Accuracy of trajectory calculations and transition state theory for

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The Journal of Physical Chemistry, Vo/. 83, No. 1, 1979

Donald G Truhlar

Accuracy of Trajectory Calculations and Transition State Theory for Thermal Rate Constants of Atom Transfer Reactions Donald G. Truhlar Chemical Dynamics Laboratory, Kolthoff and Smith Halls, Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received July 24, 1978)

The reliability of several practical techniques for computing the equilibrium rate constant of elementary atom-transfer reactions is discussed. Conventional transition state theory and two generalizations, the canonical variational theory of reaction rates (also known as the method of free energy surfaces) and the adiabatic theory of reactions (also known as the microcanonical variational theory of reactions), are all considered. For these theories the transmission coefficient is set equal to unity as usual. In addition the quasi-classical trajectory method and three extensions, the quasi-classicaltrajectory method with quantum mechanical energetic threshold, the quasi-classicaltrajectory reverse histogram method, and the classical S matrix theory, are considered. Results of the application of these theories to compute thermal rate constants for collinear and three-dimensional systems where accurate quantal calculations are available are reviewed. The systems considered are H + H2,C1+ HP, H t Clz,F + HP,Hz + I, and isotopic analogues at 300-1500 K. The comparisons discussed should allow more realistic estimates to be made of the errors in using these approximate theories to calculate thermal rate constants, isotope effects, and activation energies for chemical reactions.

I. Introduction Theoretical approaches to gas-phase bimolecular kinetics have traditionally been divided into two categories: collision theory and transition state theory. This article is concerned with the dynamical accuracy of both these approaches for predicting thermal rate constants and activation energies of atom-transfer reactions. As one attempts to make collision theory more and more realistic one eventually obtains an approach based on exact classical trajectories1 or on exact quantum mechanical calculations of cross sections,* from which exact rate constants may be obtained by Boltzmann a ~ e r a g i n g . ~ Since potential energy surfaces are not known precisely for reactive systems, rate constants calculated by exact dynamics are not completely exact, but we will be concerned here with calculations in which the dynamics is exact for a given potential energy surface. Even this kind of exact quantal rate constant is available only for the H t H2 reaction4 For many reactions exact classical mechanics, especially if employed with quasi-classical quantization of asymptotic vibrational energies, is expected to be more reliable than calculations made in a purely quantal framework with concomitant dynamical approximations to make them practical. So there has been much interest in the quasi-classical trajectory method,ls6 which combines exact solutions to the classical equations of motion for a given potential energy surface with quasi-classical quantization of initial vibrational and rotational energies. Much of this interest has been directed to state-to-state chemistry and to reaction cross sections a t fixed relative velocity, However these methods have also been applied to calculate thermal rate constants and activation energies, and it is important to have realistic estimates of the errors involved in using classical mechanics to simulate the quantum dynamics of such processes. Such estimates are difficult to obtain by comparing calculated rate constants to experiment because they may be cancelled or augmented by errors due to the use of inexact potential energy surfaces. The best way to study these errors is by comparing the results of quasi-classical trajectory calculations for a given potential energy surface to exact quantum mechanical rate constants for the same 0022-3654/79/2083-0188$01 .OW0

surface, Because the latter kind of calculation is available for only one three-dimensional reaction, such comparisons have been limited to collinear reactions. However for such reactions comparisons are now available for thermal rate constants for H t H22,637(thermoneutral), C1+ H; (slightly endothermic), F H t and H C1210-12(both exothermic), and I H213 (endothermic) plus some of their isotopic analogue^.^^^^^^ In some cases approximate thermal activation energies have also been tested.7i8J0-13J5One of the objectives of the present article is to review what can be learned from these studies about the reliability of the quasi-classical trajectory method and related trajectory methods for thermal rate constants and activation energies of atom-transfer reactions, The related trajectory methods we discuss are the quasi-classical trajectory reverse histogram methode and classical S matrix theory16 at the Bessel uniform level of approximation with real-valued (not analytically continued) trajectories only. The most important alternative to collision theory is transition state theory. (We use the terms transition state theory and activated complex theory as synonyms.) As for collision theory, there are various levels of sophistication. For bimolecular reactions between neutral species, the only version of transition state theory commonly employed for the prediction and interpretation of thermal rate constants is the conventional one in which the transition state is identified with a surface in configuration space which passes through the saddle point of the potential energy surface, However, Garrett and the author17 have recently begun a systematic study of generalized transition state theories such as the adiabatic theory of reaction^^^^^*^ and canonical variational transition state theory,21 which is equivalent t o the use of minimum-free-energy transition state^.^^^^ Such methods are practical for three-dimensional reactions (and in fact are easier than trajectory calculations) so in this article we compare their accuracy to that of trajectory calculations for the collinear atom transfer reactions for which exact quantal rate constants are available. Transition state theory rate constants always include a transmission coefficient K which could be defined as the ratio of the exact rate constant to the transition state theory one obtained with K = 1. In general, however,

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Thermal Rate Constants of Atom Transfer Reactions

accurate evaluation of K requires a collision theory treatment which, if available, would obviate the need for transition state theory. The only important effect which might be simply included in K using existing methods is tunneling, and the widespread practice is to set K -- 1when tunneling is not expected to be important. Most of the standard treatments of tunneling are not reliable,23-25but recently a promising new technique for easy calculation of tunneling probabilities has been proposed.26 We will not pursue this subject in this article; instead this article summarizes what we have learned about the dynamical accuracy of transition state theory or generalized transition state theory with K = 1 for reactions for which both quasi-classical trajectory calculations and exact quantum calculations of thermal rate constants are available. Section I1 if3 a brief review of the theory and section 111 reviews the calculated results for H Hz, C1 + H,, H + Clz, F + Hz, and I + Hz and isotopic analogues. Section IV summarizes the conclusions about how accurately these theories may be used to calculate thermal rate constants and activation energies for bimolecular atom transfer reactions.

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11. Theory

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Consider the reaction A BC AB C where (Y denotes the set of internal quantum numbers of A and BC, w and Ereldenote the reduced mass and translational energy for their relative motion, a’ denotes the set of internal quantum numbers for AB + C, and (rual(Erel) is the state-to-state reaction cross section a t a fixed energy. For conditions under which the translational and internal degrees of freedom are a t equilibrium, so that the velocity distribution is Maxwellian and the fraction f a of reactants with quantum numbers a is calculable in terms of partition functions, the thermal rate constant h(T)a t temperature T may be written3

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or some other method to predict the full transition matrix of state-to-state inelastic nonreactive and reactive transitions and then substituting these into the master equation. This is il much larger computational task, which is not generally ~ i c c o m p l i s h e d . ~ ~ ~ ~ The reactions considered here are assumed to be collinear and electronically adiabatic. Then the sums in eq 2 are just sums over the initial and final Vibrational quantum numbers n and n’, respectively. We will not include electronic degeneracy factors and branching factors for multiple potential energy surfaces33in the equations which follow. In a quasi-classical trajectory calculation of PUa4Erel), the initial conditions of the collisions are Er,l, the initial vibrational action variable NRh, the initial value xo of the atom-diatom separation x, and the initial vibrational phase q , If xo is large enough the outcome of the trajectory is independent of its exact value and depends only on IYrel, n, and the vibratifonalphase shift 4;, which may be thought of as the value of q when x has first decreased to some arbitrary fixed large value x,. For given Ereland N R one calculates a batch of trajectories corresponding to uniform sampling of 4; in its full range (0, 27r). The initial value of N R is semiclastiically restricted to the quantized values NR= n

+ ‘I2

(3)

which is used to assign an integer value of n to the batch of trajectories. The reaction probability is Pn(Ere1)

= CPrm,(Erei) n‘

(4)

and is given by the fraction of this batch of trajectories which react. The averages over n and Erelin eq 2 are t’hen performed numerically. In the quasi-classical trajectory calculations just described, the initial values N R are quantized. Thus the initial vibrational energies E v l b are also quantized. For example, for a Morse oscillator the vibrational energy Ewb is related to the vibrational action variable Nh by

and the collinear analogue of this equation isz4 h ( T ) = Cfa(T)C(27rykT)-1/2 x U

a‘

JimPuar(EreJ

exp(-E,,i/kT) dEre1 (2)

where PUu,(Erel) is the state-to-state reaction probability a t a fixed energy. The assumption that reactants are at equilibrium is valid only if transitions from states of A + BC to states of AB + C are much slower than transitions between reactant states.27)28The degree to which this is true depends on the conditions of the experiment, but transition state theory makes this assumption, and it is standard to make this assumption when calculating rate constants from quasi-classical or quantal calculated values of (ruul(Erel) or Paa,(Erel). We use this procedure here. Rate constants calculated under this assumption should be called equilibrium rate constants. They are equivalent to phenomenological experimental steady-state rate constants under conditiions of local equilibrium, Le., where the distribution of all reactant state populations is the equilibrium one and the same for products, but the overall concentration of reactants is not required to be in equilibrium with the overall concentration of products.29 The tests providedl here are thus a test of how well trajectory calculations and transition state theory can be used to predict equilibrium rate constants. T o proceed further in a straightforward way requires using trajectory calculations

where we and x, are the harmonic and anharmonic constants of the osciillator. However the final vibrational energies Ere{of the calculated trajectories form a continuous distribution. For many purposes this “sloppiness” in the treatment of product states should not seriously undermine the accuracy of the calculated Pn(Erel) because it corresponds to a sum over n’. Further the effect of product quantization may average out in the averages over n and Erel. However because of the Boltzmann factors in f,(T) and in eq 1 and 2, thermal rate constants are dominated by their threshold regions. In threshold regions, the quantization of the final vibrational state may be quite important and cause a systematic error which is not cancelled by nonthreshold contributions to h(n.For i,his reason there is some interest in treatments in which quantization of final vibrational energy is enforced. The first way to include product energy levels is to set the reaction probability equal to zero for all total energies below the zero paint of the products. This is the rigorous quantum mechanical threshold. At a more detailed level, if product-state quantization is important but reactantstate quantization is not, a simple solution is to just do quasi-classical trajectory calculations on the reverse reaction instead of the forward reaction. In this case, however, one must calculate not just total reaction probabilities but also state-to-state reaction probabilities. If reactant-state quantization is not too important it should

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The Journal of Physical Chemistry, Vol. 83,

No. 1, 1979

be sufficient to use the histogram m e t h o d l ~ ~for ~ 8this ~ purpose. According to this method the vibrational action a t the end of the trajectory is arbitrarily reassigned to the closest quantally allowed value using the correspondence of eq 3. From the calculated state-to-state reaction probabilities for the reverse reaction one can calculate PaJEreJfor the forward reaction by microscopic rever~ i b i l i t y . Then ~ ~ , ~one ~ substitutes this into eq 2. This is called the quasi-classical trajectory reverse histogram (QCTRH) method. A more consistent approach to the quantization problem in semiclassical mechanics is provided by classical S matrix theory, which is the extension of the WKB approximation to multidimensional scattering problems.16 This theory also offers a framework for the analytical continuation of classical mechanics. This analytic continuation involves trajectories wit,h complex-valued coordinates, momenta, and time scales and provides a consistent way to treat tunneling in multidimensional systems such as chemical reaction^.^^^^^ Unfortunately it is usually too difficult to calculate the required analytically continued trajectories systematically enough to calculate thermal rat,e constants.39 Using only real-valued trajectories, classical S matrix theory still provides difficult numerical problem^,^' but it also provides a framework for the consistent inclusion of both initial-state and final-state quantization. To enforce these quantizations simultaneously means that instead of basing the results on averages over a batch of trajectories with specified initial conditions, one must find the so-called root trajectories which satisfy the appropriate initial and final boundary conditions. Then one must "add" the semiclassical amplitudes for these root trajectories to obtain a scattering matrix element whose squared magnitude is a state-to-state reaction probability. When real-valued trajectories which satisfy the required double-ended boundary conditions do exist, a transition is said to be classically allowed.16 For classically allowed transitions, one generally does not obtain good quantitative accuracy by simple addition of the amplitudes of the root t r a j e ~ t o r i e s .Instead ~~ one must use a uniform procedure specially adapted to the particular configuration of root trajectories as a function of Q (or for noncollinear reactions, as a function of all the initial phase variable^).^^ For many cases a really adequate uniformization procedure is not known but the most general procedure which has been developed is the Bessel uniform a p p r ~ x i m a t i o n . ~The ~-~~ Bessel uniform version of classical S matrix theory as applied with only real-valued trajectories has been called the BUSCCA m e t h ~ d , where ' ~ ~ ~ BU ~ represents Bessel uniform, SC semiclassical, and CA classically allowed. Transition-state theory is a simpler and widely used approach to the calculation of thermal rate constants, Classically it consists in equating the rate constant for reaction to the one-way equilibrium rate constant for passage through a surface in configuration space which divides reactants from products. Since all reactive trajectories must pass through this surface at least once, the classical transition-state theory rate constant provides an upper bound to the classical equilibrium rate constant.21 The classical rate constant for the one-way equilibrium flux through a dividing surface equals ( k T / h )times the ratio of the transition-state partition function to the reactant partition function per unit volume (or per unit length for collinear reactions). The same expression can be derived by the usual postulate of a quasi-equilibrium between reactants and transition states although this makes the upper bound character of the classical result less obvious. Of course quantization of internal degrees of freedom is very important so it is customary to replace the classical

Donald G. Truhlar

partition functions by quantum mechanical ones, Since the transition state is a surface it contains one less degree of freedom than the reactants and this degree of freedom, called the reaction coordinate s, is still being treated classically when this is done. For the collinear reaction this leads to the conventional transition state theory rate c~nstant~~~~~

where Q' and Qvlbare transition-state and reactant vibrational partition functions, respectively, with their zeroes of energy a t the respective zero-point levels, Qrel is the relative translational partition function per unit length for A + BC, and V,(n=O,s=O)is the zero point energy of the transition state (including the saddle point potential energy) minus the zero point energy of reactants. Once the quantal partition functions have been substituted one loses the bounding property of transition state theory. Nevertheless by analogy with the bound obtained classically one hopes that variationally adjusting the dividing surface to minimize the calculated rate will result in improved agreement of the calculated rates with the true ones. So we consider a sequence of dividing surfaces parametrized by the value s of the reaction coordinate a t the point where it intersects the dividing surface. For conventional transition state theory one need define the reaction coordinate only in the vicinity of the saddle point and that is accomplished by a normal mode analysis. Here we require an extension. We first define the mass-scaled coordinate y by Y = (m/d1i2

where m is the BC reduced mass and r the BC internuclear distance. Then the Hamiltonian is % = (2co-1(px2 4- ps.2)+ V ( x , y )

(7) where p n and p y are conjugate momenta and V ( x , y )is the potential energy surface, so that the reduced mass is the same for motion in any direction in the ( x , y )plane. Then the reaction coordinate s for collinear reactions is defined as the signed distance from the saddle point along the minimum energy reaction path (path of steepest descent from the saddle point) in the ( x , y ) plane. (The path of steepest descent would be the same in any other coordinate system in which the kinetic energy has no cross term and has the same reduced mass for motion along both coordinates. Other choices of coordinate systems would yield different paths of steepest descent.) For collinear reaction the dividing surfaces are dividing curves, and the sequence of dividing curves considered here are straight lines in the (x,y) plane perpendicular to the reaction coordinate at this point where they intersect. Motion along any of these dividing lines is bound by an increasing potential energy for small enough vibrations. Thus we may compute quantal vibrational energies t(n,s) and generalized-transition-state partition functions QGT( T,s) for each dividing line parametrized by s. The generalized-transitionstate-theory rate constant a t a given temperature T is minimized by the canonical variational transition-state ( T ) we define choice s = S + ~ " ~where QCVT(r ) = QGT(T , S * ~ " ~ ) (84 = min QGT(T,s) (8b) S and

The Journal of Physical Chemistry, Vol. 83, No. I , 1979 191

Thermal Rate Constants of Atom Transfer Reactions

where T~,[~=O,S=S,~~~(T)] is the zero point energy of the canonical variational transition state (including its potential energy) minus the zero point energy of the reactants. We call1 this canonical variational transition-state theory, or, when we wish to emphasize the ad hoc substitution of quantal partition functions into a classical variational theory, quasi-classical canonical variational transition-state theory. Conventional and canonical variational transition-state theory are sometimes rewritten in a thermodynamic formulation. T o do this we write a generalized transition-state theory rate constant as

where V,(n=lO,s)is t.he zero point energy of the dividing line at s (including its potential energy) minus the reactant zero point energy. Then rewrite eq 10 as kT kGT(T,s)= -KGT( T,s) h where KGT(?',s) is a generalized transition-state-theory quasi-equilibrium constant. Relating this to the free energy in the usual way yields , ~ G T ( T , ~= ) -K*Oe-hGGTo(T,~)/hT kT (12) h where K'O is the reciprocal of the standard state concentration and AGGT,O(T,s) is standard-state free energy change in forming the generalized transition state located a t reaction coordinate s from reactants. Equations 11and 1 2 reduce to the conventional and canonical variational versions of transition-state theory by the substitutions s = 0 and s =: s * ~ ~ ~ (respectively. T), This shows that canonical variational transition state theory is equivalent to minimizing the free energy of the generalized transition state. T o further minimize the flux through the dividing surface, one can consider more complicated dividing surfaces. In particular it is possible to find a dividing surface in phase space such that one obtains the exact rate.21 However this is prohibitively difficult in practice. Instead we consider dividing surfaces which are still simple in configuratiion space but whose location depends on the total system energy. This leads to microcanonical variational transition state theory. For collinear reactions we again choose the dividing curve to be a straight line perpendicular to the path of steepest descent in the scaled coordinate system (x,y),i.e., perpendicular to this path where it intersects it. However now the location s*QVT(E,e) of this intersection is to be varied for each total energy E to minimize the contribution of trajectories of that energy to the thermal reaction rate. T o do this we write the thermal rate constant h(T) in terms of the microcanonical rate constant k(E). In general the relationship between these rate constants is

k ( T ) = [QR(T)]-lJ'mdEexp(-E/hT)pR(E)k(E)

(13)

where pR(E)is the density of states o i A + BC per unit energy and pier unit volume (for collinear reactions, per unit volume becomes per unit length) and where QR(T ) is the partition function for A + BC per unit volume and is given by

Q"(T)= l m d Eexp(-E/hT)pR(E) For collinear A

+ BC we have

(14)

and

where pvib(c) is the density of vibrational states per unit vibrational energy t of BC, which is given by Pvib(E) = mv,b/de ((16) where Nvib(t)is the number of vibrational energy 1evel.s of energy less than or equal to e. Since we consider the vibrational energy levels to be quantized, pvib(t) is a tium of delta functions centered a t the allowed vibrational energies. According to transition-state theory the microcanonlical rate constant is4'"-"$

where N*(E)is the number of energy levels of the transition state of energy less than or equal to E. For collinear A BC, M ( E )is the number of vibrational energy levels of the one bound degree of freedom of the transition state with energy less than or equal to E. The generalized transition state version of eq 17 is

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where P T ( E , s )iij the number of energy levels with energy less than or equal to E of the generalized dividing surface which intersects the reaction coordinate at s. Minimizing the rate with respect to s then gives the microcanonical-variational transition-state-theory microcanonical rate constant

The microcanonical-variational-transition-state-th~~ory canonical rate constant kfiw( T ) is obtained by substituting this result into oq 13 and integrating numerically. It can be shown17that the microcanonical variational transition state theory result (19) is equivalent to the prediction of the adiabatic theory of r e a c t i ~ n s . ~ JIn " ~this ~ article we sometimes call it adiabatic transition state thieory (ATST). The relationship to minimum-density-of-states method^^^,^^ is discussed e l ~ e w h e r e . ~ ~ For the generalized transition state theory results kCVT(T)and k"TST(T) presented here QGT(T,s)and NGT(E,s)were computed by fitting the potential energy along the dividing line to a Morse curve such that second derivative of the potential a t its minimum is correctly reproduced and the Morse curve dissociates to the correct dissociation energy of A + B C in every case.17

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111. Comparison of Results

In this section results for a number of representative systems are compared. A. H + H 2 . This is the one case where accurate quantum mechanical calculations are available for the three-dimension reaction. Schatz and Kuppermann3 computed accurate rate constants for temperatures up to 600 K using potential energy surface no. 2 of Porter and K a r p l ~ s This . ~ ~ will be called the PK2 surface. In general, in computing quantum mechanical cross sections and rate constants for reaictions involving identical atoms one must take quantal particle indistinguishability into a c ~ o u n t . ~ ~ , ~ - "

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Donald G. Truhlar

TABLE I: Ratio of Approximate to Accurate Rate Constantsa-c for H + H, (Using the PK2 Potential Energy Surface Except where Stated Otherwise)

T,K

300

k*(T)d/k( T ) e collinear k*(T’f/kLT)p three dimensions k Q c (7’) /k(T)I collinear kQCT(T)f/k(T)B three dimensions

0.11

400

TABLE 11: Ratio of Approximate to Accurate Rate Constantsa-d for Collinear H t H, Using the PK2 Potential Energy Surface T, K 300 400 600 1000

600

k i T “*f/k(T)g

0.23 0.40 0.05, 0.17, 0.50 0.3$ 0.52J 0.69j 0.30 0.55 0.93

kcGd(T)f/k(T)g kATST(T)f/k(T)B

&

J

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One would expect these effects to be larger for H Hzthan for other reactions. Schatz and Kuppermann found that thermal rate constants for ortho-para conversion in H H2 computed with and without assuming the protons are distinguishable differed by less than 2% at 300 K and less than 1% a t 400 K and higher. Thus the effect should generally be negligible and in the rest of this article we will discuss only calculations, quantal or quasi-classical, which assume distinguishable nuclei. The H + H2 reaction provides an opportunity to learn about the effect of reduced dimensionality on tests of approximate methods for calculating thermal rate constants. Table I compares the ratio of the conventional transition state theory rate constant to the accurate quantum one for H + Hz using the PK2 surface in one and three dimension^.^^^^^^@ This shows that conclusions drawn on the basis of the collinear test would not be misleading for this reaction, with its light masses, identical atoms, and high symmetrical barrier. Table I also compares ratios of quasi-classical trajectory (QCT) results t o accurate quantum rates in one and three dimensions. For the QCT methods, collinear tests are available only for the surface of the author and KuppermannZ4(this will be called the T K surface), and three-dimensional tests are available only for the PK2 surface. The comparison of these tests shows slightly larger discrepancies than the first comparison, perhaps because of the differences of the surfaces. Although results from one-dimensional tests can be applied quantitatively to three dimensions only with great caut i ~ n ,the ~ comparisons ~ , ~ ~ in Table I lend support t o the belief that, at least for temperatures in the range 300-600 K, many general conclusions drawn here about approximate methods should be applicable to real three-dimensional reactions. Tables I1 and I11 present a series of comparisons for collinear H Hz using the PK2 and T K surfaces. On both surfaces transition state theory underestimates the rate at all temperatures up to 1000 K. This is due to the neglect of tunneling. The TK surface is more realistic, and the PK2 surface has a barrier that is too thin.59,60Thus more tunneling occurs on the PK2 surface than the T K surface, and it is not surprising that transition-state theory without tunneling more greatly underestimates the rate for the PK2 surface than the T K surface. For both surfaces the generalized transition state theory treatments differ little from the conventional one up to 1000 K. Tunneling is expected to be more important for H + Hz than for just about any other important reaction. Thus it is interesting to see how much transition state theory without tunneling corrections underestimates the rate for collinear H + H,; this might serve as an approximate bound on the expected deviation for other collinear re-

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0.23 0.23 0.23

0.40 0.40 0.40

0.64 0.64

0.62

k* ( T )from conventional transition-state theory. kCVT( T ) from canonical variational theory. kATST(T )

a

k* T ) from conventional transition state theory. kQC ( T )from quasi-classical trajectory method. k(T ) from accurate quantum mechanical calculations. Truhlar, Kuppermann, and Dwyer, ref 55. e Schatz, Bowman, Dwyer, and Kuppermann, ref 56. Karplus, Porter, and Sharma, ref 5. Schatz and Kuppermann, ref 4. Bowman and Kuppermann, ref 7. Truhlar and Kuppermann, ref 23-25. These results are for the potential surface of ref 24. a

0.11 0.11 0.11

from adiabatic transition state theory, equivalent to microcanonical variational theory kPVT(T). d k ( T ) from accurate quantum mechanical calculations. e Truhlar, Kuppermann, and Dwyer, ref 55. f Garrett and Truhlar, ref 17. Schatz, Bowman, Dwyer, and Kuppermann, ref 56.

TABLE 111: Ratio of Approximate to Accurate Rate Constantsa-b for Collinear H t H, Using the TK Potential Energy Surface T,K 300 400 600 1000 1500 k* ( T ) C/ k ( T ) C kcvT(T)d/k(T)C kATST(T)d/k(T)C kQCT(T)e/k(T)C

0.30 0.30 0.30 0.32

0.50 0.50 0.50 0.52

0.70 0.70 0.69 0.69

0.88 0.86 0.84 0.81

1.06 0.97 0.93

a For definitions of k * ( T ) ,k c V T ( T ) kATST(T), and kQdT( T ) from quasik( T) see footnotes to Table 11. classical trajectory calculations. Truhlar and Kuppermann, ref 23-25 and unpublished, Garrett and Truhlar, ref 17, e Bowman and Kuppermann, ref 7 .

actions. For the more realistic surface this underestimation factor is about 3 a t 300 K and 2 a t 400 K. However for the PK2 surface these factors are about 9 and 4, respectively. For the more realistic surface transition state theory is accurate within 12% a t 1000 K and 6% at 1500 K. It is interesting that conventional transition state theory finally overestimates the rate a t 1500 K. At this temperature, Table 111shows that generalized transition state theory still underestimates the rate. For the T K surface there are also quasiclassical trajectory calculations available for comparison. The results of these calculations are remarkably close to the predictions of both conventional and generalized transition state theory. We will see that this is not uncommon. Of course, the transition state theory and generalized transition state theory calculations are much less expensive. This will remain true even if tunneling corrections are included in these methods. However, as discussed in section 11, incorporating tunneling into trajectory calculations involves a considerable amount of difficult numerical work. Thus it is important to further examine the accuracy of methods for including tunneling in transition state theory. This is discussed in a separate article.61 B. D D2. Results for D D2 are shown in Table IV. Because of the heavier masses, tunneling is less important than for H H2. Here the error in transition state theory is down to a factor of 2 a t about 350 K as compared to 400 K for H + H,. The use of free-energy curves makes negligible change but adiabatic transition state theory lowers the calculated rate 10% at 1000 K and 20% at 1500 K. C. C1 + H z and C1 T,. The C1 + Hzreaction and isotopic analogues have been studied using a semiempirical potential energy surface due to Baer.6z This surface has a classical endoergicity of 3.01 kcal/mol and a classical barrier height in the forward direction of 7.67 kcal/mol, so the intrinsic classical barrier height is 4.66 kcal/mol. At the saddle point, the H-C1 bond and C1-C1 bond are extended by 0.23 a. and 0.48 a. over their equilibrium distances, respectively. The ground-state-to-ground-state

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Thermal Rate Constants of Atom Transfer Reactions

The Journal of Physical Chemistry, Vol. 83, No.

TABLE IV: ]Ratio of Approximate to Accurate Rate Constantsa for Collinear D t D, Using t h e TK Potential Energy Surface

T, K k* ( T')b / k ( T ) b kCVT(T)c/k(T)b kATZ;T(T)c/k(T)b

300 0.43 0.43 0.43

600

400

1000 1500 0.78 0.98 1.14 0.78 0.98 1.13 0.76 0.88 0.95

0.60 0.60 0.59

a For definitions of k * ( T ) , k C V T ( T ) , kATST(T), and Truhlar and k ( T ) see footnotes to Table 11. Garrett and Kuppermann, ref 25 and unpublished. Truhlar, ref 1'7.

TABLE V: Ratio of Approximate to Accurate Rate Constantsambfor Collinear C1 t H, Using t h e Potential Energy Surface of Baer T, K 300 400 600 1000 1500 I _ _ _

k * ( T ) C , d / k ( T ) C , e 0.38 0.50 0.69 k C V T ( T ) d / k ( T ) C , e0.28 0.40 0.58 k A T S T ( T ) d / k ( T ) c i e 0.28 0.40 0.58 0.96 kQsx(T)C/h!(T)C 0.43 0.52 a For definition of h * ( T ) ,k C V T ( T , k A T S T ( T ) , and k C T ( T ) from quasik ( T ) see footnotes t o Table 11. classical trajectory calculations. Baer, Halavee, and Persky, ref 8. Garrett and Truhlar, ref 1 7 . e Truhlar and Gray, unpublished, based o n probabilities in A. Persky and M Baer, J. Chem. Phys., 60,133 (1974). 0.98 0.81 0.78

1.34 1.00

Q

quantal endolergicity is only 1.05 kcal/mol for C1 + H2 and 1.93 kcal/mol for C1 T2. This system more nearly resembles the symmetric H H2 reaction than it resembles the other systems considered here. Because of the small reduced masElesboth C1+ H2 and C1+ T2proceed mainly by tunneling a t low energy. For these reactions accurate quantal rate constants are compared to approximate results in Tablles V and VI. Despite the slightly higher reduced masses, the errors in transition state theory without tunneling corrections are about as large at low temperature as for H H2 on the T K surface. The C1+ T2system is interesting because the lightest atom is T, and accurate quantal results are not available for any other system with ,a lightest atom this heavy or heavier. Here, however, tunneling is still important and transition state theory and generalized transition state theory underestimate the rake by a factor of 2 a t 300 K and by 28-3370 a t 450 K. For C1 H2 and C1 T2a t 300-450 K, quasi-classical trajectory calculations are more accurate than transition state theory or generalized transition state theory without tunneling corrections. For example, for C1 Tz the trajectory calculation underestimates the rate by only 3570 a t 300 K and 25% a t 450 K. As for H + H2 though, the trajectory results, transition-state theory results, and generalized transition-state theory results agree better with one another than with the accurate results, at least up to 600 K Quasi-classical trajectory calculations are not available for temperatures above 450 K for the C1 Hz reaction and isotopic analogues. D. H + C1,. The H C12 reaction has been studied using another semiempirical potential energy surface due to Baer.63 This surface has a classical barrier height of 2.42 kcal/mol and a classical exoergicity of 48.64 kcal/mol. The saddle point corresponds to nearest neighbor distances RHCl = 4.25 a. and Rclcl = 3.81 a. compared to equilibrium = 2.41 a. and Re,c1CI= 3.78 internuclear distances ao. Thus the conventional transition state is in the entrance channel where the small reduced mass for relative motion of reactants allows for some tunneling.l0 This reaction diffors from the ones considered above not only in having an earlier, lower barrier but also in the sluggishness of the reactant diatom. The various rate constants

+

+

+

+

+

+

+

+

I, 1979 193

TABLE VI: Ratio of Approximate to Accurate Rate Constantsa-d for Collinear C1 t T, Using the Potential Surface of Baer

T,K

_-

300

400

k*(T)C*d/k(T)Cse 0.49 0.65 k C V T ( T ) d / k ( T ) " , e 0.49 0.64 k A T S T ( T ) d / k ( T ) c , e 0.48 0.61 kQCT(T)c/k(T)C 0.65 0.72 a-e See corresponding footnotes t o

600

1000 1500-

0.85

1.17 1.55 1.11 1.31 0.92 1.091

0.84 0.75

Table V.

TABLE VII: Rztio of Approximate to Accurate Rate Constantsa-c for Collinear H + C1, Using the Potential Energy Surface of Baer

T,K k* ( T ) d/ k ( T)etf k CVT (T)p/ k ( TIe k ATST ( T ) g / k ( 'T)e f k Q C T ( T ) d/ k ( T )e f kBUSCCA(T)d/k(T)e*f 9

3

300

400

600

1000

0.82 0.82 0.82 0.82 0.58

0.90 0.90 0.90 0.89 0.67

1.00 1.00 1.00 0.96 0.75

1.16 1.16

1.14 0.99 0.80

a For definitions of k * ( T ) , kCVT T k A T S T ( T ) , and k ( T ) see footnote t o Table 11. kkhLT) from quasiclassical trajectory calculations. kBU C C A ( T ) from Bessel uniform version of classical S matrix theory using Truhlar, Merrick, and Duff, real-valued trajectories. ref 10. e Essen, Billing, and Baer, ref 11. f Truhlar, Garrett and Truhlar, ref 17. Gray, and Baer, ref 12.

available are compared in Table VII. For this react ion, like the others considered so far, the approximate calculations underestimate the rate constant a t low temperature. Again this is attributed to quantum mechanical tunneling because again the fixed-energy results show reaction occurs a t a lower total energy quantally than quasi-classically, For this reaction the errors are smaller than for the more symmetric reactions because their higher more symmetric barriers are favorable for tunneling. This can be made mcwe quantitative by comparing the irnaginary frequencies of the unbound normal mode of the conventional transition states. For the reactions considered so far thetie me172189i cm-l (H + H2,PK2 surface), 1641i cm-l (H + ]I2,T K surface), 1034i cm-l (D + D2,T K surface), 14923 c13i-l (C1+ H2,Baer surface), 8653 cm-' (C1 + T2,Baer surface), and 48% cm-l (H + C12,Baer surface). Using this frequency as a guide, tunneling would be expected to be less important for H + C12than for the other reactions. Thus the less serious underestimates of the rate constants a t low temperature for this reaction are understandable. For H + C12co~nventionaltransition state theory witlhout tunneling and thie canonical variational theory are accurate within 44% over the whole 30(r1000 K temperature range. Adiabatic transition state theory improves the accuracy to 18% or better over this whole range. Quasi-classical trajectory calculations involve much more work but are no more accurate up to 600 K although they disagree with the accurate results by only 1% a t 1000 K. The more sophisticated but more difficult classical S matrix calculations with on1:y real-valued trajectories seriously overestimate the threshold energylO and hence seriously underestimate the rate constant. It is interesting to notice that for all the reactions considered so fair h * ( T ) / k ( T ) ,hCVT(T)/h(T),and kPtTST( T ) / k ( T are ) all increasing functions of temperature. In some of the cases this is due to tunneling at low i;emperatures. However, a t high temperatures this may be interpreted as the increasing ability of high-energy collisions to recro'3s a dividing line through the sacldlepoint.17,64-6sThus classical transition state theory overestimates the high-temperature classical rates and (qua-

194

The Journal of Physical Chemistry, Vol. 83, No. 1, 1979

TABLE VIII: Ratio of Approximate to Quantal Rate Constantsa-c for Collinear F t H, Using Muckerman's Potential Energy Surface No. 5

Donald G. Truhlar

TABLE IX: Ratio of Approximate to Quantal Rate Constantsa-c for Collinear F t D, Using Muckerman's Potential Energy Surface No. 5

__I_______

T,K

300

k' ( T ) d/ k ( T ) e 3.15 k CVT ( T)d/ k ( T)e 1.90 k A T S T ( T ) d / h ( T ) e 1.90 kQCT(T)"/k(T)' 1.40 k Q C T R H ( T ) e / k ( T ) e1.36 kUSC ( T ) e/ k ( 2')" 0.55

400

600 1000 1200

3.17 3.10 2.17 2.41 2.17 2.41 1.54 1.68 1.23 1.15 0.67 0.84

2.94 2.52 2.52 1.57 1.07 0.94

2.90 2.54 2.54 1.51 1.05 0.93

a For definitions of k * ( T ) , kCVT T kATST( T ) , and T ) from quasik( T ) see footnote t o Table 11. classical trajectory calculations.b it Q c k R H ( T ) from quasi-classical trajectory reverse histogram calculations. d Garrett and Truhlar, ref 17. e Schatz, Bowman, and Kuppermann, ref 9, and private communication. The accuracy of the quantal rate constants of these authors has been called into question by Connor and Jakubetz [J. N. L. Connor, W. Jakubetz, and J. Manz, Mol. Phys., 35, 1301 ( 1 9 7 8 ) J Connor and co-workers have not calculated thermal rate constants from their own quantal reaction probabilities and the correct thresholds so we compare here with the results of Schatz and co-workers; presumably the corrections would not change any of the qualitative conclusions drawn here, but the quantitative rate ratios may be less reliable than for the other systems.

si-classical transition-state theory should overestimate the high-temperature quantal rates. E. F + H2 and F + D2. The reactions F + H2 and F f D2 have been studied using Muckerman's potential energy surface no. 5.69 This surface has a classical exoergicity of 31.75 kcal/mol and a small (1.06 kcal/mol), early ( R F H = 2.91 ao, R" = 1.44 a. compared to RenFH= 1.73 ao, Re," = 1.40 ao) barrier. The imaginary frequency of the unbound normal mode of the conventional transition state is only 360i cm-I.l7 The available calculations are compared in Table VIII. For F + Ha, unlike the reactions considered so far, the fact that the quantal reaction threshold lies below the quasi-classical one is not the dominant effect even at 300 K, and the approximations all overestimate the accurate rate a t all temperatures considered here. Further h * ( T ) / h ( Tis) a decreasing function of temperature from 400 to 1500 K. The errors in all the approximate methods are much larger than for H + Clz. Conventional transition-state theory overestimates the rate constant by a factor of 3 a t all temperatures in the table. Either version of generalized transition-state theory provides significant improvement but still overestimates the rate by a factor of 2-2.5. I t would be interesting to study other heavy atom-light diatom reactions with early barriers to see if the serious overestimates of transition-state theory can be correlated with these features. The quasi-classical trajectory method is much more accurate than transition-state theory for this reaction, leading to errors of 40-51% over the 300-1500 K range. Use of the reverse histogram method improves the reaction probabilities in the threshold regiong and so improves the calculated rate constants. Generalization of this result seems unwarranted, however, since the same improvement in reaction probabilities near threshold was not obtained for H ClZ.loSince the errors in both conventional and generalized transition state theory are much larger than those in quasi-classical trajectory calculations, and since the transition state theory calculations overestimate the rate, it seems that a t least a large part of the error in transition-state theory for this reaction can be attributed to recrossing effects which are taken into account by trajectory calculations. Quantal effects are clearly important for this reaction, however, and some of the errors

+

T,K

300

400

600 1000 1200

h* ( T ) d/ k (T ) @ 1.48 k C V T ( T ) d / k ( T ) e 1.11 h A T S T ( T ) d / k ( T ) e 1.11 h Q CT ( T )" / k ( T ) e 0.86 k Q C T R " ( T ) e / k ( T ) e 1.29 k usc ( T ) e/ k ( T ) 0.64

1.51 1.52 1.65 1.73 1.01 1.32 1.50 1.60 1 . 0 1 1.32 1.50 1.59 0.93 0.97 1.03 1.02 1.10 0.96 0.93 0.93 0.72 0.78 0.89 0.91 a-d See corresponding footnotes to Table VIII. e Schatz, Bowman, and Kuppermann, ref 1 4 , and private communication,

must be due to the quasi-classical quantization aspects of both transition-state theory and the trajectory calculations. Since the approximate theories are less accurate for F + Ha,which has two hydrogen atoms, than for H + Cl,, which has one, it is interesting to consider F + Dz. Table IX shows that the approximate theories are much more accurate for F + D2 than for F + H2. Generalized transition state theory agrees with experiment within 59% over the 300-1500 K temperature range and the quasi-classical trajectory method is accurate within 14% over this range. Just as for F + H2, the trajectory calculations, which include recrossing effects, correct a large part of the overestimate provided by transition state theory. Tables VI11 and IX also include a result marked USC. These calculations by Schatz et aL9J4involved a uniform classical S matrix treatment involving some analytic continuation. However, no attempt was made to find the contributions of all important complex-valued trajectories. As for H + C12,this underestimates the rates. For F H2 the errors are less than those for the quasi-classical trajectory calculation for T L 400 K but for F + Dz the errors are larger over the whole temperature range studied. F. H 2 I . Accurate quantal rate constants for the H2 + I reaction are available for two different potential energy surfaces. The first is the semiempirical surface of Raff et al.70 and the second is the rotated-Morse-curve (RMC) surface of Duff and the author.71 The latter was designed to mimic the former in the reactants and products regions and along the conventional-transition-state dividing line but to differ elsewhere, particularly in the region of steep energy consumption where the reaction path turns the corner in the (R", R H I ) nearest-neighbors-distance coordinate plane. For both surfaces, the classical endoergicity is about 35.82 kcal/mol and the additional intrinsic barrier to reaction is 0.06 kcal/mol. For both surfaces the saddle point is located at R H I = 3.17 ao,R" = 2.42 a. as compared to R e , H I = 3.03 ao, Re," = 1.40 ao. Thus the barrier is late but it is a little more symmetric than the barriers for the H + C12 and F H2 surfaces considered above. On the Raff et al. surface the barrier is followed by a potential well 1.85 kcal/mol deep. This is too deep to be realistic for this system but it does provide an interesting topological feature for a surface to be used as a test of dynamical models, Because of the well the imaginary frequency, 13963.cm-l, of the unbound normal mode of the conventional transition state is much higher than would be expected for a system with such a small intrinsic barrier. The RMC surface has no wells and the imaginary frequency is just 333i cm-l. This too is higher than might be expected for such a small barrier, but is less surprising when it is recalled that the saddlepoint is more symmetric than might be expected. The results for these reactions are given in Tables XI and XII. Because of the small intrinsic barriers, tunneling corrections should be negligible for both surfaces.

+

+

+

The Journal of ,Physical Chemistry, Vo/. 83, No. 1, 1979

Thermal Rate Constants of Atom Transfer Reactions

TABLE X: Ratio of Approximate to Accurate Rate for Collinear H, + I Using the Potential Energy Surface of Raff, Stivers, Porter, Thompson, and Sims

TABLE XIII: Isoltope Effects for Collinear C1 + H,, D,, T,, DH, and HI) Using t h e Potential Energy Surface at IBaer

300

k*(T)"/k(T)'" 11.23 1.08 kCVT ( T ) fk/ ( T ) e 1.08 k ATST ( T)f/ k ( T ) e 27.64 kQCT ( T ) e/ k (T ) e k Q C T ( Q M ) ( T ) e / k ( T ) e 0.58

400

600 1000 1500

6.23 1.06 1.06 9.21 0.62

3.58 1.06 1.06 3.35 0.71

2.53 1.10 1.10 1.73 0.81

2.37 1.18 1.18 1.34 0.87

a For definitions of h * ( T ) ,k C V T ( T b k A T S T ( T ) ,and k( T ) see footnotes to Table 11. k Q T ( T ) from quasiclassical trajectory calculations including contributions from ener ies below product zero point energy. kQcT(Q )( T ) from quasi-classical trajectory calculations excluding contributions from below product zero Duff and Truhlar, unpublished, and point energy. Garrett and Truhlar, ref 17. e Gray, Truhlar, Clemens, Duff, Chapman, Morrell, and Hayes, ref 13. Garrett and Truhlar, ref 17.

8

TABLE XI: Ratio of Approximate to Accurate Rate Constantsa-d for Collinear H, + I Using the Rotated-Morse-Curve Potential Energy Surface of Duff and Truhlar

T,K 300 400 600 1000 1500 k* ( T ) R / k ( T ) f 12.13 6.57 3.66 2.45 2.11 kcVT (T)p/ki:T)f 1.09 1.07 1.05 1.03 1.03 kATS' (T)g/ik(T)f 1.09 1.07 1.05 1.03 1.03 kQCT( T ) f/ k i( T ) f 20.88 7.97 3.24 1.75 1.38 k Q c T ( Q M ) ( T ) f / k ( T l 0.68 0.70 0.74 0.84 0.93 kQCTRH(T)IP/k(T) 0.98 0.98 0.98 0.98 0.98 See corresponding footnotes to Table X. kQCTRH( T ) from quasi-classical trajectory reverse histogram calculations. e Duff and Truhlar, unpublished, and Garrett and Truhlar, ref 17. Gray, Truhlar, Clemens, Duff, Chapman, Morrell, and Hayes, ref 13. .e Garrett and Truhlar, ref 17. _ _ I _ _ _

TABLE XII: Isotope Effects for Collinear H t H, and D, and D t H, and D, Using the TK Potential Energy Surface kH tH , T = 300 K

accurate" TST" CVTb pVT = ATSTb

0.05 0.08 0.07 0.07

kH+H,

IzH t H ,

1.24 1.59 0.80 0.80

0.09 0.14 0.14 0.14

1.15 1.35 0.80 0.80

0.17 0.21 0.21 0.20

1.03 1.15 0.80 0.81

0.28 0.32 0.32 0.31

0.92 1.03 0.79 0.81

0.41 0.46 0.48 0.44

0.84 0.98 0.80 0.81

0.50 0.56 0.61 0.53

T = 400 K accurate" TS'P CVTb pVT = ATSTb

0.12 0.15 0.13 0.13

T = 600 K

accuratea TST" CVTb pVT = ATSTb

0.24 0.28 0.25 0.25

T = 1000 K accurate" TSTa CVTb p VT = ATST

0.40 0.46 0.42 0.42

T = 1500 K

accurate a TST" CVT pVT = ATSTb

0.50 0.58 0.55 0.54

a Truhlar and Kuppermann, ref 24, and Truhlar, Kuppermann, and Adams, ref 25. Garrett and Truhlar: ref 17.

-

1 1

--____

T,K

195

kD,/

kT,/ kDH/ kHD/ kn, k ~ , k ~-___ , k ~ , T = 300 K

accuratea TS'P CVTb p V T = ATSTb' QC'P

0.08 0.09 0.11 0.11 0.10

accurat ea TS'P CVTb p V T = ATSTt' QC'P

0.14 0.15 0.19 0.18 0.17

accuratea TSF CVTb p V T = ATSTt' QCF

0.16 0.18 0.22 0.21 0.20

0.02 0.03 0.04 0.04 0.03

0.25 0.26 0.35 0.35 0.25

0.17 0.41 0.19 0.19 0.63

0.33 0.35 0.44 0.43 0.36

0.23 0.49 0.27 0.27 0.65

0.36 0.38 0.48 0.47 0.40

0.25 0.53 0.31 0.31 0.65

T = 400 K 0.05 0.06 0.08 0.07 0.07

T = 450 K 0.06 0.08 0.10

0.09 0.09

a Baer, Halavee, and Persky, ref 8. Truhlar, ref 17.

Garrett and

TABLE XIV: Isotope Effect kF + D2/kF + HZ (Collinear) Using IMuckerman's Potential Energy Surface N ~ I5. T, K quantal0 TST~ CVTb pVT = ATST" $CY QCTRH"

usca

300

400

600

1000

1.09 0.51 0.64 0.64 0.67 1.04 1.27

1.18 0.56 0.66 0.66 0.72 1.06 1.26

1.26 0.62 0.69 0.69 0.73 1.04 1.16

1.19 0.67 0.71 0.71 0.79 1.04 1.10

-

Schatz, Bowman, and Kuppermann, ref 9 and 14, and Brivate communication; see footnote e of Table VIII. Garrett and Truhlar, ref 1 7 .

The most striking point about the results is that for both surfaces conventional transition-state theory overestimates the rate constants by over an order of magnitude a t 300 K and by over a factor of 2 for the whole 300--1500 K temperature range. Thus the improvement afforded by either version of generalized transition state theory is even more dramatic: these theories are accurate within 18% over this whole temperature range for the Raff et al. surface and within 9% over this whole range for the RMC surface. The better agreement for the RMC surface may be due to the more direct character of the dynamics on this surface (see ref 13). If attention is restricted to the 300-1000 K temperature range, generalized transition state theory is accurate within 10% or better for both surfaces compared to errors of 145-1113% for conventional transition state theory. For both surfaces the generalized transition states are found to be located at the products configuration. For three-dimensional reactions the €;eneralized transition state would never be located at infinite separation if centrifugal effects are included.72 Nevertheless we still expect dramatic improvement if generalized transition state theory rather than the conventional theory is used for this kind of reaction in three dimensions. Probably the most significant factor for explaining the large overestimate of conventional transition state theory for the I + Hz reaction is the low intrinsic barrier height (the barrier height in the exoergic direction is called the intrinsic barrier height). Because the intrinsic barrier height is small but the conventional transition state is rather tight on the surfaces employed, entropic consid-

196

The Journal of Physical Chemistry, Vol. 83,

No. 7, 7979

Donald G. Truhlar

TABLE XV: Comparison of Accurate and Approximate Energies of Activation Computed b y the Tolman Method reaction (surface)

H

i

H, (TK)

D

+

D, (TK)

C1 + H, (Baer)

C1 + T, (Baer)

H t C1, (Baer)

H,

+

H,

+ I (RMC)

I (Raff e t a l . )

E,, kcal/mol 600 K

method

300 K

accurateu TST~ CVTb pVT = ATSTb accuratea TST~ CVTb pVT = ATSTb accurateC TST~ CVTb pVT = ATSTb accurateC TST~ CVTb pVT = ATSTb accurateu TST~ CVTb pVT = ATSTb QCP BUSCCAu accuratea TST~ CVTb pVT = ATSTb QCT(QWU accuratea TST~ CVTb pVT = ATSTb QCT(QMla QCTRH'

5.3 6.7 6.7 6.7 6.8 7.7 7.7 7.6 3.0 3.7 3.9 3.9

6.3 7.0 7.0 7.0 7.5 8.1 8.1 8.0

6.8 7.7 7.4 7.4 8.0 8.9 8.9 8.5

3.3 4.2 4.2 4.2

5.0 5.6 5.6 5.6 2.4 2.7 2.7 2.7 2.7 2.8 33.2 31.8 33.1 33.1 33.3 33.2 31.7 33.2 33.2 33.2 33.2

5.5 6.2 6.2 6.0 2.8 3.0 3.0 3.0 3.0 3.1 33.4 32.2 33.5 33.5 33.8 33.5 32.2 33.5 33.5 33.8 33.5

3.6 5.0 4.8 4.6 5.8 7.1 6.6 6.6 3.2 3.4 3.4 3.3 3.3 3.4 33.8 33.1 34.1 34.1 34.3 34.1 33.1 34.1 34.1 34.6 34.1

1000 K

Truhlar and Gray, ref 15. In this reference the sum over n , in the partition function Qv was converged, but the five additional sums over n , in the translational contribution Earel(T) and the vibrational contribution EaV(T)t o the energy o f activation included on1 n - 0 2 for H t C1, and n , = 0-3 for I t H,. This systematically overestimates (Erel)Treactionsand underestimates (Evib)( e , ~ t ~ n s - m d(E,%) at high T and makes the estimated E, less accurate than necessary at high T. For H t H, and isotopic analogues we ap roximated E , as E P l . For the results in this review we used the available results for n, reactions and state-selected rate constants for higher n , for all systems, and we corrected = 1 for H t H,, we estimated an error in the H + Cl,.quantum mechanical calculations. At temperatures up to 1000 K , the corrected energies of activation differ from those in ref 1 5 by less than 0.10 kcalimol for H + H,, 0.13 kcalimol for D T D,, 0.06 kcalimol for all methods for H + Cl,, and 0.02 kcal/mol for all methods for both surfaces for H, t I. Garrett and Truhlar, ref 1 7 . Truhlar and Gray, unpublished, based o n the probabilities in A. Persky and M. Baer, J. Chem. Phys., 6 0 , 133 (1974).

5.

erations are much more important than energetic considerations even a t room temperature. In such a case the purely energetic criterion employed by conventional transition state theory for the location of the dividing surface is not reliable. As mentioned in section 11, if one merely quantizes the initial vibrational energy in a trajectory study, one might find reaction below the quantal energetic threshold. This is found to be the case for Hz I on both surfaces.13 As a consequence the simple quasi-classical trajectory (QCT) method overestimates the rate constants by factors of 21 and 28 a t 300 K. A simple remedy is to set the reaction probability equal to zero below the quantal energetic threshold but to use the QCT method otherwise. We call this the QCT(QM) method where QM denotes the use of the quantum mechanical threshold. This simple expedient decreases the error in the trajectory results for both surfaces over the whole 300-1500 K temperature range. The errors are still largest at 300 K where they are 42% for the Raff e t al. surface and 32% for the RMC surface. Quasi-classical trajectory reverse histogram rate constants are available for Hz I only for RMC surface. Almost unbelievably, they agree with the accurate results within 2% a t all temperatures from 300 to 1500 K. This

+

+

agreement for thermal rate constants is particularly amazing because it results from cancellation of much larger errors (up to 817~) in the rate constants for selected initial vibrational states.13 This provides an example of the general conclusion that it is a t least sometimes possible to predict accurate completely thermally averaged rate constants without accurately predicting all the details of the collisions. There may be an element of chance in such accurate predictions, however, and we require even more studies of the type summarized here before all the factors necessary for such success will be clear. In this case, however, the fact that quantization of the products would be more successful than quantization of the reactants might have been anticipated from the fact that the generaized transition state is located at the products' configuration. This illustrates a useful correlation between the trajectory studies and generalized transition state theory. G. Isotope Effects. In calculating isotope effects, Le., ratios of thermal rate constants for two different isotopic species, there is the possibility of even more cancellation of errors. Exact quantal isotope effects have been studied for H + Hz on the TKZ5and PK273surfaces, for C1 + H: and H + ClZ" on the Baer surfaces, and for F + H2 on

Thermal Rate Constants of Atom Transfer Reactions Muckerman's surface no. 5.9314 Some of these results are compared to approximate calculations in Tables XII-XIV. The worst failures for conventional transition state theory occur for kH+D2 and kF+D, a t 300 K where the calculated isotope effects are 0.084 and 0.51 compared to the accurate values 0.054 and 1.09, respectively. We c ~ n c l u d e d ~on ~ ~the ~ ! 'basis of Table XI1 that accurate isotope effects for reactions involving H or D cannot be calculated with unit transmission coefficients at 300 K. Tables XI11 and XIV confirm this. At higher temperatures agreement is better but quantitative cancellation of errors does not occur. For the C1 H2 13ystema t 300-450 K, conventional transition state theory even without tunneling corrections predicts more accurate isotope effects than does the quasi-classicad trajectory method. Generalized transition state theory in a little less accurate than the other methods for three of the isotope effects in Table XI11 but it is much more accurate for hHl)/hHz. Its greatest qualitative success is predicting that kC1+DH> kcl+HD, a result that both conventional transition state theory and the quasi-classical trajectory method get wrong. However, for F + D,, conventional transition state theory, generalized transition state theory, and the quasi-classical trajectory method all predict the wrong sign of the isotope effect. The two trajectory calculations employing final-state quantization do predict this inverse isotope effect correctly. H. Energies of Activation. So far we have considered rate constants a t given temperatures. The Arrhenius energy of activation is defined as

+

and is in general a function of temperature. I t can be computed by least-squares fits of In k(T) as a function of l / T or by the Tolman i n t e r p r e t a t i ~ n . l ~The ~ ~ ~later -~~ states that Ea for the equilibrium rate constant is the average energy of all reactions minus the average energy of all pairs of possibly reacting reagents. When the Tolman method is applied it has the advantage that it yields E , a t given temperatures whereas the least-squares method gives a weighted average of E , over a range of temperatures. Energies of activation computed by the Tolman method are compared in Table XV. The accuracy of computed activation energies is important for two reasons. First, these are often used for extrapolation of rate constants. Second, they are often used to estimate classical barrier heights by adjusting semiempirical potential energy surfaces until the computed activation energy agrees with experiment. For reactions such as H + H2,C1-1 H,, and H + Cl,, the largest errors in the approximate rate constants occur at low temperature where the rate is underestimated due to lack of tunneling. Since the rates are more accurate at higher temperatures, the approximate rates increase more rapidly with temperature than the accurate ones. Thus the TST and GTST activation energies are too high by 0.6-1.4 kcal/mol for H Hz, by 0.6-0.9 kcal/mol for D + Dz, and by 0.1-0.4 kcal/mol for H + Clz. For H + C12 the quasi-classical trajectory calculations predict the temperature-dependent activation enlergy within 0.1-0.3 kcal/mol. It is interesting that this err or is much less than the temperature-dependent variiation of E,. Since experimental data can generally be fit within experimental error by assuming E, is a constant, the error in the approximate theories is small enough for these theories to be useful for interpreting experimental data. For H z + I, the serious overestimate of the rate at low temperature by conventional transition

+

The Journal of Physical Chemistry, Vol. 83, No.

I, 1979 197

state theory makes the predicted E , too low. The generalized-transitifon-state-theoryE , values and those computed by the QCT(QM) method agree with experiment within 0.5 kcal/mol in the 300-1000 K range, with the largest errors a t 1000 K. For this reaction accurate results are also available at 1500 K.15 The errors in the QCT(QM) values of E, at that temperature are 0.45 kcal/mol for the Raff et al. surface and 0.75 kcal/mol for the RMC surface. The QCTRH method underestimates E, for the RMC surface by 0.05 kcal/mol a t 1500 K. For F H, and F D2 the percentage errors in the approximate rate constants are relatively slow functions of temperature, and in that sense the predicted E , values are more accurate than the predicted rate constants. However for the cases in Table XV the accuracy of E, follows from the accuracy of the rate constants rather than the constancy of their errors.

+

+

IV. Summary By comparing approximate calculations of collinear reaction rate constants to accurate quantum mechanical ones computed for the same potential energy surface one can test the dynarnical approximations in the approxiniate theories without the usual uncertainties due to the potential energy suirfaces. Limited information available for the H H2 reaction in one and three dimensions indicates that conclusions about the validity of approximate theories based on collinear tests would not be misleading. So the results of such collinear tests are reviewed. The temperature range considered is 300-1500 K and tests are available for H -t. Hz, C1 H,, H Cl,, F + Hz, I + H,, and isotopic analogues over all or part of this range. The simplest theory is conventional transition-state theory (TST) with unit transmission coefficient, i.e., without corrections for tunneling. For H + H, and D + D2 using the TK.24surface, C1 + H 2 and C1 + Tz,and H C12,the errors in TST never exceed 7070 at 300 K, 50% a t 400 K, 30% a t 600 K, 12% at 1000 K, or 44% a t 1.500 K. These rates are underestimated at low temperatures, but the biggest e'rrors a t high temperatures are overestimates. For these reactions generalized transition state theory (GTST) does not provide much improvement at low temperature, nor does the quasi-classical trajectory (QCT) method. However, at 1000-1500 K, the QCT method is much more accurate than TST for H + Clz. The classical S matrix theory calculations involving only real-vallued trajectories for H + Clz are less accurate than TST, GTST, or the QCT method. For F + H,, IF' + D,, and H, + I, T S T is much less accurate, leading to errors a t various temperatures of 190-215, 48-73, and 111-1113% for the three reactions, respectively. Thle error is always an overestimate for these cases. In these cases, GTST is an improvement and it reduces the error3 to 90-154,l-60, and 3-18% for the three reactions, respectively. This may be attributed to two factors: (i) tunneling is less important than for H t H2, C1 + H,, and H .t Cl,, (ii) the intrinsic potential energy barriers are only 1.06 and 0.06 kcal/mol compared to a t least 2.42 kcal/mol for the cases considered first. The errors in the QC'T method, with the quantum energetic threshold used in the endoergic case, are 40-51, 2-14, and 7-3270 for the three reactions, respectively. This is better accuracy than GTST for F + H2 and F + D, but not for Hz and H2 + I, the quasi-classical traH, I. For F ijectory reverse histogram (QCTRH) method is more accurate than the QCT method, but this is not so for F + Dz and H + Cl, In several cases the smallest error in transition state theory occurs at about 1000 K. In these cases the error

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a t lower temperatures is dominated by quantum effects which increase the accurate rate, and the error a t higher temperatures is dominated by recrossing effects which decrease the accurate rate. Transition-state theory and the quasi-classical trajectory method predict the correct direction of all isotope effects examined except two and generalized transition state theory predicts the correct direction of all isotope effects but one. Errors in predictions of Arrhenius activation energies are also reviewed. In many cases these errors are small, a few tenths of a kcal/mol. For the thermoneutral and nearly thermoneutral cases where tunneling is most important, they are larger, 0.6-1.4 kcal/mol. Several related questions are not discussed here. One is the use of purely classical calculationsM@to gain insight into the reasons that T S T and GTST work and do not work. These calculations are instructive, but it is sometimes hard to apply the conclusions quantitatively to real reactions because zero point energy is such an important effect. For example, in purely classical calculations the adiabatic transition state theory often predicts significantly more accurate results than canonical variational but this is not the case for the studies reviewed here. Another important question concerns accuracy of tunneling corrections to transition-state theory. This question is explored in the following paper.61 Finally to calculate accurate rate constants for real systems one must use not only accurate dynamical methods but also accurate potential energy surfaces. These are generally not available yet.

Acknowledgment. This paper is based on a lecture given a t the Symposium on Current Status of Kinetics of Elementary Gas Reactions a t the National Bureau of Standards, Gaithersburg, Maryland, June 19-21, 1978. I am grateful to Frederick Kaufman for organizing the conference and to Michael Baer, Jonathan Connor, and William H. Miller for helpful discussions at the conference. I acknowledge all those whose work is reviewed here, especially my collaborators, past and present, on ref 10, 12,13,15,17,23-25, and 53. I am very grateful to Bruce C. Garrett and Joni C. Gray for assistance in preparing this review and to George Schatz and Joel Bowman for sending extra details of their work. This research was supported in part by a grant from the National Science Foundation. References and Notes For a review see D. G. Truhhr and J. T. Muckerman in "AtomMolecule Collision Theory: A Guide for the Experimentalist", R. B. Bernstein, Ed., Plenum Press, New York, in press. For a review see D. G. Truhlar and R. E. Wyatt, Annu. Rev. fhys. Chem., 27, l ( 1 9 7 6 ) . M. A. Eliason and J. 0. Hirschfelder, J. Chem. fhys., 30, 1426 (1959). G. C. Schatz and A. Kuppermann, J. Chem. fhys., 65, 4668 (1976). M. Karplus, R. N. Porter, and R. D. Sharma, J . Chem. Phys., 43, 3259 (1965). E. M. Mortensen, J . Chem. fhys., 49, 3526 (1968). J. M. Bowman and A. Kuppermann, Chem. fhys. Lett., 12, 1 (1971). M. Baer, U. Hahvee, and A. Persky, J. Chem. fhys., 61, 5122 (1974). G. C. Schatz, J. M, Bowman, and A. Kuppermann, J . Chem. fhys., 63, 674 (1975). D. G. Truhlar, J. A. Merrick, and J. W. Duff, J . Am. Chem. SOC., 98, 6771 (1976). H. Essgn, G. D. Billing, and M. Baer, Chem. fhys., 17, 443 (1976). J. C. Gray, D. G. Truhlar, and M. Baer, J. fhys. Chem., to be published in the Bunker Memorial Issue. J. C. Gray, D. G. Truhlar, L. Ciemens, J. W. Duff, F. M. Chapman, Jr., G. 0. Morrell, and E. F. Hayes, J . Chem. fhys., 69, 240 (1978). G. C. Schatz, J. M. Bowman, and A. Kuppermann, J . Chem. fhys., 63, 685 (1975). D. G. Truhlar and J. C. Gray, Chem. Phys. Letf., 57, 93 (1978).

Donald G. Truhlar For a review, see W. H. Miller, Adv. Chem. fhys., 30, 77 (1975). B. C. Garrett and D. G. Truhlar, J. fhys. Chem., to be published in the Bunker Memorial Issue. R. A. Marcus, J. Chem. fhys., 43, 1598 (1965), 45, 2138 (1966). D. G. Truhiar, J . Chem. fhys., 53, 2041 (1970). W. H. Wong and R. A. Marcus, J. Chem. fhys., 55, 5625 (1971). For a review, see J. C. Keck, Adv. Chem. fhys., 13, 85 (1967). A. Tweedale and K. J. Laidler, J . Chem. fhys., 53, 2045 (1970). D. G. Truhlar and A. Kuppermann, Chem. Phys. Lett., 9, 269 (1971). D. G. Truhlar and A. Kuppermann, J. Chem. fhys., 56, 2232 (1972). D. G. Truhlar, A. Kuppermann, and J. T. Adams, J . Chem. fhys., 59, 395 (1973). R. A. Marcus and M. E. Coltrin, J . Chem. fhys., 67, 2609 (1977). N. S. Snider, J . Chem. fhys., 42, 548 (1965). 6.Widom, J . Chem. fhys., 61, 672 (1974). R. K. Boyd, Chem. Rev., 77, 93 (1977). L. L. Poulsen, J . Chem. fhys., 53, 1987 (1970). J. C. Keck, Adv. At. Mol. Phys., 8, 39 (1972). N. C. Blais and D. G. Truhlar in "State-to-State Chemistry", P. R. Brooks and E. F. Hayes, Ed., American Chemical Society, Washington, D.C., 1977, p 243. D.G. Truhlar, J . Cheni. fhys., 56, 3189 (1972); erratum, 61, 440 (1974). M. Karplus and L. M. Raff, J . Chem. fhys., 41, 1267 (1964). For a discussion see D. G. Truhlar, Int. J. Quantum Chem., Symp., 10, 239 (1976). J. Ross, J. C. Light, and K. E. Shuler in "Kinetic Processes in Gases and Plasmas", A. R. Hochstim, Ed., Academic Press, New York, 1969, p 281. D. G. Truhlar, C. A. Mead, and M. A. Brandt, Adv. Chem. fhys., 33, 295 (1975). T. F. George and W. H. Miller, J . Chem. fhys., 57, 2458 (1972). J. D. Doli, T. F. George, and W. H. Miller, J . Chem. fhys., 58, 1343 (1973). J. W. Duff and D. G. Truhlar, Chem. fhys., 4, 1 (1974). J. W. Duff and D.G. Truhlar, Chem. fhys., 9, 243 (1975), and references therein. H. Kreek, R. L. Ellis, and R. A. Marcus, J . Chem. Phys., 62, 913 (1975), and references therein. J. R. Stine and R. A. Marcus, J . Chem. fhys., 59, 5145 (1973). J. W. Duff and D. G. Truhlar, Chem. fhys. Left.. 40, 251 (1976). E. M. Mortensen, J . Chem. fhys., 48, 4029 (1968). H. M. Rosenstock, M. B. Wallenstein, A. L. Wahrhaftig, and H. Eyring, f r o c . Natl. Acad. Sci. U.S.A., 38, 667 (1952). J. L. Magee, Roc. Nafl. Acad. Sci. U.S.A., 38, 764 (1952). R. A. Marcus, J . Chem. fhys., 45, 2630 (1966). J. Troe in "Physical Chemistry: An Advanced Treatise", Vol. VIB, "Kinetics of Gas Reactions", W. Jost, Ed., Academic Press, New York, 1975, p 835. D. L. Bunker and M. Pattengill, J . Chem. fhys., 48, 722 (1968). 8. C. Garrett and D.G. Truhlar, J . Chem. fhys., to be published. R. N. Porter and M. Karplus, J . Chem. fhys., 40, 1105 (1964). W. H. Miller, J . Chem. fhys., 50, 407 (1969). D. G. Truhlar, J . Chem. fhys., 65, 1008 (1976). D. G. Truhlar, A. Kuppermann, and J. Dwyer, Mol. fhys., 33, 683 (1977). G. C. Schatz, J. M. Bowman, J. Dwyer, and A. Kuppermann, unpublished, quoted in ref 55. R. B. Bernstein and R. D. Levine, Chem. fhys. Lett., 29, 314 (1974). J. N. L. Connor, W. Jakubetz, and J. Manz, Chem. fhys., 17, 451 (1976). D. G. Truhlar and R. E. Wyatt, Adv. Chem. Phys., 36, 141 (1977). D. G.Truhlar and C. J. Horowitz, J . Chem. fhys., 68, 2466 (1978). B. C. Garrett and D. G. Truhlar, J . fhys. Chem., following paper in this issue. M. Baer, Mol. fhys., 27, 1429 (1974). M. Baer, J . Chem. fhys., 60, 1057 (1974). K. Morokuma and M. Karplus, J . Chem. fhys., 55, 63 (1971). G. W. Koeool and M. Karolus. J . Chem. fhvs.. 55. 4667 (1971). S.Chapman: S. M. Hornstein, and W. H. Miller; J . Am. Chem. Soc., 97, 892 (1975). W. J. Chesnavich, Chem. fhys. Le??.,53, 300 (1978). B. C. Garrett and D. G. Truhlar, J . fhys. Chem., to be published in the Bunker Memorial Issue. J. T. Muckerman, unpublished. The parameters of this surface are given in ref 9 except that the equilibrium internuclear distance of H, is misquoted as 0.7149 A instead of 0.7419 A. L. M. Raff, L. Stivers, R. N. Porter, D. L. Thompson, and L. B. Sims, J . Chem. fhys., 52, 3449 (1970). J. W. Duff and D. G. Truhlar, J . Chem. fhys., 62, 2744 (1975). See, e.g., M. Quack and J. Troe, Ber. Bunsenges. fhys. Chem., 78, 240 (1974). S.-F. Wu, 6.R. Johnson, and R. D. Levine, Mol. fhys., 25, 609 (1973). See, however, ref 55 where it is shown that some of the numerical results of this reference are inaccurate. R. C. Tolman, "Statistical Mechanics with Applications to Physics and Chemistry", Chemical Catalog Co., New York, 1927, pp 260-270. R. H. Fowler and E. A. Guggenheim, "Statistical Mechanics", Macmillan, New York, 1938, pp 491-506. D. G. Truhlar, J. Chem. Educ., 55, 309 (1978).

Thermal Hate Constants of Atom Transfer Reactions

Discussion MARTrN QUACK(Institut fur Physihalische Chemie der Universikat Gottingen). From a general point of view, quantum transition state theory should be formulated in such a way that it gives an upper bound to the exact rate constants. In your applications, very often the quantum transition state rate constants were considerably below the exact rate constants. Could your please comment on this? The fundamental assumption of DONALDG. TRUHLAR. transition state theory is that no trajectories cross the transition-state dividiing surface more than once.’ This is clearly a classical concept, and classical transition state theory leads to an upper bound on rates obtained by exact classical dynamics? There is no unique quantum mechanical analogue of this assumption; in fact, MiIler3z4and McLafferty and Pechukas5 have presented different theories which both qualify as quantum mechanical transition state theories. Neither theory has been applied yet without further approximations. Miller’s formalism does not provide an upper bound on exact quantum dynamics, but it has the virtue of suggesting semiclassical approximation^.^,^)^ These do not provide a bound, but they have been applied successfully to collinear H i t H2.S McLafferty and Pechukas’ formalism does provide a bound, but it may be a poor one. In fact, Pechukas has suggested that any transition-state-theory upper bound to the exact quantal irate will be generally inaccuratee6 The calculations reviewed in my talk are not based on these quantum mechanical or semiclassical formalisms but rather on a simpler ad hoc quantization scheme that one might describe as “quasi-classical” by analogy to the quantization of initial vibrational and rotational energies in a quasi-classical trajectory calculation. Transition state theory, in a generalized sense but involving the aesumption of a separable reaction coordinate, may be formulated in terms of a quasi-equilibrium hypothesis for the remaining degrees of or in terms of the assumption that all degress of‘ freedom except the reaction coordinate are adiabatic.’’~’2 In either formalism one simply quantizes the energy

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levels of all degrees, of freedom except the reaction coordinate as if one were quantizing a stable bound species. This is the conventional procedure and it is the one used for the calculations reviewed in my talk. Addition of quantum effects on the reaction coordinate requires further c o n s i d e r a t i ~ n s . ~ With ~ J ~ J or ~ without a quantal treatment of reaction-coordinate motion, quasi-classical transition state theory does not provide a bound on rates calculated by pure classical dynamics, by quasi-classical trajectories, or by exact quantum dynamics. The goal, however, is to use it to give good approxirnations to the exact quantal rate. If tunneling along the reaction coordinate is very important though, one expects transition state theory with quantized energy levels but a classical treatment of the reaction coordinate to systematically underestimate the rate. In fact when we do treat the reaction-coordinate motion quantum me~hanically’~ those calculated rates which were considerably below the exact rates increase significantly. (1)E. Wigner, Trans. Faraday SOC.,34, 29 (1938). (2) J. C. Keck, Adu. Chem. Phys., 13, 85 (1967). (3) W. H. Miller, J . Chem. Phys., 61, 1823 (1974). (4) W. H. Miller, J. Chem. Phys., 63, 1166 (1975). (5) F. J. McLafferty and P. Pechukas, Chem. Phys. Lett. 27, 511 (1974). (6) P. Pechukas in “Dynamics of Molecular Collisions”, Part B, W. H. Miller, Ed., Vol. 2 of “Modern Theoretical Chemistry”, Plenum Press, New York, 1976, p 269. (7) W. H. Millei-, J. Chem. Phys., 62, 1899 (1975). (8) S. Chapman, B. C. Garrett, and W. H. Miller, J . Chem. Phys., 63, 2710 (1975). (9) S. Glasstone, K. J. Laidler, and H. Eyring, “The Thleory of Rate Processes”, McGraw-Hill, New York, 1941. (10) H. S. Johnston, “Gas Phase Reaction Rate Theory”, Ronald Press, New York, 1966. (11)M. A. Eliason and J. 0. Hirschfelder, J . Chem. Phys., 30, 1426 (1959). (12) D. G. Truhlar, J . Chem. Phys., 53, 2041 (1970). (13) D. G. Truhlar and A. Kuppermann, J. Am. Chem. Soc., 93, 1841 (1971). (14) B. C. Garrett and D. G. Truhlar, J. Phys. Chem.,this issue.