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J. Phys. Chem. 1980, 84, 1730-1748
Improved Treatment of Threshold Contributions in Variational Transition-State Theory Bruce C. Garrett,+ Donald G. Truhlar,* Roger S. Grev,t and Alan W. Magnusont Chemical Dynamics Laboratory, Koithoff and Smith Hells, Depattment of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received January 1 1, 1080) Publlcation costs assisted by the United States Department of Energy
We extend the improved canonical variational theory, presented previously by us for reaction rates in a classical mechanical world, to reactions in a quantum mechanical world where internal energies of reactants are quantized. We also present a detailed discussion of vibrationally adiabatic models for transmission coefficients for conventional transition-statetheory and three versions of variational transition-statetheory. We show the relationship of the improved canonical variational transition-statetheory to the classical limit of these transmission coefficienta, The improved canonical variational theory and the new quantal and classical transmission coefficients are illustrated and tested by applications to quantal collinear and three-dimensional reactions rates for several reactions. We also examine semiclassical approximations to the quantal transmission coefficients. Applications considered are collinear H + H2 and isotopic analogs, C1 + Hz and isotopic analogs, and I + H2 and threeI + Dz, 0 + Hz, and F + H2. dimensional D + H2, C1+ HD, I + Hz,
I. Introduction We have recently applied microcanonical, canonical, and improved canonical variational theories (pVT, CVT, and ICVT, respectively), to calculate classical mechanical rate constants for chemical reactions.lI2 (These are all versions of variational transition-state theory; references to earlier work on this and closely related subjects have been given in a previous artic1e.l) We found these theories to give significantly better agreement with numerically computed accurate classical equilibrium rates than is afforded by conventional transition state theory. The improvement is due to the variational optimization of the location of the generalized-transition-state-theory (GT) dividing surface for each total energy in pVT and for each temperature in CVT. In ICVT spurious contributions from energies below the pVT threshold are removed, and the location of the GT dividing surface is reoptimized for the remaining contributions. We have also tested the microcanonical and canonical variational theories by applying them to collinear reactions for which accurate quantum mechanical equilibrium rate constants have been computed for given potential energy s u r f a ~ e s . We ~ found that it was possible to use these theories to obtain more accurate predictions than are afforded by conventional transition-state theory but that their accuracy is limited at low temperatures by quantum effects. In this article we attempt to treat the threshold contributions to these rate constants more accurately and to incorporate quantal effects on reaction-coordinate motion more consistently. We apply the improved theory to several collinear systems to test the consequences of these improvements. The improvements discussed in the present article are of two kinds. (i) We discuss the quantization of vibration and rotation in the ICVT and we extend this theory to both collinear and three-dimensional reactions with quantized internal degrees of freedom. (ii) We present a vibrationally adiabatic model for inclusion of quantum mechanical effects on reaction-coordinate motion. This model is also applied to conventional transition-state theory and the canonical variational theory. In the classical limit it still includes an important threshold correction for Battelle Columbus Laboratories, 505 King Ave., Columbus, Ohio 43201. t Lando Summer Undergraduate Research Fellow, 1979. 0022-3654/80/2084-1730$01 .OO/O
these theories. The resulting quantal and classical transmission coefficients provide a way to improve the treatment of the threshold region without performing a full ICVT calculation. The ICVT and the new quantal and classical transmission coefficients are illustrated and tested by applications to quantal collinear and three-dimensional reaction rates. Semiclassical transmission coefficients are also tested. 11. Theory The classical mechanical theory of the pVT,lt4CVT,'p4 and ICVT2for collinear reactions and the quantized theory of the CVT for collinear3and three-dimensional6reactions have been presented in detail elsewhere. (Additional applications of the CVT to three-dimensional reactions have also been The present development builds on these presentations and uses a notation consistent with them. We consider the reaction A + BC AB C. A. Collinear Reactions with Quantized Vibrationsand Classical Treatment of Reaction-Coordinate Motion. Remove the center-of-mass motion, and consider a scaled coordinate system in which the kinetic energy has no cross terms and the reduced mass is 1.1 in both directions. Let distance along the minimum-energy reaction path (MEP) be measured by the reaction coordinate s, and for each s define a rotated-translated coordinate system (zB,us) such that zsis tangent to the MEP a t s and zero at the point of tangency, and usis normal to z8 and zero at the intersection of the MEP. Let the GT dividing surface be a straight line zs= 0 perpendicular to the MEP so that the minimum potential energy in the dividing surface is V(z8=0,u8=O). This may also be called Vmp(s). Let N8h be the local vibrational action for motion in the uscoordinate at s, and e(NB,s)is the vibrational energy for a given NEand s. Then the classical vibrationally adiabatic potential curve is (1) va(Ns,s)= VME€'(s)+ e(Ns,s) Let E be the total energy with respect to the zero of energy at the minimum of potential energy for reactants. For a dividing surface at s the GT approximation J'@T(E,s) to the classical cumulative reaction probability is the volume of phase space, in units of h, in the dividing surface with energy less than or equal to E for the bound degree of freedom transverse to the reaction coordinate. Quantization of this bound degree of freedom is accomplished in
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0 1980 American Chemical Society
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The Journal of Physical ChernWy, Vol. 84, No. 13, 1980 1731
Improved Variational Transition-State Theory
the usual quasi-classical manner by rounding this unitless quantity to the closest integer. For energies below dissociation, this yields the quantized cumulative reaction probability (eq 1 of ref 3)
to be the location sYAGof the maximum of the ground-state vibrationally adiabatic curve. Thus the threshold for reaction in the microcanonical variational theory is VAG,and the microcanonical variational theory rate constant is given by
n=O
where O(z) is the H[eaviside step function, nmax(s)is the highest vibrational (quantumnumber of the bound degree of freedom in the dividing surface, and the quantized adiabatic potential curve is given in terms of the classical adiabatic potential curve of eq 1 by
V,(n,s) = V,(AP=n+f/,,s)
(3)
+
= V(Z~:=O,U~=O)@=n+l/,s)
(4)
When eq 1 and 22 of ref 3 are used, the generalized-transition-state-theory rate expression for a dividing surface at s is
kGT(T,s) = [ haR(V 1 - l
J0 mdEe-PEPT(E,s)
(5)
where aR(T)is the partition function per unit length for reactants. In eq 5 we have extended the integral from the dissociation energy to infinity to simplify the subsequent algebra; this causes negligible difference in the result^.^ From eq 2 above it is evident that the threshold for reaction implicit in eq 5 is the height V,(n=O,s) of the ground state adiabatic potential curve at the location s. Using eq 2 and 5 we obtain
where the microcanonical-variational-theorycumulative reaction probability is
N"T(E) = IVGT[E,S=s*fiVT(E)] (14) The goal of the improved canonical variational theory is to treat the threshold region as accurately as in the microcanonical variational theory. Thus we write an improved generalized-transition-state-theory rate expression as kIGT(T,s)= [h@R(T)]-lJvIGdEe-OENGT(E,s) (15) where we have obtained eq 15 by introducing the microcanonical variational threshold into eq 5. Then the improved canonical-variational-theory dividing surface is located to minimize the improved generalized-transitionstate-theory rate constnt, i.e.
Substituting eq 2 into eq 15 yields where the generalized-transition-state partition function for a dividing surface at s is given by n,(s)
QGTW,s) = C exp(-p[t(n,s) - e(~,s)Ii n=O
(7)
The tilde-on the generalized-transition-state partition function QGT(T,s)dienotes that its zero of energy is now the ground state vibrational energy ~(0,s). This allows us to explicitly display the threshold of reaction as a Boltzmann factor in the generalized-transition-state-theory rate expression, eq 6. Canonical variational theory locates the dividing surface to minimize the generalized-transition-state-theory rate constant
dk'TT,s)l as
=
s=s*CVT(T)
as
=
n=nf+l
(18)
and n< is the highest quantum number for which the height of the adiabatic curve at the location s is less than or equal to the maximum of the ground-state adiabatic potential curve. Combining eq 17 and 18 we obtain k'GT(T,s) =
(8)
The threshold for reaction in the canonical variational theory is therefore 'Va[n=O,s=sICVT(T)].Microcanonical variational theory locates a different dividing surface for each total energy to minimize the generalized cumulative reaction probability
aNTE,s)l
where the improved generalized-transition-state-theory sum over states with zero of energy at VAGis
Thus we see that the improved generalized-transitionstate-theory rate expression can be obtained from kGT(T,s) by replacing the adiabatic potential at the dividing surface by the ground-state adiabatic barrier height whenever the latter exceeds the former. The generalized-transition-state-theoryrate expressions, eq 6 and 7, can be rewritten as
(9)
s=s*YVT(E)
We define the maximum of the ground-state vibrationally adiabatic potential (curve as VAG= max V,(n=~,s) S
(10)
= V,[n=O,s=s*AG] (11) For energies less than or equal to VAG,the location of the microcanonical variational dividing surface can be taken
When eq 19 and 20 are compared it is evident that kGT(T,s) I kIGT(T,s) (21) Furthermore since the improved canonical variational theory minimizes the integral in eq 15 whereas the microcanonical variational theory minimizes the integrand of eq 15 we obtain the relationship
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Garrett et al.
The Journal of Physical Chemistry, Vol. 84, No. 13, 1980
hCVT(T)1 kICVT(T)2 kWvT(T)
(22)
Thus, just as in the classical case: the improved canonical variational theory is intermediate between the canonical variational and microcanonical variational theories. The relationship between the theories can also be understood in terms of the theoremlJ that microcanonicalvariational theory is equivalent to the adiabatic theory of reactionsg12 when the reaction coordinate is treated classically. Because of this theorem, microcanonical variational theory is equivalent to locating the dividing surface to maximize the value of each adiabatic potential, i.e., to minimize the terms of eq 20. In contrast, canonical variational theory is equivalent to minimizing the whole sum. Improved canonical variational theory does not minimize every term of the sum but it does require that no term be larger than the largest term (the first term) in the microcanonicalvariational-theory sum. B. Three-Dimensional Reaction with Quantized Vibrational and Rotational Energies and Classical Treatment of the Reaction-Coordinate Motion. We consider reactions in which the minimum-energy reaction path is collinear. If we assume a unit transmission coefficient, the generalized-transition-state-theory rate expression is given in eq 12 of ref 5 as
where cr is the statistical symmetry factor and aR(r)is the partition function per unit volume for reactants. The lack of a tilde on QGT(T,s) indicates that the zero of energy for this partition function is at the minimum value of the potential energy in the dividing surface, as in ref 1 , 3 , and 5. The generalized-transition-state partition function is approximated as a product of partition functions for the stretching, bending, and rotational degrees of freedom QGT(T,s)= Q~~(T,s)[QBT(T,s)12&PT(T,s) (24) where each partition function involves a sum over the energy levels for the corresponding bound degree of freedom and the energy levels are relative to the bottom of the potential well. We can rewrite eq 23 as kGT(T,s)=
where the adiabatic potential curve is given by V,(n,i,i’,J,s) = V,,p(s) + egtr,n(S) + %,i(s)+ %,jf(s) + h2J(J 1)/[21(s)] (26)
+
where E ~ ~ , q&), ~ ( s ) and , €b,i’(s) are the energy levels for the stretching and the two bending degrees of freedom of the generalized transition state, respectively, and I ( s ) is the moment of inertia at the generalized transition state. The improved generalized-transition-state rate expression for a three-dimensional reaction is then obtained in an equivalent manner that took us from eq 5 to eq 15 and 19 in the collinear case. Thus we define kT P T ( T , s ) = {C ’ (2J + 1) exp[-pVAG]+ ha’( T) n,i,i’,J C ” ( 2 J + 1) exp[-/3Va(n,i,i’,J,s)l) (27) n,i,i’,J
where VAGis the maximum of the ground-state, zero-angular-momentum adiabatic potential curve, i.e. VAG= max V,(n=O,i=O,i’=O,J=O,s) (28) S
and where the single and double primes on summations indicate sums over only those states for which the value of the adiabatic potential curve V,(n,i,i’,J,s) is less than (or equal to) or greater than VAG,respectively. Equation 27 can be written as
kT QIGT(T,s) kIGT(T,s)= exp(-PVAG) h QR(T)
(29)
where the improved generalized-transition-state sum over states with zero of energy at VAGis $IGT(T,s) =
C ’ ( 2 J + 1) +
n,i,i‘,J
C ” (2J + 1)exp(-P[V,(n,i,i’,J,s)
-
VAGI)(30)
n,i,i‘,J
vAG]}
= Q G T ( ~ , s )exp(-P[V,(n=O,i=O,i’=O,J=O,s)C ’ ( 2 J 1) [exp(-P[V,(n,i,i’,J,s) - VAG]{- 11 (31) n,i,i‘,J
+
The quantized microcanonical variational threshold for reaction is explicitly displayed in the Boltzmann factor of eq 29. By using eq 29 and 31 we found that the improved generalized-transition-state rate expression is only slightly more difficult to evaluate than the generalized transition-state expression 23. In particular one needs to calculate the ground-state, zero-angular-momentumadiabatic barrier height VAGand perform an additional summation over a presumably small number of states. The improved canonical variational theory rate constant is obtained by minimizing the rate constant of eq 29 with respect to the location of the transition state, and it satisfies eq 22.
111. Incorporation of Adiabatic Ground-State Transmission Coefficients into Variational Transition-State Theory with Quantized Vibrational and Rotational Energies In section I1 we showed how the quantized canonical variational transition-state theory rate expression could be improved to give a threshold for reaction consistent with the adiabatic theory of reactions. The resulting improved canonical-variational-theory rate expression has the appealing feature of being intermediate between the canonical variational and microcanonical variational rate expressions (see eq 22 and the associated text). In sections 1II.A and 1II.B we introduce an approximate transmission coefficient (Le., a multiplicative factor) for canonical variational theory which provides an alternative way to improve the theory. Another version of the transmission coefficient is given for use as a multiplicative factor with conventionaltransition state theory. Unlike the improved canonical variational theory, the transmission coefficients are not based on a classical variational bound. However, they are easier to calculate than the improved canonical variational rate constant, and under a variety of circumstances they may be about as accurate or even more accurate, In sections 1II.C and 1II.D we discuss the inclusion of quantum mechanical effects on the reaction-coordinate motion for conventional transition state theory, canonical variational theory, and the improved canonical variational theory. A. Classical Treatment of Reaction-CoordinateMotion for Collinear Reactions. In the adiabatic theory of reactions with quantized vibrations and classical treatment of the reaction coordinate, a threshold for reaction is identified for each initial internal state of reactants. This threshold VA(n)is the maximum of the adiabatic potential curve V,(n,s) for the particular initial internal state n. Since the rate constant k A ( T ) of the adiabatic theory of reactions with classical reaction-coordinate motion is
The Journal of Physical Chemistry, Vol. 84, No. 13, 1980
Improved Variational Transition-State Theory identicalla with the irate constant hfiw(T) of microcanonical variational theory, it can be obtained from eq 20, which yields
by s = 0 in eq 35 and 37 yields the classical adiabatic ground-state transmission coefficient for conventional transition-state theory K * / ' ~ ' ( T )= exp(--P[VAG- V,(n=O,s=O)])
where nmURis nmU(.s=-m),The exact transmission coefficient for canonical variational theory is defined as Kcv" ( T ) = h ( T )/ hCYT( T) (33) where h ( T ) is the exact rate constant. We could use microcanonical variational theory to compute an approxi~ ( exact transmission coefficient mation K ~ ~ / T") to~ this as follows K C V T / ~ V Tr( )
= k"T( T ) /hCVT( T )
(34)
Use of eq 34 entails a full microcanonical variational calculation but we now present an approximation to it which is much simpler. We use eq 8, 20, and 32 to write eq 34 as nmaxR
C exp[-PVA(n)1
KCVT/CAG(T) (36) where we have defined the classical adiabatic ground-state (CAG) transmission coefficient for canonical variational theory as
K ~ ~ ~ =/ exp(--P( ~ ~ ~VAG ( -TV ,)[ ~ = O , S * ~ ~ ~ ( T ) ] (37) ]) The approximation iin eq 36 consists of replacing each sum in eq 35 by its first and largest term. This approximation is a good one under more than one possible set of circumstances. First the approximation is valid at low temperature when only the first term contributes appreciably to each sum. Second it is valid when VA(n)- VA(n=O)equals Va[n,s*cw(T)]- V,[~=O,S,~~(T)] for all important n. This would be true if the quantized energy level spacings were independent of s, but in that case all adiabatic potential curves would necessarily have the same shape as the classical potential barrier and be shifted up by some constant amount. Then canonical variational theory would agree with conventional transition-state theory and both would be in perfect agreement with the microcanonical variational theory. 'Thus the classical vibrationally adiabatic transmission coefficient, eq 37, would be unity. A third, less obvious, condition for the validity of the approximation is that the difference, VA(n)- Va[n,s'CVT(T)], between the maximum of the adiabatic potential curve and value of the adiabatic potential curve at the canonical variational dividing surface is the same for each state n. If this is true then multiplying the generalized transition-state rate expression 20 by the transmission coefficient of eq 36 will give the microcanonical-variational-theoryrate expression. Thus we have defined the classical adiabatic ground-state transmission coefficient for canonical variational theory so that it corrects the reaction rate out of all states by the error in the ground-state reaction rate as compared to the ground-state reaction rate of the adiabatic theory of reactions. Since the adiabatic theory of reactions is equivalent to microcanonical variational theory, this correction is variationally motivated for the ground state but not for the excited states. Clearly, the same sort of correction can be applied to conventional transit ion-state theory. Replacing s * ~ ~ ~ (
1733
(38)
Generalizing the above discussion we define the classical adiabatic ground-state (CAG) transmission coefficient for an approximate rate constant kapp(T)as
where the numerator is the thermally averaged classical transmission probability for the adiabatic ground-state potential energy barrier and the denominator is the thermally averaged classical transmission probability for a one-dimensional potential energy barrier of height Et{!. We choose E#!' to make ,pPP/CAG( T)kapp(r ) -+ hd'T( T ) (40) T-0 and this requires that E$: equals the threshold energy for the approximate theory being considered when reactioncoordinate motion is treated classically and vibrations are quantized. In section I1 we showed that, for this definition
E;hr = V,(n=O,s=O) E&YT= V , [ ~ = O , S = S , ~ ~ ~ ( T ) ]
EfgYT = VAG
(41) (42) (43)
EtXT = VAG (44) Equations 39 and 41 yield eq 38, and eq 29 and 42 yield e 37. Equations 39, 43, and 44 yield the trivial results K ~ = 1 and ~ ~ / p = 1. The ~~ interpretation /~ ~ ~ of the ~ latter two results is that adiabatically there is no classical recrossing correction for the ground state in ICVT and pVT because the GT dividing surface at threshold in these theories corresponds to the position of the maximum in the ground-state adiabatic potential curve. B. Classical Treatment of Reaction Coordinate Motion for Three-Dimensional Reactions. The extension of the above arguments to three-dimensionalreactions is obvious so we state the results without derivation. The classical adiabatic ground-state transmission coefficient for canonical variational transition state theory is KCVT/CAG(T) = exp(-pi VAG- Va[n =0,i =0,i '= O,J=O , S * (T ~ )~3 )) (45)
~
The classical adiabatic ground-state transmission coefficient for conventional transition state theory is ,*lCAG(T) = exp{-P[ VAG- V,(n=o,i=o,i'=O,J=~,s=~)]] (46)
We note that, as for the collinear case, K ~ ~ ~ and / ~ ~ ~ ( K * / ~ ~ ~ can ( Tbe) considered to be approximations to K ~ ~ ~ /(defined ~ ~ ~by( eq T 34) ) and K * / " ~ ~(defined (T) analogously), respectively. C. Quantal Treatment of Reaction-Coordinate Motion for Collinear Reactions. Our use of generalized transition-state theory has retained the assumption of a separable reaction coordinate. Although there has been some work on nonseparable transition-state theory,13J4 or nonseparable treatments of the transmission coefficient,15-19that is beyond the scope of the present work. TWithin ) the separable context, the vibrationally adiabatic
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The Journal of Physical Chemistry, Vol. 84, No. 13, 1980
modelz0provides a framework for consistently including quantum mechanical effects on reaction-coordinate motion into transition-state theory. Equations 45 and 46 of ref 3 give the quantum mechanical vibrationally adiabatic rate constant as kVA(T) = KVA( T )k p V T ( T ) (47) where k”(T) is given in eq 32 above, and the vibrationally adiabatic transmission coefficient is nmuR
dE e-PE
Jrn
nmii Jm
dE e-PE C B[E - VA(n)] n=O
= ’f
dE e-PEPVA( n =O,E)
sv:G
,aPP/VAG(T)ppp(T)+ k V A ( T ) (54) T-+O This leads to the same identification of E!{: as we made in section 1II.A above. Thus we obtain the quantal vibrationally adiabatic ground-state transmission coefficient for canonical variational theory KCVT/VAG
s 0
(48)
PVA(n,E)is the quantal vibrationally adiabatic transmission probability from an initial quantum state n at total energy E. Its calculation is discussed in section IV. Use of eq 48 entails the calculation of transmission probabilities for all important initial vibrational states n. Here we approximate the quantal vibrationally adiabatic transmission coefficient by the quantal ground-state vibra~ ~ ( tionally adiabatic transmission coefficient K ~ Tj defined as ,VAG
We choose E#‘: so that
exp(ova[n = O , S * ~T)])P ~ ~ ( -dE e-PEPVA(n=O, E) (55)
C PVA(n,E)
n=O
KVA(T)=
Garrett et al.
(49)
dE e-oE
= e ~ v A G f l ~ e-PEPVA(n=O,E) rndE (50)
which we recognize as a practical approximation to the quantal vibrationally adiabatic transmission coefficient for canonical variational theory ,CVT/VA
=
kVA/kCVT
(56)
Using eq 37, 50, and 55, we can also write ,CVT/VAG = KCVT/CAG(TjKVAG
(58)
Similarly the quantal vibrationally adiabatic ground-state transmission coefficient for conventional transition-state theory is given by ,*/VAG = ,I/CAG,VAG = exp[/3V,(n=O,~=O)]fl~~dE 0 e-PEPVA(n=O,E) (59)
This approximation is valid at low temperatures when only the ground state contributes in eq 48 and also if the adiabatic barriers for low n have approximately the same shape. We can generalize this argument to include tunneling in other approximate theories by rewriting eq 47 with the approximation of K ~ ~ for ~ (KT” )~ (as T) kfiVT/VAG( T ) = KwVT/VAG( T)kpVT(T ) (51) where we have written the new approximation as kpVTIVAG(Tj to emphasize that it is the microcanonical variational theory rate constant as corrected for tunneling using the transmission coefficient calculated from the quantal vibrationally adiabatic ground-state transmission probability. We use VAG to denote quantum scattering in this approximation and CAG to denote classical scattering by the vibrationally adiabatic potential-energy is clearly just another name barrier, In eq 51, ,pVTIVAG(T) for K ~ ~ ~ The ( T )importance , of eq 51 is that is transcribes r ) to the quantal the low-temperature approximation vibrationally adiabatic transmission coefficient K ~ 7‘)~ into ( the language of variational transition-state theory. We then use eq 49 to obtain &rnPvA(n=O,E)exp(-PE) dE ,rVT/VAG
=
~
and it is clear that KICVT/VAG(Tj
=
,pV T /V A G ( T ) e
KVAG(T)
(60)
D. Quantal Treatment of Reaction-Coordinate Motion for Three-Dimensional Reactions. To apply the above methods to reactions in three dimensions we simply replace PvA(n=O,E)by PVA(n=O,i=O,i’=O,J=O,E), which is the quantum transmission probability for the vibrationally and rotationally adiabatic, ground-state, s-wave potential energy barrier. E. Comparison of Improved Canonical Variational Theory and Classical Adiabatic Ground-State Transmission Coefficient. First consider the collinear case. Equation 18 may be written QIGT(T,s)= QGT(T,s)exp(-P[V,(n=O,s) - VAG])ni
C (exp(-P[V,(n,s)
n=O
-
VAGI]- 1) (61)
The ratio of the improved generalized-transition-statetheory rate constant to the original3 generalized-transition-state-theory rate constant is given by kIGT(T,s) - Q ’ G T ( ~ , s ) exp (-0VAG) (62) hGT(T,s) QGT(T,s)exp[-PV,(n=O,s)l nf
= 1 - [QGT(T,s)]-’C (exp{-P[V,(n,s) - V,(n=O,s)l}-
The generalization to quantal vibrationally adiabatic ground-state transmission coefficients for other approximate theories is &mPVA(n=O,E) exp(-PE) dE (53)
n=n
exp(-P[VAG-- V,(rt=O,s)])) (63) If we assume that only one vibrational state is important, as is often the case, we obtain kIGT(T,s)/hGT(T,s) expl-P[ VAG- Va(n=O,s)l] (64) In comparison we consider the CAG transmission coefficient of eq 37, 38, and 39, which can be written
The Journal of Physical Chemistry, Vol. 84, No. 13, 1980 1735
Improved Variational Tiransition-StateTheory
x*ICAG(T)=: exp(-P[ VAG- Va(n=O,sl)]) (65)
We see that the CA,G transmission coefficient yields the same result as the improved canonical variational theory for collinear reactions when only one vibrational state is important. The situation is more complicated for three-dimensional reactions, even when only one vibrational level is important. Equation 31 may be written Q I G T ( ~ , s )= QGT exp{-p[V,(g,J=O,s) - vAG]) C ’ (2J + l)(exp{-P[V,(n,i,i’,e~,s)- VAG]) - 1) (66) n,i,i‘,J
where g is an abbreviation for n = i = i’ = 0. The ratio of rate constants mlay be written k’GT(T,s)/kGT(T,s)= 1 tQGT(~,s)l-lC ’ (2J + l)(exp(-P[V,(n,i,i’,J,s) n,i,iJ
V,(g,J=O,s)]]- expl-PIVAG- V,(g,J=O,s)l])(67) Now assume that only one vibrational level, the g level, is important. Then eq 67 becomes Ji
k’GT(T,s)/kGT(T,s) = 1 - [&T(T,s)]-l
C (2J + 1) x
J=O
(exp(-P[ Vs(g,J,rr)- V,(g,J=o,s)])- exp(-P[ VAGVa(g,J=O,s)l))(68) where Jf is the largest J for which
By treating rotation classically, the sum in eq 68 may be approximated as an integral and the inequality in (69) may be taken as an equality. Then eq 68 becomes kIGT(T,s)/kGT(T,s) = 1 - exp(-P[VAG- V,(g,J=O,s)])X (exp(P[VAG- V,(g,J=o,s)])- 1- p[VAG- V,(g,J=O,s)]) (70)
We assume now that P[VA,“- V,(g,J=O,s)l
