10893
J. Phys. Chem. 1992, 96, 10893-10898 (15) Chen, Y.-M.; Armentrout, P. B. Work in progress. (16) Clemmer, D. E. Ph.D. Thesis, University of Utah, 1992. (17) Fontijn, A,; Felder, W. J. Chem. Phys. 1979, 71, 4854. (18) Johnston, H. S . Science 1971, 173, 517. (19) Dotto, L.; Schiff,H. I. The &one W w ,Doubleday: New York, 1978. (20) Chase, M. W.; Davies, C. A.; Downey,J. R.; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. J. Phys. Chem. Ref. Data 1985,14, Suppl. 1 (JANAF Tables). (21) Dagdigian, P. J.; Cruse, H. W.; Zare, R. N. J . Chem. Phys. 1975, 62, 1824. (22) Pastemack, L.; Dagdigian, P. J. J. Chem. Phys. 1977, 67, 3854. (23) Hildenbrand, J. L. Chem. Phys. Lett. 1973, 20, 127. J. E.; Liebman, J. F.; Holmes, J. L.; Levin, R. (24) Lias, S.G.; Bart”, D.; Mallard, W.G. J . Phys. Chem. Ref Dora 1988, 17, Suppl. 1. (25) Pcdley, J. B.; Marshall, E. M.J. Phys. Chem. Ref. Data 1983, 12, 967. (26) Schamps, J. J . Chem. Phys. 1973,2, 352. (27) Moore, C. E. Natl. Stand. Ref Data Ser. Natl. Bur. Stand. 1970, 34.
(28) Ervin, K. M.;Armentrout, P. B. J . Chem. Phys. 1985, 83, 166. (29) Schultz. R. H.; Armentrout, P. B. Int. J. Mass Spectrom. Ion Processes 1991, 107, 24. (30) Chen, Y.-M.; Clemmer, D. E.; Armentrout, P. B. J . Chem. Phys. 1991, 95, 1228. (31) Armentrout, P. B. In Advances in Gas Phase Ion Chemistry; Adams. N. G., Babcock, L. M., Eds.; JAI: Greenwich, 1992; Vol. 1, p 83.
(32) Vibrational frequenciesfor NO2 are 1357.8,756.8, and 1665.5 cm-l, for N 2 0 are 1276.5, 589.2 (2), and 2223.7 cm-I, and for C02are 1384.86, 667.70 (2), and 2349.30 cm-’ as given in ref 20. (33) Gioumousis, G.; Stevenson, D. P. J. Chem. Phys. 1958, 29, 292. (34) Calculated from P(OAl+-O) D ( O M - 0 ) + IE(A10) - IE(Al0J where DO(OAl-0) = 98.3 f 6.5 kcal/mol and IE(AI02) = 10.5 i 1.0 eV are taken from Ho and Burns, ref 35, and IE(Al0) is taken from Hildenbrand, ref 23. (35) Ho, P.; Burns, R. P. High Temp. Sci. 1980, 12, 31. (36) Armentrout, P. B. In StructurelReactivity and Thermochemistryoj Ions; Ausloos, P., Lias, S. G., Eds.;D. Reidel: Dordrecht, 1987; p 97. (37) Farber, M.; Srivastava, R. D.; Uy, 0. M. J . Chem. Soc.. Faraday Trans. I 1972,68, 249. (38) Paule, R. C. High Temp. Sci. 1976, 8, 257. (39) Smoes, S.; Drowart, J.; Meyers, C. E. J. Chem. Thermodyn. 1976, 8, 225. (40) This value is calculated by explicitly averaging the rotational and vibrational energies assuming Boltzmann distributions(constants from ref 20). (41) Lessen, D. E.; Asher, R. L.; Brucat, P. J. J. Chem. Phys. 1991,95, 1414. (42) Herzberg, G. Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules; Van Nostrand New York, 1966. (43) Troe, V. J.; Wagner, H. G. Ber. Bunsenges. Phys. Chem. 1967, 71, 946. (44) Lorquet, A. J.; Lorquet, J. C.; Forst, W. Chem. Phys. 1980,51,253.
-
Angular Momentum Coupling in SimpleFission Transition States Andrew J. Karas and Robert G. Gilbert* School of Chemistry, University of Sydney, New South Wales 2006, Australia (Received: August 21, 1992)
An efficient means is deduced for calculating RRKM microscopic reaction rates for simple-fission transition states by using a Hamiltonian consisting of the separate Hamiltonians for each moiety (A and B), hard-sphere interactions between A and B, and symmetric tops connected by a light rigid rod; the reaction coordinate is the breaking bond. This extension of the simple Gorin model takes exact account of the angular-momentumcoupling between A and B and of the transition-state requirement that the density of states of the activated complex be evaluated with the reaction coordinate held fmed. Evaluation for typical systems shows that the simpler treatment where A and B are treated as independent rotors is both accurate and in accord with experiment, if both are small but breaks down when A and/or B are large. In the latter case, the variational calculation givcs a rate coefficient that is highly sensitive to the assumed hard-sphere radii; although the model cannot then be used meaningfully for a priori prediction of experiment, it provides a useful means of fitting extant data to predict falloff behavior.
Introductioa The calculation of rqte coefficients for reactions proceeding through simple-fission transition states (such as radical-radical recombinations, dissociations, and ion-molecule reactions) is made complex by the fact that some modes (the “transitional” modes) change from vibrations to free rotations along the reaction coordinate from reactant to product. This paper is a contribution to the often-addressed questions of devising a suitable Hamiltonian to describe this situation and devising computational means to describe the dynamics dictated by this Hamiltonian so as to lead to an evaluation of the rate coefficient. Given a form of the Hamiltonian, evaluation of the requisite rate coefficient is usually camed out by using a statistical approach such as RRKM theory or the SACM (for an overview, see, e.g., Gilbert and Smith’), employing the basic formula
where k(E,J) is the rate coefficient for reaction from a reactant state with energy E and angular momentum, J, p ( E ) is the density of states of reactant, and the sum of states W’(E,J) is the total number of states that can become product. Author to whom correspondence should be sent.
RRKM theory starts with the dynamical assumption that there exists a hypersurface in phase space such that all trajectories passing through this hypersurface (defining the transition state or activated complex) from the direction of the reactant valley go on irreversibly to product without recrossing. This model therefore gives an exact upper bound to the exact classical rate coefficient and thus may be used variationally: the transition state can be located as that hypersurface along the reaction coordinate (rt) where the RRKM rate coefficient is a minimum. Practical means of evaluating the RRKM expression for W(E,J) may be for simplicity divided into two categories, on the basis of the assumptions made as to the form of the Hamiltonian of this transition state. (1) The f m t category assumes the Hamiltonian to be separable into harmonic terms for bends, stretches, etc. and separable terms for internal rotations; variants of the directaunt algorithm’ then enable W’(E,J) to be evaluated in a few seconds of computer time. The assumed form of the Hamiltonian is readily transferrable: Le., it can be easily written down for any system and the requisite parameters estimated by straightforward recipes (whose accuracy must of course always be open to question). However, the assumption of a separable Hamiltonian is clearly not an accurate representation of reality: e.g., the fact that the form of the Hamiltonian changes qualitatively between reactant and product (because of the transitional modes) says that the separability assumption cannot generally be valid. (2) The second
0022-3654192 1209610893S03.00 IO- 0 1992 American Chemical Society , ~, ~. ~. .- ._ --,
10894 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
category invokes a nonseparable Hamiltonian that takes much better account of the transitional modes. Because of this nonseparability, conventional direct-count techniques cannot be employed to evaluate the sum of states, and the requisite calculationsZ4 are much more complex than those for category 1. Moreover, the assumed form of the Hamiltonian is not such as to be readily transferrable, and thus calculations on a new system require much greater human effort than does category 1. It is the objective of the present paper to examine an RRKM approach for calculating k(E,J) for simple-fwion transition states that combines some of the desirable features of the two categories given above; viz., a method that (a) has a Hamiltonian whose functional form and component parameters are readily written down for a given system, while at the same time retaining certain essential features of the transition state; (b) avoids some of the unphysical approximations inherent in the assumption of a completely separable Hamiltonian; and (c) is still readily evaluated with computational resources comparable to those required for a conventional RRKM calculation. Our starting point is the Gorin model,5 whose basic physical approach is to take the transition state for the reaction AB A + B to consist of the (vibrational and rotational) Hamiltonians for the two isolated A and B moieties plus an appropriate intermoiety interaction (usually6 taken to be that of hard spheres representing the atoms of the component moieties). Next, we identify the reaction coordinate as the distance between the atoms of the breaking bond, a choice that is both physically reasonable and has been shown to be optimal in a more sophisticated a p p r ~ a c h .Thus ~ we treat the rotational motions as those of two symmetric tops connected by freely rotating joints on a light inextensible rod. This Hamiltonian is a small but significant improvement over that used in a number of previous applications of the basic Gorin idea,6*8-Io which assume that the rotations of each moiety are separable. However, in the derivation of RRKM theory, W(E,J) must be evaluated with the reaction coordinate held fixed. It is apparent that the assumption of separable rotations cannot be valid if the reaction coordinate is the breaking bond, since the rotation of one moiety will influence the motion of a second one to which it is connected by a rigid rod. The major innovation in the present paper is to show how this coupling, which arises from the kinetic energy component of the Hamiltonian, can be readily taken into account. The model thus preserves the appealing simplicity and transferrability of the Gorin approach, while removing a restriction that could cause both quantitative and qualitative inaccuracies. We restrict our attention to systems where the intermoiety interaction is that of hard spheres (including being zero if the moieties are sufficiently far apart). It will be seen that the present approach reduces the calculation of k(E,J) to a multidimensional integral that can be evaluated in a few seconds of computer time by a Monte Carlo method, producing a result that can then be trivially incorporated into conventional RRKM programs (e.g., ref 11). The present approach differs from but is parallel to the work of Smith,12 who devised an elegant method of evaluating the rotational component of the sum of states analytically but with the restriction that the moieties rotate only about their centers of mass: i.e., intermoiety coupling was properly taken into account but the reaction coordinate was taken to be the distance between the centers of mass of the moieties rather than the breaking bond (these two will only coincide in cases of high symmetry). Our approach is also related to important work by Klippen~tein,~ which enabled the reaction coofdinate to be generalized to the distance between any two points in the fragments. Moreover, Klippenstein explicitly showed the sum of states for a coupled oscillatory system, which must be evaluated by a Monte Carlo integration with a general method for weighting and sampling. The present paper differs from the work of Klippenstein by including the holonomic constraint of a constant value of the reaction coordinate in a different (although equivalent) way that completely decouples the center-of-mass translational motion; the present method simplifies the calculation of the rotational partition function. We also address issues such as the effects of the assumed potential function and of simplifying
-
Karas and Gilbert the kinetic energy to a more readily computable form. The improvements on the simple Gorin approach deduced here permit investigation into the following disturbing feature of the simple approach. If one or both of the moietia are bulky (in which case there is considerable hindrance in the rotations of the moieties at typical transition-state intermoiety distances) then the calculated rate coefficient is found to be extremely sensitive to the assumed hard-sphere radii of the atoms comprising each moiety:I0 e.g., the simple approach gives a change of 3 orders of magnitude in the high-pressure frequency factor for the decomposition of neopentane when the van der Waals radii are changed by 7%! Some of this explosion must arise from the sensitivity of the density of states to the amount of hindrance experienced by the rotation of a bulky moiety in close proximity to another species. However, the kinetic coupling that is the subject of the present paper must also play a role: because one of the moieties in the transition state (the tert-butyl radical) has a center of mass located at some distance from the carbon atom that defines one end of the breaking bond and, moreover, has a much greater mass than that of the other moiety (the methyl radical), uncoupling the rotations of each moiety will clearly be invalid. The present study will reveal whether eliminating these approximations will obviate the extreme sensitivity to the assumed van der Waals radii. Matbematical Development In this section we show how rate coefficients can be obtained by applying RRKM theory to the transition state described above. The RRKM treatment of eq 1 leads to1J3
where the total angular momentum has been approximated as the orbital angular momentum, R = BJ(J + l), B being the rotational constant of the reactant, Rt = (Z/fl)R,where I and are the moments of inertia of the inactive (conserved) external rotations of reactant and of activated complex, p(E) and pt(E) are the densities of states of reactant and activated complex, and V(rt) is the interaction potential between the two moieties com rising the simple-fission transition state at the bond distance r . The densities of states of reactant and transition state can be evaluated by using the convolution theorem, which states that if the corresponding Hamiltonian is separable into HI and H2, then
P
(3) where p, and p2 are the densities of states for the two terms in the Hamiltonian. The vibrational component of the density of states can be calculated quantum mechanically from the usual direct-count algorithm,14 and the problem is then to determine the rotational density of states or equivalently the partition function: Q = Jmp(E)e-E/kBTdE
(4)
The rotational component of a partition function can for most casea of interest be evaluated by using the classical expression
Q = hN
... s d p dq exp(-H/kBT)
where N is the number of degrees of freedom involved. As a preliminary to the exact treatment of kinetic energy for two rotating connected moieties, we first give the results from the simpler treatment of the transition state: the basic Gorin model5 and its further The transition state here is approximated as that of two moieties (each of which has the vibrational and rotational Hamiltonian of the isolated molecular or atomic fragment) rotating independently; the interaction is supposed to be hard-sphere. The independent rotors are each taken to be symmetric tops. The reacrion coordinate is supposed to be the length of the breaking bond: Le., each moiety pivots inde-
Angular Momentum Coupling pendently about its component atom belonging to the breaking
bond. These assumptions imply that the kinetic energy is then given by
T = y2Zf.piu(b,2+ I $ , ~sin2 e,) + ~ 2 Z f - p i u ( ~+l 6,cos 1 9 , ) ~+ y2~5j.piu(B22 + d22 sin2 e,) + y2~5-piu($2 + d2 cos e2i2+ ‘/21a.piu(62 + & sin2 e) + KZ;ol.piu(+ + d cos e)2(6) where Bi, &, and qi are the Euler angles of moiety i; 8, a, and \k are those of the combined moieties (which are assumed to
approximate an symmetric top when combined), cPiu, cpiu, and c.piu are the three principal moments of inertia of moiety i about
the pivot point of that moiety (Le., about the atom in the breaking bond), and &. and ~ o l , i u are those of the combined moieties arountthe overall center of mass (Le., each moiety and the overall activated complex arc approximated as symmetric tops). For the transition state, the nine spatial coordinates are restricted by the holonomic constraint that the reaction coordinate remain constant: in this case, that the two pivots remain at a fixed distance: i.e., in this approximate Hamiltonian, there are only eight generalized coordinBtt8. Given this assumptionof a separable kinetic energy and assuming that the intermoiety potential is also separable (i.e., that the rotations of the moieties, including any hindrance, are independent of each other), the dynamics become those of the following independent entities: two two-dimensional rotors (one for each moiety), one one-dimensional rotor (the torsional motion of the moieties counterotating), and the overall rotation of the combined moieties (one two-dimensional and one one-dimensional rotation). The classical contribution to the transition-state partition function of these degrees of freedom is then (e.g., ref 1) given by
where the Bs are the rotational constants corresponding to the moments of inertia and the up the symmetry numbers for each rotation; the Bp in the second product are about the two (nearly) degenerate axes of the fragment, while the B@in the first product are about the remaining (single) principal axis. By inverse Laplace transform, the corresponding density of states is
We now develop equivalent expressions for simple-fission transition states where now proper account is taken of rotational coupling between the moieties: Le., taking into account that the transition state assumption involves evaluation of the density of states with the reaction coordinate (in this case,the breaking bond) held fixed. The description adopted here for the kinetic energy terms in the simple-fission transition state is shown in Figure 1: two symmetric tops representing the two combining moieties, c o ~ a c t e dby an inextensible light rod. The pivot points of these tops are not fmed in space, and the tops are therefore coupled in their motions. The variables required to define the Hamiltonian are the following: hi = distance from pivot point to center of mass of moiety i; mi = mass of moiety i; rt = distance between pivot points (the reaction coordinate); Qi = vector giving the position of the center of mass of moiety i; r = vector connecting centers of mass of each moiety; (e, qi) = Euler angles of moiety i; (e,@)= Euler angles of the rod (note that is absent since the rod has no thickness); and 4 , = principal moments of inertia of moiety i about center of mass of that moiety (as distinct from around the pivot point as in the simpler Gorin model given above). This description is similar to that of Klippenstein,’ with the addition that hl and h2 are explicitly included, which will be seen to simplify the calculation of Q!o,, and also without the requirement that a coupled translation be introduced in order that the center of mass remain fixed. The kinetic energy of this system can be written down from the theoremls that states that the kinetic energy of a body is the sum of the kinetic energy of its center of mass plus the contributions from the motion of each particle of
c+,,,ccm,
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10895 the body with respect to the center of mass of the body. Applying this to each moiety, we have T = Trel+ + pt (9) where is the rotational kinetic energy of moiety i about its own center of mass and the contribution from the relative motion of the two moieties is given by
rot
(10) TreI = Y2mlQl2 + h ~ 2 Q 2 ~ Trelis transformed to kinetic energy relative to the center of mass of the whole system by the usual techniques to become
= 1/ZIL(Qt- Q2)’ = ’/2ctfZ (11) where p = mlm2/(ml m2). The rotational terms can be written in terms of the Euler angles a d 5 Trel
+
= ! I ~ I ~ , , ( B ~+ : 6 sin2 e,) + y2~f.cm($i + & cos ei)z (12) The next step is to find TWl.Defining el, e2, and e as unit vectors along h , , h2, and rt, respectively, one then has r = -Alel + h2e, + rte -hl sin 0, cos 4, + h2 sin e2 cos 42 + rt sin e cos -hl sin e, sin 4, h2 sin e2 sin 42 + rt sin e sin -hl COS e, h2 COS e2 r+ COS e =( (13) After differentiating and taking the scalar product, one obtains (after some algebra) qol
+
+
)
+
+
T,,, = f/2p[h12(8,2 + I $ , ~sin2 e,) + h22(822 t$22 sin2 e,) + (rt)2(62 + 42 sin2 e) - 2rthI(B,{6[sine, sin e cos e, cos e cos (4, - o)]- 4 cos e, sin e sin (a - 4,)) 4,sin e,@ sin e cos (4, - a) - 6 cos e sin (4, - a))) 2rth2(e2{6[sine2 sin e + cos e2 cos e cos (42- a)] b cos e2 sin e sin (O - 42))+ h2 sin e2{4sin e x cos (42- O) - 6 cos 8 sin (42- a))) 2hlh2(8,82[sin8, sin e2 cos 8, cos 62 cos (4, - 42)] sin (4, - 42)(81~2 sin e2 cos 0, - d2& sin 8, cos e,))] (14)
+
+
+ +
+
To evaluate the partition function, the Hamiltonian, and hence T, needs to be expressed in terms of the conjugate momenta. In the Lagrangian formalism, the momentum p conjugate to a coordinate q is given by
where L = T - V and H = T + K This leads explicitly to Pe, = ZT-~,,,~,+ p[hl2BI2- rth,(6[sin 8, sin e + COS e, COS e cos (9, - o)]- b cos e, sin e sin (a - 4,)) hlh2(b2[sin8, sin e2 + cos 8, cos 62 cos (4, & cos 8, sin B2 sin (& - #,))I (16) pe, = I$.c,b2 + p[h2*8,- rth2(6[sin e2 sin 8 + COS e2 cos e cos (42 - O)] - 4 cos e2 sin e sin (a - 42)) h,h2(8,[sin sin e2 + cos e, cos e2 cos (4, - 42)] 6,cos O2 sin el sin (4, - 42))] (17)
+ z:.~~($, + & cos e,) cos 8, + el sin e cos - a) 6 sin 8, cos e sin (4, - a)] - hlh2[42sin e, sin e2 cos (4, -
pQ, =
sin2 e,
p [ h 1 2 & sin2 6 , - rthl[4 sin
$1~11- 6, sin 8, cos O2 sin (4, -
(18) pQ, = Zf.cm42 sin2 B2 + I$.cm($2 + & cos 6 2 ) cos O2 + p [ h 2 2 d 2sin2 e2 - rth2[4 sin e2 sin e cos (42 - @) 6 sin e2 cos e sin (42 - e)] - hlh2[,$2sin e, sin e2 cos (4, d2)] - 8, sin e2 cos el sin (42- +,)I (19) p e = p[(rt)26 - rth,(jl,[sin e, sine + cos e, cos e cos (4, o)] - d, sin e, cos e sin (41- a)) rth2(d2[sine2 sin e + COS e2 COS e cos (e, - a)] - h2 sin e2 cos e sin ($2 - o))] (20)
+
10896 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
Karas and Gilbert
p o = p [ ( r + ) % sin2 e - r+hl(4,sin e, sin e cos (4' - a) 8, cos sin 0 sin (@ - 4,)) + rth2(d2sin e2 sin 8 cos (42@) - f12 cos e2 sin 0 sin (@- &))I (21)
+ 41 0,) = G - c m ( 4 2 + d2 cos 82)
P+, ~
$
IT.cmt4,
2
(22) (23)
The total classical rotational partition function of the transition state can be deduced as follows (or alternatively as in the work of Asron and Eidinoff and of R t z ~ r ~ ~ JOne ' ) . changes integration variables to obtain
F i 1. Variables in the description of a simple-fission transition state as two symmetric tops joined by a light inextensible rod.
where the elements of the Jacobian matrix J are given by Jij
= aPi/a4j
(25) 0.7
The elements of J follow from eqs 16-23. Since the Jijdepend only on qi and are independent of the
4 I
os4 3
4
r+
In eq 26, the limits of integration have been written down for nonoverlapping moieties; in the case when rf is such that the moieties may overlap for certain values of the three sets of Euler anglca, these limitschange appropriately (seebelow). If a n o m 0 potential is included, then eqs 16-23 are unchanged, from eq 15. This elimination of the momenta at this stage avoids the complication' of having to introduce a coupled translation in order that the center of mass remain fixed. Equation 26 gives the required result for the exact rotational contribution to the density of states of a transition state consisting of two vibrating moieties where the reaction coordinate is the breaking bond. Note that it makes no restriction on the overall rotation of the conjoint entity: Le., the overall rotation is not approximated to be that of a symmetric top. Equation 26 does not admit an analytic solution for the general case where the two tops are not the same (although we note that the equivalent expression for certain cases of higher symmetry has led to an analytic solution,'* this being for cases where couplings involving the reaction coordinate are absent). However, it is easily evaluated numerically by Monte Carlo integration (including numerical evaluation of the determinant). For the case of nonoverlapping moieties (when the limitsof integration are those of eq 26) an accuracy of a few percent is obtained from a Monte Carlo evaluation of 103-104 samples, with minimal computer time: about the same as for a direct-count RRKM calculation. Evaluation for the case when the moieties may overlap is also straightforward, although more time consuming, since each must be tested to see random selection of (e,,4,,~,,e2,4,,h,0,a) if overlap occurs. Moreover, unlike eq 1 (which contains the rotational energy difference between reactant and transition state in the limits of integration), it incorrectly includes trajectories which pass through the transition state but which have insufficient energy to surmount the subsequent centrifugal barrier. Lastly, it is useful to note that eq 26 has the functional form
this holding whether or not the moieties overlap; the factor I (evaluated by numerical integration) is dependent only upon the moments of inertia, p, hi, h2,and rt. This has the same dependence on temperature T as does q 7;hence the density of states obtained from the inverse Laplace transformation of eq 26 will be identical with that from eq 7, viz.
5
(A)
F @ e 2. Ratio of exact total rotational partition function, eq 26, to that calculated with the approximate Hamiltonian, eq 7, for methyl recombination with planar and with trigonal-pyramidal methyl moieties and for the neopentane dissociation. This last result means that standard RRKM programs, which take appropriate account of rotations for uncoupled motion (e.g., ref 1 1 ) can be used to implement the new method deduced in the present paper without recoding. However, this simplicity only holds when the interaction potential between the moieties is hard-sphere; in other cases (e.g., sinusoidally-hindered rotorslO) both the form of the partition function and the Monte Carlo integration are more complex. Applit~ti~~ We give here the results of illustrative calculations using the formulas derived above. These represent instances where the effects of intermoiety kinetic coupling and hindrance become progressively more important. The results given in Figure 2 are the ratio of the partition function calculated by using the physically correct expression, eq 26, to that for the simple expression, which ignores coupling, eq 7. This has been done for the following three cases, in each of which there has been taken to be no hindrance between the moieties: (1) 2CH, C2Hs,where the transition state consists of planar methyl moieties. Here, for a given moiety, the center of mass and the pivot point are the same, and thus any difference between the exact and simple calculations arises solely from kinetic coupling between the moieties. The parameters for the calculation are hi = h2 = 0,I;l-plu = = qScm = &.cm = 1.76 amu A2, and = &u = If.em = &,, = 3.52 amu A2, with other quantities required in eq 26 deduced from the masscs and distance between the C-C bond. (2) 2CH3 C&, where the transition state now consists of trigonal-pyramidal methyl moieties with bond angles of 109.3' (the planarity of the preceding model is in fact a better representation of physical reality, but these calculations are purely illustrative). For a given moiety, the center of mass and the pivot point are no longer coincident, and this calculation shows the effects of both kinetic coupling between the moieties and the constraint that the length of the breaking bond be held f ~ e d The . rameters are hi = h, = 0.07024 A, q.piu = = 1.931 amu fi.cm= I;.c = 1.857 amu A2,and If, = F& = If, = Is, = 3.123 amu fz, with other quantities required in eq 26 deduced from the masses and distance between the C-C bond. (3) The decomposition of neopentane: CSHlz-c C4H9+ CH3. One of the moieties (tert-butyl) is highly asymmetric and bulky, and this will show to a greater extent than the preceding case the
-
c.plu
-
5;
c-piv
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10897
Angular Momentum Coupling 20
19
log A-Is-1 18
17
3
4 rtlh
5
Figure 3. Calculated transition-state high-pressure rate coefficient for neopentane dissociation at 1100 K, calculated by using eq 26 with hindrance included (RRKM parameters otherwise from Jordan et a1.I0), as function of intermoiety distance, and the corresponding hi h-pressure frequency factor. Variational minimum in krnioccurs at rf = 3.44 A, when the two moieties become locked (100% hindrance).
effects of fued reaction coordinate. The parameters are hl = 0, h2 = 0.2 A, I;l-p,v= I;l-cm = 1.76 amu A2, = = 3.52 amu = 63.27 amu A2,and G.piu= = 65.549 amu A2, A2, = 116.253 amu A2,with other quantities required in eq 26 deduced from the masses and distance between the C-C bond. In each case, taking proper account of angular-momentum coupling and of fixed rt leads to a rotational partition function smaller than that from the simpler, but less accurate, treatment of uncoupled motion around the center of mass of each moiety. This is because these couplings restrict the motion of each moiety, and thus reduce the volume of phase space available to the system. It is apparent from Figure 2 that for small and/or highly symmetric moieties there is relatively little effect of keeping the reaction coordinate truly fued, compared with the results of the simpler calculation where rotation of the moieties is about the center of mass rather than around the atoms forming the breaking bond. Concomitantly, the coupling between the angular momenta of these moieties and the total angular momentum does not have a large effect on the calculated micrampic rates. However, with large asymmetric moieties, these effects are much larger: for the neopentane case, almost a factor of 2 in the partition function at a bond distance corresponding to that typical of the location of the transition state (ca. 3 A). We next turn to the effect of incorporation of hindrance (Le., n o m interaction between the nonbonded atoms in each moiety) into the calculation. It has been shown1° that the simple Gorin description of eq 7, together with a sinusoidally hindered interaction between the two moieties to account for nonbonded effects, gives good accord with experimental data prouided both moieties are small (e.g., CH,). However, when one or both moieties are large (e.g., tert-butyl) there is extensive hindrance, and results are extremely sensitive to the assumed van der Waals radii of the moieties; use of the model to predict experimental data meaningfully is therefore injudicious. Indeed, the high-pressure frequency factor for the neopentane dissociation found from the variational calculation shown in Figure 2 is almost 1020s-I, which is 3 orders of magnitude greater than e ~ p e r i m e n t . ~The ~ J ~calculations of Figure 2 indicate th?t it is just such a case when the proper incorporation of angular momentum conservation and the correct reaction coordinate might make a large difference compared to the predictions of the simpler treatment. It is apparent that the effects of hindrance (ignored in the preceding calculations) will be large when one or both moieties are large. Figure 3 shows the result of incorporating intermoiety hindrance into the calculation (using "standardm6values of the van der Waals radii of the atoms in each moiety) for the neopentane dissociation. The variational minimum now moves into the region where there is extensive hindrance (incidentally, when hindrance is extensive, the number of function evaluations in the Monte Carlo integration increases enormously, being approximately inversely proportional to the final calculated value of the frequency factor, since many randomly selected values of the Euler
c.pivc.cm
angles are not permitted). Indeed, the calculated rate coefficient is still decreasing when the two moieties become completely locked (100% hindrance): Le., the variational minimum is where the two moieties completely lase their rotational degrees of freedom, when the calculated value of A, is 1018.6 PI. The value of A. obtained from the variational calculation is also sensitive to the assumed van der Waals radii. Hence the extreme sensitivity of the variational rate coefficient in this model to the assumed van der Waals radii cannot be obviated by taking exact account of angular momentum coupling and constant reaction coordinate: it will be recalled that investigation of this possibility was one of the objectives of the present study. The results of eq 26 in systems with large hindrance must moreover be viewed with caution, since the classical partition function (which goes to zero with large hindrance) becomes inaccurate compared to the quantum one (which approaches unity under the same circumstance). Furthermore, this largehindrance situation must be where the assumed potential function (hard-sphere interactions) is inapplicable, since the transitional modes are some hybrid between a collection of rotors and a collection of (anharmonic and coupled) oscillators at such a small intramoiety distance. Indeed, if one takes the extreme case when the five transitional modes all become high-frequency harmonic oscillators (which would only be expected to be the case at the reactant equilibrium geometry, at a much smaller intramoiety distance), the calculated value of A, at this value of rt is SI. Since the experimental value of A, is 10'7.2M.4 s-l,18J9 it is indeed apparent that physical reality lies between these two (easily evaluated) extremes. These last results indeed indicate that the potential assumed in the current treatment, while very convenient to implement in RRKM theory, is inadequate if one or both of the moieties are large. The model clearly reflects the correct physical description of the process, since in such cases, physically reasonable values of the van der Waals radii can be chosen, which give accord with experimental frequency factors, with the variational minimum at physically reasonable intermoiety distances. However, this sensitivity means that the treatment cannot be used for a priori prediction in such circumstances; this would require a better knowledge of the potential functional for the transitional modes than afforded by the present treatment. It is clear that when the moieties are highly hindered, a hard-sphere interaction is physically incorrect for the potential, and some means of allowing the potential to progress to harmonic oscillators is necessary (a number of methods of doing this have been given in the l i t e r a t ~ r e ) . * ~ - ~ - ~ * ~ ~ However, given such a functional form, the methods developed in the present paper, or more complex treatments by Uppenstein,' could then be employed to compute the rate coefficient, but with much more computational labor than that afforded by the present reductionist approach. We note in closing that calculations using the present model show that the value of the frequency factor is insensitive to the assumed interaction potential between the two bond-forming atoms.
Conclusions In the present paper, we have explored a model that enables exact account to be taken of the full coupling between the rotations of the two moieties comprising a simple-fission transition state, where the reaction coordinate is the breaking bond. This description is a rigorous quantification of the intuitively appealing Gorin model and gives all the essential physical features of the behavior of simplefission transition states. The method involves a Monte Carlo integration which, if the interaction between the moieties is hard-sphere, is easily appended to a conventional RRKM program and requires about the same computation time. Importantly, the necessary parameters for the calculation are easily written down for any system; i.e., the model is transferrable. The present technique can be seen as complimentary to more laborintensive calculations with more physically realistic Hamiltonia n ~ and ~ - to ~ analytic solutions applicable to more restricted systems.I2 The results show that for systems where both moieties are small, the much simpler approximate treatment, which ignores
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the kinetic coupling between the moieties, is quite accurate. This is especially convenient, since the latter is readily implemented (e.g., ref 11) and has (with appropriate extensions) been shownI0 to give a good quantitative prediction of rate coefficients for reactions of small species. While the present method enables RRKM calculations with these Hamiltonians to take precise account of the complexity of the kinetic energy in the quantification of these Gorin-style models, a major unsolved problem is in the potential term between the moieties when one or both are bulky. The variational calculation then gives a frequency factor that is highly sensitive to the assumed hard-sphere radii and the model cannot be employed meaningfully for a priori prediction of experiment: more sophisticated treatment of the transitional modes needs to be employed in such circumstances, clearly a fruitful area for future study. However, the present treatment could be used to give a physically meaningful fit (by scaling the van der Waals radii) of experimental data for reactions involving large moieties over a limited pressure and/or temperature range, and the k(E,J) so obtained could be used to predict the entire pressure and temperature dependence by using standard' methods.
Acknowledgment. The financial support of the Australian Research Grants Committee is gratefully acknowledged. References and Notes (1) Gilbert, R. G.; Smith, S.C. Theory of Unimolecular and Recombinarion Reacrions; Blackwell Scientific: Oxford and Cambridge, MA, 1990.
(2) Wardlaw, D. M.; Marcus, R. A. J. Chem. Phys. 1985, 83, 3462. (3) Klippenstein, S. J.; Marcus, R. A. J . Phys. Chem. 1988, 92, 5412. (4) Wardlaw, D. M.; Marcus, R. A. Adu. Phys. Chem. 1988, 70, 231. (5) Gorin, E. Acra Physiochim. URSS 1938, 9, 691. (6) Benson, S. W. Thermochemical Kinetics, 2nd ed.; Wiley: New York, 1976. (7) Klippenstein, S. J. J . Chem. Phys. 1991, 94, 6469. (8) Pitt, I. G.; Gilbert, R. G.; Ryan, K. R. Ausr. J . Chem. 1990,43, 169. (9) Greenhill, P. G.; Gilbert, R. G. J . Phys. Chem. 1986, 90, 3104. (10) Jordan, M. J. T.; Smith, S. C.; Gilbert, R. G. J . Phys. Chem. 1991, 95, 8685. (11) Gilbert, R. G.; Smith, S.C.; Jordan, M. J. T. UNIMOL program suite (calculation of fall-off curves for unimolecular and recombination reactions); 1992; available directly from the authors: School of Chemistry, Sydney University, NSW 2006, Australia. (12) Smith, S. C. J . Chem. Phys. 1991, 95, 3404. (13) Forst, W. Theory of Unimolecular Reactions; Academic Press: New York, 1973. (14) Beyer, T.; Swinehart, D. F. Commun. Assoc. Comput. Machin. 1973, 16, 379. (1 5 ) Goldstein, H. Classical Mechanics; Addison-Wesley: Reading, MA, 1980. (16) Eidinoff, M. J.; Aston, J. G. J. Chem. Phys. 1935, 3, 379. (17) Pitzer, K. S. J . Chem. Phys. 1937, 5, 469. (18) CGme, G. M. In Pyrolysis: Theory and Indusrrial Practice; Albright, L. F., Crynes, B. L., Cocoran, W. H.,Eds.; Academic: New York, 1983. (19) Baldwin, A. C.; Lewis, K. E.; Golden, D. M. Inf. J. Chem. Kinei. 1979, 11, 529. (20) Klippenstein, S. J.; Marcus, R. A. J. Phys. Chem. 1988, 92, 3105. (21) Wardlaw, D. M.; Marcus, R. A. J. Phys. Chem. 1986, 90, 5383. (22) Troe, J. Z . Phys. Chem. N.F. 1989, 161, 209. (23) Troe, J. J . Chem. Phys. 1987,87, 2773. (24) Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 242. (25) Quack, M.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 240.
Origin of the Electrocatalytic Properties for O2 Reduction of Some Heat-Treated Polyacrylonitrile and Phthalocyanine Cobalt Compounds Adsorbed on Carbon Black As Probed by Eiectrochemlstry and X-ray Absorption Spectroscopy M. C.Martins Alves, J. P. M e l e t 2 D. Guay, M. Ladouceur, and G. Tourillon* LURE, Batiment 209 D, 91405 Orsay, France (Received: February 27, 1992)
Electrochemicaland X-ray absorption techniques have been used to determine the influence of heat treatment in electrocatalytic activity for O2reduction for two cobalt catalysts. The catalysts are cobalt phthalocyanine (catalyst 1) and polyacrylonitrile + cobalt acetate (catalyst 2) adsorbed on carbon black and heat treated at several temperatures. A maximum far the catalytic activity was obtained for PcCo at 850 "C and for the PAN + Co catalyst at 950 "Cwith subsequent decrease. The results obtained by XANES and EXAFS data clearly show that metallic cobalt aggregates with different size are synthetized in the range of increased activity. In the region of highest activity were observed the smallest cobalt clusters (20 A). For higher temperatures these cobalt aggregates became bigger (100-200 A), which corresponds to the decrease in the catalytic activity. TEM was utilized as a complementary technique and it confirms the influence of the annealing temperature in the size of the cobalt aggregates obtained. XANES measurements at the Co and N K edges confirm that CON, centers and nitrogen atoms are no longer detected after heat treatment in the region of increased activity.
Introduction In fuel cell technology the search for efficient and cost-effective electrocatalysts for cathodic oxygen reduction is a crucial prob1em.I With the knowledge that platinum is one of the best electrocatalysts for oxygen reduction, the investigation of compounds that do not use precious metal is very important from practical and theoretical points of view. In particular, N4-metal chelates appeared to be good candidates as catalysts for the oxygen reduction r e a c t i ~ n . ~These , ~ chelates became more interesting when it was demonstrated that the heat treatment of these materials adsorbed on high-area carbons improves their stabilities and activities for O2r e d u ~ t i o n . ~ ~ Several authors have attempted to explain the origin of the high activity and ~ t a b i l i t y ~and - ~ especially the nature of the active 'On sabbatical. Permanent address: INRS, Energie, C.P. 1020, Varenes,
F Q J3X 1S2, Canada 11.
0022-3654/92/2096-10898$03.00/0
species for the oxygen reduction after the thermal treatment: Wiesenerg proposed initially that during the heat treatment, a special "kind of carbon" with fupional chemical surface groups is synthetized on the substrate. The role of the metal ion is to influence the thermal formation of the active catalyst. According to others,'*I2 the annealing treatment does not lead to the complete destruction of the chelate macrocycle, but rather to a ligand modification which preserves the central N4-metal part. The electrocatalyticactivity would come from this N,-metal group. Gupta et al.I3demonstrated that the N4-metal centers are not essential to the electrocatalysis. They studied a system composed of a mixture of cobalt or iron salts and polyacrylonitrile adsorbed on carbon black (Vulcan XC-72) and annealed up to 1000 OC. The catalytic activities of such compounds are identical to those of the corresponding transition metal-N4 macrocycles. They proposed a model where the active species is a modified carbon 0 1992 American Chemical Society
