et al.' on the Hirst potential surface over the 200-1000 K temperature range. khowas obtained by treating the two transitional modes as quantum harmonic oscillators and khr by treating them as a classical hindered rotor. kMomand kLlppwere computed by King et a1.I' at T = 300, 500, and 1000 K using a simplified potential surface based on Hirst's ab initio calculation^.^^ Specifically, they modeled the transitional modes as a two-dimensional hindered rotor with a sinusoidal potential depending only on the as either a Morse (kMorse)or a angle x and modeled V(rCwH*) Lippincott (kLlpp)function; it is further assumed that the conserved mode frequencies are independent of R. It can be seen in Table VI11 that k; and khr agree within the error bound bc and that k$ exceeds khofor T = 300-1000 K; at 200 K kg is slightly smaller than khrand agrees with kho. Agreement between kg and khr is to be expected since the treatment of the transitional modes is effectively the same, whereas the small numerical differences are presumably attributable to differences in the treatment of angular momentum conservation and to slightly different reaction coordinates ( R versus rc-H*). An extensive comparison of FTST and the canonical VTST of Hase and co-workers is provided in ref 25. On the 300-1000 K temperature range kMorseand kLlppsignificantly exceed k$, kho,and khrby 40-60%, indicating that the collective assumptions of ref 1 1 effectively result in a different VTST model for this recombination reaction. The transition-state locations R:, Rh2,RhJ,RMon2,and R,,,: corresponding to kg, kho, kh,, kMone,and k,,,, respectively, are given in Table IX. The decrease in Rt with increasing temperature is a feature shared with other barrierless association reaction^.'^,^^^ RTt, RhJ, and Rhrtare in excellent agreement with each other whereas RMorset and RLIp; exceed RTt, Rh2,and Rh: by 1-4 A, depending on the temperature. The semiclassical correction (eq 10) to NE, has a discernible effet on k: only at the lowest temperatures studied. Comparing k r (fourth column of Table VII) to kz reveals that they agree within the uncertainty 6, for T 1 500 K and that k r slightly exceeds k: at T = 300 and 400 K. These observations are consistent with the expectation that PiJshould generally exceed NE, but that as the values of E used in eq 7 increase, Le., as T increases, the difference between PkJ and N E j decreases to the point where k: 1: k:. An assessment of the validity of the usual assignment of the transition-state symmetry number u to the loose value of u, = 6 is provided by comparing k i (column 5 of Table VII) to k:. Over b,, the entire 300-2500 K temperature range, k,"' 5 kUu5 k,"' a result that may be contrasted with the corresponding FTST rate constants for CH, CH, C,H,, which were found to diverge from a common value as T was increased from 300 K, differing by -40% at 2000 K. Conserved mode anharmonicity was found to have a larger effect on the microcanonical rate constant than either the semiclassical correction or the symmetry correction discussed above. Calculations of kyh,' (column 6 of Table VI1 and open circles in Figure 2) were confined to the temperature range 300-1000 K for reasons discussed in footnote 3 I . In this range kFh,' exceeds k: with the relative difference between them increasing with temperature (kyhsiexceeds kz by --13%, -19%, and -23% at T = 300, 600, and 1000 K, respectively). The latter relationship can be understood in terms of the R-dependence of the anharmonic vibrational levels of the C H 3 umbrella motion, as illustrated by Table IV. Both at R = m, where Q, is evaluated, and in the range 2.2-4.1 %, of transition-state locations REJt,the anharmonic level spacings are greater than the corresponding harmonic spacings so that Q, and NE,(&>) will be less than their harmonic

+

-

+

Rate Constants for CH,

+H

-

CH4

The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3123

equivalents. Since the anharmonicity of the umbrella motion is more pronounced at R = m than at the RE> values, Q, decreases proportionately more than NEJ(REJt) with the result that kYh*I > k:. The decrease in the average value of RE> with increasing temperature (cf. Table IX) and the associated decrease in the anharmonic shifts of the umbrella harmonic levels (see Table IV) explains the observation that kFh9'increases more rapidly with temperature than k:. Including anharmonicity of the C H 3 stretching modes, in addition to anharmonicity of the CH, umbrella motion, had no further effect on the microcanonical rate constant in the 300-1000 K temperature range, Le., krh,' agreed with krh3"within the associated Monte Carlo uncertainties. The effect of quartic anharmonicity on transition-state bending partition functions for a large number of A BC reactions has been investigated by Garrett and T r ~ h l a r .These ~ ~ authors found that a purely harmonic treatment of the (ABC)+ bending mode leads to a significant and systematic overestimate of the canonical transition-state theory rate constant. The important difference between the umbrella motion considered here and the bending motion considered in ref 35 is that the former is a conserved mode whereas the latter is a transitional mode that does not affect the vibrational part of the reactant partition function. Comparison with Experiment. Four experimentally derived for the CH, H high-pressure rate constant in the 300-600 K temperature range are plotted as solid symbols in Figure 2. A variety of measurement techniques, reactant precursors, and buffer gases were used in these experimental studies. In all but one study,)' values of k were obtained by extrapolating a fit of measured pressure-dependent recombination rate constants to the high-pressure limit. In ref 37 the CH3 + H reaction was judged to be very close to its second-order limit, and no extrapolation was therefore performed. Not plotted is the room-temperature rate constant obtained by Patrick et al.39since their low value (a factor of 3 lower than the recent values of Brouard et al.) is primarily attributable to known errors in their kinetic analysis.22 Available estimates of k at temperatures above 600 K are based on conversion of experimentally derived rate constants for the dissociation reaction CH, CH, + H40 and are not considered here. In addition to uncertainties associated with measurement and extrapolation, such recombination rate constants are sensitive to uncertainties in the thermochemistry involved in the conversion of the dissociation rate constants. The scatter of the experimental data in Figure 2 and the size of the associated error bars make even a qualitative assessment of the absolute accuracy of the theoretical results impossible. Instead we focus on the recent (1988) work of Brouard et a1.,22 whose experimental technique for the measurement of k in the pressure-dependent regime is the most direct available. Their prediction of a k that is essentially independent of temperature in the 300-600 K range (solid circles in Figure 2) is based on the extrapolation of measured pressure-dependent rate constants at each of four temperatures spanning this temperature range. An energy-grained master equation calculation, with microcanonical dissociation rate constants determined by a variational RRKM procedure, was fit to the pressure dependence of the measured rates and provided the high-pressure limiting rate constants. In Figure 2 the Brouard et al. rate constants are seen to exceed all of the FTST predictions in the 300-600 K temperature range, contrary to the expectation that transition-state theory should provide an upper bound to the actual rate. Possible origins of this discrepancy are listed below. Perhaps more interesting is an apparent isotope anomaly in the rate constants for C H 3 H/D. Brouard et a1.21s22 have measured rate constants for CH, + D for a 289-401 K temperature range and a 50-600-Torr pressure

+

+

-

+

(35) Garrett, B. C.; Truhlar, D. G. J . Phys. Chem. 1979, 83, 1915. (36) Cheng, J.; Yeh, C. J . Phys. Chem. 1977, 81, 1982. (37) Sworski, T. J.; Hochanadel, C. J.; Ogren, P. J. J . Phys. Chem. 1980, 84, 129. (38) Brouard, M.; Macpherson, M. T.; Pilling, M. J.; Tulloch, J. M.; Williamson, A. P. Chem. Phys. Letf. 1985, 113, 413. (39) Patrick, R.; Pilling, M. J.; Rogers, G. J. Chem. Phys. 1980, 53, 279. (40) For example: Chen, C. J.; Back, M. H.; Back, R. A. Can. J . Chem. 1975, 53, 3580. Cobos, Skinner, and Troe, as cited in ref 10.

range. They conclude that the rate constant is essentially independent of pressure and effectively corresponds to the high-pressure limiting value under their experimental conditions. The reported high-pressure rate constant of (1.1 f 0.03) X 10" L mol-' s-l for the 289-401 K range is therefore free of the uncertainties associated with an extrapolation procedure. The ratio pcxptlof their experimentally derived rate constant for CH3 H (solid circles in Figure 2) to their directly measured rate constant for CH, D is 2.5. The FTST rate constants k:,D and k;$' for CH, D are listed in Table X. In the 300-400 K temperature range, k& almost agrees with, and k;f$' exceeds, the measured value of 1.1 X 10" L mol-' s-'. The ratios p: and pYh,' of corresponding pairs of rate constants are seen, in Table X, to be -1.4 = 21/2 = (mD/mH)'I2, as expected for an effectively loose transition state. The canonical VTST calculations of Hase et a1.l also yielded ratios of 1.4. The "anomalous" discrepancy between pcxptl 2.5 and pthmry 1.4 could be attributable to any combination of the following: experimental error, extrapolation of the CH, H rate constants over an apparently large pressure range, the potential energy surfaces used in the theoretical treatments, or the Occurrence of isotope-specificdynamical effects. In any case further study, both experimental and theoretical, is clearly needed. Measurement of rate constants over a broader temperature and pressure range would further constrain any parameters involved in fitting the theory to experiment. Various theoretical approaches are possible. A classical trajectory study of C H 3 H / D on the Hirst surface would shed light on dynamical effects (if any) and the suitability of transition state theory for this system. If the available measured rate constants22can be fit to a reasonable theoretical model that extends the present FTST treatment to the pressure-dependent regime, then the isotope anomaly would be resolved. This approach is under study; the analogous extension of FTST for 2CH3 C2H6has recently been reported by Wagner and ward la^.^' The pronounced effect of anharmonicity of the conserved umbrella mode on the magnitude and temperature dependence of the microcanonical rate constant suggests that accurate implementation of VTST for this system may require consideration of other diagonal, as well as all off-diagonal, conserved mode anharmonicities. Treatment of the anharmonicity of the doubly degenerate CH, bending motion would be straightforward but tedious, as described in section 111; several approaches to the treatment of intermode coupling are reviewed in ref 42. VTST rate constants for this system are known to be quite sensitive to the CH,-H potential curve at intermediate separations,I and inaccuracies in this part of the Hirst surface cannot be ruled out. Furthermore, since none of Hirst's geometries17correspond to B configurations, the accuracy of the Hirst surface in this region is unknown. However, B configurations are more repulsive than A configurations at smaller R values and accordingly contribute less to NEJ(R),thus reducing the effect of any uncertainties in this part of the potential on the calculated rate constants. Since the transition state is effectively quite loose for this system, a transition-state theory ratio p i= 2'12 is expected regardless of any potential surface modifications that might be required to adjust the absolute values of the theoretical rate constants.

+

+

+

-

-

+

+

-

Acknowledgment. This research was supported by NSERC of Canada. We are grateful to Professor Bill Hase for helpful discussions and for supplying us with the computer code used in the minimum-energy path analysis and to the reviewers whose suggestions improved the manuscript. Appendix: Action-Angle and Internal Coordinates The internal coordinates rC-H*,C$14, C$24, and C$34 in which the

transitional mode potential V, is expressed are determined from the action-angle coordinates as follows. A set (xy,z) of Cartesian coordinate axes is fixed in the CH3-.H system, with the z axis ~~~~

~~

(41) Wagner, A. F.; Wardlaw, D. M. J . Phys. Chem. 1988, 92, 2462. (42) Truhlar, D. G.; Isaacson, A. D.; Garrett, B. C. In Theory of Chemical Reacfion Dynamics; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. IV, p 65.

3124

J . Phys. Chem. 1989, 93, 3124-3129

chosen to lie along i and the x axis along i X 7. Atomic position , ;7 7?, and ic”for the three H atoms and the C atom vectors 7]”, of CH3 are given by the R-dependent CH3 equilibrium structure, as specified by the MEP analysis, and assigned in a CH3-fixed (x”,y”,z’’) Cartesian system chosen so a_sto diagonalize the CH3 inertial tensor. The separation vector R“’is chosen to lie along the axis of a (x”’”’’’,~’’’) system whose axis coincides with the z axis of the molecule-fixed (x,y,z) system. The origins of the unprimed, singly primed, and triply primed Cartesian systems are_chosen to be the center of mass of CH3. By transforming each of R’”and 7,“ (i = I,?, 3, C) to the (x,y,z) system, the coordinates rC+* = l?C-H*l = IR - 7c1, and COS 414 = Z;C-H;Z;c-H* (i = 1-3) can be obtained, where Z;c;c-H, and ZcC-H*are unit vectors pointing from the C atom to the indicated H atom. For given R, J, and _six-dimensional Monte Carlo point ( l j , K , ( Y ~ , ~ ,the , Y )vectors R and 7,(i = 1, 2, 3, C) are obtained by application of the inverse rotation matrix A-i:43 7, = A-’(a,,O,,,O) A-1(y,0K,,0)7,”, (i = 1, 2, 3, C) ( A l )

d = A-’(a,,O,O)d’’’ (‘42) The first application of A-’ in eq A1 yields the intermediate v-ector 7: in a CH,-fixed (x’,y’,z? system whose z’axis lies alongj and which is rotationally related to the (x”,y”,z’? system by the Euler angles (y,OKj,O):y (conjugate to K ) is the angle between the x ” axis and the x’axis, the latter lying along the line of nodes ii X 3 OKj = c o d ( ~ / j )is the angle contained between K and j (Le., between the axis and the z’axis). The second set of Euler angles (a,@,O) in eq A1 connects the (x’,y’,z? and (x,y,z) systems: aj (conjugate to j ) is the angle between the x l axis and the x axis, the latter lying along the line of nodes 1 X j ; O,j = c0s-I ((3 - l2 -j2)/2/j) is the angle contained between 1 and j (i.e., between the z’axis and the z axis). The Euler angle aIin eq A2 is conjugate to 1 and is the angle between the x”’ axis and the x axis. Registry No. CH,, 2229-07-4; H, 12385-13-6; D2,7782-39-0. (43) Goldstein, H. Classical Mechanics; Addison-Wesley: Reading, MA, 1950.

Chemiluminescence Mapping: Rate Constants for Formation and Relaxation in the HBr System F iL. L. Feezelt and D. C. Tardy* Department of Chemistry, University of Iowa, Iowa City, Iowa 52242 (Received: July 7, 1987; In Final Form: October 11, 1988)

The spectrally and time resolved chemiluminescencemapping technique is extended to isobaric conditions in which the atomic reactant is formed by the rapid multiphoton dissociation of SF6designated as chemiluminescence mapping-laser pulse (CM-LP). Nascent product vibrational energy distributions are obtained by extrapolation of the relative vibrational populations to zero time. Total reaction and microscopic relaxation rate constants are computed from the population-time profiles for each vibrational energy level populated. For vibrational levels below the maximum vibrational level populated, it is necessary first to decouple input due to relaxation of the upper levels from input due to reaction. It is shown that reaction and deactivation rate constants obtained by analysis of specific vibrational levels are more accurate than those obtained by analyzing total emission intensity-time profiles. The validity of the CM-LP method is demonstrated by the agreement of experimental results obtained for the F + HBr reaction to previous studies. The rate constant for relaxation of HF(v=3) by HBr is 7.1 x cm3 molecule-’ s-l.

Introduction The need to understand elementary energy-disposal and energy-transfer processes is fundamental to the understanding of complex reaction systems; this has also been emphasized by the recent report of Pimente1.l Relative microscopic reaction rate constants have been obtained from steady-state (Le., arrested relaxation,2 measured r e l a ~ a t i o n fast , ~ flow4) and pulsed (Le., chemiluminescence mapping5-’) experiments in which the infrared chemiluminescence of the product of exoergic reactions has been monitored. Absolute reaction rate constants and relaxation rate constants have been obtained from fast-flows experiments and time-resolved infrared chemiluminescence experimentssi2 (either the total intensity9*10or the emission from the highest populated vibrational level~i~12 is measured). Molecular beam techniquesi3 provide detailed reaction cross sections for energy disposal and energy transfer of state-selected molecules under single-collision conditions. The fast-flow experiments4 provide reaction rate constants relative to a reference reaction and are based on the emission of HF(vL 1); due to potential population in the ground vibrational level, a lower limit for the reaction rate constant is measured. Kaufman et al.’ has used the fast-flow experimental technique ‘Present address: Argonne National Laboratory, Chemistry Division, Argonne, IL 60439.

to obtain relaxation rate constants as a function of collider and vibrational level by using different generating reactions and by monitoring the emission from the highest populated level. This technique is limited to colliders that do not react with the atomic precursor of the generating reaction. Chemiluminescence measurements of total intensitygpiOfrom pulsed reactions have been used to measure both reaction and relaxation rate constants; these results are questionable because the total intensity is a composite of different vibrational levels with different relaxation depen-

‘.

(1) Omorruniries in Chemisrrv: Pimentel. G. C . . Chairman: National Academy Press: Washington, D.C:, 1983. (2) Polanyi, J. C.; Woodall, K. B. J. Chem. Phys. 1972, 57, 1574. (3) Jonathon, N.; Melliar-Smith, C. M.; Slater, D. H. Mol. Phys. 1971, 20, 93. (4) Sung,J. P.; Malins, R. J.; Setser, D. W. J. Phys. Chem. 1979,83, 1007. (5) Bittenson, S.; Tardy, D. C.; Wanna, J. Chem. Phys. 1981, 58, 313. (6) Tardy, D. C.; Wanna, J. T.; Bittenson, S.;Nelson, T. Proceedings of’ rhe Infernational Conference on Lasers, 1981; STS Press; 1982. (7) Wanna, J. T. Ph.D. Thesis, University of Iowa, 1982. (8) Berquist, B. M.; Bozzelli, J. W.; Dzelzkalns, L. S.; Piper, L. G.; Kaufman, F. J . Chem. Phys. 1982, 76, 2972. (9) Heidner, R. F., 111; Bott, J. F.; Gardner, C. E.; Melzer, J. E. J . Chem. Phys. 1979, 70, 4509. (10) Wurzburg, E.; Houston, P. L. J . Chem. Phys. 1980, 72, 4811. (11) Smith, I. W. M.; Wrigley, D. J. Chem. Phys. 1981, 63, 321. (12) Frost, R. J.; Green, D. S. Osborn, M. K.; Smith, I. W. M. Inr. J . Chem. Kinet. 1986, 18, 885. (1 3) Bernstein, R. B. Chemical Dynamics uia Molecular Beam and Laser Techniques; Oxford University Press: New York, 1982. ~

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0022-3654/89/2093-3 124$01.50/0 0 1989 American Chemical Society