J. Phys. Chem. 1989, 93, 8142-8148
8142
partition functions, since we neglected anharmonicity and mode-mode coupling. Thus, we do not believe that the above adjustment is sufficient evidence for estimating the barrier height to better than f l kcal/mol. It does indicate, however, that the MP2/6-31G** value of 4.5 kcal/mol, while probably on the high side, is a more reasonable estimate of the barrier height than the 8-10 kcal/mol values in Table 11. In order to assess the importance of vibrational zero-point and quantization effects on this reaction, we also evaluate the completely classical TST rate constant at 300 K. This calculation differs from the calculation discussed above only in that the quantum mechanical vibrational partition functions are replaced by their classical analogues. (The zero-point energy difference between the transition state and reactants is not included in the classical calculation.) We find that the classical rate constant is only 20% slower than the rate constant computed with quantized vibrations at 300 K for this reaction.
sufficiently reliable scaled frequencies and energies for creating a potential energy surface which will be useful in quantitative dynamical studies. The frequencies were improved somewhat by including a 6% correction at all geometries. This correction brings the calculated frequencies for CH3CI to within 2.5% of the experimental value for each mode. We saw that, for this reaction, the MP2 calculations account for a large fraction of the geometry-dependent part of the correlation energy (9, = 0.917), and that MP-SAC2 corrected energies, relative to the reactants energy, are very similar (-0.01 kcal/mol at the transition state) to the uncorrected MP2 values. Transition-state-theory rate constants calculated at 300 K and compared with experimental yield a semiempirical estimate of the barrier height of 3 kcal/mol, indicating that the MP2/6-31G** value of 4.5 kcal/mol is reasonable, though probably a little high. Finally, we have tabulated our best estimates of geometries, charges, energies, and frequencies at the three stationary points on the reaction path.
7. Conclusion On the basis of a comparison of calculated transition-state frequencies at seven ab initio levels for the gas-phase SN2reaction of C1- with CH3C1, we conclude that second-order Mailer-Plesset perturbation theory calculations with a 6-31G** basis set provide
Acknowledgment. We are grateful to Thanh N. Truong for valuable assistance with the calculations. This work was supported in part by the National Science Foundation and the Minnesota Supercomputer Institute. Registry No. CI, 16887-00-6; CH3CI, 74-87-3.
Microscopic Reaction Rates in Ion/Molecule Reactions: Effects of Uncoupled Modes Sean C. Smith, Murray J. McEwan, Department of Chemistry, University of Canterbury, Christchurch 1 , New Zealand
and Robert G. Gilbert* Department of Theoretical Chemistry, University of Sydney, NS W 2006, Sydney, Australia (Received: March 29, 1989; In Final Form: May 25, 1989)
A new expression for microscopic rate coefficients is deduced for reactions with a simple-fission transition state wherein certain modes become uncoupled from the reaction coordinate at long range. The new expression, which involves a modification to the standard RRKM result, takes account of this uncoupling, thereby incorporating features inherent in capture models. Application to the case of ion/dipole reactions is considered, where only the hindered dipole rotation is coupled with the reaction coordinate at large separations. When implemented variationally, the new expression is shown analytically to produce in the high-pressurelimit a capture-rate expression equivalent to that of Chesnavich,Su, and Bowers. This provides an important conceptual connection between the RRKM approach to ion/dipole reactions, which includes all degrees of freedom and may be applied at any pressure, and that of capture theories (which apply only in the high-pressure limit) in which only those degrees of freedom that are coupled with the reaction coordinateat long range are included. Neglect of uncoupling in conventional RRKM theory can lead to an overestimate (typically by ca. 50%) of ion/molecule rate coefficients at high pressures; the error is however negligible at low pressures.
Introduction The high-pressure rate coefficient for ion/molecule association reactions can be successfully calculated (except at very high temperatures) from expressions for the capture rate based on the electrostatic forces acting between the moieties. The simplest of such models is the well-known Langevin formula, which supposes that the capture rate is that of crossing the centifugal barrier between a point charge and an induced dipole. There are a number of more sophisticated treatments, which take the structure of the reacting moieties into account in varying ways: average dipole orientation (ADO) models,Is2microcanonical ~ a r i a t i o nthe , ~ statistical adiabatic channel model (SACM),4 and the centrifugal ( I ) Su, T.; Su, E. C. F.;Bowers, M. T. J . Chem. Phys. 1978, 69, 2243. (2) Bates, D. R . Proc. R. SOC.London, A 1982, 384, 289; Chem. Phys. Lett. 1983, 97, 19. (3) Chesnavich. W . J.; Su,T.; Bowers, M . T. J . Chem. Phys. 1980, 72, 2641.
0022-3654/89/2093-8 142$01.50/0
sudden approximati~n.~It is generally found that the more sophisticated of these treatments gives good agreement with a wide range of experimental data on high-pressure capture rates and presumably therefore encompass a proper physical description of the process. However, such a high-pressure calculation cannot describe the pressure dependence of the association rate, when collisional deactivation and activation also become rate-determining. To take these "fall-off" effects into account requires solution of the master equation for reactive and collisional processes,involving a microscopic reaction rate k(E,J) (which depends on total energy E and angular momentum J) and the probability distribution function for collisional energy and angular momentum transfer (see, e.g., ref 6). The capture-rate methods only find (4) Troe, J. J . Chem. Phys. 1987, 87, 2773. ( 5 ) Clary, D. C. Mol. Phys. 1984, 53, 3; Mol. Phys. 1985, 54, 605; J . Chem. SOC.,Faraday Tram. 2 1987,83, 139. Clary, D. C.; Smith, D.; Adams, N . G. Chem. Phys. Lett. 1985, 119, 320.
0 1989 American Chemical Society
Uncoupled Modes in Ion/Molecule Reaction Rates
The Journal of Physical Chemistry, Vol. 93, No. 25, 1989 8143
a transition state chosen by canonical or microcanonical variation the thermal average of k(E,J) in the high-pressure limit. give results for the high-pressure association rate that are sigA convenient means to find the pressure dependence of an nificantly greater than those found variationally from the capture association rate coefficient is to calculate that for the reverse method3 with the same form of the potential function (e.g., ref reaction (the unimolecular dissociation); this gives the association rate with use of the microscopic reversibility relationship (e.g., 10). This discrepancy can be as much as a factor of 2. Since ref 6-9 and see eq 15 below). This relationship has been shownIO it is found that variational capture rates give agreement with experiment, it is necessary to improve upon the RRKM approach to be extremely accurate for such systems throughout the fall-off for these systems and to examine further the relationship between regime. The unimolecular dissociation rate, in turn, is most commonly calculated through RRKM theory. This yields a simple these two approaches. The primary difference between RRKM theory and statistical expression for k(E,J) that is suitable for use in calculating fall-off capture models (e.g., ref 3, 4,and 13) is the number of degrees effects by solution of the master equation.I0JT For ion/molecule reactions, the RRKM transition state consists of the separate ion of freedom included in the calculation. Implicit in RRKM theory is the ergodic hypothesis, which assumes that energy is randomized and molecular moieties at some distance rt. The value of rt can rapidly among all internal degrees of freedom of the molecule. be chosen by canonical or microcanonical variation (the latter The RRKM expression for k(E,J) therefore necessarily includes being the preferred method).I2 all degrees of freedom of the reacting system. On the other hand, How does microcanonical variational RRKM theory compare theories of electrostatic capture include only those “pertinent” with capture model approaches? For the simple case of an degrees of freedom that are coupled with the reaction coordinate ion/induced-dipole system, RRKM theory gives results for the at long range on the electrostatic part of the potential surface. high-pressure association rate that are identical with those from the simple Langevin model for capture of an isotropically poInherent to this choice of pertinent degrees of freedom is the understanding that there is a difference in the dynamical interlarizable neutral by a point charge provided the transition state action of coupled modes with the reaction coordinate as opposed is chosen microcanonically as being at the top of the centrifugal to that of uncoupled modes. Conventional RRKM theory makes barrier. This result has been demonstrated by Troet3using the no distinction between coupled and uncoupled modes in the SACM and by Chesnavich and Bowers;I4 we give an alternative proof in Appendix I, using RRKM theory. This identity between transition state. In this paper, we present a modification to conventional RRKM RRKM and capture rates for this system is to be expected, since the two models involve the same assumptions; however, the identity theory for application to processes where certain modes are is not immediately apparent because of the quite different starting identical in the transition state and products: these modes are points of the two techniques. A capture model calculates the rate completely uncoupled to the reaction coordinate from the transition state out to products. This uncoupling is a special type of adiabatic of electrostatic capture of the neutral by the charged species and therefore explicitly excludes all degrees of freedom within the constraint that affects the dynamics in the transition-state region molecules that are not involved in the dynamics of electrostatic and is likely to apply in reactions where the interaction between capture. In the Langevin capture model for an ion/induced-dipole moieties is long range, e.g., ion/molecule systems. We deduce a formula for k(E,J) that can be evaluated by microcanonical system, this means that one simply deals with a point charge and variation. This expression combines the ideas of RRKM and an induced dipole: reaction is assumed to occur if and only if a capture-rate theories and is expected to be accurate for describing trajectory surmounts the centrifugal barrier. In RRKM theory, the transition-state assumption that all trajectories crossing the the dynamics of systems with such long-range interactions. transition-state surface are reactive will therefore be equivalent A number of workersT6T9have highlighted the importance of to the capture assumption under the conditions stated above. adiabatic considerations, Le., the change in certain degrees of freedom as the system goes from reactant, through the transition RRKM theory, being a theory of unimolecular decay, must state, and on to product. Foremost among such theories is the necessarily include all degrees of freedom of the system. However, Our new expression applies ideas implicit in work by as is shown in Appendix I and has been indicated p r e v i ~ u s l y , ~ ~ ~ ’SACM.]’ ~ Troe20*2T and by Marcus and c ~ - w o r k e r sbut ~ * ~is ~much ~ simpler the contribution of these degrees of freedom to the association due to the restricted nature of the adiabatic constraint considered, rate cancels out in the high-pressure limit. This result emphasizes Le., complete uncoupling. In the high-pressure limit, our result that, for systems without a barrier to recombination, the optimal transition state is that chosen microcanonically. Microcanonical reduces analytically to an expression for the capture-rate coefficient choice of the transition state (variationally or at a barrier) is always that is equivalent to that of Chesnavich et al.,3 except that whereas the Chesnavich result does not explicitly conserve angular moto be preferred over the semiempirical choice of a transition state mentum, the present work includes this constraint (under the found by comparison with experimental data (e.g., as in the simple approximation, commonly used in RRKM theory, that the total Gorin modelTs)since such a semiempirical choice may not be angular momentum J may be replaced by the orbital angular transferable to systems differing significantly from those for which the semiempirical evaluation was carried out. momentum L). This equivalence illustrates clearly the relationship between RRKM theory and statistical capture theories that deal The foregoing exact accord between high-pressure ion/molecule only with degrees of freedom that are “active” on the long-range association rates calculated from RRKM and capture methods electrostatic region of the potential ~ u r f a c e . ~ ~ ~ ~ * ~ ~ no longer holds when the interaction potential becomes more complex. RRKM calculations for an ion/dipole reaction using Microscopic Rates Involving Uncoupled Modes We first give a brief overview (in eq 1-5) of RRKM-type (6) Smith, S. C.; Gilbert, R. G. Int. J . Chem. Kinet. 1988,20, 307; 1988, 20, 979. theories for k(E,J), since it is necessary to examine the funda(7) Keck, J. C.; Carrier, G. J . Chem. Phys. 1965, 43, 2284. mental assumptions involved and then make appropriate ad(8) Gilbert, R. G.; McEwan, M. J. Ausf. J. Chem. 1985, 38, 231. justments in deriving the result for our special case (an excellent, (9) Dodd, J. A.; Golden, D. M.; Brauman, J. I. J . Chem. Phys. 1984,80, more detailed, overview has been given by Wardlaw and Mar1894. Olmstead, W. N.; Lev-On, M.; Golden, D. M.; Brauman, J. I. J . Am. Chem. SOC.1977, 99, 992. Jasinski, J. M.; Rosenfeld, R. N.; Golden, D. M.; Brauman, J. I. J. Am. Chem. SOC.1979, 101, 2259. Chang, J. S.; Golden, D. M. J. Am. Chem. SOC.1981, 103, 496. (IO) Smith, S. C.; McEwan, M. J.; Gilbert, R. G. J . Chem. Phys. 1989, 90, 4265. (1 1) Smith, S. C.; McEwan, M. J.; Gilbert, R. G. J . Chem. Phys. 1989, 90, 1630. (12) Truhlar, D. G.; Garrett, B. C. Annu. Reo. Phys. Chem. 1984, 35, 159. Garrett, B. C.; Truhlar, D. G. J . Phys. Chem. 1980, 84, 805. (13) Troe, J. Chem. Phys. Leu. 1985, 122, 425. (14) Chesnavich, W. J.; Bowers, M. T. Prog. React. Kinet. 1982, 11, 137. ( I 5 ) Benson, S.W. Thermochemical Kinetics, 2nd ed.; Wiley: New York, 1976.
(16) Eliason, M. A,; Hirschfelder, J. 0. J . Chem. Phys. 1959, 30, 1426. Hofacker, L. Z. Naturforsch. 1%3,18A, 607. Marcus, R. A. J. Chem. Phys. 1965, 43, 1598. Marcus, R. A. J. Chem. Phys. 1966,45, 4493. (17) Quack, M.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 240; 1975, 79, 170; 1977, 81, 329. (18) Wardlaw, D. M.; Marcus, R. A. Ado. Chem. Phys. 1987, 70, 231. (19) Klippenstein, S . J.; Marcus, R. A. J . Phys. Chem. 1988, 92, 3105. Klippenstein, S. J.; Khunkar, L. R.; Zewail, A. H.; Marcus, R. A. J . Chem. Phys. 1988,89,4761. (20) Troe, J. J . Chem. Phys. 1983, 79, 6017. (21) Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 242.
8144 The Journal of Physical Chemistry, Vol. 93, No. 25, 1989
cus18). We change variableszz from (E,J), the total energy E and total angular momentum J , respectively, to (t,Em), the total energy in active modes t and the inactive rotational energy Erot; here, E = c E , and E , = BJ(J 1) ( B being the external rotational constant of the molecule and I the corresponding moment of inertia). This change of variables is not exact but is an excellent approximation and invokes the common assumption that the total angular momentum J can be approximated by the orbital angular momentum. It should be stressed that this does not mean that the angular momenta of the separating (or associating) fragments are taken to be always zero. In the usual RRKM framework, these rotations are treated as active internal degrees of freedom that may exchange energy with the reaction coordinate throughout the dissociation process. Rather, changes in the angular momenta of these fragments as the dissociation (or association) proceeds are assumed to be a minor perturbation on the orbital angular momentum L. Given the ergodic hypothesis (that the time average can be replaced by an ensemble average), the exact classical expression for k(E,J) for unimolecular reaction (e.g., for an ion/molecule system, AB+ A+ B) can be written as follows:
+
+
-
k(t,Erot)
Smith et al.
+
=
s...sdI'
G(t-s)S[P,] x(F) b(Hi,,+ZEr0,/mr2-E)
1...
JdI'
b(Hint+Erot-E)
(1)
The numerator of eq 1 counts trajectories passing through a dividing surface from reactant to product regions in phase space. The denominator counts the classical ensemble of reactant molecular states. The microscopic rate coefficient is given by the ratio of reactant flux to ensemble population. With the approximation of replacing (E,J) by (€,Erot),the orbital angular momentum is conserved, and so the external (orbital) rotational degrees of freedom do not enter into eq 1 explicitly: they simply affect the energy available to the internal modes for barrier crossing. The remaining one-dimensional external rotation is treated as active (see, e.g., ref 22). Hi,, is the total Hamiltonian for active degrees of freedom in the system, and r is the complete set of momenta and positions involved in Hint.One of the coordinates in I', denoted r (the reaction coordinate) is such that a surface defined by r = s completely separates reactant and product regions in phase space; for reactions involving a simple-fission (or "orbiting") transition state, r is usually taken as the separation between the dissociating or recombining fragments. p , is the momentum conjugate to r. The delta function b(r-s) evaluates the multiple integral on the surface defined by r = s. The function G(Hi,,+lEro,/mr2-E)ensures that trajectories counted on the dividing surface correspond only to molecules with active internal energy t = E - Era;one has Hint+ IErot/m3= E at all positions along the reaction coordinate, m being the reduced mass of the two moieties. S is a unit step function [S(x) = 0, x 6 0, and S(x) = 1, x > 01; it thus counts only trajectories with positive pr, i.e., those traveling from reactant to product and not vice versa. The characteristic function x(r) = 1 if the trajectory with position and momenta values I' on r = s started in the reactant valley and goes directly on to products without recrossing the dividing surface; otherwise x = 0. Equation 1 is a surface integral (Le., it is evaluated on the surface r = s); hence, Cauchy's theorem shows that it is invariant to the choice of this surface, provided this surface completely separates reactant from product.23 Equation 1 reduces to the RRKM expression if one assumes that a surface r = s exists such that every trajectory crossing the surface does so only once on the way from reactant to product. To account for angular momentum conservation, it is also necessary to ensure that the trajectory has enough energy to overcome the centrifugal barrier, which for a given E , will be the maximum ( 2 2 ) Forst, W. Theory of Unimolecular Reactions; Academic: New York, 1973. (23) Pechukas, P . In Dynamics of Molecular Collisions; Miller, W. H., Ed.; Plenum: New York, 1976; Part B.
r=s
rqm
r
Figure 1. Effective potential V,&) as a function of the reaction coordinate r , for low and high values of the total angular momentum J. s denotes the transition state.
of the effective potential Vefdr)along the reaction coordinate r (see Figure 1): V d r ) = V(r) + (Z/ms)Erot (2) V(r)is the interaction potential between the moieties (which will be the electrostatic interaction in the case of an ion/molecule system); r, (a function of rotational energy E,) denotes the value of r where Vcffis a maximum. Thus in RRKM theory, x(r) is approximated by the function x+(I'),where x+(I')is given byI2 x+(I') = S[E - Vedrm)l (3) (note for later reference that the location of the centrifugal barrier at r = r, may be different from the location of the transition state at r = s). Equation 1 yields the conventional RRKM expression when one notes that the denominator is h" times the classical density of states for active modes of the reactant p(t) ( n being the dimensionality of the system); the numerator is then converted to a sum of states of the transition state by carrying out the integration over r (Le., evaluating the integrand at the transition state r = s) and changing one variable of integration from pr to E+, where E+ = p,2/2m. One also assumes that there is negligible curvature of the reaction path at the transition so that the Hamiltonian Hi,, on the surface r = s can be written Hint = @:/2m) + V(r) + H,,..l(r,I',,-l),where I',,.., refers to all the other degrees of freedom. One then has the RRKM expression
k(crErot) = w(c>Erot)/hp(€) (4) where w ( t , E r o t )is the sum of states of the transition state:z2 t+Em-V&)
w(c,Erot) =
S,
dE+ p+(E+)?
+ Erot >
Veff(rm)
(5)
Here, pt is the density of states of the transition state, Le., the system with Hamiltonian Hrrl(r=s,I',,..l).The upper limit of the integral arises from the energy conservation requirement that Hht = E - ZEr0,/m6 throughout the reaction; hence the available energy for the barrier crossing is Hint- V(s)= E - Vc&). The condition t + E,,, > Veff(rm) arises from the definition of x+ in eq 3, which requires that trajectories have sufficient energy to overcome the centrifugal maximum. Because RRKM theory assumes that every trajectory through r = s is reactive, it is an upper bound to the exact classical rate coefficient, and hence eq 4 can be used ~ariationally.~~ Microcanonical variation will result in a transition state at a point r = s (chosen to minimize p)that may not lie at the position rmof the centrifugal maximum. This is because the decrease in the density of states pt(E+) with decreasing r can cause the minimum of eq 5 to lie at a position s
< r,.
Having summarized RRKM theory, we now consider the incorporation of the effect of uncoupled modes into the standard RRKM derivation presented above. Suppose that a certain group (24) Marcus, R. A. J . Chem. Phys. 1966, 45,4500. ( 2 5 ) Bunker, D. L.; Patengill, M. J . Chem. Phys. 1968, 48, 772.
Uncoupled Modes in Ion/Molecule Reaction Rates of modes, described by the Hamiltonian H,, with coordinates and momenta ru,becomes completely uncoupled from the reaction coordinate at long range, whereas the remaining degrees of freedom, described by the Hamiltonian H, with coordinates and momenta l’,, remain coupled with the reaction coordinate out to products. If curvature in the reaction coordinate is neglected,24 the Hamiltonian is then written as follows: Hint= p?/2m + Hc(rc;r)+ H,,(F,,)+ V(r) = Ha + Hu + V r ) (6) The total energy available for barrier crossing is Ha = p:/2m + H,. The energy in the uncoupled modes, H,, is under our assumption a conserved quantity from transition state to products. Thus, H,, contributes to the effective potential, being unavailable for motion along the reaction coordinate. We define an effective potential V e i ( r )that applies on the long-range electrostatic region of the potential surface: Veri(?) = Vcff(r)+ Hu (7) V e f i ( r )correctly describes the adiabatic long-range potential that the system must surmount in order to react [note that the maximum in Vefi(r)occurs at r,]. The construction of an effective potential in this fashion is in the spirit of the SACM of Quack and T r ~ e . l ’ * However, ~~ it should be noted that the uncoupling approximation of this work applies only under much more restricted conditions. Quack and Troe construct adiabatic potentials, or “channels”, by correlating eigenstates that are calculated parametrically as a function of r, the Hamiltonian for which includes all degrees of freedom (except r and p,), Le., including both H, and H,. Their model assumes that a trajectory must follow the adiabatic channel so constructed, and hence an additional condition for x = 1 in the SACM is that the trajectory must have sufficient total energy to overcome the maximum in the adiabatic channel potential. This additional restriction gives a lower value for k(E,J) and has been used to produce a variety of expressions and algorithms for determining this quantity.“I3J7J6 Thus, in the SACM, a distinction between coupled and uncoupled modes is not made: all modes are considered adiabatic in a more general sense. We now rewrite the variables of integration in eq 1 explicitly as r , p,, rc,and ruand define the quantity k’(c,Erol):
where
x+(r)= S [ E - Vet~’(rnJ1 (9) k’(c,Ero,)is an upper bound to the true rate coefficient k(c,Erot) in the case where the Hamiltonian at large r is separable into coupled and uncoupled terms. Equation 8 may be used variationally and will best approximate the true rate coefficient when the value r = s is chosen so as to minimize k’(t,Em). Substitution for the total energy E (=Hint+ IEro,/mr2) and Vefi(r,) in eq 9 allows x+ to be expressed alternatively as x+(r)= S[Ha - AVeffl where AVeff= V&,) - Vefr(r). Substituting eq 10 into eq 8 and noting that Ha = H, E,, where E , = p?/2m, we obtain
+
k’(t,Erot)= ( I d r 6[Hin,+Er,t-E])-1( I d r ~ ( P s )X dE, L E ‘ d E r S d r , 6[HC-(E,-E,)]X
(26) Troe, J. J . Chem. Phys. 1987,86,6171. Troe,J. J . Phys. Chem. 1986, 90, 3485. Cobos. C. J.; Troe, J. J . Chem. Phys. 1985,83, 1010. Cobos, C. J.; Hippler, H.; Luther, K.; Troe, J. J . Phys. Chem. 1985, 89, 4332. Cobos, C. J.; Troe, J. Chem. Phys. Lett. 1985, 113, 419. Cobos, C. J. Znt. J . Chem. Kinet. 1989, 21, 165.
The Journal of Physical Chemistry, Vol. 93, No. 25, 1989 8145 The limits on the integral over E, arise from the fact that (given the zero level for the potential term in H, is defined at the minimum in the potential) both H, and E, must be positive definite quantities. In order to achieve the final result, it remains to evaluate the integral over rC.We therefore seek to identify the coupled and uncoupled modes. For the purposes of the present paper, we confine ourselves to the simple case where the Hamiltonian as expressed in eq 6 has a fixed form at long range, Le., the number c of coupled modes versus the number u of uncoupled modes is fixed on the long-range part of the potential surface: such is likely to be the case for ion/molecule reactions. In such reactions, the optimal choice for the surface r = s is at a large separation of the A and B moieties. It is likely that the vibrational and some rotational degrees of freedom will satisfy the criteria for having a Hamiltonian that is completely unchanged between r = s and products. Certain rotational degrees of freedom, however, must be coupled to the reaction coordinate: as has been pointed o ~ t ,if~the , ~ neutral moiety B has a dipole moment, then its internal rotation makes it a two-dimensional hindered rotor that couples to the reaction coordinate at long range. In addition, the torsional degree of freedom of the separate moieties (a type of internal rotation) may couple to the reaction coordinate if the charge on the ionic moiety is not situated at its center of mass. Thus for ion/dipole reactions, the term H,, corresponds to all vibrations of the separate moieties, the active external rotor, the internal two-dimensional rotation of the ionic moiety, and (except under the above conditions) the torsional internal rotation of the two moieties. In other words, H , will include only the two-dimensional hindered dipole rotation, precisely those degrees of freedom that are considered “pertinent” in the capture models for ion/dipole r e a c t i o n ~ . ~ ? ~ J ~ Our approach to incorporating angular momentum conservation needs some further explanation. The angular momentum of the dipole rotor, Jd, is taken as a small perturbation on the orbital angular momentum L, so that the total angular momentum J = L and thus the orbital angular momentum is essentially conserved. This does not mean that the dipole rotor is required always to have exactly zero angular momentum as is assumed, for instance, in the “frozen rotor” capture model of Dugan (see, e.g., ref 14 and 27). In the present model, the inclusion of the hindered dipole rotation in H, implies that the dipole rotor is able to exchange energy with the reaction coordinate at large separations and hence may provide energy for barrier crossing; this leads to an upper bound to the microscopic rate coefficient. On the other hand, the frozen rotor modelz7 requires that Jd always be zero so that the dipole rotor is unable to provide energy for barrier crossing. In such a model, energy terms involving the dipole are included in the effec?ive potential, leading to a one-dimensional problem with a &dependent effective potential (6 being the ”frozen” angle of orientation of the dipole with respect to the ion/neutral axis). As has been pointed out,I4 neglect of interaction between the dipole and the reaction coordinate in the frozen rotor model leads to a significant underestimate of the microscopic rate coefficient. Given that the coupled degrees of freedom in the ion/dipole transition state will comprise just the two-dimensional hindered dipole rotation, we now proceed to evaluate the integral over I?, in eq 1 1 . This is carried out by a transformation from the Euler coordinates (8,cp,pe,p,),in which the Hamiltonian for the dipole rotation is given by H, = (1 /21d)@:
+ p:/sin2
6)
+ ~71~/4.rrt~rZ( 1 - cos 6)
(12)
to the set of coordinates {e,(P,{,JdIin which the Hamiltonian is written as H, = J.j2/21d
+ qpd/4*€,39( 1 - COS 6)
(13)
Here, q is the electronic charge, p the dipole moment, and co the permittivity of free space. Note that the dipole rotor is approximated as a symmetric top with degenerate moments of inertia Id and that the one-dimensional rotation about the symmetric axis (27) Dugan, J. V., Jr. Chem. Phys. Lett. 1973, 21, 476.
8146
The Journal of Physical Chemistry, Vol. 93, No. 25, 1989
is not included since this is a free rotation and is not coupled with the reaction coordinate. The angle {specifies the orientation of the angular momentum vector Jd in the plane perpendicular to the symmetry axis of the dipole rotor; {has limits 0 - 2 ~ . The angles 6 and v, with limits 0 - A and 0 - 2a, respectively, specify the orientation of the dipole axis in space. The Jacobian of the transformation from the variables of eq 12 to those of eq 13 is Jd sin 6. Substituting for rCin eq 1 1 with this set of coordinates gives k'(t,E,,,) = ( S d r 6[Hint+Eroi-E])-I(S d r 6 ( m )
X
dE, S E0 ' d E , s 20 * d p S 20 " d ( S e0 m d 6 sin 6 &=dJd JdX V(S)]} S d r , 6{H,-[E- Vef,-(r)-t- V(B)-E,]
6(t- [ E,+,-
where t = Jd2/21d,V(6)= q/ld/4??t$( 1 - cos e), and 6, as follows: v(6max) = El - En E, - Er
emax = 7,
}) ( 14)
is defined
E1 - E, 3 q/ld/2~tor'
(15)
X
The integral over rumay be identified as hW3p:[E - Ve&) -El], where n is the total number of active modes and put is the density of states of uncoupled modes at the transition state. Since (assuming that the zero level for the potential in Huis taken at the minimum value of the potential) H, must be a positive definite quantity, we must have El < E - Veff(r). The integral in the denominator is hnp(t), where p is the density of states of the molecular reactant (recall that t = E - E,,[). The integrals over p, {, and 6 may be evaluated to yield h2p2(El - Er), where p 2 is the density of states of the two-dimensional hindered dipole rotor. Hence, eq 16 becomes
1 2mor2E q/ld =-, E
