Generalized Transition State Theory. Classical Mechanical Theory

generalizations of the conventional transition state which may usefully be labeled ...... and lower of the three solid curves are Pci(E), PCfivT(E), a...
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1052

The Journal of Physical Chemistry, Vol. 83, No. 8,

B. C. Garrett and D. G. Truhlar

1979

denominator in eq 34 should be squared. (22) See, for example, ref 4, 5, and 20; R. L. Wilkins in "Handbook of Chemical Lasers", R. W. F. Gross and J. F. Bott, Ed., Wiley-Interscience, New York, 1976, p 551; D. G. Truhlar and D. A. Dixon in "Atom-Molecule Collision Theory: A Guide for the Experimentalist", R. B. Bernstein, Ed., Plenum Press, New York, 1979, in press. (23) B. C. Garrett and D. G. Truhlar, unpublished resutts. The transition-state parameters are rtW, = 4.254 a,, rtCG,= 3.811 a, V t = 2.420 kcal/mol, hu,sb = 1.478 kcal/mol = 523.9 cm-', x,Sb = 6.731 X

h d = 1.3953 kcai/mol = 487.93 cm-'. The reactant parameters are r:lCl = 3.779 a, ho:"' = 1.606 kcal/mol = 561.9 cm-', xc :II = 6.919 X (24) E. C. Garrett and D. G. Truhlar, J . fhys. Chem., following paper in this issue. (25) E. P. Wigner, Z . fhys. Chem. B , 19, 203 (1932); H. S. Johnston, "Gas-Phase Reaction Rate Theory", Ronald Press, New York, 1966, pp 133-136; D. Rapp, "Statistical Mechanics", Holt, Rinehart and Winston, New York, 1966, pp 133-136.

Generalized Transition State Theory. Classical Mechanical Theory and Applications to Collinear Reactions of Hydrogen Molecules Bruce C. Garrett and Donald G. Truhlar" Chemical Dynamics Laboratory, Kolthoff and Smith Hails, Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received August 9, 1978)

We consider classical versions of three generalizations of transition state theory: microcanonical variational transition state theory, canonical variational transition state theory, and Miller's unified statistical theory. We prove that microcanonical variational transition state theory is identical with the adiabatic theory of reactions, Le., to adiabatic transition state theory. To test the validity of these approximate theories, we present calculations for several collinear reactions of hydrogen and halogen atoms with hydrogen molecules. Average reaction probabilities are computed using conventional and microcanonical variational transition state theory and the unified statistical theory and are compared with those of exact classical dynamics for seven cases. These results confirm the general validity of the fundamental assumption of transition state theory at low energy and show that the variational method can be used to extend the range of validity to higher energies. Thermal rate constants are calculated by these methods and by canonical variational transition state theory for nine systems. Using a Morse approximation involving the second and third derivatives of the local vibrational well at its minimum, the average absolute value of the error and range of absolute values of the errors at 600 K for the seven cases where we computed exact classical canonical rate constants are 28 and @78% for conventional transition state theory, 10 and 1-37% for the microcanonical variational theory or adiabatic transition state theory, 7 and 1-22% for the unified statistical theory, and 15 and 0-41% for the canonical variational theory.

I. Introduction Recent attempts to develop a useful quantum mechanical version of transition state theory have led to a more careful examination of the dynamical foundation of transition state the0ry.l The fundamental dynamical assumption may be expressed unequivocally only in classical mechanics, where it may be stated as follows: Transition state theory with unit transmission coefficient is exact if and only if all trajectories through the transition state, which is a surface dividing reactants from products, cross this surface only ~ n c e . ~Further -~ if trajectories do recross this surface, transition state theory with unit transmission coefficient overestimates the rate. Transition state theory may also be derived on the basis of a quasi-equilibrium postulate; this is the traditional basis of the theory,6-10 but it does not lead to the same kind of interpretation as the dynamical justification. One approach to reconciling the quasi-equilibrium assumption with a dynamical model is the adiabatic theory of reactions; the adiabatic theory assumes that all the degrees of freedom which remain bound throughout the course of the reaction adjust adiabatically to progress of the system along a separable reaction coordinate.lGZ0It is now realized that the adiabatic assumption and the quasi-equilibrium assumption do not lead to identical results,20but the adiabatic assumption still provides a useful framework in which to include quantum mechanical effects on motion 0022-3654/79/2083-1052$0 1,0010

along a reaction coordinate which is assumed separable.10721-24Another advantage of the adiabatic theory is that one can easily include total angular momentum c o n ~ e r v a t i o n . l ~ J ~The J ~ -adiabatic ~~ theory of reactions is also caIled adiabatic transition state theory;28 with additional assumptions about product states it is called the statistical adiabatic channel The dynamical justification of transition state theory in terms of classical trajectories not recrossing a dividing surface leads to the classical variational versions of transition state theory.1~4J3~29-34 In these versions, the position of the dividing surface is varied so that the calculated rate is minimized. The adiabatic theory of reactions and variational transition state theory lead to generalizations of the conventional transition state which may usefully be labeled generalized transition state^.^^,^^ The adiabatic theory of reactions is not really transition state theory. Although the generalized transition states of the variational transition state theories are still dividing surfaces in phase space between reactants and products, in the adiabatic theory they are not.37 Nevertheless we show in this article that adiabatic transition state theory leads to the same rate constants, at least classically, as one version of variational transition state theory. The dynamical justification of transition state theory with unit transmission coefficient in terms of the recrossing criterion is expected to be most valid at low energy.3,413B-40 0 1979 American

Chemical Society

The Journal of Physical Chernjstry, Vol. 83, No. 8, 1979

Generalized Transition State Theory

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To study the exti?nt to which this criterion is satisfied, it Pechukas have tested microcanonical variational transition is useful to consider microcanonical transition state theory, state theory against exact classical microcanonical rate i.e., the theory of reaction rates in microcanonical enconstants for collinear H H2and F + H2.85There have been no previous tests of exact classical canonical rate sembles. Although transition state theory was originally developed for canonical rate constants,6-10 it has been constants against classical transition state theory. (Anextended to micirocanonical rate constants on the basis of derson and c o - ~ o r k e r s ~have ~ @ compared exact classical the quasi-equilibrium p o s t ~ l a t e , alone ~ ~ ~ or ~ ~combined -~~ canonical rate constants to the equilibrium flux across with adiabaticity assumptions for other degrees of freedividing surfaces in the interaction region but these didom,41-44on the basis of the inverse Laplace transform viding surfaces were not conventional or variational relation of microcanonical rates to canonical rate^,^^,^^-^^ transition states; they were chosen for convenience.) The and on the basis of the recrossing c r i t e r i ~ n . ~It?is~also ?~~ present study encompasses conventional transition state possible to apply the quasi-equilibrium assumption to theory and five generalized versions of it, it considers ensembles of molecules with not only fixed total energies microcanonical and canonical rate constants, and it inE (like a microcanonical ensemble) but also fixed total cludes a systematic series of applications to several reangular momentum J.36954-67 By averaging over the apactions. It is hoped that this more comprehensive propriate distribution of J values one can then calculate treatment allows for a greater understanding of the relative a microcanonical rate constant. The canonical rate merits of the various generalized theories when they are constant is then a thermal average of the microcanonical applied to reactions with varying characteristic features. ones. (In practice the treatments of the E,J ensembles are In addition this is the first numerical test against exact often augmented with additional assumptions to yield dynamical results of two important theories: the canonical product state distributions, which are not of interest here.) variational transition state theory9J0~87-92 (as discussed in section I11 this is identical with what is also called the When one attempts to include quantum mechanical minimum free energy method or the method of free energy effects in transition state theory, the methods to be emsurfaces or curves; it is also simply called the variational ployed are much less clear. It is of course necessary to theory of reaction rates) and Miller’s unified statistical include quantum effects if calculations are to be compared theory. to experiment since, for example, many of the most useful qualitative predictions of transition state theory depend It may be helpful to point out early in this paper one simply on the calculated zero-point energy of the transition possible point of confusion. There are two viewpoints on states. Also tunneling is important for many reactions. the role on equilibrium in reaction rate theory. From one However to study the basic assumptions of transition state viewpoint the observed phenomenological rate constant theory and its generalized versions, a purely classical is to be calculated directly. Then one must question treatment offers some advantages. In this paper we discuss whether there is an equilibrium distribution among all conventional transitioin state theory and various generpossible reactant states, including very reactive ones, and alized versions of it: the adiabatic theory of reactions, the transition state. Such discussions are traditional in microcanonical variational transition state theory, Miller’s textbook coverage of transition state theory. From another unified statistical theory,68 and canonical variational viewpoint, taken here and in several other papers in the transition state theory. last few years, the problem is broken into two parts. One All these methods are applied to calculate canonical rate would first calculate the equilibrium reaction rate. This constants and all1 but the last are applied to calculate is an average over an equilibrium distribution of reactants microcanonical rate constants for a series of collinear of the state-selected rate constants for given states. Then reactions. For comparison we calculate these same rate one would use perturbation theory or some other method constants by exact classical mechanics and by conventional to treat activation and depletion steps and thus to relate transition state theory. In a companion paper69we include the phenomenological steady-state rate constant to the quantum effectc; on the bound degrees of freedom and equilibrium one.93199From this viewpoint, one uses colcompare the results of conventional and generalized lision theory to first calculate the equilibrium rate constant, transition state theory to exact quantum mechanical rate for which the equilibrium assumption holds by definiconstants. In both papers we limit ourselves to election.loO Thus the equilibrium assumption need not be tronically adiabatic collinear reactions of the form A BC questioned in this step. It is only this step, the calculation AB + C. The primary reason for this is that, with the of the equilibrium rate constant, which is considered by exception of the H ]-I2 reaction,70this is the only case the theory of reactive scattering and in this article. Even for which accurate quantum mechanical rate constants are for purely classical systems though transition state theory available for testing the t h e o r i e ~ . ~The ~ - limitation ~~ has is not generally exact even for the equilibrium rate conanother advantage for the present paper in that it allows and in section 111, for stant. As discussed el~ewherel-~ us to focus more clearly on the basic relationships between classical systems this is due entirely to recrossing effects the various theories and approximations without conwhich are neglected when the transmission coefficient is sidering all the practical complications. set equal to one. However, to confuse matters even more, even for the equilibrium rate constant there is a sense in Two other approaches related to those considered here which one could instead blame the inexactness of transition are the minimum-density-of-states methods of Bunker and Pattengill and Wong and M a r ~ ~ sThey . ~are ~not~ ~ state ~ , theory ~ ~ on~the~ equilibrium ~ ~ ~ assumption rather than the transmission coefficient.lol applied here because, as discussed elsewhere,s2 they are less accurate than (or in some cases equivalent to) miIn section I1 we review the exact classical expression for crocanonical variational calculations, but they involve AB + C reaction. We the rate of a collinear A + BC about the same computational effort. compare various forms of this expression which are useful Karplus and co-workers, Chapman, Hornstein, and for our classical trajectory calculations reported here and Miller, and Chesnavich have tested conventional transition for understanding the dynamical foundation of transition state theory against exact classical microcanonical rate state theory. In sections I11 and IV we present the conconstants for a few ~ o l l i n e a rand ~ ~ three-dimensional ~~~,~~ ventional and various generalized transition state forr e a c t i o n ~and, ~ ~ ,concurrently ~ with our work, Pollak and malisms as they apply to the collinear A + BC reaction

+

-

+

+

-

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The Journal of Physical Chemistry, Vol. 83, No. 8, 1979

B. C. Garrett and D. G. Truhlar

with classical mechanics. In section V we discuss the details of the calculational procedures. The rest of the paper gives the results and further discussion.

-

11. Exact Classical Dynamics We consider the collinear reaction A + BC AB + C. Let r be the BC distance with conjugate momentum pr and x be the distance from A to the center-of-mass of BC with conjugate momentum px. The Hamiltonian as a function of the phase space coordinates (p,q) is *(P,q) = (~P)-'P,'

+ (2m)-lpr2 + V(x,r)

(1)

We will also find it useful to consider a scaled coordinate y such that

SR(q)is a special choice of S(p,q) which is located sufficiently toward the reactants direction that H(p,q) may be approximated accurately by HR(p,q)where HR(p,q)= lim H(p,q)

(10)

,-m

A convenient choice of SR(q)would be Sxo(q)= xo - x, where xo is some arbitrarily large value. x(p,q) is the characteristic function3 of reactive trajectories: x(p,q)is unity if (p,q) lies on a trajectory which was on the R side of the surface in the asymptotic past and will be on the P side of the surface in the asymptotic future. The physical content of eq 5 is more clear if it is written as a single average hC(E) = ( R ( P , d X(P,q))E,SR

(11)

where

so that s ( P , q ) = (2P)-YPx2+ Py2)+ V(X,Y)

(4)

Then motion in the (x,y) plane is mathematically identical with the motion of a single point of mass p governed by the potential V(x,y). The phase space (p,q) can be taken as either (Px,Pr,x,r) or ( p x t p y ~ , ~ ) . A . Microcanonical Rate Constants. Consider a total energy E less than the threshold energy D for dissociation to A + B + C. Then phase space integrals will converge in the y direction because of the potential. To make them converge in the x direction we put the system in large box of side length 2L. We choose L large enough so that the trajectories of interest never hit the walls and so the rate constant is independent of L. Then the exact classical rate constant kc(E)for an equilibrium microcanonical ensemble of A, BC pairs a t total energy E may be written in terms of the average of the reactive flux over the appropriate phase space as f o l l o ~ s ~ ~ ~ ~ ~ ~ ~

kc(E) =

(F(p,q)x(P,q))E (s[sR(q)l)E

(5)

where we have introduced a shorthand notation for the microcanonical phase space average of a phase function of (p,q),i.e.

and (f(p,q))E,SR =

J J Jd' 6[E H(p,q)l 6[SR(q)lf(P,4) (13) h-'J J J J d r 6[E - H ( ~ , q )8[SR(dl l

h-21

-

where we have used eq 9b and the fact that eq 5 is independent of surface. It is convenient to rewrite ( 5 ) with the aid of (6) as

M E ) = Nc(E)/[h+cR(E)l

(14)

where

NcW) = 2nhh-'J

J J JdT

a[E - H ( ~ , q )&,q) l x(P,q) (15)

and +CR(E)is the reactant's density of states per unit energy per unit length, i.e.

+ c R W = h-'J

J J Jdr

6[E - HR(p,q)l6[SR(q)I (16)

4CR(E)is reduced to a practical form for the collinear A N c ( E )is unitless and is called the cumulative reaction probability. A more in-

+ BC reaction in Appendix A.

tuitive form of the cumulative reaction probability can be obtained from eq 15 as follows. Choose the dividing surface as S"0. Motion in y is bound and we transform (p,,y) to action-angle variables (JVR,aR) so that

where

d r = dp, dp, dx dy

(8)

and h is Planck's constant. In these equations pc3Yy"(E)is the system's classical density of states per unit energy which cancels out in the numerator and denominator of eq 5, and F(p,q) is the flux through any surface S(p,q) = 0 which separates reactants (R) from products (P)

In a Cartesian coordinate system such as ( x , y ) and for a surface that is a function only of q, not p, this becomes

6[E - (2P)-1Px2- V,(NR,-m)l (-P,/d x(NR,px,qR,xo) (17) where V,(NR,-a) = V(x=xo,r=re)- c(NR,-m) (18) q R = 2naR

(19)

NR = J v R / h

(20)

and c(NR,-a)is the vibrational energy of the BC diatom for action NRh. The integration over px is performed by virtue of the delta function resulting in

The Journal of Physical Chemistry, Vol. 83,No. 8, 1979

Generalized Transition State Theory

where

1055

JDdE e-PEPcSys(E) (f(P,d)E

Pc(NR,E) =

(29)

( f ( p , d )T,D =

-(2*)-'J2rdqR

[ x ( l ~ , l , Nx~0,,4

JDdE e-flEpCSys(E)

1 - X(-lPxl,NR,~o,4R)I (22)

&lpxl = 4 2 p [ E - Va(NR,-m)]j1'2

(23)

Using eq 6 , 7 , and 28 a canonical phase space average may also be written

and NcR(E)is the value of NRsuch that Va(NR,-m) equals E . Pc(NR,E).has the interpretation of a classical reaction probability for a total energy E and for a given initial value of the action A%. The quantum mechanical a n a l ~ g u e ~ ~ ~ ~ ' ~ ~ of eq 21 is a basic relation given in eq 2.30 of ref 102 and eq 2.2 of ref 1103. In addition to the cumulative reaction probability Nc(E), it is useful to define the average reaction probability Pc(E), which is P c ( m = Nc(E)/NcR(E)

(24)

It is useful to define the classical number of states for one degree of freedom as the area in phase space in units of Planck's constant. 'Then NcR(E)is the number of energetically allowed vibrational states of the reactants at total energy E and can ble written L

NcR(E) = h-lspd.,$ -m

-L

dy

8[E- (2p)-lpy2 - V(x=xO,y)] (25)

= J m d P O[E - Va(NR,-m)]

where the Heaviside step is defined by e(x) = 0 x c 0 =1 2 1 0

(26)

where we used (7) to obtain (34). Using the definition (32), the cutoff canonical analogue of eq 5 is (35)

(27)

B. Canonical Rote Constants. The canonical rate constant for a temperature T can be obtained using the relationship between canonical and microcanonical ensemble averages: J m d E e-flEpsys(E) (f(p,q)) E (28)

( f ( p , d) T =

J m d E e-flEpsYY"(E) where P is (k7J-l and h is Boltzmann's constant and where we assume f .- 0 if E C 0. For classical calculations the density of states psYs(E) in (28) is pC:ys(E) although (28) itself is more general. In the previous discussion of the microcanonic(a1rate we restricted the total energy to be below the dissociation energy D. Above this energy it is possible for reaction to occur stepwise by dissociation of BC followed by recombination of the A and B. At such energies there would be contributions to the reactive flux for values of x and y outside of the box of side L even for arbitrarily large L. This would cause the cumulative reaction probability to diverge with increasing L instead of becoming iridepenldent of L when L gets large. To avoid this problem we have considered only energies below the dissociation threshold in the microcanonical case. Higher energies get weighted less in the canonical average due to the Boltzmanri factor, therefore energies less than but near the dissociation contribute negligibly when the temperature is not too high. However contributions from energies above D still cause the Boltzmann average to diverge. We circumvent this problem by truncating all energy integrations at the dissociation threshold in a completely consistent way. Thus we define a cutoff canonical phase space average by

1J Jd'

= (h-21

ex~[-PH(~,qX )l

- H(p,q)l F(p,q) X(P,Cl)J/@cR(T,D) (36)

where the cutoff partition function per unit length for reactants is V(T,D)

= h-zJ

J J Jd'

~~P[-PHR(P,x ~)I

e[D - HR(p,dl8[SR(q)l(37) and using eq 16 @cR(T,D)= JDdE exp(-PE)4cR(E)

(38)

where the lower limit of integration is determined by the fact that henceforth we assume V(x=m,r=rJ 1 0 where re is the classical equilibrium internuclear distance of BC. The reactant cutoff partition function is reduced to a practical form in Appendix B. The cutoff canonical analog of eq 11 is kC(T,D) = (p(P,q) X(P,d)T,D,SR

(39)

where we have defined (f(P,d)T,D,SR= Ih-ZJ

J J Jd'

exP[-PIfR(P,dl x

e[o- HR(P,d1 s[sR(s)l f(P'cl)Jl (h-.'J

11I d s exp[-PHR(p,s)l x e[o - HR(p,s)1J[sR(q)l] (40)

Equation 39 is clearly equivalent to (35) because (35) is

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The Journal of Physical Chemistry, Vol. 83, No. 8, 1979

independent of the choice of dividing surface S(p,q) in F(p,q). We note that eq 14-16 can be used to rewrite (35) in the form DdE exp (-@E)4cR(E)kc ( E ) Kc(T,D) =

(41)

d E exp(-PE)4cR(E)

The threshold for the reaction will be the saddle point energy which we denote V. Therefore the lower limit of integration in the numerator of eq 41 can be replaced by V allowing the rate to be rewritten as

LD-'dc e-b'Nc(t) -

where E is the total energy above the barrier. A more suggestive form of the thermal rate constant is obtained by integration by parts

where Qc(T,D) = s m d Nexp[-P~(N)]O[D - V 0

-

4N)]

-

V)] (45) and e ( N ) is defined implicitly by N = NC(e+ V). The surface term Nc(D) exp[-P(D - V)]will be negligible for N d D ) exp[-P(D

-

most temperatures of interest. It should be noted that in general e ( N ) is not a single-valued function of N and care must be exercised in the use of eq 44 and 45. C. Other Characteristic Functions. The rate expressions 5, 14,36, and 42 are independent of the location of the dividing surface as long as it divides reactants from products. One convenient location is a surface which passes through the saddle point of the potential surface (when a saddle point exists). In particular, let the saddle point normal coordinates, scaled for reduced mass p, be z and u where z is the unbound degree of freedom and u is the bound degree of freedom, and let the dividing surface be the line z = 0 with positive z on the product side. Then the cumulative reaction probability of (15) becomes

This expression may be arranged as follows:

6.C.Garrett and D. G. Truhlar

form for evaluating Nc(E) [and hence, using (14) and (42), kc(E) and hc(T,D)] by numerical quadrature using characteristic functions x(flp,l,p,,O,u) obtained by integrating trajectories forward and backward in time with conditions specified on the dividing surface a t time t = 0. Equation 48 is also a convenient form for illustrating the equivalence of several forms which have been suggested for the characteristic function x(p,q). The definition of x(p,q) already given is due to Pechukas and McLafferty3 and is used in eq 3.7 of ref 4. Using this definition, every crossing of a trajectory through the dividing surface contributes +1, -1, or 0 to the integral (48). Consider a general trajectory which crosses the dividing surface m times. If m is even or if the trajectory originates from the P side it contributes 0 every time it crosses. If it originates from the R side and m is odd, it contributes +1 for crossings toward products and -1 for crossings toward reactants, giving a net contribution of +l. Thus the net contribution is '/'[ 1 - (-l)m] for trajectories originating on the R side and 0 for trajectories originating on the P side. Miller, in eq 4.1 of ref 4, suggested a slightly different definition. He suggested letting the contribution be zero for all crossings of trajectories originating on the P side and letting it be fl, depending on the sign of p a ,for all trajectories originating on the R side. This clearly leads to the same net contribution, +1 for reactive trajectories originating on the R side and zero for all other trajectories, but the weighting of individual crossings of trajectories originating on the R side and crossing an even number of times is different. A third method of counting was suggested earlier by Anderson and ~ o - w o r k e r s . ~ In ~ ,this ~~J~~ method, like the method of Pechukas and McLafferty but unlike the modification of Miller, the contribution of a crossing is nonzero only for nonreactive trajectories. In particular a crossing toward products contributes +l if it is part of a trajectory which originated from reactants and which proceeds directly to products without an additional recrossing. Other crossings contribute 0. Thus the net contribution is again +1 for all reactive trajectories originating on the R side and 0 for all others but the weighting of some individual crossings is different and there are no negative contributions to the integral. There is an obvious alternative version of Anderson's procedure, namely, to let each crossing contribute +1 if it is a crossing toward products which is part of a trajectory which originated from reactants with no previous crossings and which proceeds eventually to products. Other crossings contribute 0. We have considered so far four ways to count the equilibrium reactive one-way flux from reactants and products. All four methods give nonzero weight only to (some or all) crossings associated with R P trajectories. Four more methods are obtained by simply reversing the role of reactants and products since at equilibrium the flux from products to reactants equals that from reactants to products. Four more methods could then be obtained by weighting crossings associated with both R P and P R trajectories but dividing all weights by two. Finally we consider a thirteenth method, due to Kecklo5and incorporated in eq 3.9 of ref 4. This was the first method to be suggested; like methods 9-12 it gives nonzero weights to both R P and P R trajectories. In this method we again consider individual crossings of trajectories which cross the dividing surface m times. All crossings in the direction of products of R P and P R trajectories contribute rn-l. Crossings toward reactants and crossings of nonreactive trajectories contribute 0. Reactive trajectories originating on the R side make (rn + 1 ) / 2 crossings toward products and so make a net contribution

-

-

-

where i.e., there are two possible trajectories through the dividing surface for given pu,u, and E. Equation 48 is a convenient

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The Journal of Physical Chemistry, Vol. 83, No. 8, 1979

Generalized Transition State Theory

of (rn + 1)/(2m). Reactive trajectories originating on the P side make (m - 1)/2 crossings toward the products and so make a net contribution of (rn - 1)/(2m). There are an equal number of R P and P R reactive trajectories at equilibrium since each P R trajectory is the time P trajectory, therefore the net contrireversal of a El bution is again 1 per R P trajectory. The equivalence

-

--

-+

+

-

of methods 1-4 and 13 is even more clear if no trajectory has rn > 1. Then, in all five of these methods, every R P crossing contributes 1 and every I' R crossing contributes 0.

-

111. Approximate Microcanonical Rate Constants A . Conventional Transition State Theory. According to eq 5-7 and 14-16, after cancelling [pcSYy"(E)]-l in the numerator and denominator of (5), the exact classical microcanonical rate constant kc(E) is the net density of forward reacting stales per unit energy and time divided by the density $CR of reactant states per unit energy and length. In conventilonal transition state theory the net density of forward reacting states per unit energy and time is replaced by the density of forward crossing transition states per unit energy and length each weighted by the speed of crossing the transition state. The density of forward crossing transition states is one-half the density of transition states. The density of transition states per unit energy and length at total energy E is obtained by compounding the density pc'(el) per unit energy of vibrational statles of vibrational energy e* with the density Y ~ " ' ( E- V - €2) per unit energy and length of translational states of translational energy E - Vi -- &. For a state with this translational energy the speed of crossing the transition state is [2(E - Vi - e*)/p]1/2where we assume, as in section 11, that all coordinates have been scaled to a common reduced mass p. Thus the classical conventional transition state theory microcanonical rate constant is

1057

where E(#,S=O) is the vibrational energy of the conventional transition state for a given value of the classical transition state vibrational action N'h. Thus Nct(E) is the number of energetically available vibrational states of the transition state. These results can also be obtained from the exact expression (46) for the cumulative reaction probability. To do so we approximate the exact characteristic function x(pz,pu,O,u) by x+*(p,,p,,O,~) where x+*counts all phase points representing forward crossings as unit contributions and those representing reverse crossings as zero, i.e. X+'(Pz,Pu,O,U) =

O(Pz)

(57)

Using this approximation and eq 14, eq 46 can be cast into the form of eq 52 and 54 as follows:

and

kc'(E) = pCf(&yCrel(E - V - et)[2(E- Vi - e2)/p]1/2

LE-'det

Substituting yCrel(E-

Vt

- €1) =

[2p/(E - Vi - ~ ' ) ] " ~ / h (51)

yields

hcJ(E)= ['-'de* u o

~~*(e*)/[h$~~(E)]

+

Let Nc*(c V ) hbe the classical action variable for the transition state vibration of energy E * . Then pc*(d) = dNc'/dt'

(53)

Substituting (53) into (52) yields hc*(E) = Nc*(E)/[h4cR(E)l

(54)

which is the standard r e s ~ l t . ~ ~Comparing - ~ J ~ (54) to (14) shows that the scaled action variable Nc:(E)may also be interpreted as the conventional transition state theory approximation to the cumulative reaction probability. A third useful way to think about this quantity is

NcYE) = h-l f m d p , , l L d uB[E - (2p)-'pU2- V(z=O,u)] (6

-m

-L

(55) =

J"w O[E

-

V(z=O,u=O) - e(M,s=O)] (56)

we obtain eq 50 again, and eq 52 and 54 follow as before. As discussed at the end of section 11, x(p,q) = x+(p,q) if m 5 1 for all trajectories. Thus the second derivation shows that conventional transition state theory is exact if no trajectory crosses the conventional dividing surface through the saddle point more than once. Further, using any of the first four exact prescriptions for x(p,q) from section II.C, it is easy to see that x+(p,q)2 x(p,q) for any phase point. Thus transition state theory yields an upper bound on the rate constant. These properties of classical transition state theory are now well known. The second derivation of this section, combined with the third method34~86J04 of section 1I.C for assigning x(p,q), illustrates another important point. The transition state assumption is that all transition states crossing in the forward direction contribute to the net reactive flux. One need not separately assume an equilibrium distribution of transition states; the distribution of transition states is an equilibrium one because we are calculating the equilibrium rate constant. However the equilibrium distribution of forward crossing transition states includes forward crossing transition states which originated as products and forward crossing transition states which will recross the dividing surface (i.e., not proceeding directly to products). These are not to be counted in the net forward reactive flux but transition state theory counts them. Thus, as emphasized by Anderson,@transition state theory is just the assumption that the transmission coefficient (Anderson@calls it the conversion coefficient) is unity. The exact transmission coefficient is the ratio of the net reactive flux to the flux of forward crossing

1058

The Journal of Physical Chemistry, Vol. 83, No. 8, 1979

transition states. The exact classical transmission coefficient is always less than or equal to unity. It is less than unity when trajectories which originated as reactants recross the dividing surface or when some forward crossing transition states originated as products. The first of these two effects is the one usually mentioned in discussing the transmission coefficient. The second effect is usually not mentioned and is sometimes overlooked. In this article we set the transmission coefficient equal to one. The exact transmission coefficient is defined so that its use as a multiplying factor makes the theory exact. Equation 57 shows that the transition state approximation is a dynamical one. If one identifies the reactive the exact rate expression is obtained flux as F(p,q)X(p,q), by averaging the reactive flux over an equilibrium distribution of classical states (p,q). Thus the transition state approximation can be interpreted as a dynamical approximation to the reactive flux. A second interpretation involves identifying a distribution of forward reactive states as 6[E - H(p,q)]x(p,q). The exact rate expression is obtained by averaging the flux through a dividing surface over this distribution of forward reactive states. The transition state approximation can now be interpreted as assuming that the equilibrium distribution of forward reacting states is the same as the equilibrium distribution of forward crossing states on a dividing surface. This is sometimes called the quasi-equilibrium hypothesis. If any trajectories recross the dividing surface, this assumption causes errors even though the distribution of all phase points on the dividing surface is an equilibrium one. The basic connection between the two viewpoints just discussed is most clear in Miller's projection operator formalism where the transition state approximation is the replacement of an exact projector P by an approximate one The exact rate involves tr[FPD6(E - H ) ] and P , the projector onto states which originated as reactants, may be considered to operate to the left as a transmission-coefficient effect or to the right as a nonequilibrium effect. B. Adiabatic Theory of Reactions. The adiabatic model has been very useful in the interpretation of transition state theory and is well documented in the l i t e r a t ~ r e . l l - ~ ~ The adiabatic approximation is that motion along a reaction coordinate can be separated from the other degrees of freedom and that these bound modes adjust adiabatically to the system's motion along the reaction coordinate. In the collinear A BC reaction, the bound degrees of freedom are the electronic degrees of freedom (which are already understood to be adiabatic when the Born-Oppenheimer adiabatic approximation is made) and one vibrational degree of freedom. Thus the adiabatic theory in this case involves just the assumption of vibrational adiabaticity. In order to separate motion along the reaction coordinates from the vibrational coordinate us,it is convenient to neglect the curvature of the reactive coordinate and to lets and usbe natural collision c o o r d i n a t e ~ . ~(Although ~~J~~ reaction path curvature can be included in an approximate way,107-109 that is beyond the scope of the present work.) In the vibrationally adiabatic zero-curvature (VAZC) approximation, the Hamiltonian can be expressed as

+

% = (2y)-l(ps2 + p / )

+ V(S,US)

= (2y)-'ps2 + Va(N,s)

(62) (63)

where

V,(N,s) = V(s,uS=O)+ c(N,s) (64) and where c(N,s)is the local vibrational energy for a given s and a given classical action Nh. In the adiabatic ap-

B. C. Garrett and D. G. Truhlar

proximation N is a conserved variable. Thus the scattering problem is reduced to a one-mathematical-dimensional motion of a mass y on an adiabatic potential curve Va(NR,s) where NRdetermines the initial vibrational state of the diatom. The microcanonical adiabatic rate constant is given by eq 14 and 21 where the reaction probability Pc(NR,E)is now replaced by its adiabatic approximation PCA(NR,E), i.e.

Evaluation of PCA(NR,E)is trivial, since it equals unity if the local translational energy, E - Va(NR,s), is positive a t all points along the reaction coordinate and equals zero otherwise:

PCA(NR,E)= B[E - V,*(NR)]

(66)

where

V/(NR) = max Va(NR,s)= V,[NR,~*A(NR)] (67) S

and s , ~ ( Ndenotes ~) the location along the reaction coordinate where V/(NR,s)is the absolute maximum. Thus the classical adiabatic microcanonical rate constant is given by

kcA(E)= NcA(E)/[h$cR(E)l

(68)

where

NcA(E)= =

JNcR(E)

&NcR(E)

dNRPCA(NR,E) dNR B[E - VaA(NR)] (70)

At low energies only small values of NR contribute to the adiabatic rate expression in eq 68. For low NR, Va(NR,s)is dominated by the behavior of the potential along the reaction coordinate so its maximum occurs at or near the saddle point which we define as having s = 0. If we assume that s,*(NR) equals zero for all NRfor which V,A(NR)5 E , the classical adiabatic cumulative reaction probability of eq 70 becomes identical with the conventional transition state expression of eq 56. Clearly the integrands are the same and the upper limit of integration in eq 56 can be replaced by NcR(E)since s , ~ ( Nequal ~ ) to zero implies V,(N,s=O) I Va(N,s=-m).In other words, if the barriers of the adiabatic potential curves for all contributing states are located at the conventional transition state, then the adiabatic approximation is sufficient for the validity of the quasi-equilibrium hypothesis. However it is not necessary, Le., the quasi-equilibrium hypothesis might be valid to a good approximation even when the system is not a d i a b a t i ~ . ~ ~ ? ~ ~ ~ , ~ ~ ' The adiabatic theory identifies the threshold for the reaction for each initial vibrational state, specified by NR, as the maximum in the vibrationally adiabatic potential curve V,(NR,s)for that NR. Assuming the maximum to always occur at the saddle point leads to classical conventional transition state theory. If this is not a valid assumption the conventional method gives nonzero weights to values of N in eq 56 that do not contribute in the classical adiabatic theory. Therefore the conventional transition state theory rate constant is always greater than or equal to the adiabatic theory one. Using eq 24, 26, and 70 we can calculate the adiabatic theory average reaction probability which is

Generalized Transition State Theory

The Journal of Physical Chemistry, Vol. 83,

[NcR(E)dNRO(E - Va[NR,s**(NR)]) = -v u (72) &NcR(E)dNROIE - Va(NR,s=-a)]

No. 8, 1979 1059

same as zs for s = 0) and u (which is the same as us for s = 0). Thus a set of dividing surfaces normal to the reaction coordinate at various values of s is defined by zs = 0. Analogous to the derivation in section 1II.A we can derive the generalized transition state theory rate constant

Because Va[NR,s*A(NR)] IVa(NR,s=-m),PcA(E)is always less than or equid to one. By comparison the conventional transition state approximation to the average reaction probability may be obtained using eq 24, 56, and 64. It is

Pc’(E) = .NcyE)/NcR(E)

(73)

[IrndN OIE - Va(N,s=0)] =-

v IJ

(74)

iNcR’@)dNR OIE - Va(NR,s=-a)] Whenever V,(I\J,s=O) < V,(N,s=-m) this exceeds unity, which is unphysical. The analogue of these unphysical reaction probabilities in excess of unity for the case where vibrational energy is quantized was discussed previously.20 C. Microcanonical Variational Transition State Theory. As discussed in section 1II.A classical transition state theory for any choice of dividing surface provides an upper bound to the exact classical rate constant. This leads naturally to Variational approaches in which the dividing surface is positioned to minimize the transition state rate constarnt.1~4,13~25~z~~,68~1 One may minimize either the microcanonical or the canonical rate constant. In this subsection we consider minimizing hc(E);we call this classical microcanonical variational transition state theory and the variationally determined dividing surfaces are called classical microcanonical variational (generalized) transition states. In principle, one should consider arbitrary variations of the dividing surface in phase space. If one does this one can obtain the (exact result.32 In the present work we do not consider such arbitrary variations. For the collinear A + BC reaction the dividing surface in configuration space is a curve and we consider only dividing curves which are straight lines in the (x,y) plane, where x and y are the reactant Jacobi coordinates scaled to achieve a common reduced mass as in eq 2-4. Further we require the dividing line to be perpendicular to the path of steepest descent in the x,y plane at the point where the dividing line intersects the path of the steepest descent. We vary the location of this intersection. Koeppll has considered a different kind of variation of the dividing line, namely, he required the dividing line to pass through the saddle point but he varied its ~ r i e n t a t i o n .Complicated ~~ variations of the dividing surface are not practical, and the variational freedom allowed here is sufficient to illustrate the power of the variational methods. Pollak and Pechukas have identified the best transition state in the microcanonical variational sense with a bound trajectory of the three-body system.@ This method offers a possible way to circumvent the difficulties associated with finding generalized transition states. With the above constraint on the variational freedom of the dividing surface the generalized transition state rate expression can be derived straightforwardly. A set of orthogonal coordinate systems, differing from ( x , y ) by a translation of origin and a rotation of axes, is defined by (zs,us), where zs is tangent to the reaction coordinate at s and is zero a t tlhe point of tangency, and us is normal to zs and zero at its intersection with the reaction coordinate. These coordinates are generalizations of z (which is the

where we have introduced the shorthand notation Vs for V(zs=O,us=O);i.e., Vs is the value of the potential at the bottom of the local vibrational well for a value s of the reaction coordinate. The local number of energetically available states is

= &E-v’dc pCGT(c,s)

From (80) it is clear that NcGT(E,s)also can be defined implicitly by Va[NcGT(E,s),s]= E

(82)

The classical microcanonical variational transition state is the minimum value of hCGT(E,s) rate constant hCfivT(E) which can be obtained by varying s, i.e. hC”VT(E) =

NC”VT(E)/[h+CR(E)]

(83)

where

NcPvT(E)= min NcGT(E,s)= N C ~ ~ [ E , S * ~ ~ ~(84) (E)] S

6NcGT(E,s)= - GV,(NS,s)/ 6 ~ ~(87) 6s 16 Va(Ns$) / aNsIN S , N ~ G T ( E , ~ )

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The Journal of Physical Chemistry, Vol. 83, No. 8, 1979

The microcanonical variational criterion locates the dividing surface at the position of the minimum number of states in the bound degree of freedom for a specified total energy E. In comparison the adiabatic theory predicts a threshold for each value of NR. However Va[NR,S*A(NR)]I V,[NR,s*”VT(E)]

(88)

therefore eq 66 is equivalent to

PCA(NR,E) = O(E - V,[NR,~*”VT(E)]) (89) and eq 69 yields Va[NR,~*VT(E)] (90) which is equivalent to NcWw(E) as given by eq 80 and 84. In other words, NcA(E)is independent of the location s * ~ ( Nof~all) the adiabatic barriers and barrier heights except for the one with barrier height equal to E. However, as discussed following eq 87 this barrier is located at s = S**[NC”~(E)], which is the position of the dividing surface of generalized transition state theory which minimizes the microcanonical rate. Therefore, even though microcanonical variational transition state theory and the adiabatic theory are in principle very different, they predict the same classical microcanonical reaction rates. Furthermore since the microcanonical variational transition state theory is an upper bound to the rate constant both theories provide upper bounds, and since the adiabatic theory assures average reaction probabilities less than unity the microcanonical variational transition state theory will also guarantee average reaction probabilities less than unity. However although the two theories predict the same classical reaction rates they are amenable to different types of approximations and different kinds of improvements when quantum mechanical effects are included. Thus the rate constants predicted by the two theories need not remain identical when quantal effects are included. The theories will also differ if the constraints on the shape of the dividing surface are removed for the microcanonical variational calculation. D. Unified Statistical Theory. Miller has proposed a unified statistical theory which tends in the limit of a single dominant saddle point to the conventional transition state theory and in the limit of two or more loose bottlenecks to the statistical model of Pechukas and Light55 and Nikitin.56 Using a probability branching analysis’’ for a system with two bottlenecks, Miller derived a more general result for the cumulative reaction probability, i.e.

B. C. Garrett and D. G. Truhlar

If we again assume that the approximation of straight-line dividing surfaces is exact (that is, straight-line dividing surfaces are the result of arbitrary variations of the dividing surface) then Pcus(E)is exact if only generalized transition state exists. For energies at which there are two or more local minima in NCGT(E,s), Pc”w(E) overestimates the exact results. Pcuswill always be lower than PefiVT(E) at these energies and therefore may offer a better approximation to Pc(E). IV. Approximate Canonical Rate Expressions A. Conventional Transition State Theory. The canonical rate constant of classical conventional transition state theory can be written by using relation 42 as

J D d E exp(-PE)Nc’(E) kc’ (T,D) =

(94)

h@cR(T,D) Substituting eq 56 for Nc’(E) gives k&(T,D) = [hPCR(T,D)]-’JDdE 0 e-bEJmdN 0

OIE - Vt. - E(M,s=O)] (95)

O(D - E ) OIE - V - E(M,S=O)] (96) (97) where the cutoff partition function for vibrational motion at the saddle point is defined by

QcTT,?) OID -

= l m d Nexp[-Pt(M,s=O)] X 0

v - t(N,s=O)]- Nc’(D) exp[-P(D - v)](98)

and t(N,s=O)is defined by eq 64. B. Adiabatic Theory of Reactions. The canonical rate constant of the adiabatic theory is likewise obtained by employing relation 42 and expression 70 to obtain NcR(E)

kcA(T,D)= [h@cR(T,D)]-’iDdE n e-oEJ n

dNR

exp(-PE) O(D - E)O[E- VaA(NR)] (100)

A more compact form is

NcUS(E)= Nc””T(E)Ncmin(E)NcmaX(E)

+

[NcFVT(E) Ncmin(E)]Ncmax(E) - NcFVT(E)Ncmin(E) if sernin(E) exists = NCpVT(E) if not (91) where NCmln(E) = second lowest local min NcGT(E,s) S

QcA(T,D)= N~*(D)

dNR e~p(-/3{V,[iV~,s*~(N~)] - VI) iVcA(D)exp[-P(D - VI1 (102)

= NcG*[E,S*min( E )1

(92) Ncmax(E)= max NcGT(E,s)

where we have identified a generalized cutoff vibrational partition function QCA(T,D) as

(93)

S*I”V*SS