the reactive flux correlation function for collinear ...transition-state theory’ is, without a...

Faraday Discuss. Chem. SOC., 1987, 84, 441-453

The Reactive Flux Correlation Function for Collinear Reactions H + H 2 , Cl+HCl and F+H2

John W. Tromp and William H. Miller” Department of Chemistry, University of California, and Materials and Chemical Sciences

Division, Lawrence Berkeley Laboratory, Berkeley, California 94720, U S . A.

The thermal rate constant for a chemical reaction is given, in its full quantum- mechanical form and without approximation, as the time integral of the flux-flux autocorrelation function. This provides an efficient way for comput- ing rate constants because the correlation function decays to zero in times much shorter than that required for the full state-to-state reactive scattering process, thus requiring one to determine the quantum dynamics of the system only for short times. Reactive flux correlation functions have been calculated for the three title collinear reactions in order to illustrate the various generic behaviour expected for reactive flux correlation functions. H + H2 is an example of simple ‘direct’ dynamics, C1+ HCl has more complex dynamics because of the heavy + light-heavy mass combination, and F + HI has effects due to the formation of a short-lived collision complex. The reactive flux correlation functions have also been computed using classical mechanics to show the nature of the classical-quantum comparison for them.

1. Introduction

Transition-state theory’ is, without a doubt, the most commonly used approximate method of determining the thermal rate constant for a chemical reaction from a given potential-energy surface. Furthermore, within the realm of classical mechanics, there are many situations for which transition-state theory is actually exact.2 This is the case when the dynamics are ‘direct’, there being no trajectories that recross the transition-state dividing surface more than once.

Molecules, however, and their dynamics obey quantum mechanics, so that one desires a rigorous quantum-mechanical version of transition-state theory. Unfortunately, there does not seem to be a unique, rigorous quantum generalization of classical transition-state theory,’ but one can approach it perhaps most closely by using the quantum reactive flux correlation function introduced by Miller and The rate constant for a reaction can be expressed, without approximation, as the time integral of this correlation function,

k = Q-’lOa Cf( t) d t

where Q is the partition function (per unit volume) of reactants and C,( t ) is the flux-flux autocorrelation function defined in the next section. For the case of direct dynamics, the classical limit of Cf( t) is a delta function at t = 0,

kT h

CFL( t) = - Q”26( t )

so that eqn ( 1 ) gives classical transition-state theory. [Q’ is the partition function of the activated complex and 2 I: S( t) d t = J:m 6( t ) d t = 1.1 Thus only ‘zero-time’ dynamics i.e. no dynamics) is required classically in the case of direct dynamics to evaluate the

44 1

442 Reactive Flux Correlation Functions

integral in eqn (1 ) . Quantum-mechanically, however, it requires a time of order A/? = A / k T (=25 fs at T = 300 K) for C,( t ) to decay to zero even if the quantum dynamics are ‘direct’, so that one must determine the quantum dynamics of the system for times at least this long in order to obtain a result that one would wish to call transition-state-like. Thus, although there seems to be no quantum-transition-state theory in the sense of a ‘zero-time’ dynamical approximation, there is if one is willing to define it as quantum dynamics for times of order AP.

From a practical point of view the reactive flux correlation function is useful for calculating rate constants because, as noted above, it requires that one determine the quantum dynamics only for short (if not zero) times, of order AP. It would in general be necessary to determine the dynamics for times much longer than h p to obtain state-to-state properties of the reaction.

In this paper we report calculations of the quantum-mechanical reactive flux correlation function for three different collinear atom-diatom reactions over a range of temperatures. The three systems we study are the H + H 2 reaction on the Porter-Karplus surface,6 the Cl+HCl reaction on a LEPS surface’ and the F + H , reaction on the Muckerman-5 The purpose of this work is to illustrate the generic behaviour of the quantum correlation function for different kinds of reaction dynamics. We also calculate classical reactive flux correlation functions for the same potential surfaces. This allows us to interpret features in the quantum correlation function in terms of classical features that can be explained in terms of individual trajectories.

In the next section of the paper we briefly review the reactive flux correlation function formalism and outline our computational approach for obtaining these correlation functions for the collinear surfaces. In section 3 we present the results of our calculations: for the H + HI system the dynamics are direct; flux recrossing is observed in the C1+ HCl system and in the F+ H2 reaction we see effects of the well known scattering resonance, i.e. formation of a short-lived collision complex. Section 4 summarizes our results and conclusions.

2. Theory and Method

Quantum Theory

The thermally averaged flux-flux autocorrelation function ( = the reactive flux correlation function) as defined in ref. (4 j is

C,( t ) = tr [ F exp ( - P H / 2 ) F ( t ) exp ( - P H / 2 ] . (3 )

where H is the Hamiltonian of the system, p = l/k7’, F ( t ) is the Heisenberg representa- tion of the flux operator F, i.e.

F ( t ) =exp ( i H t / A ) F exp (-iH/A) (4)

and tr denotes a quantum-mechanical trace. The operator F measures the quantum flux through a dividing surface in configuration space which separates reactants from products. If f ( q ) is the function of coordinates which defines this dividing surface through the relation f ( q ) = 0, then the flux operator F is

where h ( 5 ) is the step function

w = 0 , 5<0

1, p o .

J. W. Tromp and W. H. Miller 443

If the commutator in eqn ( 5 ) is evaluated [and if H is of Cartesian form, H = p 2 / 2 m + V ( q ) ] then one obtains

Further, if f ( q ) is given by

f(4) = s - SO

where s is the reaction coordinate, then F has the simple form

S(s - so) + S(s - so) - m (7)

where p is the momentum operator conjugate to s. Different choices for the dividing surface give different correlation functions, though

the integral of these yields the same rate ~ o n s t a n t . ~ The optimum choice of dividing surface is one for which C,(t) decays to zero most rapidly,’ so that only the shortest dynamics are necessary to determine the net reactive flux, i.e. the rate constant. This is the quantum analogue of the variational aspect of classical transition-state theory. ’ Since the H + H 2 and Cl+HCl reactions are symmetric, the optimum choice of the dividing surface here is the symmetric line which divides the reactant and product valleys from one another; this is the conventional transition-state dividing surface. For the F+H2 reaction we have chosen the dividing surface at the location determined by classical transition-state theory, i. e. to minimize the classical transition-state rate.

In this paper we compute Cf( t ) by evaluating the trace in eqn (3) in a discrete basis ~ e t . ~ ’ ~ Since a collinear A+BC system consists of only two degrees of freedom, this is a very straightforward and efficient calculation. (For more complex systems, i.e. those with more degrees of freedom, path integrals method^^"'"^ for representing the time evolution and Boltzman operators, and evaluating the trace, are generally expected to be efficient.) Thus if Inm) = 1n)lm) is the basis set, the Hamiltonian matrix Hn,mf,,m in this basis is diagonalized to produce eigenvalues { E k } and eigenvectors { U n m , k } . The flux correlation function is then given by

where the matrix elements of the flux operator are

Alternatively, one can use eqn ( 5 ) and evaluate the matrix elements of F as

(k lFlk’ )= ( i / f i ) ( E k -~!?k,)(klhlk’). ( 8 4

[ Eqn (8c) is analogous to the ‘length’ us. ‘velocity’ expression for dipole matrix elements.] The particular basis set we have used is the vibrationally adiabatic basis in polar (a.k.a. hyperspherical) coordinates, which has been frequently used for carrying out reactive scattering calculations for these collinear atom-diatom

Classical Theory

We have also computed the reaction flux correlation functions for the three title reactions using classical mechanics. This is useful to help in identifying which features of the correlation functions are due to quantum effects and which are classical dynamical effects.


If s is the reaction coordinate and Q the coordinate orthogonal to it, then the classical expression for the reactive flux correlation function for a dividing surface at s = so is

C:'"(t) = ( 2 7 ~ h ) - ~ [ ds, dp, [ d Q 1 dP, exp[ -p(d+s+ 2m 2m V ( s , , Q , ) ) ]

PI P( t ) x - S( s, - s,) a[ s( t ) - so] - m m (9)

where ( p, P ) are the momenta conjugate to (s, Q ) and s( t ) and p( t ) are the values of s and p that have evolved classically from the initial conditions (sl, p,, Q,, P , ) . (One sees that a phase-space average over initial conditions has replaced the quantum- mechanical trace.) The integral over s1 can be evaluated by virtue of the delta function, giving

CF"(t) = ( 2 d - l [ dQ, [ dP, exp [ --p(%+ V(so , Q , ) ) ] 2m

x ( 2 d i - I 5 dp, p1 exp ( -p ":) S[s( t ) - so] P( t ) -. m 2m m

It is illustrative to consider eqn (10) in the short-time limit. In this limit s ( t ) and p( t ) are given by a free-particle trajectory (independent of Q, and PI),

P1 m

s ( t ) = so+- t

so that eqn (10) becomes

where QIL is the classical partition function of the activated complex on the dividing surface

QgL= (2vh)-'[ dP, 1 exp [ -/3(%+ 2m V(so , Q ) ) ] dQ,.

Since

and

-&) I p1 1 = 2 - kT 2m m h

eqn (12) becomes

kT h

CF"( t ) = - Q*2S( t ) (14)

as noted in eqn (2). The analysis in the above paragraph shows that the classical correlation function

will always have a short-time, free-particle delta function at t = 0. If s( t ) is never equal to so for t > 0, then eqn (14) is the complete classical correlation function, uiz. classical


transition-state theory is exact if no trajectories recross the dividing surface. If s ( r ) does equal so at some time t > 0 , then p ( t ) will have the opposite sign of p , if this is the first recrossing time for this trajectory; if it is the second recrossing, p ( t ) and p , will have the same sign, and so forth. Thus negative and then positive regions of C,(t) are identified as classical recrossing effects.

The quantum correlation function, of course, has no delta function singularity at t = 0. Specifically, the quantum free-particle correlation function for a one-dimensional reaction coordinate is4

which as one can readily show, approaches the classical result [eqn (14)] in the limit AP -+ 0. To minimize this well understood difference between the classical and quantum correlation functions we have therefore chosen to 'smear out' the classical delta function by averaging the classical correlation function over a time interval of order hP/2. Specifically, we have modified the classical expression according to

CFL( t ) -+L[a hPJ7r -a exp [ -( t - t ' ) 2 / ( -32] CF"( t ' ) dt'.

The primary effect of this averaging is to replace the delta function in the classical correlation function by a peak whose width is hip/2.

Another source of disagreement of the purely classical correlation function from its quantum counterpart is the quantization of the 'activated complex, e.g. within the short-time approximation Q' is a quantum vibrational partition function in the quantum case and a classical one for the classical case, and these can differ substantially in their numerical values. To patch up this defect we have used the often-applied quasiclassical model. Thus the initial values (PI, 9,) are specified in terms of initial values of the action-angle variables for this degree of freedom. For a harmonic oscillator, for example,

P, = [ ( 2 n , + 1)Amw]1/2 coslq,.

[In the applications presented herein the Q-oscillator is taken to be a Morse oscillator, for which the equations analogous to eqn (17) are more complicated, but still known."] The (PI , 9,) * ( n , , q , ) transformation is canonical, so an integral over PI and Q, is identical to an integral over n , and 41, but the quasiclassical model sums over integer values of n,; thus eqn (10) is modified as follows:

r r

(It should be noted that this same type of quasi-classical initial condition for the activated complex has also been used for full classical trajectory calculations for reactive scattering.'") For the temperature range of our calculations, essentially only the n , = O ( i e . the ground state of the activated complex) contributes significantly. Also, by definition of the action-angle variables, one has

where E , , is the vibrational energy level of the activated complex.


Table 1. Comparison of the initial decay of the different quantum correlation functions

200 300 600

1000

200 300 600

1000

200 3 00 600

1000

( a ) H+H,-* H,+H 410 790 0.52 345 526 0.66 210 263 0.80 125 158 0.79

( b ) C1+ HCl + ClH + C1 630 790 0.80 420 526 0.80 210 263 0.80 120 158 0.76

( c ) F + H , + F H + H 590 790 0.75 390 526 0.74 220 263 0.84 160 158 1.01

46 8.7 2.5 1.5

0.99 0.69 0.42 0.27

0.8 1 0.57 0.43 0.40

In summary, then, with the modifications implied by eqn (16)-(19), and with the choice so = 0, the classical expression for the reactive flux correlation function is

where s( t ) and p ( t ) are the classical trajectory determined by the initial conditions ( n , , q l , pl, s1 = 0) and { t k } are the times for which s( t ) = 0.

3. Results

In this section we compare the quantum and classical correlation functions for the three title reactions to see to what extent the various dynamical features observed in the quantum correlation functions can be understood classically. To this end we display the correlation functions all normalized to their value at t = 0, i.e. the quantities plotted are C,( t ) / C f ( 0 ) . We note that the integrals of the quantum correlation functions yield the correct quantum rate constants in all cases.

Table 1 gives various quantities which characterize the quantum correlation functions and also gives (in the last column) the ratio of the correct quantum rate to the non-tunnelling, conventional transition-state theory rate (with quantum Q' ). This ratio is the historical' K, which corrects all the defects of conventional transition-state theory (i.e. neglect of tunnelling and recrossing effects).

H+H,-+ H,+H

This is the simplest known chemical reaction, and it has been studied extensively. In classical mechanics it is known that microcanonical transition-state theory is exact for


I I

500 1000 If

I I I I

0 500 1000 1500 I I

t I I 0 500 1000 1500

0 500 1000 1500 t l au

Fig. 1. The quantum-mechanical reactive flux correlation function for the H + H2 -P H, + H reaction at ( a ) 1000, ( b ) 600, (c) 300 and ( d ) 200 K. The displayed correlation functions are all normalized by unity at t = 0 , i.e. the quantities shown are actually C,(t)/C,(O). Note that 1000

atomic units of time =24 fs.

energies up to ca. 0.4eV above the reaction threshold.” Since the thermal energy at lOOOK, the highest temperature considered, is only 0.09eV, one would not expect to see any effects of recrossing dynamics. That this is true can be seen by noting that none of the H+H, correlation functions, quantum or classical, displayed in fig. 1 and 2 is ever negative, i.e. all reactive dynamics are ‘direct’.

The differences between the classical and quantum correlation functions in fig. 1 and 2 are therefore due solely to tunnelling effects. To quantify the discussion somewhat, we define tllZ as the time at which Cf(t) has fallen to half its value at t=0. If the behaviour is free-particle-like [as are all of the classical correlation functions for short times because of the averaging implicit in eqn (16)] then this half-time would be (2’13 - 1)l” A P / 2 = 0.77 ( A p / 2 ) . Free-particle, non-tunnelling short-time behaviour is thus characterized by the ratio f , 1 2 / ( A p / 2 ) = 0.77. Table 1 lists these values for all the reactions.

At the highest temperature, 1000 K, the effect of tunnelling is small ( K = 1.5) and the half-time for the decay of the quantum correlation function is essentially the classical value (cJ: table 1). The half-time increases with decreasing temperature, but not as fast as Ap (cJ: table l), and appears to be reaching a limit, i.e. the width of Cf( t ) in fig. 1


t I I I 0 500 1000 1500

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500 t / au

Fig. 2. The classical correlation functions for H + H2 ; see also notes for fig. 1

is essentially the same for T = 200 and 300 K. This can be understood because the reaction is dominated by tunnelling at T = 200 K ( K = 46), and it was shown in ref. (4) that the correlation function for a parabolic barrier, the generic tunnelling system, decays as exp(-2ubt) , where 6& is the imaginary frequency of the barrier, i.e. the decay is temperature-independent when tunnelling dominates. For this system ub ' =: 200 au, which is seen to be the order of magnitude observed for the decay of C,(t) in the low-temperature limit.

One notes, therefore, that when tunnelling effects dominate, the time averaging of the classical correlation function, eqn (16), does not describe the quantum short-time dependence correctly.

C1+ HCI -+ CIH + C1

The C1+ HCl example is a typical example of a heavy/light-heavy system where a small mass is transferred between two large masses. The quantum and classical and correlation functions for this system are shown in fig. 3 and 4. Classically such a system can exhibit multiple recrossings, since motion through the dividing surface is in the same direction as vibrational motion of the reactants and products. Quantum-mechanically, such recrossing is indeed observed at higher temperatures.


I 1 I

0 1000 2000 3000

h

I I I I 0 1000 2030 3000

n I I I I

0 1000 2000 3000

1 I 0 1000 2000 3000

t / au

Fig. 3. The quantum-mechanical correlation functions for the CI + HCI + CIH + CI reaction; see also notes for fig. 1 .

First, consider the quantum correlation function at 200 K. It shows almost no recrossing and the decay rate of the correlation function is almost exactly the same as that for a free particle [see table l (b)] . Thus, for this case we expect conventional transition-state theory to be quite accurate, and it is indeed correct to 1 % .

Note in table 1 that all the correlation functions for Cl+HCl have essentially the same ratio t , / * / ( h p / 2 ) as the free-particle value. Thus, for all temperatures considered here tunnelling is not very important, and the errors in transition-state theory are solely due to the classical recrossing dynamics. As temperature increases, the recrossing becomes much more pronounced and thus the transition-state theory rate (which omits the effects of recrossing) increasingly overestimates the rate constant; cJ: the last column in table 1 . For a system such as C1+ HCl with a high skew angle, the best variational dividing surface occurs at the symmetric location only for low energies. Above a certain critical energy, the symmetric periodic orbit dividing surface becomes unstable and bifurcates into two symmetrically equivalent dividing surface^.'^"^ From the point of view of transition-state theory, it is clear that a better short-time estimate of the rate constant could be obtained by moving the dividing surface to the variational location. However, since we are interested in obtaining the exact rate constant in the shortest possible time, it is more advantageous to place the dividing surface in the symmetric location. It is clear that this should give the quickest convergence, since the rate is


0 1000 2000 3000

I I I ~~

0 1000 2000 3000

0 1000 2000 3000

t I I

0 1000 2000 3000 t/au

Fig. 4. The classical

determined by dynamics

correlation functions for C1+ HCl; see also notes for fig. 1.

through both bottlenecks and the symmetric location treats them equivalently. Note also that the classical correlation function accurately mimics the oscillatory features of the quantum function at temperatures where recrossing occurs.

F+F,-*FH+H

Finally, we consider a more realistic example of a chemical reaction. This potential is no longer symmetric, so chemical change occurs during the reaction. The FH molecule is much more strongly bound than the H2 molecule, and so F+ H, ( v = 0) can react at thermal energies to produce products in any of the four lowest states of FH. The probabilities of forming H F ( v = 0) and H F ( v = 1 ) are almost negligible, but HF ( v = 2) and HF(v=3) are both produced at thermal energies.'

The quantum and classical correlation functions for this system are displayed in fig. 5 and 6, and the half-times for the decay of the correlation functions and the comparison to transition-state theory are shown in table 1 ( c ) . Note at this point that in order to obtain the quantum results shown, it was necessary to change the definition of the flux operator so that states that lack sufficient energy to exist asymptotically as reactants are not included in the trace. This point has been thoroughly discussed in ref. ( 5 ) . The only change from a computational point of view is that in eqn ( 8 a ) the sum is restricted to eigenvalues with energies greater than the zero-point energy of H,.

J. W. Tromp and W. H. Miller 45 1

n t I I 0 1500 3000 45 30

I I I I

0 1500 3000 4500

I I 1 I 0 1500 3000 4500

h

I I 0 1500 3000 4500

t l a u

Fig. 5. The quantum-mechanical correlation functions for the F + H2 --* FH + H reaction; see also notes for fig. 1.

Note first in fig. 5 and 6 (and from table 1) that the short-time decay of the quantum and classical correlation functions is essentially the same, indicating that tunnelling has a minor effect for this reaction. This is expected, of course, because the barrier is quite low and flat. The next most obvious feature is that there are significant recrossing effects in the correlation functions at all temperatures, both quantally and classically. At sufficiently low temperatures one knows that transition-state theory must become exact classically,* so that recrossing effects must disappear; it is apparent in this case that 200 K is not yet low enough.

Although the classical recrossing effects seen in fig. 6 are in rough agreement with the quantum behaviour in fig. 5 , there are differences: the quantum correlation function appears to oscillate about the classical value. This is most apparent at 1000 K, although it is also recognizable at other temperatures, and the spacing between minima is At=500au. This behaviour can be understood by noting that there is a scattering resonance, i.e. a short-lived collision complex, for this system at a collision energy of 0.015 eV.9920 The Boltzmann population will thus access this resonance at all the temperatures considered here. Since the dividing surface in this case is in the entrance valley, the part of the collision complex that breaks up non-reactively will recross the dividing surface on its way back to reactants. This will lead to negative contributions to C,(t) spaced by time intervals that correspond roughly to the vibrational period of


1 I 0 1500 3000 4500 I I

I I I 0 1500 3000 4500

I I I

0 1500 3000 4500

I I 0 1500 .3000 4500

t/au

Fig. 6. The classical correlation functions for F+H,; see also notes for fig. 1.

the classical motion of the collision complex, i.e. the complex can break up non- reactively, with various probabilities, after one oscillation in the complex, after two oscillations in the complex etc.; c j the semiclassical description of this by Waite and Miller.21 One thus identifies the spacing A t with 2 7 ~ / 0 , where w is the vibrational frequency of the collision complex. The observed value A t == 500 au gives w = 2800 cm-', a sensible value.

4. Concluding Remarks

The reactive flux correlation function is a useful way to characterize chemical reaction rates since it bridges the gap between transition-state theory (through its short-time behaviour) and the dynamically exact rate constant. This paper has considered three different A + BC reactions to illustrate the way various dynamical phenomena are manifest in this correlation function.

H + H2 shows the simplest, transition-state-theory-like dynamics, its only complicat- ing factor being quantum tunnelling at the lower temperatures. C1+ HCl shows transition-state-theory-violating dynamics, i.e. recrossing flux, but this is well described within classical mechanics. Finally, F+ H2 shows non-classical recrossing effects that one can identify with the formation of a short-lived collision complex, i.e. a scattering resonance.


The basis-set method we have used to compute the quantum correlation functions is quite straightforward and efficient for systems with a few degrees of freedom. More generally we expect path-integral methods to be more useful, especially for systems with many degrees of freedom. This methodology is currently undergoing rapid develop- ment,22-25 providing considerable cause for optimism in this regard.

We gratefully acknowledge support from the National Science Foundation Grant CHE84- 16345.

References

1 See e.g.: S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes (McGraw-Hill, New York, 1941); P. Pechukas, in Dynamics of Molecular Collisions, ed. W. H. Miller (Plenum, New York, 1976), p. 269; D. G. Truhlar, W. L. Hase and J. T. Hynes, J. Chem. Phys., 1983, 87, 2264.

2 P. Pechukas and F. J. McLafferty, J. Chem. Phys., 1973, 58, 1622. Chem. Phys. Lett., 1974, 27, 511. 3 W. H. Miller, Acc. Chem. Res., 1976, 9, 306. 4 W. H. Miller, S. D. Schwartz and J. W. Tromp, J. Chem. Phys., 1983, 79, 4889. 5 J . W. Tromp and W. H. Miller, J. Phys. Chem., 1986, 90, 3482. 6 R. N. Porter and M. Karplus, J. Chem. Phys., 1964, 40, 1105. 7 D. K. Bondi, J. N. L. Connor, B. C. Garrett and D. G. Truhlar, J. Chem. Phys., 1983, 78, 5981. 8 P. A. Whitlock and J. T. Muckerman, J. Chem. Phys., 1975, 61, 4618. 9 G . C. Schatz, J. M. Bowman and A. Kuppermann, J. Chem. Phys., 1975, 63, 674.

10 B. C. Garrett, D. G. Truhlar, R. S. Grev, A. W. Magnusson and J. N. L. Connor, J. Chem. Phys., 1980, 73, 1721.

11 R. Jaquet and W. H. Miller, J. Phys. Chem., 1985, 89, 2139. 12 K. Yamashita and W. H. Miller, J. Chem. Phys., 1985, 82, 5475. 13 A. Kuppermann, J. A. Kaye and J. P. Dwyer, Chem. Phys. Lett., 1980, 74, 257. 14 J. Romelt, Chem. Phys. Lett., 1980, 74, 263. 15 C. C. Rankin and W. H. Miller, J. Chem. Phys., 1971, 55, 3150. 16 ( a ) I. W. M. Smith, J. Chem. Soc., Faraday Trans. 2, 1981,77, 747; ( b ) R. J. Frost and I. W. M. Smith,

17 S. Chapman, S. M. Hornstein and W. H. Miller, J. Am. Chem. Soc., 1975, 97, 892. 18 E. Pollak and P. Pechukas, J. Chem. Phys., 1978, 69, 1218. 19 E. Pollak, M. S. Child and P. Pechukas, J. Chem. Phys., 1980, 72, 1669. 20 See e.g. J. Romelt and E. Pollak, in Resonances in Electron- Molecule Scattering, van der W a d s Complexes

and Reactive Chemical Dynamics, ed. D. G. Truhlar (Am. Chem. SOC., Washington, D. C., 1984), ACS Symp. Ser. No. 263, p. 353.

21 B. A. Waite and W. H. Miller, J. Chem. Phys., 1982, 76, 2412. 22 R. D. Coalson, D. L. Freeman and J. D. Doll, J. Chem. Phys., 1986, 85, 4567. 23 J. Chang and W. H. Miller, J. Chem. Phys., in press. 24 J. D. Doll, R. D. Coalson and D. L. Freeman, J. Chem. Phys., in press. 25 W. H. Miller and N. Makri, Chem. Phys. Lett., in press.

Chem. Phys., 1987, 111, 389.

Received 22nd May, 1987

the reactive flux correlation function for collinear ...transition-state theory’ is, without a...

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