200
The Journal of Physical Chemistry, Vol. 83,
No. I , 1979
B. C.
Garrett and D. G. Truhlar
Accuracy of Tunneling Corrections to Transition State Theory for Thermal Rate Constants of Atom Transfer Reactions Bruce C. Garrett and Donald G. Truhlar" Chemical Dynamics Laboratory, Koithoff and Smith Hails, Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received Juiy 24, 1978)
U'e consider quantum mechanical effects on transition-state-theory transmission coefficients and on generalized transition-state-theory rate constants for elementary atom-transfer reactions in the gas phase. We examine two treatments of tunneling in the context of conventional transition-state-theory, the lowest order correction of Wigner and the full solution of the one-dimensional scattering problem for the classical potential energy barrier. We also examine two additional methods of including quantum effects on the reaction-coordinate motion in the context of adiabatic transition-state theory. One corresponds to taking the reaction coordinate as the minimum-energy reaction path, and the other uses the Marcus-Coltrin tunneling path. To test the accuracy of these approximate theories we present calculations for four collinear reactions: H + HP,D D2, C1 H2, and C1+ TP.Thermal rate constants are computed using these approximate theories and compared with those calculated using conventional and adiabatic transition state theory in which no quantal correction is made to the separable reaction-coordinate motion. The results are also compared with thermal rate constants obtained from accurate quantal calculations. We find that for the H + H2 and D + D2systems the method using the Marcus-Coltrin path is very accurate, leading to errors less than 17% over a temperature range 300-1500 K. However the simple Wigner transmission coefficient is found to give the best overall agreement for the four systems at low temperature,
+
1. Introduction The most serious failing of transition-state theory for many atom-molecule reactions at the usual temperatures of interest is its neglect of tunneling. This causes the predicted rate constants at and around room temperature to be too 10w.l There are many ways to incorporate one-dimensional tunneling corrections into transition-state theory. When the tunneling correction is small and the potential energy surface in the whole region important for tunneling is well approximated by the saddle point normal-mode expansion a simple quantal correction due to Wigner2 is justified. More generally the theoretically best justified way to include tunneling involves quantal transmission through vibrationally adiabatic potential barrier^.^-^ However, it has been shown that if the vibrationally adiabatic barrier is measured along the minimum-energy path and if all numerical approximations in carrying out such a one-dimensional correction are eliminated, it does not lead to accurate predictions of collinear rate constants for the H H2 or D D2 reactions."1° In addition the less well justified but more commonly used method of making tunneling corrections in terms of the classical potential energy barrier also fails.8-10 From this one might have concluded that no one-dimensional tunneling correction would be adequate and that the theory can only be remedied by incorporating quantum mechanical corrections in a framework which does not assume separation of variables at the transition state. This approach was quite successful" for H + H2but the calculations required were harder than desirable for routine application to interpret experimental measurements of rate constants. Recently Marcus and Coltrin proposed a new one-dimensional tunneling path within the framework of the vibrationally adiabatic theory.12 This was tested successfully for collinear microcanonical reaction probabilities for H + H2.I2 If this method proves generally successful it will be of great use for practical work. This article presents a more complete test of the Marcus-Coltrin tunneling path for calculating thermal reaction rates. We consider canonical rate constants for
+
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0022-3654/79/2083-0200$01 .OO/O
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two collinear reactions and isotopic analogues, and we compare the predicted rates to exact dynamical ones as well as to the predictions of the older methods of including tunneling in transition-state theory. 2. Theory The canonical rate constant for a collinear reaction A + BC AB C a t temperature T may be written N T ) = Cf,(;r?Mn,T) (1)
-
+
n
where f,(T ) is the fractional occupation of vibrational state n and h(n,T) is the state-selected rate constant at translational temperature T. In the vibrationally adiabatic theory we define a set of potential curves Va(n,s)which are functions of the reaction coordinate s and the vibrational quantum number n. In the zero curvature approximation they are defined by va(n,s) = v(s,us=o)+ tvlh(n,S) (2) where V(s,us=O)is the potential energy along the reaction coordinate and tvih(n,S) is the vibrational energy of the degree of freedom transverse to the reaction coordinate in vibrational state n at location s. We choose a coordinate system so that the reduced mass for motion along s is equal to p , the reduced mass for relative translational motion, and the vibrationally adiabatic collinear state-selected reaction rate at translational temperature T is9
P ( ~ , T=)
( L ~ ~ ~ T ) - ~exp(-E,,l/kn / ~ S ~ ~dEre1A ( ~ , E 0
(3)
where PA(n,Erel)is the transmission probability for scattering by the one-dimensional barrier V,(n,s) - V,(n,s =--a) where s = --m corresponds to reactants. If we approximate this probability by P*(n,EreJ= O[E,,, - V,(n,s=O) + Va(n,s=-m)l (4) where O(x) is the Heaviside step function O(x) = 1 x >0 =o x 0.1 and of In PA(n,Erel)for smaller probabilities. The spline interpolation is checked against two-, three-, andl four-point Lagrangian interpolation to be sure it causes negligible error. The thermal rate constants were obtained by calculating the n = 0 and n = 1 terms in eq 1 with a quantum mechanical treatment of the reaction coordinate motion. Terms with n I2 were included for corivergence but were calculated with a classical treatment of reaction-coordinate motion, eq 6. Terms with n I2 contribute no more than 1% for H H2, D + D2, and C1 + H, and no more than 3.5% for C1
+
+
+ Tz,
For comparison with the Marcus-Coltrin-path vibrationally adiabatic (MCPVA) calculations we also made calculations using the minimum-energy-path vibrationally adiabatic (MEPVA) barrier. For these the procedure is the same except for computation of s as explained already. As mentioned above, both vibrationally adiabatic approximations involve the assumption of zero curvature (ZC) of the reaction path, i.e., no internal centrifugal terms13 are included in eq 2 or elsewhere. Since all the tunneling calculations involve the ZC approximation we do not explicitly include ZC in the superscripts. If desired, transmission coefficients K ( 7')for conventional transition state theory could be computed by dividing; the calculated thermal rate constant by h*(T). Finally we compared with some even simpler tunneling corrections. The most commonly used tunneling correction involves a quantal correction for one-mathematical-dimensional motion through the classical potential energy barrier along the minimum energy path, Le., the transmission coefficient is approximated by JapvE(EreJ
,CVE(n
=
__
f
.o
exp(-Erei/hT) dEre1 (8)
mo(Ere~ - Eb)
exp(-Erei/hT) dEre1
where pVE(Erel)l is the quantal transmission probability
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The Journal of Physical Chemistry, Vol. 83, No. 1,
7979
B.
TABLE I: Energetics (kcal/mol)
H+ classical endoergicity classical barrier height VA endoergicity VA barrier height VA endoergicity VA barrier height
TABLE 111: State-Selected Vibrationally Adiabatic Rate Constants k A ( n , T )(cm molecule-' s-')
D+ D, C1 + H, C1 + T,
H,
0.00 9.79 n=O 0.00 6.37 n = l 0.00 4.60
0.00 9.79
3.01 7.67
3.01 7.67
0.00 7.35
1.05 3.58
1.93 5.25
0.00 5.02
-2.59 0.95
-0.13 2.19
T,K 300 400 600 1000 1500 300 400 600
TABLE 11: State-Selected Vibrationally Adiabatic Rate Constants k A ( n , T )(cm molecule-'s-')
T, K
H MEPVA
+ H,
300 1.84 x 10' 400 2.79 x 10' 600 4.80 x 10' 1000 5.32 x 103 1500 1.93 x l o 4 300 400 600 1000 1500
2.80 x 10' 7.49 X l o 2 2.84 x 103 1.30 x l o 4 3.36 x l o 4
MCPVA
1000
~~
D + D2
MEPVA n=O 5.18 x 10' 3.17 x 10.' 4.93 x 10' 7.06 x 10' 6.37 x 10' 1.73 x l o 2 6.07 x 103 2.52 x 103 2.08 x l o 4 1.05 x lo4 n = l 5.38 x l o 2 2.20 x 10' 1.36 X l o 3 1.41 x 10' 4.24 x 103 1.12 x 103 1.55 x l o 4 7.32 x l o 3 3.67 x l o 4 2.09 x l o 4
1500
MCPVA
6.18 x 1.05 x 2.14 x 2.80 x 1.11x
10.' 10' 10' 103
lo4
3.61 x 10' 2.06 x 10' 1.41 x 103 8.19 x l o 3 2.24 x l o 4
for scattering off this barrier and Ebis the classical barrier height. To emphasize that the effective barrier is not lowered by release of some of the reactant vibrational energy we call this the conservation-of-vibrational-energy ( W E ) method. Although it i s widely used as a correction to k*(T),it has no justification in terms of a dynamical model (i.e., there is no reason to expect vibrational energy to be conserved throughout the collision) or in terms of the adiabatic derivation of transition-state theory. For illustrative purposes we show the effect of multiplying either h'(T) or kATST(T) by K ~ " ~ ( T ) . T h e simplest tunneling correction is the Wigner transmission coefficient, which is simply given by2,14b16 ~ ~ ( 7=' 1 ) + (hl~1I/RT)~/24
C.Garrett and D. G.Truhlar
(9)
where a* is the imaginary frequency of the unbound normal mode a t the saddle point. This may be derived from (8) by approximating the potential energy barrier by an inverted parabola and assuming hlw*l/hT
