An exact quantum mechanical transition state theory. 1. An overview

Quantum Mechanical Transition State Theory

The Journal of Physical Chemistry, Vol. 83, No. 1, 7979 171

An Exact Quantum Mechanical Transition State Theory. 1. An Overview? Aron Kuppermann Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 9 1 125 (Receiwed November 17, 1978)

Starting with the collision theory expression for the rate constant h of a thermal gas phase bimolecular reaction in terms of reaction cross sections, we derive a modified exact expression for h which has a mathematical form identical with that provided by transition state theory (TST). No approximations are introduced in this derivation and the only difference between this exact expression and the TST one is in the transmission coefficient K . The exact one, K ~ is~ expressed , in terms of the correct reaction probakiilities, whereas the TST one, K ~ involves ~ , transmission probabilities across appropriately defined one-mathematical-dimensional adiabatic barriers. It is shown that K~~ can be larger or smaller than unity, and need not approach one as the temperature of the reaction tends to infinity. Comparison of the expressions for K~~ and K~~~ leads to a necessary and sufficient set of conditions for validity of TST. These involve only dynamical assumptions regarding reaction probabilities, and do not require any additional equilibrium condition among activated complexes or between them and reagents; the only equilibrium assumption made is the one among reagents, which is inherent in the definition of a thermal rate constant. A comparison of usual TST rate constants with the exact one for the H + H2,F + H2, and F + D2 systems is given, the validity of the dynamical assumptions of that theory is examined, and suggestions for its improvement are presented and tested.

1. Introduction Eyring's formulation' of transition state theory (TST) stresses the thermal equilibrium among the activated complexes and between them and reagents. The theory invokes the evaluation of the concentration of those complexes by the methods of equilibrium statistical mechanics. 'The transmission coefficient, which is considered in the classical mechanical case as a factor smaller than but close to unity, is introduced in order to take into account the possibility that an activated complex may cross the activation barrier and return without decomposing into products. It is definedl as the reciprocal of the average number of crossings of the barrier required for each complex which reacts, but no explicit prescription is given for its evaluation. The stress on the structural aspects of the activated complex, needed to evaluate its partition function, and the lack of emphasis on the dynamical aspects of the theory, are the basis for its past success. This makes it practicable to apply TST to a fairly large variety of reactions and to temperature ranges over which the rate constants may change by many orders of magnitude. On the other hand, this structural focus makes TST unsuitable for accurate rate constant calculations which do depend on the dynainical aspects of the reactive process. Another conceptual difficulty of TST is that its development is classical in nature, insofar as treatment of the motion along the reaction coordinate is concerned. Quantum corrections to the resulting rate constant have been treated by Wigner,2aHirschfelder and Wigner3, and Eyring, Walter, and Kimball,4 and more refined versions have been suggested more r e ~ e n t l y . ~It- ~is usual to apply to the classical partition functions of the activated complex (the pseudo-partition functions in the nomenclature of Johnston') the same quantum corrections as for true molecular partition €unctions and to replace the classical motion along the reaction coordinate by the corresponding quantum motion. According to Johnston, such procedure 'Research fiinpportedin part by the United States Air Force Office of Scientific Research (Grant No. AFOSR-77-3394). t Contribution No. 5742

"appears to ble better chemical engineering than philosophy".8 Major developments of exact calculation procedures during the last decade and a half have permitted the solution of the classicalg and quantum mechanicallo equations of motion for reactive collisions on realistic electronically adiabatic potential energy surfaces, from which accurate" thermal rate constants were obtained. The availability of the results of such dynamical calculations permit a very intimate test of the foundations of TST. Tests of this type are more sensitive than a comparison between theoretical and experimental rate constants, because of the lack of accurate knowledge of' the potential energy surfaces which describe the experimental reactive systems$,and because of the much greater level of detailed dynamical information provided by the scattering calculatialns. Examples of such tests of the basic assumptions of ?'ST for the cIassica19J2and q u a n t u r ~ i ' ~ J ~ versions of the theory are available. Tests of this kind require a quantitatively more precise formulation of TST, with a more explicit specification of its dynamical aspects than given Eliason and Hir~chfelderl~ have developed a very clear collisional approach to TST', in which the dynamics are also contained in a transmission coefficient not explicitly related tlo reaction cross sections. Marcus has formulated important generalizations of TST.15 He develops these generalizations using curvilinear coordinates for finding separable approximations to the potential energy function. The quantum mechanical version of the resulting generaked expression for the rate constant does not refer explicitly to the activated complex but contains instead a generalized transmission coe€ficient. It reduces to the usual expression when a Cartesian metric is employed. Several choices have been proposed for the dividing surface in configuration space which defines the activated complex. 16-19 They usually involve flux criteria associated with the classical mechanical versiion of the theory. Some of these methods have recently been tested against the results of accurate collinear quantum calculations by Truhlar and co-worke m z o Miller21 has recently developed a generallized

0022-3654/79/2083-0171$01.00/0 @ 1979 Amerlcan Chnmlcal Society

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quantum version of TST devoid of any assumptions (such as the separability of the Hamiltonian in the reaction coordinate and transverse coordinates) other than that implied in the choice of the dividing surface, and considered its semiclassical and classical limits. For a recent review of the assumptions of TST and its tests, see the two papers in the present issue by Marcuszzand by Truhlar.20e In the present paper we present a formulation of TST for gas phase bimolecular reactions with a strong focus on dynamical quantities, namely, reaction probabilities. The starting point is the collision formulation of rate constants in terms of reaction cross sections of Eliason and Hir~chfe1der.l~ The expression for the rate constant is transformed, without introducing any approximations, by changing the origin of measurement of translational energies from the separated reagent region to the strong interaction (i.e., transition state) region of configuration space, resulting in a rigorous expression for the rate constant which strongly resembles the TST expression. In this rigorous expression, the generalized transmission coefficient is given explicitly in terms of reaction probabilities. Obvious approximations to the latter lead to the usual TST. Modifications of these approximations are then suggested which can improve this theory. In section 2 the derivation of the exact generalized TST is given for collinear atom-diatom reactions. The necessary and sufficient conditions for validity of the usual TST are then obtained in section 3, and some improvements in the latter are given in section 4. A generalization of these ideas to arbitrary gas phase thermal bimolecular reactions is outlined in section 5. The usual TST for such reactions is derived in section 6, improvements in its dynamical assumptions are considered in section 7 , and state-to-state rate constants are considered in section 8. A summary and conclusions are given in section 9.

In this expression, pAac is the reduced mass of the colliding partners, k is the Boltzmann constant, and FT(Etr)is normalized according to

It is convenient to rewrite the mass-containing factor of eq 2 as 1/(2w*,BCkT)1/z = (1/ h ) @ / f l I % c )

(3)

where h is Planck's constant and f$ (X = A,BC,I) is the translational partition function per unit length of a particle of mass mx given by

f"$

(l/h)(2rmXkT)' '2

The symbol mi represents the total mass of the system (mA + mp, + mc). Equation 3 already contains all translational partition functions of transition state theory which, therefore, stem directly from the assumption of a Boltzmann distribution of the reagent relative translational energies. Replacement of eq 3 into ey 1 yields

U T ) = (kT/h)(f:'/fiftB'c)P, (4) where P, is the average reaction probabilityz4defined by P,(T) =

Lop,exp(-Etr/kT) dEtr/

lm exp(-Et'/kT) dEtr = 0

( l / k T ) ~ m Pexp(-E"/kT) , dEt' (5) 0

2. Derivation of an Exact Transition State

Theory for Thermal Gas Phase Collinear Atom-Diatom Reactions The ideas presented in this paper have general validity for thermal bimolecular gas phase reactions. We present them initially for electronically adiabatic collinear reactions of the type A + BC AB + C because these contain the essence of the physical ideas involved, without being incumbered by partial wave expansions, bifurcation problems, degeneracies, electronic transitions, or configuration spaces of high dimensionality. Once these ideas are grasped for this mathematically simple example, their generalization is reasonably straightforward. Throughout this paper, effects associated with perturbation by the reaction of the thermal equilibrium among the reagents will be ignored, as these have been extensively considered elsewhere.z3 Let P,(Etr)be the cross section for reaction of BC, initially in internal vibrational state u, with atom A, at an initial relative translational kinetic energy Et'. The products are assumed to be AB + C. This cross section, which is dimensionless in a collinear world, is also a probability in this world. In three dimensions there will be a difference between probabilities and cross sections, which is considered in section 4, but which does not concern us here. The thermal rate constant k,(T) corresponding to a Boltzmann distribution F,(Et') of the translational energies Et' at absolute temperature T but for a state-selected initial vibrational state u is given by14

-

h ( T ) = JYF~(Etr)VP,(Et') 0 dEtr = (VP,)

collinear case being considered13b

(1)

where V is t,he relative collision velocity and, for the

The universal frequency factor, k T / h , appearing in eq 4 is a trivial mathematical consequence of the definition of P, and of the translational partition functions. It will be preserved intact in the final generalized TST rate constant expression, but no particular physical meaning need be attributed to it, since it already appeared in an expression which makes no reference so far to a transition state. All of the information about the dynamics of the reaction are contained in P,, the remaining factors having to do with details of the thermal distribution function of the reagents. Let us now obtain the full thermal rate constant k(T) by averaging h,(T) over the thermal distribution function of initial BC vibrational energies. Denoting the vibrational energy of state u of BC (assumed to be nondegenerate) with respect to its own ground vibrational state by E\, we have14

k ( T ) = ( l / f ~ ~ ) C e x p ( - E ' , / k T ) k , ( T ) (6) L

where f?; is the internal (Le., vibrational) partition function of BC with respect to its own zero point energy (ZPE): f & = Cexp(-E',/kT)

(7)

L

Since A is an atom, its total partition function per unit length f'A is the same as its translational partition function per unit length pi, The total partition function per unit length flBC of BC with respect to its own ZPE is /&fT&. These considerations and the last two equations yield k (TI = (kT/h)

(g/f 'J'Bc)

Cexp (-E ',/kT)P, u

(8)

This collision theory expression is closely related to one

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


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'I

AB+C nrnducts

A+BC rezgenls

I 1

t
1, even in the absence of tunneling. For example, even under conditions for which quantum effects (other than the presence of zero point energy in the reagent) are negligible, and quasi-classical trajectory methods can be used to evaluate P,, a value of K~~ larger than unity is expected, due to reagent zero point energy contributions to these P,. Furthermore, P, can be

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_ _ 1

1.01l-

H +H2 ( v - o ) - - H ~ +H

0.2 I

collinear scaled SSMK s u r f a c e

E'YeV Figure 3. Reaction probabilities as a function of translational energy for the H H2(I/ = 0) H2 H reaction on the collinear scaled SSMK dashed curve), classical vibrationally adiabatic 1 MD surface: probability; fhD (dashed-double-dottedcurve), uantum vibrationally adiabatic probability without curvature correcton; (dasheddotted curve), quasi-classical trajectory collinear (2 MD) probability; Po (solid curve), exact quantum collinear (2 MD) probability.

+

6'

--+

+

4""'

-

T/ K ,I,Op,O, , , 500 400

300

200

c3iiinear s c o l e d

e'

either smaller, equal to, or larger than even if quasiclassical P, are used In addition, there is no requirement 1 alsymptotically as T a,since even a t high that K energies P, can differ appreciably from The rest of eq 20 involves only quantities which resulted from the assumption of thermal equilibrium among the reagents and definitions of energy levels associated with the adiabatic potentials. A knowledge of P,(Etr)permits an accurate evaluation of k ( T ) either by using the collision expression (eq 8) or the generalized T S T expression (eq 20). The former is usually much easier to apply, and the main reason for considering the latter altogether is to derive physical insight and/or to introduce simplifying approximations. Such approximations necessarily involve approximations in K~~ and therefore in P,, and can be introduced in eq 8 instead, if desired. It is interesting to note that Eyring, Walter, and Kimbal14 presented a derivation of usual TST similar to the derivation of eq 20 above, in that they started out obtaining an expression analogous to eq 8 (their eq 16.15). However, the crucial transition to eq 20 through the use of eq 14 was not made and, therefore, the explicit expressions for f \$and K~~ were not obtained. In addition, as pointed out above, the explicit dependence of K~~ on P, is essential for analyzing the conditions for validity of TST.

-

Figure 4. Same a!; Figure 3, for the collinear PK surface.

3. Necessary and Sufficient Conditions for Validity of Usual TST for Collinear Atom-Diatom Reactions 3.1. Unit Transmission Coefficient. We can now obtain the usual T S T by introducing appropriate dynamical approximations into eq 21. Let us first consider the version of the theory without "tunneling" corrections, for which K ~ ~ =( T1.) A sufficient condition for eq 21 to yield this result is that

P,(Etr)= P",'(Etr) (26) It should be remembered that P,(Etr)is the correct quantum mechanical reaction probability from state u of the reagent 13C to all states of the product AB. It can be calculated by solving the quantum mechanical problem in two mathematical dimensions (2 MD) describing the motion in R,,,r, configuration space of a point of m a s s p (given by eq 10) subject to the potential V. On the other hand, p",'(Et")is the Heaviside step function (defined by eq 16) which is the solution of the classical mechanical

IOOOK/T

Flgure 5. Transmiiesion coefficients and rate constant ratios as a function of reciprocal temperature for the H f H, --+ H, H reaction on the collinear scalled SSMK surface: (a) K~~ (dashed horizontal line), (dashed-double-dotted classical transmission coefficient (= 1); K;!& curve), quantum vibrationally adiabatic transmission coefficient "ithout curvature correctbn; K~~~~~~ (dashed-dotted curve), quasi-classical collinear transmission coefficient; K" (solid curve), exact quantum approximate rate collinear (2 MD) transmission coefficient; (b) kapp, (d,ashed constant k;exact quantum collinear (2 MD) rate constant; kTST curve), TST rate constant with unit transmission coefficient; other rate constants k are T8T rate constants with transmission coefficients indicated by superscirnpt and/or subscript. The rate constant ratios plotted are equal to the corresponding ratios of transmission coefficients.

+

problem of the tiransmission of a particle of mass p through a 1 MD barrier (of height E:. If the condition K ( T )= 1 is to be valid a t all temperatures, eq 21 implies that Pu = at all temperatures and, therefore, that elq 26 be valid at all translational energies.29 Therefore, eq 26 is a necessary and sufficient condition for the validity of K ( = 1at all temperatures. This equation represents a dynamical assumption which can easil!y be tested, if P,(@) i.s known. Three such tests are presented. In Figure 3 we have plotted the H + H,(u = 0 ) --* W, C H reaction probabilities for the scaled13 Shavitt, Stevens, Minn, and Karplus (SSMK) potential energy surface.30 The exact collinear quantum results are those of Truhlar and K ~ p p e r m a n n . In ~ ~Figure ~ ~ ~ 4 the corresponding results31 for the same reaction on the Porter and Karplus (PK) surface32a:re given. In these figures, in addition to VELEtrcurves, we display other curves to be the Po and 6'

in

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Aron Kuppermann

T/K

.........

...

+D

L

F+D2 (v.0) -Fa

0.1

1.0

_ _ _ I _ _ L

0.2

E'YeV

-

Figure 8. Reaction probabilities as a function of translational energy for the F -D2( l v = 0) FD 4- D collinear reaction (from ref 33c,d). Same notation as for Figure 7. 0.05[-

t

\

T/ K 500 400 300

IO00

I

I I ~ I I II

Figure 6. Same as Figure 5, for PK surface. Included are curves for the semlclassical transmission coefficient K'" (dotted curve in panel from the cal(a)) and the corresponding rate constant ratio kscl/keY culations of ref 47.

I

1

,

I

200

,

I /

kZMDQCT/k

-.-._

-,-.-.

......................

.................................................

0.5

RU"/t?

k 2 M D Q CT/k

:j

I.-'-'-'---'-

1

-.

..................

............. .......... ....

0'5 F + H 2 - F H + H

-.\.

.................. ...... ......... ......

IOOOX/T Figure 7. Reaction probabilities as a function of translational energy FH H colllnear reaction (from ref 33b,d): for the F f H2(v = 0) P"' dashed curve), classical vibrationally adiabatic 1 MD probability; @6,T (dasheddotted curve), quasi-classical trajectory collinear (2 MD) probability; Ffsc (dotted curve), uniform semiclassical probability; P o (solid curve), exact quantum collinear (2 MD) probability.

+

described in sections 3.2 and 4.1. It can be seen that the exact quantum probabilities differ appreciably from step functions. The effect of this difference is to make the usual TST transmission coefficients ( K = 1)and rate constants significantly smaller than the exact ones for temperatures below 800 I(,as shown in Figures 5 and 6. The reason is that over this temperature range the Boltzmann distribution of translational energies samples the energy range below E);more than that above E:, thereby enhancing the contribution of tunneling to this reaction. In Figures 7 and 8 we comparePC,'(Etr) with P,(Etr)for the F + Hz(u = 0) F H + H and F + Dz(u = 0) FD + D collinear reactions on the Muckerman 5 potential energy surface.33 For these cases also, these two functions differ significantly. (The additional curves displayed are described in section 4.1.) The corresponding rate constant ratio kTST/k is displayed in Figure 9 and, as can be seen, is appreciably greater than unity. We conclude that, although eq 26 is acceptable on physical grounds, it is really an assumption involving

-

-

Figure 9. Relative rate constants for the F f H,(v = 0) FH 4- H and F D2(v = 0) FD -t D rate constants as a function of reciprocal temperature. Same notation as for Figures 5 and 7.

+

-

----,-

k"SC/R

-+

minimum knowledge about the reaction dynamics. Several improved assumptions suggest themselves outright, some of which are considered in sections 3.2 and 4. I t is important to notice that the approximation described by eq 26 is physically reasonable on vibrationally adiabatic g r ~ ~ n donly s ~if ~the~ barrier J ~ ~ height involved in eq 16 is E: rather than the Et of eq 23 employed in the usual TST.1-5 Therefore, the partition function fttof eq 18 rather than the usual one fTt of eq 25 should be used. These same conclusions were arrived a t by T r ~ h l a as r~~ part of an explanation of why Morokuma, Eu, and K a r p l u ~found ~ ~ that for many cases the usual TST implies reaction probabilities which exceed unity. He suggested a corrected TST in which eq 16 and 18 rather than eq 25 and the usual = H(Et' - E;)

+

are used, and showed that it eliminated the Morokuma, Eu, and Karplus anomaly. The derivation of the generalized TST given above, coupled with the physically reasonable approximation given by eq 26 and 16, is in complete agreement with Truhlar's suggestion. The concept that the configuration of the transition state may depend on its internal quantum numbers was also proposed by M a r ~ ~and s Elliason ~ ~ ~ and , ~ Hirschfelder,14 ~ but



not used in actual calculations other than T r ~ h l a r ' s .In~ ~ the rest of this paper, unless otherwise stated, in refering to the usual ?'ST, we will mean its corrected version34in which the Et are replaced by the Eh. 3.2. Vibrati;onally Adiabatic Transmission Coefficient without Curvature Corrections. As pointed out a t the beginning of section 3.1, the accurate evaluation of P,(Et') involves the solution of a quantum mechanical scattering problem in 2 MI) corresponding to the collinear reaction which occurs in one physical dimension (PD). If we can replace that 2 MD problem by a 1 MD one, a considerable decrease in computational effort will result. The simplest way of achieving this is by setting

P"(Et') = PAMD(Et')

(27)

where PLMn(,Ftr) is the quantum mechanical transmission probability of a particle of mass p across the 1 MD barrier E,(q), ignoring curvature effects (see eq 44, section 4.2). The resulting vibrationally adiabatic zero curvature (VAZC) transmissioii coefficient

K#&T)

=

(28)

(P;MD/P;C')b;

is a slight generalization of the usual vibrationally adiabatic TST transmission coefficient given by

~$i\zc(T)= PiMD/%'

(29)

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make K~~ >> 1. At sufficiently high temperatures, rlelatively high valuer3 of Et' for which Po(Et')is significantly smaller than unity contribute significantly to K~~ and make it smaller than one. There is a range of Et', from about 0.3 to about 0.5 eV, for which Po N 1for both the PK ,and scaled SSMK surfaces. Associated to it there should be a temperature range for which K~~ N 1, and above which it continues to decrease. We wish to reiterate that nothing in the dynamics of collinear chemical reactions requires that a t high temperatures K = 1since nothing requires that at high energies ZJ"(Et')= p",'(Et').

4. Improvements in Collinear TST Since the explilcit expression (eq 21) for ~ ~ ' ( involves, 7') in addition to the energies E $ evaluated a t the barrier maxima, only the 2 MD reaction probabilities P,, any approximations to the latter which are improvements over eq 26 and 27 should lead to improved TST's. The nature of these approximations is exclusively dynamical, and will, in general, be quite different in character from the assumptions made in the early justifications of TST1-4but not used in deriving the exact expressions 20 and 21. Two straightforward dynamical approximations leading, to improved versions of collinear TST are given below, as illustrations of the possibilities permitted by the present formulation, and several others are mentioned in section 4.3. 4.1. Quasi-Classical Trajectory Transmission Coofficient. If, in eq 21, we use the correct 2 MD quantum mechanical probalbilities P,, eq 20 is exact. An obvious approximation is to set

where the u = 0 (i.e., ground state) vibrationally adiabatic probabilities are used in calculating the transmission coefficient. This generalization was used previously by Truhlar and K ~ p p e r m a n n . ~ ~ ~ It is expected that eq 27 should be a better ap roxi] ~ " ( ~ t r= ) PZMDQCT(E~~) (30) mation to the reaction dynamics than eq 26. The PA vs. where PiMDQCT(Etr) is the quasi-classical trajectory colliEt' curves of Figures 3 and 4 show that this is in fact so for the H + H2 Ha -I- H reaction on the ~caled-SSMKl~~9~near (i.e., 2 MDJ reaction probability. Although this probability does not contain any tunneling corrections, it and the P K s ~ u r f a ~ erespectively. s,~~ The resulting effect includes 2 MD d,ynamical effects such as the interaction on the 1 ransmission coefficient is shown in Figures 5 and between the q arid p degrees of freedom, omitted in the 6. We see that for the scaled SSMK surface, using eq 27 vibrationally adiabatic approximation of eq 27. Therefore, produces somewhat worse transmission coefficients than using eq 26, for temperatures above 275 K. The reason eq 30 is not only expected to be an improvement oveir eq is that the coirresponding v = 0 adiabatic barrier is fairly 26 but, under conditions for which tunneling is not important, it is also expected to be an improvement over eq flat and thick., resulting in 1 MD adiabatic probabilities 27 and over eq 31 of section 4.2 which includes curvature that do not deviate strongly from a step function. However, to the extent that such a deviation does occur, corrections. The use of eq 30 in conjunction with eq 21 is equivalent to using it with eq 1and 6 which are the exact it involves amounts of 1 MD tunneling at energies below collinear collision theory expressions. In other words, the the barrier top which are in average less than the amounts of 1 MD reflection a t energies above the barrier use of 2 MD quasi-classical trajectory reaction probabilities Since, for this surface, initial states with u > 0 contribute in the generalized T S T expression (eq 21) is exactly no more than 2% t o the rate constant for temperatures equivalent to the quasi-classical trajectory theory of rate constantsg as particularized to collinear systems. It should up to 1200 K , the net result is that K;?& < 1 over the temperature range 275 K to a t least 1200 K. This is a be noticed that this equivalence requires the use of striking manifestation of the fact that K is not merely a quantized initial reagent states. tunneling correction but, as stated toward the end of In Figure 3 we display the PZMDQCT vs. Et' curve,39tosection 2, contains the entire information about the regether with the olther ones mentioned above for the 13 action dynamics. For the P K surface, the difference Hz(u = 0) H2 + H collinear reaction on the scaledbetween the PAM"and vs. he curves is more appreciable SSMK surface. A s can be seen, it is fairly close to the p"d (see Figure 4),because the corresponding v = 0 adiabatic and PAMD curves, probably because of the thickness of the barrier is thinner than the scaled-SSMK one, permitting scaled-SSMK b a ~ r r i e r . ~ For ~ ~this particular potential more 1 MD tunneling and generating dMD values37closer energy surface, neither the PAMD nor the PZMDQCT aPto K ~ The ~ . reason that K ~ ~ differs / K so~ appreciably ~ from proximations produce major improvements over the p"d unity at. room temperature and below, is that tunneling one, as can be seen by comparing the corres onding ~(7') in 2 MI), as determined from an analysis of the streamlines of Figure 5. For the P K surface, the PfMgCT curve of of probability current density,38occurs mainly through Figure 440differs from the 6' curve somewhat more than cutting of thi. minimum energy path corner, an effect in the scaled-SSMK case and leads to a more significant associated with a breakdown13d of the vibrational adiaimprovement in the corresponding classical transmission baticity assurription. coefficients and rate constant^;^^ K ~ as shown ~ ~ in Observatioii of the Pxvs. 1 / T curves of Figures 5 and Figure 6, is appreciably greater than unity a t room 6 seems t o inldicate that as T 00. K~~ 1. This is not temperature. (Other results for the P K surface are actually the cdse. At low temperatures, tunneling effects considered in section 6.)

R

-

-

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+

Q

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Aron Kuppermann

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In Figures 7 and 8 we also display the PPMDQCT vs. Et' curves for the F + Hz(u = 0) F H + H and F + Dz(u = 0) FD + D reactions.33 It can be seen that in the F + Hz case there is a significant difference between this curve and the corresponding exact collinear 2 MD Po and also the p"d one, resulting in a 50% deviation of K ~ ~ ~ from unity, a t temperatures above 300 K, as shown in the lower anel of Figure 9.33b-d This difference between PiMDaR and Po is due not only to tunneling, but also to the other quantum effects associated with the fact that, in these highly quantum systems, the potential energy varies significantly over a distance of one local de Broglie wavelength. Conceptually, the approximation represented by eq 30 illustrates the important point that the transmission coefficient K in eq 20 is neither a tunneling correction nor even a quantum correction to the usual TST for which K = 1. It is instead a full dynamical correction to the simple approximation of eq 26, and may be greater than unity even if quasi-classical collision dynamics is used to approximate it, due to the vibrationally nonadiabatic contribution of the reagent vibrational energy to the reaction probability. In the same vein, the correct quantum partition function of the transition state given by eq 18 did not result from the "chemical engineering" approach mentioned in the Introduction but from a rigorous handling of collinear quantum reactive systems. Therefore, Johnston's criticism8 of the quantum version of the usual TST is not applicable to the present formulation. 4.2. Vibrationally Adiabatic Transmission Coefficient with Curuature Correction. An improvement over the vibrationally adiabatic approximation to P, without curvature effects, given by eq 27, is to include such effects, Le., to use

-

P,)(Et')= PAMDC(Et')

hb,(p;q) = E,(q)4, Q

The~ solution / K ~of eq ~ 36 furnishes the vibrationally adiabatic barriers depicted in Figure 2 and used for lhe derivation of eq 20. Let $(q3p)be the bolution of the collinear three particle Schroedinger equation

~

HICi(q,p) = E$(q,p)

(37)

where E is the total (translational plus vibrational) energy of the system. The correct reaction probabilities P, are obtained from solutions of this equation which satisfy appropriate reactive scattering boundary c o n d i t i ~ n s . ~ ~ , ~ ~ In order to solve eq 37 it is convenient to expand $ ( g y p ) in terms of the &,(p;q), ICi((7,P)=

c, g, ( q b , ( p ; q )

(38)

Substitution of eq 38 into eq 37. multiplication by qq39p4:,(p;q),integration over p , and use of the orthogonality relation

J $ ; b d h (P;d'IqvpdP

= 6,,,

(39)

leads, after u and u' are interchanged. to the set of caupled-channel second-order differential equations

where g is the column vector whose elements are the coefficients g,(q) of the coupled-channel expansion (eq 38) and the W, U, and 6 matrices are defined by

(31)

where PAMDC is the transmission probability for a 1 MD barrier which includes curvature corrections as specified below. The appropriate formalism involves the use of properly defined curvillinear coordinates; the use of such coordinates in TST has been investigated by Marcus.15c,16a Since for the present purposes we are considering reaction probabilities only (none of the other aspects of eq 20 are affected by the approximations to PL),we will define such curvature-corrected probabilities by a somewhat different and more direct route than previously.16c,16aLet us consider the system of orthogonal curvilinear distance coordinates (q,p)introduced after eq 10, and used to define the vibrationally adiabatic energies E,(q). An example of such coordinates is displayed in Figure 1 and is defined in greater detail toward the end of this section. The Hamiltonian for the motion of the effective particle of mass p describing the collinear ABC system is, in this or any other orthogonal system of curvilinear coordinates42a c

where42b

In eq 40, which is exact, the g, are coupled by the offdiagonal elements of W, U, and t, In the adiabatic approximation with curvature corrections, we assume that the off-diagonal elements of all three of these matrices can be neglected. This assumption decouples the g, and is equivalent to taking only one term in eq 38, namely, the one Corresponding to the initial state of the reagent

-

It is informative to examine the behavior of the W, U , and e matrices as q f m , Le., in the separated reagent and separated product regions. In the former, r becomes independent of q , R becomes independent of p , aR/aq I , and g),(p;q)becomes independent of q. Therefore, qq 1,aqp/dq 0 and a @ / a q 0. Conspquently, W 0, U 0, and t,,, ( E - EL)6,,,,. This means that the g, of eq 40 automatically become decoupled in the separated reagent region. The same thing happens in the separated product region. In the adiabatic approximation we assume that the equations remain decoupled for a11 q. It is also interesting to examine the nature of the curvature correction implicit in eq 42. The curvature of the lines of constant q and of constant p are given respectively by42c

- --

-+

.rlq(q,p)= [ ( a R / w I 2+ (ar/ad211'2

(36)

(33)

and vP(4,p)= [(aR/apI2 + ( a r / ~ ) ~ l ~ ' ~(34)

As in section 2, let h(p;y) be the 1 MD Hamiltonian obtained from H(q,p) by considering q to be a parametric variable, and let $,(p;y) and E,;(q)be its eigenfunctions and eigenvalues:

-

-

---f


The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 179

-

The lines of constant p, such as the minimum energy path (for which p = 0) become straight m q fw, in which case C,, 0, q 4 / a p 0, and 1~~ 1. The presence of curvature in those lines (Le,, C p f 0) in the interaction region manifests itself as an 77 which is a function of p and which, therefore, affects the &agonal matrix elements W,,, Vu,, el,,, which appear in eq 42. In this adiabatic approximation with curvature correction we seek solutions of the 1 MD eq 42 which behave, when q --a7 as the superposition of an incident plane wave with unit coefficient and a reflected plane wave and when q 4-m as a transmitted plane wave. The square of the absolute value of the coefficicnt of the latter times the ratio of final to initial velocities of the effective particle of mass p is piMDC. We may further simplify this equation by dropping W,,, and U,,, which is reasonable if 9,and a$,/aq change sufficiently slowly with q. The vibrationally adiabatic approximation without curvature corrections is obtained by further omitting the vq2from the integrand of e,, thereby setting E,,equal to E - E,. 'Then eq 42 becomes

-

+

-

-

-

(44)

which is the 1 MD Sahroedinger equation for this model. In this zero-curvature approximation, we may, in addition, simplify the evaluation of E,(q) by setting in eq 35 vP = vy = 1which iinplies that the lines of constant p (including the minimum energy path, p = 0) are straight and parallel and that the lines of constant q are also straight and perpendicular to the minimum energy path. If this is done, eq 20 and 21 are still exact as long as we use these modified E , in both of them. However, the subsequent approximations, introiduced into eq 21, such as eq 26, will lead to different results depending on which E,(q) are used. A different but closely related way of introducing the full curvature correc1,ions is to define the used in the expansion (38) as the eigenfunctions of the one-dimensional Hamiltonian h(p;q) = vy(q,p)h(p;q)and E,(q) as the corresponding eigenvalues. The associated orthogonality relation differs from eq 39 by the absence of the 7, factor in its integrand. The resulting coupled-channel equations are formally identical with eq 40 but the corresponding W, U, and E matrices are slightly different, the integrands in their elements, being equal to those in eq 41 divided by vq. As previously, the adiabatic approximation with curvature corrections is derived by neglecting the off-diagonal elements of these matrices. The results of this modified theory are not expected in general to be very different from the previous one. The implementation of the full vibrationally adiabatic approximatiori with curvature effects implied in eq 42 requires a calculation of qq and T~ and, therefore, a knowledge of the transformation of variable equations q = q(R,r) and p = p(h',r) or their inverses. These can be obtained from the following defining conditions: (a) the minimum energy path (Le., the line of steepest ascent and descent) is a line of constant p = 0; (b) the line passing through the saddle point of the potential energy function and orthogonal to all equipotentials it intersects away from that point is a line of constant q = 0; (c) we assume that V +a as either the&AB or the BC distances approach zero and require that the V f m equipotential be a line of constant p , an assumption which is wrong for H Since this assumption was not made in our (derivation of a corrected version of the usual TST, 1;he QLJ(Et')given by eq 53 does not diverge as Et' 00, nor does the Q(E) weighted average which results from it. Lin, Lau, and E ~ r r i n ghave ~ ~ also considered the relation between TST ratie constants and cross sections by a Laplace transform method different from that of Morokurna, Eu, and K a r p l u ~ . ~They ' pointed out that in going to 1;he Et' 03 limit it is important to exclude vibrational states above the dissociahion limit, and therefore that harmonic oscillator-type aplproximations to the partition functions appearing in the 'I'ST rate constant expression should be replaced by truncai,ed harmonic oscillator ones. When this was done for the 3 PD A + BC AB C reaction, it vvas found that Q(E)-3.0 as E m. In our formulation of the exact generalized 'IFST these difficulties are bypassed and eq 52 is a dynamically accurate necessary and sufficient condition for usual TST to be valid; it must be satisfied by all states u which contribute to the rate constant beiing considered, over the temperature range of interest.29 6.2. One-Mathematical-Dimensional Transmission Coefficient. This approximation consists in replacing the ~ ~ ~ of( 7eq' 51 ) by a 1 MD expression in which all angular momentum, curvature, and bending energy effects are ignored. For the A + BC electronically adiabatic system whose saddle point configuration is collinear this would mean replacing ~ " ( 5 " )of eq 47 by the K $&( T ) of eq '28. The barrier E,(q) used for the evaluation of the F'LIMD(Etr) whose thermal average appears in that expression is the E,(q) obtained for the collinear system, without any curvature corrections, as described after eq 44. This barrier not only omits any effects of the angular momenta ,associated with the rotation of BC and the orbital motion of A with respect t o BC, but also excludes the effect of zero point energy of the bending motion of ABC at the saddle point configuration. As a result, the height of this collinear barrier, E!, will, in general, be lower than the height of Ihe 3 PD barrier even for the case J = j = 1 = 0, since the latter includes the ABC bending mode zero golint energy omitted in the former. Consequently, the PA vs. Et' curves will be shifted toward lower Et' compared to the PJuJlcurves, resulting in PMD > PJLJi.A similar shift occurs for the Corresponding P" Y probabilities, resulting in > I . The underlying dynamical physical basis for this 1 M b approximation is the hope that the net effects of these two shifts cancel each other out, resulting in the relation +

+

Figure 10. Cross section Qoofor a 3 PD one-reaction-path model H -t H,(v= 0,TjC= ' 0) H2 -k H reaction cross section vs. translational energy: Q, (solid curve), cross section implied by TST with unit transmission coefficient (eq 59). This curve is limited by the two depicted dashed curves, which tend to the asymptotic value 3 a r t (indicated by the horizontal dashed line). @?I2 (dashed-dotted curve), quasiclassic trajectory two-reaction-path cross section, divided by 2 (and (dashed-double-dotted curve), multiplied by 10 before plotting): @!/2 accurate quantum mechanical distinguishable atom two-reaction path cross section, divided by 2 (and multiplied by 10 before plotting). The arrow on the abcissa labeled indicates the height of the J = v = j = I = 0 adiabatic barrier. -+

occurs at the saddle point of V5),eq 55, 56, and the 3 PD counterpart of' eq 12 permit us to write

E>,,jl = Y(o)- h(u - u J ( u

+ y2) + hvb(n+ 1) +

where u and r,, are the separated diatom reagent fundamental vibration frequency and ground state equilibrium internuclear distance, respectively, and us is the symmetric stretch fundamental vibration frequency a t the saddle point configuration of' H,. Replacement of eq 58 into eq 53 furnishes, for u = .j = 0

2 (2J + l)H k"J=o

-

Qoo(Etr)= ;:

h2

Et' - E$ - -J(J -t 1) 21: (59) where E! is the barrier height for J = u = j = 1 = 0 given by Eij = V(0)- ( h / 2 ) ( u- v,) + h v b (60) This is the form of the Qoo cross section for the one-reaction-path of H + Hzimplied by the dynamical assumption of usual TST described by eq 52. We have plotted this cross section as a function of Et' in Figure 10. As can be seen, it has a serrated appearance, with a threshold of E[;,and tends to an asymptotic value of 37rr: = 27.3 (bohr radius)2as Et' m. The two limiting dashed curves between which Qoo remains confined tend, as Et' a,to two line of center hard sphere curves having the same 3nro2asymptotic value. Equation 59 assumes reaction with one atom of the diatomic target only. For reaction with both ends, the limiting Qoo cross section implied by TST with K ~ ~ D (=T1) is therefore 54.6 (bohr radius)2. This is a factor of 12 greater than the asympotic value 4.5 (bohr radius2) obtained by the quasi-classical trajectory m e t h ~ d .We ~ have also plotted in Figure 10 the QCT9 and accurate 6 MD quantum mechanical valueslO

-

-

-

-

--

-

/(byh(c(?3=

(~tlMD/~l)bt

+

KeX(T) =

(pJuJl/fi\,l)b$

The validity of this approximation for the H + H,(para) H2(ortho)+ H reaction is tested37in Figure 11, where we plot K$~ZD,K ~and~ their , ratio as a function of 1/T. This ratio varies from 0.027 a t 200 K to 0.78 a t 600 K. The ~ ~ from ~ 2.3 X to O.!jO. corresponding ( K ~ ' = 1 ) /varies

104


No. I , 1979

Aron Kuppermann

minimum as a function of the ABC bend angle), it may be reasonable to solve instead the collinear A BC reactive scattering problem which involves only 2 MD. The corresponding ~ ~ ~ l lof( Teq) 21 is then used to replace the 6 MD PX(T)of eq 47. This approximation takes into account the nonadiabatic mixing of the motion along the reaction K coordinate with the symmetric-stretch motion transverse to it, but not with the other internal motion (hindered IO rotation) associated with the ABC angle. In Figure 11 we plot tiCol1(T) and its ratio to ~ ~ ~ (as7 a' )function of the reciprocal of the absolute temperature for the H + H2 para ortho reactiod7 on the PK surface. It can be seen that ticoll/ Kex equals 0.11 a t 200 K and increases monotonically with increasing T , reaching the values 1.02 a t 500 K and 1.23 a t 600 K (the maximum temperature for which accurate 3 PD quantum mechanical calculations on the PK surface are availablelo). There is no indication that this ratio tends to unity as T m, in agreement with the remarks in sections 2 and 5.1. Comparison of the variation ~ KCou/ K~~ with temperature indicates of tiC1/tiex, K ~ % & / K ~ and that up to 500 K, replacement of K~~ successively by xC1, I I I I &,: K; and tic0l1 results in improved approximations. 0.001 2 3 4 5 However, because of the different rates of change of these 1000K / T transmission coefficients with temperature, this situation Figure 11. Transmission coefficients as a function of reciprocal does not prevail a t higher temperatures. All of these K o-H, + H reaction on the PK temperature for the 3 PD H + p-H, decrease as the temperature increases and a t 600 K the potential energy surface: (a) K" hdashed horizontal line), classical effective "activation energies" for K ~ I K, $ & , K ~ and ~ K~~ ~ are , (dashed-double-dotted curve), transmission coefficient (= 1); K& respectively 0, -0.033, -0.055, and -0.11 eV, whereas those quantum vibrationally adiabatic 1 MD transmission coefficient without I curve), quantum curvature correction; K ~ (double-dasheddoubledotted of kTsT (with K = 1) and the exact rate constant k are 0.3g9 collinear (2 MD) transmission coefficient; K ~ (shortdashed-longdashed I and 0.27 eV,I0 respectively. The basic hope behind the curve), quantum coplanar (4 MD) transmission coefficient; K~~~ TST of reaction rates was that the rate of change of tiex (doubleddashed-singledotted curve), three-dimensional quasi-classical with temperature would be very slow compared with those trajectory (6 MD) transmission coefficient; tiex (solid curve), exact of kTSTand k so that the latter two quantitites would have three-dimensional (6 MD) quantum transmission coefficient; (b) kapp, essentially the same temperature dependence. The acapproximate 3 PD rate constant; k , exact three-dimensional (6 MD) quantum rate constant; kTsT(dashed curve), 3 PD TST rate constant tivation energies just given indicate that for the 3 P D H with unit transmission coefficient; other rate constants are 3 PD TST + H2para ortho reaction on the PK surface this is not rate constants with transmission coefficients indicated by superscript the case (at least up to 600 K) and that the approximations and/or subscript. The rate constant ratios plotted are equal to the K~~ = 1,K~~ = titggc and tiex = ticou at best are adequate over corresponding ratios of transmission coefficients. limited temperature ranges. At 200 K they lead to rate Therefore, although the use of$!$K in lieu of Kcl= 1 is constants that are too low by factors of 427, 25.6, and 9.0, a substantial improvement, even at 600 K it is 22% below respectively. Kex. As described in section 4.3, several semiclassical methods ~ ' have recently been d e v e l ~ p e d that ~ , ~ furnish ~ ~ , ~ K~ ~~ for 7. Improvements in Three-Dimensional H + H2 which are too low by a factor of about 2 a t 200 Transition State Theory K,21e,47b Since K c ~ lis l itself too low by a factor of 9 a t this One obvious improvement in usual TST is to replace temperature, the corresponding rate constants are too low P,(Etr)in eq 51 not by P",'(Etr)but by PiMD(Etr) where the by a factor of about 18. As a matter of fact, approximate latter is the 1 MD transmission probability of a particle approaches for calculating ticoi1,no matter how successful, of effective mass p across the 1 MD adiabatic barrier E,(q). will not significantly improve the situation. What is We can do this in a zero curvature approximation, or needed are improved approximate methods for calculating include curvature corrections by an extension of the K ~ since ~ , tico1' is not a sufficiently good approximation to methods described in section 4.2 for the collinear A + BC it under the conditions considered. AB + C reaction. This would require a calculation of Another approximation is to replace the P, in eq 51 by all of the contributing E,(q) barriers, which would be its 3 P D quasi-classical trajectory counterpart. This is rather laborious, I t may be interesting to do this, nevequivalent to replacing the Q in eq 49 (or its generalization ertheless, for bench mark cases, in order to test this for reactions between polyatomic molecules) by its quaadiabatic model for P,. In particular, the barriers obtained si-classical value. Although this approximation does not by Wyatt and co-workers for the three-dimensional H + contain any tunneling contributions, it does permit inH2 system5l>j3 could be used for this purpose. Another kind of approximation is to replace the ~ ~ ~ ( 2 ' ) teraction between all the degrees of freedom of the system. In Figure 11 we display the resulting tiQCT(T) and its ratio of eq 51, which involves in general the reaction probato K ~ ~ as ( Ta) function of reciprocal absolute temperature.lob bilities of a 3N - 3 MD scattering problem, by one inIt can be seen that this quasi-classical approximation is volving a lower dimensionality scattering problem. For better than the usual ( K = 1) TST and than the 1 MD example, if we consider the electronically adiabatic 3 P D VAZC one, but not as good as the full 2 MD quantum A + BC AB + C reaction, the full scattering problem mechanical one. which must be solved to obtain the PJL;jlinvolves 6 MD. For A + BC systems which are not collinearly domiIf the potential energy function is collinearly dominated nated, it is possible to develop a 2 MD approach which is (i.e., if for fixed AB and BC internuclear distances near of the same level of approximation as tiex = K ~ O " for the the saddle point configuration the potential has a narrow 1000

500 400

T/ K

300 I

+

200

I

I

-

-

~

-

-

-

-

~

The Journal of Physical Chemistry, Vol. 83,No. 7, 1979 185


collinearly dominated one. Let the potential energy function V have, in the strong interaction region, a minimum as a function of the ABC bend angle at a value of that angle different from 180’ (i.e., a bent transition state). The approximation under consideration consists in “freezing” that angle a t that value and ignoring the rotations of the ABC system. The only remaining degrees of freedom are the AB and BC distances. We solve the associated 2 RiID scattering problem and use the resulting tizMD as an approximation for K ~ If, ~ instead, , V is rather independent of this bend angle (as it is, for example, in some van der Waals r n o l e ~ u l e s ~we ~ ~replace ~ ~ ) , V by its bend angle average, and ignore other rotations, again reducing the wattering problem which must be solved to a 2 MD one. In a similar vein, for a reaction between two polyatomic molecules, we may be able to identify two relative coordinates which1 dominate the reactive scattering problem. In other words, the reaction may proceed, to a good first approximation, along a path for which one of these variables decreases from infinity to a finite value, whereas the other change? in the opposite direction, while the remaining internal coordinates do not change. In the example of the previous paragraph, the constant coordinate was the bend angle and the variable ones were the AB and BC distances. Under these conditions, we replace the 3 N - 3 MD scattering problem by the 2 MD one involving those two coordinates, and use the resulting K~~~ as an approximation for tiex. A less drastic approximation is to allow more than two coordinates to vary during the reaction. In the case of the electronically adiabatic 3 PD A + BC reaction, for example, we may replace the 6 MD scattering problem by a 4 MD one, in which the three atoms are permitted to move on a space-fixed From the solution of the corresponding scattering problem, we obtain the coplanar Pop1 which we use as an improved approximation to tiex. This approximation not only permits the bend angle to vary and allows for nonadiabatic interations between the motion along the reaction Coordinate and both internal motions transverse to it (the symmetric stretch and the bend), but even takes into account the coupling between the in-plane triatom rotation and the other internal degrees of freedom. It is, therefore, expected to lead to a significant improvement over the collinear-type (Le., 2 MD) approximations. In Figure 11 we display K ~ and ~its ratio ~ to~ the exact 3 PD transmission coefficient as a function of reciprocal abslolute temperature for the H + p-H2 o-H, + H reaction on the PK surface. As can be seen, although these results are significantly better than the ticnuones, they are still off by a factor of 3.2 at 200 K and 1.6 a t 300 K. Therefore, the neglected interaction between the tumbling of the triatomic plane and the motions in that plane is still sufficiently layge to significantly affect the tunneling and the low temperature rate constants for this reaction. These results seem the best that can be achieved for this highly quantum system short of performing an exact 3 P D (i.e., 6 MD) calculation. We now extend these ideas to more general bimolecular reactions. Let us consider a rate constant knMDfor an n MD problem ( n = 6 for the 3 P D A + BC system), for which the exact transmission coefficient and usual (Le., K = 1) TST rate constants are K~~~ and h@$?, respectively. Let the index in label the corresponding quantities for the lower dimensionality Scattering problem by which we are replacing the full one (Le., m = 2 for the collinear approximation and m = 4 for the coplanar one). If we assume K~~~ K “ ‘ ~ ~then , we have, in general

-

hmMDlm

(61) Use of this improved TST requires solving the lower m-dimensionality problem to obtain hmMD,as well as a use of the usual (ti = 1) T S T to obtain k?gD and h;:!. Since it is relatively easy to calculate the latter two quantities, the numerical e€fort associated with eq 61 involves essentially a solutiton of the mMD scattering problem. Since, as pointed out in section 2, K~~ contains the full dynamics of the system we may, for systems in which quantum effects are small, use quasi-classical trajectory probabilities to obtain a good approximation to the lower dimensionality niMD problem if the nMD one is tool laborious to solve. This approach permits us to focus attention on the degrees of freedom which contribute most importantly to the reaction, and still obtain in the end rate constants for the full reaction. These extensions of TST have so far not been tried out for systems other than the H + H2 one. 8. State-to-Stale Transition State Theory The essential features of the present generalized TST are (a) attention is focused on the role of reaction probabilities and transmission coefficients in the determinal ion of rate constants and (b) in performing the Boltzmann averages required for obtaining fully thermal rate constants, the internal energies of the system are refered to the saddle point region of the potential energy surface. This averaging process, and the associated summing over the final states of the products, can be restricted to a subset of the quantum states involved, if desired. Let us consider, for example, the electronically adiabatic 3 PD A BC AB + C reaction. It may be desirable to obtain the rate constant h”,(T)for reagents in vibrational state u t o form products in vibrational state u’, summed over all rotational states of the latter and averaged over all rotationaly states of the former. We consider the collision theory expressi~nsl~,~~

+

~

’

~

-

~

where fTi is the internal (i.e., rotational) partition function of the vibrational state u of the reagent diatom

fTf = C ( 2 j + 1)e &LjlkT I

(163)

Qijl is the cross section for reaction from state uj1 of the approaching rea5ents to state u l j l ’ of the receding products, and PuJ$ is the corresponding reaction probability for total angular momentum quantum number (].e., partial wave) J . The latter two quantities are related to the Qbl and PJLllof eq 48 by

The fully thermal rate constant k ( T ) is related to k:’(7‘) by (64)

180

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

where f d n t is the total internal (Le., vibrational) partition function of the reagent diatom. We can now derive a generalized (exact) TST expression for hi'(T)by starting with the first part of eq 62 and the 3 PD version of eq 14 and proceeding as in the fully thermal theory. The result is

&'(T) = K k ' e x ( T ) ( k T / h ) ( f ' b t c L ) /exp(-E$/kr) f~~c~~~~) (65) where K ; ' ~ ~=( T ) C e x p ( - E j , , l / k T ) ( ~ [ , l / ~ ~C~ ~exp(-E!ju,l/kT) ,)/ = JMil

JMil

In these expressions, fkrcu)and fbC(u) represents respectively the partition functions per unit volume of the state u of the barrier maxima and of the diatomic reagent and are given by tf and &f $! where

ft$(u)

=

Jg,lexp(-E3ujl/kT)

(67)

In addition, Pyljl is the sum of P'?"' over j'and l', a similar ,.$vil definition holding for Edjl' being defined by

e$,

pyJ'c1

= @"'

W(E1' - EtjUjl)

-

(68)

Aron Kuppermann

bypassing the TST formalism. Alternately, assumptions can be made concerning the generalized transmission coefficient itself, in which case the TST language is more appropriate. The generalized K can be greater than unity, even when quasi-classical trajectories are used to obt,ain approximate reaction probabilities. There is furthermore a;it can, no requirement that K approach unity as T on the contrary, become significantly smaller than that value if the reaction probabilities become sufficiently small a t high translational energies. The validity of some approximations for the H H2 three-dimensional reactive system are examined. They indicate that replacement of the full three-dimensional transmission coefficient K by the exact collinear one ~~~~l is unsatisfactory at room temperature and below. This is the case even though xc0l1 is obtained by solving the collinear quantum mechanical problem accurately, and includes therefore full curvature effects and nonadiabaticity effects between motion along the reaction coordinate and the symmetric stretch motion transverse to it. At 300 and 200 K the resulting rate constants using this approximation are too low by factors of 2.2 and 9, respectively. Using the exact coplanar transmission coefficient K ~ O P ' as an approximation for K improves the situation but the resulting rate constants are still too low by factors of 1.6 and 3.2 at these respective temperatures. Additional improvements for general bimolecular reactions, allowing for increasingly realistic dynaniical behavior, are suggested.

-

+

Equations 65 and 66 are the u L)' state-to-state counterparts of the 3 PD version of eq 20 and eq 47 respectively. Analogous state-to-state counterparts of eq 50 and 51 and of their generalizations given in section 5.2 can also be obtained using similar reasoning. These state-to-state rate References and Notes constants involve a Boltzmann average of VQf,where Q f H. Eyring, J . Chem. fhys.. 3, 107 (1935). is the reaction cross section from state i of the reagents (a) E. Wigner, Z . fhys. Chem., 919, 203 (1932); (b) H. Pelzer and to state f of the products, over some (rather than all) of E. Wigner, ibid., 815, 445 (1932); (c) M. G. Evans and M. Polanyi, the degrees of freedom of the reagents and a sum over Trans. Faraday Soc., 31, 875 (1935); ( d ) E. Wigner, ibid., 34, 29 (1938). some of the final quantum states of the products. The J. 0. Hirschfelder and E. Wigner, J . Chem. Phys., 7, 616 (1939). extent of this summing and averaging in the theory can H. Eyring, J. Walter, and G. E. Kimball, "Quantum Chemistry", Wiley, be chosen to fit any corresponding set of experimental New York, N.Y., 1944, Chapter XVI. H. J. Johnston, "Gas Phase Reaction Rate Theory", The Ronald Press, conditions. As in the fully thermal generalized TST rate New York, 1966, pp 190-196. constant expressions, eq 65 and 66 and their generaliza(a) D. G. Truhlar and A. Kuppermann, J . Am. Chem. Soc., 93, 1840 tions are exact. They can be approximated by making (1971); (b) Chem. Phys. Lett., 9, 269 (1971). R. A. Marcus and M. E. Coltrin, J . Chem. Phys., 67, 2609 (1977). assumptions about f'$jl or K$ or about their generalized Reference 5, p 135. versions. These approximations are guided by considM. Karplus, R. N. Porter, and R. D. Sharma, J . Chem. fhys., 43, erations similar to those which were used for the fully 3259 (1965). (a) A. Kuppermann and G. C. Schatz, J . Chem. fhys., 62, 2504 thermal case and discussed in sections 6 and 7. In par(1975); (b) G. C. Schatz and A. Kuppermann, ibid, 65, 4668 (1976). ticular, the usual TST expression for hi (T)56is obtained The accuracy referred to is that of the dynamical calculations, once by assuming that P$uj,= E':$ and therefore that K;'~""(T) the potential energy surface is assumed. K. Morokuma and M. Karplus, J . Chem. Phys., 55, 63 (1971). = 1. 9. Summary and Conclusions We have derived an exact generalized transition state theory (TST) of thermal gas phase bimolecular reaction rate constants, in which the generalized transmission coefficient K is related explicitly to the partial wave reaction probabilities. All of the effects of the reaction dynamics on the rate constant are contained in this relation, which furnishes a detailed translation dictionary between the language of collision theory and that of generalized TST. The usual TST is then obtained by making specific dynamical approximations to those reaction probabilities and related transmission coefficient, without any assumption being made concerning an equilibrium between reagents and transition state configurations. Improvements to the usual TST results by making less restrictive assumptions regarding the reaction probabilities; these can be translated into approximations to the corresponding reaction cross sections, and collision theory expressions can be used to evaluate the rate constants if desired, thereby entirely

(a) D. G. Truhlar and A. Kuppermann, Chem. Phys. Lett., 9, 269 (1971); (b) J . Chem. fhys., 56, 2232 (1972); (c) J . Am. Chem. SoC., 93, 1840 (1971); (d) J. M. Bowman, A . Kuppermann, J. T. Adams, and D. G. Truhlar, Chem. Phys. Lett., 20, 229 (1973). M. A. Eliasonand J. 0. Hirschfelder, J. Chem. Phys., 30, 1426 (1959). (a) R. A. Marcus, J . Chem. fhys., 41, 2614 (1964); (b) ibid., 41, 2624 (1964); (c) ibid., 46, 959 (1967). (a) R. A. Marcus, J . Chem. fhys., 43, 1598 (1965); (b) W. H. Wong and R. A. Marcus, ibid., 55, 5625 (1971); (c) R. A. Marcus in "Techniques of Chemistry", Vol. 6, Part 1, E . S. Lewis, Ed., New York, 1974. J. C. Keck, Adv. Chem. fhys., 13, 85 (1967). D. L. Bunker and M. Pattengill, J . Chem. fhys., 46; 772 (1968). (a) A. Tweedale and K. J. Laidler, J . Chem. fhys., 53, 2045 (1970); (b) P. Pechukas and F. J. McLafferty, ibid., 58, 1622 (1973); (c) F. J. McLafferty and P.Pechukas, Chem. Phys. Lett., 27, 511 (1974). (a) J. A. Merick, J. W. Duff, and D. G. Truhiar, J . Am. Chem. Soc., 98. 6771 (1976); (b) B. C. Garrett and D. G. Truhiar, to be published; (c) J. C. Gray, D. G. Truhlar, L. Clemens, J. W. Duff, F. M. Chapman, Jr., G. 0. Morrell, and E. F. Hayes, to be published; (d) D. G. Truhlar and J. C. Gray, to be published; (e) D. G. Truhiar, J . Phys. Chem., this issue. (a) W. H. Miller, J , Chem. fhys., 61, 1823 (1974); (b) ibid., 62, 1899 (1975); (c) ibid., 63, 1166 (1975); (d) ibid., 65, 2216 (1976); (e) S.Chapman, B. C. Garrett, and W. H. Miller, ibid., 63, 2710 (1975). R. A. Marcus, J . fhys. Chem., this issue. See, for example, B. Shizgal and M. Karplus, J . Chem. Phys., 54,

Quantum Mechainicai Transition State Theory

4357 (1971),and references therein. (24)Although this average involves the Boltzmann weighting function e-Ew'/k , it differs from the distribution function F,(E") because of the omission of the V-' factor.

(25) L. M. Delves, Nucl. Phys., 9, 391 (1959);20, 275 (1960). (26) D. Jepsen and J. 0. Hirschfelder, Proc. NatI. Acad. Sci. U . S . A . , 45, 249 (1959). (27) (a) A. Kuppermann, Chem. Phys. Lett., 32, 374 (1975): (b) A. Kuppermanin, G. C. Schatz, and M. Baer, J . Chem. Phys., 65, 4596 (1 9'76). (28) For some systems E,b(q) may have a maximum elsewhere, such as in the reagent (as displayed by the curve at the top of Figure 2) or product regions. The present considerations include such special cases, which are further discussed in section 3. (29) This mathematical conclusion implicitly assumes that the effects of electronicallly nonadiabatic processes and of break-up collisions are unimportani at all temperatures or energies. Over the temperature range of interest for many bimolecular chemical reactions and the corresponding range of €"affecting the rate constant, these condtions are satisfied. The condition K( T ) = 1 then implies that eq 26 for collinear atom-diatom reactions and eq 52 for general three-dimensional reactions iare essentially (rather than exactly) valid over the range of E" for which Pv(Etr)contributes most significantly to K( T ) for the temperatures of interest. (30) 1. Shavitt, R M. Stevens, F. L. Minn, and M. Karplus, J. Chem. Phys., 48,2700 ('1968);see errata in I.Shavitt, bid., 49, 4048 (1968). (31) (a) D. J. Diestler, J. Chem. Phys., 54, 4547 (1971);(b) G. C.Schatz and A. Kuppermann, Phys. Rev. Left., 35, 1266 (1975). (32) R. N. Porter and M. Karplus, J. Chem. Phys., 4Q, 1105 (1964). (33) (a) G.C. Schatz, J. M. Bowman, and A. Kuppermann, J. Chem. Phys., 58, 4023 (1973);(b) bid., 63, 674 (1975);(c) ibid., 63, 685 (1975); (d) J. Kaye

An exact quantum mechanical transition state theory. 1. An overview

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