Generalized Transition State Theory. Quantum ... - ACS Publications

Generalized Transition State Theory (126) M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions”, Dover Publications, New York, 1965, p 298.

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

1079

(127) D. G. Truhlar and C. J. Horowitz, J. Chem. Phys., 68, 2466 (1978). (128) G. W.Koeppl and G. Stein, J. Chem. Phys., 63, 4081 (1975).

Generalized Transition State Theory. Quantum Effects for Collinear Reactions of Hydrogen1 Molecules and Isotopically Substituted Hydrogen Molecules Bruce C. Garrett and Donald G. Truhlar“ Chemical Dynamics Laboratory, Kolthoff and Smith Halls, Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received October 27, 1978)

We consider several approximate methods in an attempt to correct two major deficiencies of the conventional transition state theory of chemical reactions. One deficiency is its overestimating thermal rate constants due to ai breakdown of the fundamental transition state approximation that classical trajectories do not recross the transition state dividing surface. This error classically becomes more important at high temperatures. The other shortcoming of conventional transition state theory is the neglect of quantal effects on the reactioncoordinate motion. This becomes more important at low temperature, and when tunneling is important conventional transition state theory underestimates the low-temperature rates, sometimes severely. We consider first three generalizations of transition state theory in which no quantal correction to the reaction-coordinate motion is included but bound motion normal to the reaction coordinate is quantized. These are canonical variational transition state theory, microcanonical variational transition state theory, which is equivalent to the (vibrationally)adiabatic theory of reactions, and Miller’s unified statistical theory. We also consider several ways to make tunneling corrections. One is the Wigner transmission coefficient,and the other three involve one-mathematical-dimensional quantal scattering calculations performed numerically. The barriers for these calculations are the conservation of vibrational energy barrier, the minimum-energy-path vibrationally adiabatic barrier, and the Marcus-Coltrin-path vibrationally adiabatic barrier. To test the accuracy of these approximate methods, we present calculations for several collinear reactions of H, D, C1, or I with five isotopes of hydrogen molecules and compare these results with those from accurate quantal calculations of the reaction probabilities as functions of energy and of the thermal rate constants as functions of temperature. Our results show that the variational theories and the unified statistical theory offer useful improvements over conventional transition state theory at high temperature although bounding inequalities are not satisfied as they are in purely classical transition state theory. No method of including one-dimensional quantal corrections for the reaction-coordinate motion was found to be adequate for all the systems studied. In fact for the I + Hzreaction on each of two potential energy surfaces the variational methods with classical treatment of the reaction-coordinate motion yield excellent results while all quantum corrections to the reaction-coordinatemotion fail. For the other systems the vibrationally adiabatic method using the Marcus-Coltrin tunneling path gives the most consistently accurate results overall, although the Wigner correction gives more accurate results in some instances. A quantally unified statistical model with quantal corrections computed from the Marcus-Coltrin vibrationally adiabatic barrier predicts the most consistently accurate isotope effects for the H -t Hz and C1 H2systems,

+

I. Introduction Transition state theory incorporates the basic factors controlling the rates of most chemical reactions. Two of its most basic deficiencies, however, are the necessity of extra assumptions to include quantum mechanical tunneling effects and the fundamental assumption that trajectories crossing a dividing surface in phase space proceed directly to products. The neglect of tunneling is very importamt a t low temperature and often causes the transition state theory rate constant to be too low. The assumption of direct reaction is classically most in error a t high temperature and causes transition state theory to overestimate the rate a t high temperature. In this article we test some approaches designed to correct these deficiencies. In the preceding paper1 we discussed generalized transition state theories from a classical mechanical point of view. This viewpoint best elucidates the foundations of the various theories. We also performed calculations for collinear A + BC reactions using classical transition state theory, three different classical generalized transition 0022-3654/79/2083-1079$01 .OO/O

state theories, and exact classical trajectories. The comparisons illustrated the ability of generalized transition state theories to improve on the accuracy of conventional transition state theory. Of course quantum effects such as zero point energies and tunneling are important for many chemical reactions and the results of purely classical calculations often cannot be usefully compared to experiment. It is difficult to accurately incorporate quantum effects on all the degrees of freedom into transition state theory2 although recently there has been encouraging p r o g r e ~ s . ~ - ~ Traditionally quantum effects are incorporated into transition state theory as follow^.^ First a reaction coordinate is separated out and treated classically. Then the remaining degrees of freedom of the transition state are quantized like those of a bound state. The reactant bound degrees of freedom are also quantized; thus the transition state theory rate constant is evaluated using quantal partition functions. Optionally some attempt may also be made to treat quantum effects on a reaction coordinate which is assumed separable, but there is no consensus of

0 1979 American Chemical Society

1080

B. C. Garrett and D. G. Truhlar


opinion about how to do this.1° Even without including quantum effects on the reaction coordinate or correcting the theory for nonseparability of a reaction coordinate, this traditional method has been quite successful. However, it is often difficult to judge the theory by comparing its predictions to experiment because the required accurate information about the potential energy surface is not known. Generally it must be estimated, usually using the kinetic data against which the theory is being tested. This procedure may cause dynamical errors in transition state theory to be compensated by errors in the semiempirical or empirical potential energy surfaces. In the last 10 years it has become possible to calculate exact quantum mechanical equilibrium rate constants for collinear chemical reactions using given potential energy surfaces.11-20This kind of calculation has even been extended to one three-dimensional chemical reaction.21 In some cases, transition state theory has been applied to predict the rate constants using the same potential energy surfaces.8J-16~22~23 In such a comparison any error in the surfaces is irrelevant and, as long as the surfaces are realistic, interesting conclusions about the validity of transition state theory for treating dynamics of chemical reactions can be drawn. In the present article we apply this kind of test of conventional transition state theory for the first time to the I + H2 collinear reaction. We also make a systematic test of quantized generalized transition state theory and quantal treatments of the reaction coordinate by applying three different versions of the former and combinations of these with four of the latter to I + H2, H + H2 and three isotopic analogues, and five isotopic examples of C1+ H2. For I + H2 and H + H2 we consider two different potential energy surfaces. This extends our earlier short paper8 which included only one version of quantized generalized transition state theory and four quantal treatments of the reaction coordinate for H + Hz, D Dz, and two isotopic examples of C1+ Hz, using one potential energy surface in each case. In section I1 we present transition state theory expressions for microcanonical and canonical rate constants for which the bound degrees of freedom are quantized but the separable reaction coordinate is treated classically. In section I11 several methods of including quantal corrections on the reaction coordinate motion are discussed. Section IV is a description of the computation procedures, section V is the results, section VI is a discussion of the results, and section VI1 is a summary.

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11. Quantized Generalized Transition State

Theory with Classical Treatment of the Reaction Coordinate Purely classical versions of conventional and generalized transition state theories for collinear atom-diatom reactions have been presented in detail in the preceding paper.' In this section we indicate how quantum mechanics is incorporated into the treatment of the bound degrees of freedom. This is accomplished in the usual ad hoc way. We again restrict our discussion to the collinear reaction of an atom with a diatom. The notation used here is consistent with the preceding article, to which the reader is referred for an introduction to the various methods. A. Microcanonical Rate Constants. The generalized transition state theory approximation to the cumulative reaction probability at total energy E is the number of vibrational states, of energy less than or equal to E , for the degree of freedom transverse to the reaction coordinate s. Classically this is a phase space integral and it is a of E and s. Quantization continuous function NCGT(E,s) of this result is accomplished semiclassically by requiring

the number of states to be an integer, Le., by replacing NCGT(E,s) by the nearest integer n = flT(E,s).The removal of the subscript C denotes the removal of the classical assumption which made the number of states continuous. Then integrals over N become sums over n. The resulting expression for the generalized transition state theory cumulative reaction probability is n,(s)

flT(E,s)= C B[E - V,(n,s)] n=O

(1)

where nm(s) is the quantum number of the highest bound state of the local vibrational well at s, O(x) is the Heaviside function, Va(n,s)is an adiabatic potential curve

Va(n,s)= V(s,uS=O)+ Evib(n,S)

(2)

usis the local vibrational coordinate, V(s,us)is the potential energy as a function of both coordinates, and tvib(n,s)is the local vibrational energy for motion normal to the reaction coordinate with vibrational quantum number n. The quantized sum of states can also be written as P T ( E , s ) = IFIXINCGT(E,s)- 721 + 1 (3)

where IFIX(x) equals x if x is an integer and otherwise it truncates x to the nearest lower integer. Notice that semiclassically

Va(n,s)= V,(N=n+'/,,s)

(4)

and

evib(n,S)= tyib(N=n+l/,~) (5) where the right-hand sides use the notation of the preceding paper. It will be convenient to use the new notation involving quantum number n instead of classical action variable Nh throughout the present paper. The generalized microcanonical transition state theory rate constant becomes

where the reactant density of states per unit energy per unit length is

$R(E)=

nRmaxis nmax(s=-m)and p is the reduced mass for A-BC relative translation. Notice that hGT(E,s)and 4R(E),like NGT(E,s), but unlike the three classical counterparts, are not continuous functions of energy and s. The conventional transition state theory microcanonical rate constant expression is obtained by evaluating eq 1 at the saddle point:

/$(E) = [h$R(E)]-lNGT(E,s=O)

(8)

= [h f p (E ) ]-1Ni ( E )

(9)

where w(E)is the cumulative reaction probability of conventional transition state theory. The variational theories arise from the fact that classical transition state theory provides an upper bound to the exact classical equilibrium rate constant. Although quantized transition state theory does not give an upper bound to the exact quantum mechanical equilibrium rate constant, we still locate the dividing surface in microcanonical variational transition state theory (WVT)such that eq 1yields an absolute minimum. Because of relationship (3) this location is identical with that [s = sJ'"'(E)] ob-


Generalized Transition State Theory

tained in the purely classical theory and the cumulative reaction probability becomes

1081

where in eq 21 we neglect terms of order exp(-PD). Substituting eq 1 into eq 19 yields the generalized transition state theory canonical rate constant

n=O

According to the adiabatic theory of reactions, the microcanonical reaction rate and quantized cumulative reaction probability are given by

k A ( E ) = [h+R(E)]-lNA(E)

(11)

and

where, again neglecting terms of order exp(-PD), the generalized transition state partition function is

nR-

NA(E)= C PA(n,E) n=O

(12)

n,(4

QGT(T,s)c

respectively, where the classical transmission probability in the adiabatic approximation for an initial quantum state n a t total enery E is

PA(n,E)= 8[E - VaA(n)]

(13)

V,"(n) = Va[n,saA(n)]

(14)

C exp{-P[V,(n,s)

n=O

-

V(s,us=O)ll (24)

Conventional transition state theory is obtained by evaluating eq 23 and 24 at s = 0:

and = max Va[n,s]

and n,(s=O)

(15)

QYT)

S

= VaA(N=n+l/,)

(16)

where s*A(n)is the absolute maximum of the adiabatic curve Va(n,s) (and the right side of (16) is in the notation of the preceding paper. It can easily be shown that

C

n=O

exp(-Pcn9

(26)

and €2is the vibrational energy for the motion transverse to the reaction coordinate at the saddle point. In eq 26, and in eq 28 and 30, the "approximately equals" sign is used to denote the neglect again of terms of order exp(-PD) The canonical rate constant of microcanonical variational transition state theory or the adiabatic theory of reactions is obtained by evaluating each term in the generalized partition function, eq 24, at s = ~ * ~ ( This n). gives I

s F T [VaA(n)]= s*A(n)

(17)

from which it follows that the adiabatic theory and microcanonical variational transition state theory give the same result for the rate constant. Miller's unified statistical is quantized, following his in the same manner to give the following approximation to the cumulative reaction probability

N""E) = N,"T(E)N"'"(E)N""(E) [N"v"(E) N"'"(E)]N""(E) - N " T ( E ) P ' " ( E ) if S * ~ ~ exists ~(E) -- NG"'~(E) if not (18) where ~ * ~ l ~is(the E l location of the second lowest local minimum, as (9 function of s, of PPT(E,s)and P a x ( E is ) the largest maximum of P T ( E , s )between s = s,fiVT(E) and

where

+

s=

S*mln(E).

B. Canonical Rate Constants. In the preceding paper we carefully separated out all contributions to the thermal rate constant from total energies greater than the dissociation energy. Then canonical rate constants are obtained from the cumulative reaction probability using the following relatiomship:

nRm

QwvT(T) E

C exp(-P(V,[n,saA(n)l- VI)

n=O

(28)

We have made use of the equivalence of the adiabatic and microcanonical variational transition state theory rate constants to obtain this result. Canonical variational transition state theory (CVT) specifies the location s = S * ~ " ~of( Tthe ) dividing surface to minimize kGT(T,s).The value of S * ~ " ~ is ( Tthe ) same in the quantized theory as in the classical one. The rate constant is given by

(29) where

where we now specify the zero of energy to be the classical minimum of the reactant vibrational well, D is the energetic threshold for production of three dissociated atoms, 0is (AT)-', and the partition function per unit length for reactants is

V [ S = S , ' ~ ~T( ),us=O]]) (30) We also note the connection of the canonical variational transition state theory to the maximum free energy criterion. Equation 23 may be rewritten as kT hGT(T,s)= --KO exp[-AG(T,s)/RT] (31) h The standard state free energy change for formation of the

1082


generalized transition state is AGGTgo(T,s) but we write it as

Since S * ' ~ ( Tis)located to minimize kGT(T,s)it maximizes AG(T,s). The canonical rate constant of the unified statistical theory is obtained by substituting eq 18 into eq 19. This yields the following canonical rate constant (33) where


The most commonly used model for the transmission coefficient involves tunneling through the one-dimensional potential energy barrier V(s,us=O)along the reaction coordinate. This would be the potential energy for reaction-coordinate motion for a collinear collision if the vibrational energy in the transverse coordinate were conserved. Therefore we call this the conservation-ofvibrational energy (CVE) approximation. The CVE approximation is inconsistent with the adiabatic energy release implied by transition state theorylO although the difference of the CVE and vibrational adiabaticity assumptions in the saddle point region is not large for some reactions with early saddle points. The transmission coefficient in the CVE model is given by the ratio of the thermally averaged quantal transmission probability for this barrier to the thermally averaged classical one, i.e. kiiCvE(T) = KCvE(T)kI(T)

The weighting function W(m) is given by

Jm

W(m) = Nus(emuS)- Nus(em-lUS) m>o = Nus(eous) m = o (35) and emUS are the energies at which W s ( E )changes its value discontinuously. If NGT(E,s)has only one local minimum for all energies, W(m) equals 1 for all m and emUS equals V,[m,s=s**(m)].In this case kus(T) equals kfivT(E).

111. Quantal Corrections on the Reaction Coordinate The quantization of the bound degrees of freedom in transition state theory is accomplished straightforwardly. In conventional transition state theory the resulting rate constant h i ( T ) may then be multiplied by a transmission coefficient K(T).In principle this transmission coefficient should correct for all the errors in transition state theory. Thus it should correct for the fact that classically some trajectories do recross the dividing surface and also for all quantum effects not included by just using quantized energy levels for the bound degrees of freedom. In practice, however, no attempt short of a full trajectory calculation is usually made to estimate the recrossing correction and, except for one study6 in which no separation of variables was made, all attempts to include the remaining quantum effects have involved estimates of quantal transmission through and across one-mathematical-dimensional potential barriers. In this article we systematically test several such approaches. All have been described but we next summarize them in the present notation. Wigner has worked out the leading quantal correction to quantized conventional transition state theory to lowest order in This procedure is valid only if the correction is small. When this lowest-order correction is valid only the region of the potential energy surface in the vinicity of the saddle point need be considered and the potential can be approximated by a quadratic expansion around the saddle point. The resulting corrected canonical rate constant is ~

5

.

~

1

~

~

hW(T) = KW(T)ki(T)

(36)

where K~(T = )1 + ( h l ~ ~ l / k T ) ~ / 2 4

pvE(Ere1)

= Jm

fl(Ere1 -

(38)

exp(-PErei) dEre1

v )exp(-PErei) dfir,,

(39)

where pCvE(EreJis the quantum mechanical transmission probability for the classical potential energy barrier of height Vt. at relative translational energy ErelaNotice that the thermal averages in eq 39 are those appropriate for collinear collisions. In eq 39, as well as eq 46, we have written the upper limit of the integral in the customary way as m; strictly, as discussed above, it should be Do, but contributions from E > Do are negligible for the examples considered here. Complete specifications of the classical barrier V(s,u8=O) requires a definition for s; conventional transition state theory merely requires that s reduce to the unbound normal-mode coordinate at the saddle point. Elsewhere it is usually taken as the distance along the path of steepest descent in some coordinate system. This is the definition we use where the coordinate system is the (x,y) scaled and skewed coordinate system of the preceding paper. See the Appendix. The vibrationally adiabatic model provides a framework for a more consistent and, one has a basis for hoping, a more accurate inclusion of quantal effects on the reaction coordinate. The adiabatic model is more consistent because the adiabatic theory of reactions reduces to conventional transition state theory when the microcanonical variational transition state is located at the saddle point at all energiesaZ6Further, when this reduction does not occur, the adiabatic theory is more accurate classically than conventional transition state theory, so it should provide a better starting point for including quantum effects. The adiabatic theory with tunneling based on a vibrationally adiabatic model may be more accurate than other models because the adiabatic theory, being equivalent classically to microcanonical variational theory, includes classical recrossing effects and because the vibrationally adiabatic assumption is often justified dynamicallyz7at a detailed single-collision dynamical level. The quantal vibrationally adiabatic cumulative reaction probability is obtained by replacing the classical transmission probability for total energy E for the vibrationally adiabatic barrier V,(n,s) corresponding to quantum state n in eq 1 2 by the quantal counterpart PvA(n,E)to get

(37)

and at is the imaginary frequency of the unbound normal mode at the saddle point. Strictly speaking, the conditions for validity of the assumptions made in deriving eq 37 are seldom satisfied.2>s

n=O

The quantal vibrationally adiabatic canonical rate constant is then given by eq 19 as

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


where PVA(n,E)is considered as a function of total energy in (41) and as a function of relative translational energy in (42). Equation 4 2 may be rearranged to

~ v A ( T=)

flPffl(T>hVA(n,T)

(43)

fl=O

which proceed via long-lived complexes. In section I1 it is quantized in such a way that for direct reactions at low energy it tends to the quantized microcanonical variational theory with classical treatment of the reaction-coordinate motion. However, a t low energy the reaction-coordinate motion should be treated quantum mechanically. Thus, we present here a quantally unified statistical model which for direct reactions in the low-energy limit becomes either the MEPVA or MCPVA version of quantally corrected microcanonical variational theory and at higher energies tends to the same limit as the quantized expression of section 11. We begin by rewriting eq 18, the quasi-classical unified statistical theory approximation to the cumulative reaction probability, as follows

f,(n

where is the fractional occupation of vibrational state n and kVA(n,T)is the quantal vibrationally adiabatic approximation to the state-selected thermal rate constant and is given lby kVA(n,T)=

NUS(E) = N"VT(E)R(E)

kv"(T) = KVA(T)kA(T) where the tramsmission coefficient is

(45)

(47)

with 1

R(E)= -

NpVT(E) NfiVT(E) I+---N"'"(E) N"a"(E)

=1

Finally we note that eq 41 can be rearranged to

1083

if S * " ~ ( Eexists )

if not

where N"(E) is the quasiclassical quantity corresponding to quantized u motion and classical s motion and R(E) can be viewed as a recrossing fact,or. Since

NpvT(E)IWin(E)I Nm"(E)

(49)

we have

KVA(T)=:

J

0

SomIdE

'>

fl=O

exp(-PE)

R(E) I 1 (46)

O[E - VaA(n)]

fl=O

This equation should be compared to eq 39. Complete specification of the quantal vibrationally adiabatic model requires specification of the reaction coordinate s along which distance is measured. We consider two choices. First, as for the CVE model, we measure distance along the minimurn-energy path in the (z,y) plane. This yields the minimum-energy-path vibrationally adiabatic (MEPVA) rate constant kmpvA( T).lo Recently Marcus and Coltrin7 have suggested a new one-dimensional patlh, different than the minimum energy path, for vibrationally adiabatic tunneling in the interaction region. Based upon semiclassical arguments they find the variaitionally best path for vibrationally adiabatic tunneling, i.e., for places where Erelus less than V,(n,s) Va(n,s=--co), is close to the path along the outer'turning points of the vibrational motion transverse to the reaction coordinate. Therefore their new path is on the concave side of the reaction coordinate in the (x,y) plane, i.e., it "cuts" the corner in going from reactants to products. This effectively cointracts the reaction coordinate thus giving a thinner barrier and an increase in tunneling. Using this measure of s yields the Marcus-Coltrin-path vibrationally We note adiabatic (MCPVA) rate constant kMCPVA(T).7~6 that the variational criterion used by Marcus and Coltrin in selecting a tunneling path is the analytic continuation of the classicall principle of stationary action, which should not be confused with the classical minimum-flux variational principle for locating the dividing surface. Finally we consider the incorporation of quantal effects on the reactilon coordinate into the unified statistical theory. Recall that tlhe classical unified statistical theoryz4 correctly incorporates the limits of classical microcanonical variational theory for direct reactions at low energy and the classical statistical phase space theory for reactions

and the effect of recrossing is to decrease the rate. The quantally unified statistical theory is defined by replacing the quasiclassical cumulative reaction probability Npw(E) in eq 47 by the cumulative reaction probability corresponding to the adiabatic theory of reactions with quantized energy levels and quantal treatment of the reaction-coordinate motion, i.e., by NVA(E)=

PVA(n,E)

(51)

fl=O

to give NQ"S(E) = NVA(E)R(E) (52) For direct reactions at low energies, i.e., below the classical threshold, sqmin(E) does not exist and e U s ( E ) NVA (E). At high energy, quantal corrections to the reaction-coordinate motion become less important and NVA(E) NWVT(E),and we regain the high energy limit of the quasiclassical unified statistical theory.

-

-

IV. Calculations A . Systems Studied and Potential Energy Surfaces. For H + Hz and D + D, using a reasonably accurate rotated-Morse-curve surface13(called the TK surface2') and for C1+ Hz and C1+ T2using an extended LEPS surface and used by Baer,29we have already due to Stern et alez8 reported calculations using conventional transition state theory, microcanonical transition state theory, and the four quantal treatments of reaction-coordinate motion explained in section III.8 That study extended some earlier tests for these ~ y s t e m s ~ and J - ~ in ~ the present article we further extend these tests to include canonical variational transition state theory, with both classical and quantal treatments of reaction-coordinate motion, the unified statistical theory, and the quantally unified statistical theory and to include H + H2using surface no. 2 of Porter and K a r p l u ~called , ~ ~ the PK2 surface, two more isotopes

1084


of H + Hz using the T K surface, and three more isotopic examples of C1+ Hz. In addition we present calculations on systems for which quantum mechanical thermal rate constants are available, but no version of transition state theory has yet been tested. These systems are the I Hz reaction using the surface of Raff et aL31 and also a rotated-Morse-curve surface.32 B. Generalized Transition State Theory Calculations. Calculations of the generalized transition state theory cumulative reaction probability were based upon eq 3. The calculation of NCGT(E,s)is described in detail in the preceding paper and the interested reader is referred there. Recall that the vibrational potential for the degree of freedom transverse to the reaction coordinate is fit to a Morse potential to facilitate computations. In the purely classical calculations two methods were used to fit the potential. The differences between the two methods were generally small so in this article we will give most results only for Morse approximation I. In this model the Morse classical dissociation energy parameter is set equal to the reactant molecule's dissociation energy minus the local minimum of the vibrational well, i.e. DM(S)= D - v(S,Us=o) (53)

+

where the zero of energy is V(s=-w,us=O), and the Morse range parameter aM(s)is fit to give the correct second derivative of the local vibrational well at the minimum. When not stated otherwise, we used Morse approximation I1 for this paper. The method of computing the thermal rate constants is sufficiently different from the purely classical calculation that we describe it briefly here. For conventional transition state theory the thermal rate was computed two ways to provide a check. First N C f ( Ewas ) computed directly and inverted numerically by fitting E as a function of Nc' to a cubic spline. Evaluation a t half-integer values of NcJ gives eni as required for evaluation of ht(7') by eq 25 and 26. As a check we also computed kJ(T) using eq 31 with s = 0. This requires evaluating the free energy curves at s = 0. Evaluation of the free energy curves was accomplished by computing the generalized transition state partition-function using eq 24 and substituting into eq 32. This method was also used for the canonical variational theory thermal rates by finding seCvT( 7') as for the classical calculations. The microcanonical variational transition state theory thermal rates were computed straightforwardly from eq 27 and 28. The unified statistical theory approximation to the thermal rates was obtained using eq 33-35. The three extrema of NGT(E,s)were numerically inverted by fitting E as a function of NcFvT,NC"", and Ncminto cubic spline functions. Evaluation at half-integar values gives the energies at which the quantized functions change discontinuously. The three sets of energy values for each of the three sum-of-states functions are used to construct one monotonically increasing sequence (m = 0, 1,2, mmax) of energies emus. Each subscript m corresponds to a new set of NFw, p, N"" values and thus a new value for Nus by eq 18. The calculation was then completed using eq 33-35. C. Quantal Treatment of the Reaction-Coordinate Motion. The simplest quantal treatment of the reaction coordinate is Wigner's. The saddle point is found by a two-dimensional root search for the point where aV/ax = aV/ay = 0 and a normal-mode analysis is performed. The frequency ut is the square root of the product of /.-l and the negative eigenvalue of the force constant matrix. The other quantal treatments involve quantum mechanical transmission probabilities for one-dimensional


barriers. These are computed numerically using for each probability three calculations and the h4-extrapolation method described earlier.1° Table I contains the computational details for the finite difference solution of the one-dimensional Schroedinger equation. The boundary values of the reaction coordinate on the finite difference grid are given as well as the corresponding maximum values of the R- and RBCcoordinates. The values of the ground state adiabatic potential curves (relative to the reactant or product ground state energy levels) at the boundaries are also given. Corresponding values for the first excited state are comparable in magnitude and are approximately a factor of 3 larger. The same values for the classical potential barriers as used in the CVE method are not given because the CVE method was found to converge with respect to widening the boundaries much more rapidly than the adiabatic methods. The three grid sizes imUJof the three-finite difference calculations are also given. The finite-difference step size for the jth calculation is (s, - .s 0 on the concave side of the reaction coordinate, this is given by

where r)

(55) 0

and aM(s) and D v ( s ) are Morse parameters defined above. Then xt(n,s) and yt(n,s) are obtained by an orthogonal transformation from u:(n) and z:(n) = 0. The turning point was computed at regular intervals along s and yt(n,s) was fit to a local quadratic in x,(n,s) using three consecutive points. The distance s along the path of outer classical turning points from (xl,yl) to (x2,y2)is then given by

...)

which can be evaluated analytically from the quadratic fit. Marcus and Coltrin computed PMCPVA(n,Erel) semiclassically and so did not need to describe the path in classically allowed regions. However since we compute PMCPVA(n,Erel) quantum mechanically we need to define the path for all values of n, Erel,and s. We define the complete tunneling path such that it is along the minimum energy path in classically allowed regions, Erel> Va(n,s), and along the path of outer classical turning points in classically forbidden regions, Erel< V,(n,s). Therefore if Erel> VaA(n)the minimum energy path and MarcusColtrin path coincide. Further discussion is given elsewhere.s

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


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1085

The Boltzmann averages of the quantum mechanical transmission probabilites in eq 39 and 42 were performed as follows. The relative translational energy range was divided into two parts. The first part, from zero to the barrier height, was subdivided into seven intervals of equal length or intervals of length 0.063 kmol/mol, whichever was largest. The part above the barrier height was subdivided into intervals of length 2kT. Within each interval the numerical integration was performed by Gauss-Legendre quadrature. Integrations were extended to sufficiently high energies to converge the rate constants to at least five significant figures. Fifteen-point GaussLegendre quadrature in each interval was found to be sufficient to give four significant figures for the CVE method and for the adiabatic methods for the ground state. For the adiabatic methods for the first excited state up to 35-point Gauss-Legendre quadrature per interval was required to converge the rate constants to 1%accuracy. The required probabilities at the quadrature nodes were obtained by nine different methods of interpolation from those calculated on a grid of energies. Three different interpolation schemes (three- and four-point Lagrangian interpolation and cubic spline interpolation) were applied to interpolate (i) the probability directly, (ii) the logarithm of the probability, and (iii) the probability or the logarithm of the probability depending whether the probability was greater than 0.1 or less than 0.1, respectively. The grid of energy points was optimized until all nine methods agreed with one another to within 2.6% or better for all cases except the MCPVA method for n = 1 for H D2 where the greatest deviation was 4.8%. The results in the tables are for cubic spline interpolations of quantity (iii), which is probably much more accurate than the greatest deviation of all nine methods. Canonical rate constants for the quantally unified statistical theory are obtained by substituting eq 52 into eq 19. The integration was performed as follows. The energy integral was divided into segments defined by the energy levels emus as defined following eq 35. Between energy levels R ( E ) is constant. Each energy interval, if wider than 2.0 kcal/mol, was subdivided into 2.0 kcal/mol segments or the remainder of the energy interval, whichever was smallest. Within each subinterval numerical integration was performed by Gauss-Legendre quadrature. Fifteen-point quadrature was used and found to be sufficient to give four significant figures. Integrations were extended to sufficiently high energies to converge the rate constant to five significant figures. Interpolation of NVA(E) to the quadrature nodes was accomplished by using the nine methods described above to interpolate PVA(n= 0,E) and PVA(n=1,E) individually, NVA(E) being given by eq 51.

+

V. Results Results for twelve systems are given in Figures 1-17 and Tables 11-XXV. Figures 1, 3, 5-12, 14, and 16 have four parts each. In the upper left part of each of these figures, potential energy contours are plotted in the scaled and skewed coordinate system. The reactant region is at the lower right of this plot. The saddle point is denoted by a + sign, and the solid curve passing through it is the minimum energy path (MEP). The two dashed curves are the paths of outer classical turning points for the vibrational motion normal to the minimum energy path; the one closer to the MEP is for the ground vibrational state, and the farther one is for the first excited state. These dashed curves are only shown for those portions of the paths which are actually used in the MCPVA calculation. These are the

1086



TABLE 11: State-Selected Rate Constants (Units of cm molecule-’ s - ’ ) for t h e Collinear Reaction H t Using the PK2 Potential Energy Surfaceamc

n=O T, K

kA(n,T)

200 300 400 600

2.03(- 2) 3.63 5.05(1) 7.47(2) 7.07(3) 2.34( 4) 6.26( 4) 1.33(5)

1000 1500 2400 4000

kMEPVA(n,T)

n=l

kMCPVA(n,T)

1.26(-1) 8.57 8.56(1) 9.86(2) 8.07(3) 2.53(4) 6.53(4) 1.36(5)

H, -+ H, t H

7.45(- 1)

3.38(1) 1.68( 2 ) 1.44( 3 ) 9.78( 3) 2.84(4) 6.99(4) 1.41(5)

k(n,T)d

kyn,T)

9.42(- 1) 3.16(1) 2.23(2) 1.87( 3) 1.15( 4 )

8.50 2.03( 2) 1.03(3) 5.58( 3) 2.36( 4) 5.24(4) 1.03(5) 1.80(5)

k M E P V A ( n , T )kMCPVA(n,T) 6.18( 1) 3.72(2) 1.11(3) 4.45( 3) 1.82(4) 4.25( 4) 8.93(4) 1.63(5)

1.07(2) 6.03(2) 1.62(3) 5.52(3) 2.01(4) 4.47(4) 9.16(4) 1.66(5)

a kA(n,T ) is the state-selected rate constant calculated b the vibrationally adiabatic theory with classical treatment of the reaction coordinate motion. kmPVA(n, T ) and k M C p J A ( nT, ) are the state-selected rate constants using the vibrationally adiabatic theory with quantal treatment of the reaction coordinate by the MEPVA and MCPVA methods, respectively. k ( n ,2’) is the accurate quantum mechanical state-selected rate constants. Number in parentheses are powers of ten by which the preceding number should be multiplied. Accurate quantum mechanical state-selected rate constants are available only for n = 0. Schatz, Bowman, Dwyer, and Kuppermann (ref 40).

portions of the paths for which the vibrationally adiabatic potential curve is greater than the initial and final vibrational energies. At the lower left are plots of the potential energy along the MEP and the vibrationally adiabatic potential curves as functions of the distance along the MEP (solid curves) and the vibrationally adiabatic potential curves as functions of the distance along Marcus-Coltrin path (MCP, dashed curves). Because the MCP’s are energy dependent we plot the MCP vibrationally adiabatic barriers for one selected energy for each quantum state, as explained for each figure in its caption. The flat sections at each side are the asymptotic values for s = fm. The curves are labeled at the right by the quantum number n. The lowest curve is the potential barrier. At the upper right of each of these figures PMEPVA(n,Erel) (solid lines) and PMCPVA(n,Erel) (dashed lines) are shown for the ground and first excited state of the reactants. Accurate quantal results are shown as symbols when they are available. The lower right part of each of these figures contains plots of average reaction probability as a function of total energy according to the quantized adiabatic theory without quantal corrections to the reaction-coordinate motion and with corrections by the MEPVA and MCPVA methods. The solid piecewise constant curves are the result of the vibrationally adiabatic theory in which the reaction coordinate motion is treated classically. This microcanonical quantity is defined by nR,a.

PA@) = [NR(E)]-l

c

n=O

PA(n,E)

= NA(E)/NR(E)

(58) (59)

where NA(E),defined by eq 12, is the quantized cumulative reaction probability at total energy E in the adiabatic theory of reactions with classical treatment of reactioncoordinate motion, and @(E) is the number of vibrational states of the reactant diatom of energy less than or equal to E . PA(E)increases discontinuously for energies corresponding to maxima in the adiabatic barriers and decreases discontinuously at the reactant vibrational energy levels. The solid and dashed curves are the vibrationally adiabatic microcanonical quantities defined by nR,.

PVA(E) = [NR(E)]-l C PVA(n,E) n=O

(60)

NVA(E)/NR(E) (61) where f l A ( E ) ,defined by eq 40, is the cumulative reaction probability at total energy E in the adiabatic theory of reactions including quantal treatment of reaction-coor=

dinate motion. For appropriate choices, PMEPVA(n,E) or PMCPVA(n,E), of PVA(n,E), the average reaction probability P’JA(E)is called PMEPVA(E) (solid) or PMCPVA(E) (dashed). Accurate quantal results are shown as symbols when they are available. For all systems except the H + D2 system and the two I + H2 systems, the left bound of the abscissa in this kind of plot is chosen to correspond to the product or reactant ground state vibrational energy, whichever is greater and the right bound is the second excited state energy of the reactant. For the H + D2 system and the highly endoergic I + H2systems the left bounc; -emains the same, however the right bound is taken as the maximum of the reactant and product second excited state energies. The zero of energy is the bottom of the reactant potential curve. Figures 2 , 4 , 13, 15, and 17 are plots of the generalized free energy curves AG(T,s) = [AG(T,D,s)]as functions of distance s along the reaction coordinate for four temperatures. For 200 and 1500 K the ordinate is at the left; for 600 and 4000 K the ordinate is at the right. The flat section at both sides of the curves are the asymptotic values for s = f m . The solid and dashed curves are for Morse approximations I and 11, respectively, as used in the previous paper.l These are used as a measure of the validity of the Morse approximation used in these calculations. For all but the highest temperature the results using the two Morse approximations are in excellent agreement and agreement between the absolute maxima in these curves is within 2% for the five systems plotted. Except for the free-energy curves Morse approximation I is used throughout the present paper. Tables 11-XI11give state-selected thermal rate constants as calculated from the adiabatic theory with a classical treatment of the reaction coordinate and with quantal treatments of the reaction coordinate by the MEPVA and MCPVA methods. They also give, when available, accurate quantal results k(n,T). Tables XIV-XXV give the canonical rate constants for 13 methods: conventional transition state theory without quantal correction to the reaction coordinate motion, with Wigner’s tunneling correction, and with tunneling correction by the CVE method; canonical variational transition state theory without quantal correction to the reaction coordinate and with the CVE tunneling correction; microcanonical variational transition state theory (adiabatic transition state theory) with a classical treatment of the reaction coordinate, with quantal correction of the reaction coordinate by the CVE method, and with a quantal treatment of the reaction coordinate by the MEPVA and MCPVA methods; unified statistical theory with a classical treatment of the

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

Generalized Transitiion Statle Theory

TABLE 111: State-Selected Rate Constants (Units of cm molecule-' Using the TK Potential Energy Surfacea$b

k A ( n , T ) k M E P V A ( n , TkMCPVA(n,T) )

200 300 400 600 1000 1500 2400 4000

6.94(-3) 1.17 2.95(1) 5.22(2) 5.70(3) 2.03(4) 5.72(4) 1.26(5)

1.31(-2) 1.84d 2.79(1)d 4.80(2)d 5.32(3)d 1.93(4)d 5.52(4) 1.23(5)

2.36(-1) 5.18d 4.93(1)d 6.37(2)d 6.07(3)d 2.08(4)d 5,77(4) 1.26(5)

k(n,T)

H,

2.07(-1)' 5.87C 5.93(1)' 7.58(2)' 6.77(3)' 2.16(4)e

6.01(-1) 3.47(1) 2.75(2) 2.31(3) 1.39(4) 3.68(4) 8.30(4) 1.57(5)

4.99(1) 2.80(2)d 7.49(2)d 2.84(3)d 1.30(4)d 3.36(4)d 7.69(4) 1.49(5)

9.57(1) 5.38(2)d 1.36(3)d 4.24(3)d 1.56(4)d 3.67(4)d 8.04(4) 1.53(5)

n=O

+

D, + D,

kA(n,'T)

kMEPVA(n,T)

kMCPVA(n,T)

200 300 400 600 1000 1500 2400 4000

4.11(- 4) 2.40(-1) 6.04 1.61(2) 2.46(3) 1.03(4) 3.29(4) 7.87(4)

8.20(-4) 3.17(-1Ie 7.06" 1.73( 2)" 2.52( 3)e 1.05(4)" 3.31(4) 7.90(4)

3.98(- 3) 6.18(-1)" 1.05( 1)" 2.14( 2)" 2.80( 3)" 1.11(4)" 3.44( 4 ) 8.06(4)

k ( n ,T ) 2.68(-3)d 5.54(-1)d 1.02(1)d 2.13( 2)d 2.78( 3)d 1.1O( 4)f

kA(n,T) 1.46(-1) 1.20(1) 1.14(2) 1.14(3) 7.95(3) 2.26( 4 ) 5.37( 4) 1.06(5)

7.61(-1) 2.20( 1)" 1 , 41( 2)" 1.12(3)" 7.32(3)e 2.11( 4)e 5.07(4) 1.02(5)

1.41 3.61(l)e 2.06(2)e 1.41( 3)e 8.19( 3)e 2.24(4)" 5.26(4) 1.04(5)

Accurate quantal state-selected rate constants are available only for n = 0. 8. Truhlar and Gray (ref 43).

" These results are also published in ref

n=O 200 300 400 600 1000 1500 2400 4000

l.S8(- 4) 1.26(- 1) 3.97 1.33(2,) 2.42( 3) l.ll(4.) 3.80( 4) 9.50(4)

D

kMEPVA(n,T) kMCPVA(n,T)

TABLE V: State-Selected Rate Constants (Units of cm molecule-' s-I) for the Collinear Reaction H t D, Using t h e TK Potential Energy Surfaceavc kA(n,T )

+

n= 1

T, K

a , b See corresponding footnotes t o Table 11. Truhlar, Kuppermann, and Adams (ref 14).

2.6(3)C 5.6(3)' 1.4(4)' 3.4(4)' 5.78(4)e

These results are also published in

TABLE IV: State-Selected Rate Constants (Units of cm molecule-' s - ' ) for the Collinear Reaction D Using the TK Potential Energy Surfaceaec

T, K

H, t H

-+

k A ( n , T ) kMEPVA(n,T)kMCPVA(n,T) k ( n , T )

Truhlar and Kuppermann (ref 13).

See corresponding footnotes t o Table 11. " Truhlar and Gr,ay (ref 43).

--

+

n=l

T,K

a,b

for the Collinear Reaction H

n= 0

-.

ref 8.

s-l)

1087

kMEPVA(n,T) k M C p V A ( n , T )

+

HD t D

n=l

k ( n ,T )

kA(n,T)

kMEPVA(n,T)kMCPVA(n,T)

3.32(- 4) 1.68(- 1) 4.50 1.35(2) 2.35( 3) 1.07(4) 3.64( 4 ) 9.14(4)

1.99(-3) 1.27(-3)d 3.17(-4) 3.52(- 3) 8.88(-1) 1.05(-2) 4.87(- 1) 3.36(-1) 3.17(-l)d 2.19(-1) 6.72 6.93d 6.03 8.27 1.16(1) 1.66(2) 1.75(2)d 1.76(2) 1.80(2) 2.08( 2) 2.60( 3 ) 2.7 3( 3)d 2.58( 3) 2.86( 3) 2.74( 3 ) 1.09(4) 1.t1( 4)e 1.24(4) 1.13(4) 1.13(4) 3.56(4) 3.76(4) 4.07(4) 3.62(4) 8.78( 4) 9.32( 4) 9.90(4) 8.86(4) a , b See the corresponding footnotes to Table 11. Accurate quantal state-selected rate constants are available only for n = 0. Truhlar, Kuppermann, and Adams (ref 14). " Truhlar and Gray (ref 43). TABLE V1: State-Selected Rate Constants (Units of cm molecule-' s - ' ) f o r the Collinear Reaction D Using the TK Potential Energy Surfacea-c

_-

n=0

+

H,

-+

DH t H

n = l

T, K kA(n,T) kMEPVA(n,T) kMCPVA(n,T) k(n,T)d k A ( n, T ) kMEPVA( n,T) kMCPVA(n,T) 200 5.54(- 3) 1.88(-2) 1.85(- 1) 2.14(-1) 4.55(-1) 1.80(2) 4.44(2) 300 1.42 2.49 5.53 7.24 2.69(1) 3.81(2) 8.69(2) 400 2.38( 1) 3.40(1) 5.30(1) 6.82( 1) 2.16(2) 8.36(2) 1.49(3) 600 4.23(2) 5.17(2) 6.46(2) 7.61(2) 1.84(3) 3.33(3 ) 4.19(3) 1000 4.63( 3) 5.15( 3) 5.73( 3) 6.31 ( 3 ) 1.12( 4) 1.48(4) 1.58(4 ) 1500 1.65(4) 1.77(4) 1.88(4) 2.98(4) 3.53(4) 3.64(4) 2400 4.66( 4 ) 4.85(4) 5.03(4) 6.74(4) 7.45(4) 7.55(4) 4000 1.03(5) 1.05(5 ) 1.07(5 ) 1.28(5) 1.36(5 ) 1.37(5) a , b See the corresponding footnotes t o Table 11. Accurate quantal state-selected rate constants are available only for n = 0. Truhlar, Kuppermann, and Adams (ref 14).

reaction coordinate and with a quantal treatment of the reaction coordinate by the MEPVA and MCPVA methods; and accurate quantum mechanics.

VI. Discussion A. H + H2(Porter-Karplus Surface No. 2). The results for the H + Hz reaction using the PK2 surface are summarized in Figures 1 and 2 and Tables I1 and XIV. The

PK2 surface is thermoneutral with a classical barrier of 9.13 kcal/mol situated symmetrically between reactants and products. Only the adiabatic theory yields stateselected rate constants so we will discuss it first. Figure 1 and Table I1 give some of the results of the calculations using the adiabatic theories. The contour plot shows the tunneling paths used in the MEPVA and MCPVA methods. The path of outer turning points for

1088



TABLE VII: State-Selected Rate Constants (Units of cm molecule-' s - ' ) for the Collinear Reaction C1 t H, + ClH Using t h e Extended LEPS Potential Energy Surface Used b y Baera,b

n=O T, K

k A ( n , T ) kMEPVA(n,T) hMCPVA(n,T)

k A ( n , T ) kMEPVA(n,T) kMCPVA(n,T) k(n,T)'

k(n,T)

3.47( 3) 9.38(3) 1.61(4) 2.93(4) 5.18(4) 7.44(4) 1.06(5) 1.48(5) (ref 48).

5.48( 3) 1.24(4)" 1.97(4)" 3.33(4)" 5.59(4)" 7.81(4)" 1.09(5) 1.51(5)

5.59( 3 ) 1.26(4)" 1.98(4)" 3.35(4)e 5.60(4)" 7.83(4)" 1.09(5) 1.51(5)

8.81(3) 1.46(4) 1.92(4) 2.60(4) 3.40(4) 3.94(4)

Baer, Halavee, and Persky (ref 15).

TABLE VIII: State-Selected Rate Constants (Units of cm molecule-' s - l ) for t h e Collinear Reaction C1 t D, Using the Extended LEPS Potential Energy Surface Used by Baera,b

a,b

k A ( n , T ) kMEPVA(n,T) kMCPVA(n,T) k(n,T) 2.06(-1) 3.01(- 1) 5.65(-1) 9.39(-1)d 1.28(1) 1.44(1) 2.04(1) 3.15(1)d 1.06(2) 1.07(2) 1.36(2) 1.98(2)d 1.00( 2) 1.36( 3)' 9.22(2) 8.68(2) 5.73(3) 5.19( 3) 5.61( 3) 6.77( 3)' 1.54(4) 1.39(4) 1.45(4) 1.57(4)' 3.51(4) 3.18( 4 ) 3.27(4) 6.71(4) 6.16(4) 6.26(4)

See the corresponding footnotes t o Table 11.

k A ( n , T ) kMEPVA(n,T)kMCPVA(n,T) 4.00(2) 5.56( 2) 5.27(2) 2.00(3) 2.36(3) 2.30(3) 5.19(3) 4.65(3) 5.08(3) 1.15(4) 1.21(4) 1.22(4) 2.61(4) 2.67(4) 2.69(4) 4.23(4) 4.29(4) 4.31(4) 6.60(4) 6.65(4) 6.67(4) 9.81(4) 9.77(4) 9.78( 4)

' Gray and Truhlar (ref 48). ~~~~~

~~

k A ( n , T ) kMEPVA(n,T) kMCPVA(n,T) k ( n , T ) 4.14(-12) 1.20(- 1) 6.33(-2) 1.54(-l)d 4.15 6.76" 8.55' 4.75e 4.33(1) 5.63(1)" 4.44( l)e 7.06( 1)' 4.80(2) 5.20(2)" 4.52(2)" 6.31(2)' 3.61(3) 3.50( 3)" 3.25( 3)" 3.98( 3)' 1.07(4) 9.53( 3)e 9.98( 3)" 1.04(4)' 2.61(4) 2.41(4) 2.34(4 ) 5.24(4) 4.84(4) 4.76(4)

kMEPVA ( n , T )

+

CIT t T

k A ( n , T ) kMEPVA(n,T) hMCPVA(n,T) 9.32(1) 1.15(2) 1.47(2) 7.13(2) 7.35(2)" 8.23(2)" 2.06( 3) 2.03( 3)" 2.18( 3)e 6.29(3) 6.12(3)" 6.34(3)" 1.69(4) 1.65(4)" 1.68(4)e 2.99( 4) 2.93( 4)" 2.96(4)e 4.97(4) 4.90(4) 4.93(4) 7.71(4) 7.64(4) 7.67(4 )

kMCPVA(n,T)

k ( n ,T ) d

h(T)' 7.86(2) 2.54(3) 4.79( 3 ) 9.55(4) 1.78(4) 2.54(4)

Baer, Halavee, and Persky (ref 15).

TABLE X : State-Selected Rate Constants (Units of cm molecule-' s - ' )for the Collinear Reaction C1 + DH Using the Extended LEPS Potential Energy Surface of Baera-c n=O n = l kA(n, TI

1.08(3 ) 3.34(3) 6.08(3) 1.16(4) 2.07(4) 2.91(4)

n= 1

a , b See corresponding footnotes to Table 11. ' Gray and Truhlar (ref 48). These results have also been published in ref 8.

T, K

h(n,T)C

~~

n=O

e

ClD t D

Baer, Halavee, and Persky (ref 1 5 ) .

TABLE IX: State-Selected Rate Constants (Units of em molecule-' s - l ) for the Collinear Reaction C1 t T, Using the Extended LEPS Potential Energy Surface Used b y Baeraib

T,K 200 300 400 600 1000 1500 2400 4000

+

n= P

n=O

1000 1500 2400 4000

H

n= 1

200 4.62 6.45 1.09(1) 3.35( l)d 300 1.13(2) 1.26(2)" 1.68(2)" 3.97(2)d 400 5.87(2) 6.01(2)" 7.28(2)" 1.45(3)d 600 3.22( 3) 3.10(3)" 3.46(3)" 5.57(3)' 1000 1.38(4) 1.28(4)" 1.36(4)e 1.78(4)' 1500 3.08(4) 2.85(4)" 2.96(4)" 3.29(4)' 2400 6.10(4) 5.67(4) 5.80(4) 4000 1.06(5) 9.98(5) 1.01(5) a , b See the corresponding footnotes to Table 11. Gray and Truhlar e These results have also been published in ref 8.

T, K 200 300 400 600

t

kA(n,T)

hMEPVA( n , T )

+

C1D + H

hMCPV*(n,T)

200 1.05 1.65 3.74 5.19 1.09(3) 1.92(3) 2.01(3) 300 3.96( 1) 4.64(1) 7.24(1) 9.89(1) 4.06( 3 ) 5.77( 3) 5.92( 3 ) 400 2.54(2) 2.70( 2) 3.62(2 ) 4.77(2) 8.18(3) 1.05(4) 1.07(4) 600 1.73(3) 1.71(3) 2.03(3) 1.75(4) 2.06( 4) 2.08(4) 3.91(4) 3.53(4) 3.88(4) 1000 8.81( 3) 8.46( 3) 9.25( 3 ) 5.78(4) 5.41(4) 5.75( 4 ) 2.05(4) 2.16(4) 1500 2.14(4) 8.42(4) 8.08(4) 8.40(4) 2400 4.53(4) 4.34( 4) 4.48( 4 ) 1.20(5) 1.17(6) 1.19(5) 7.94(4 ) 8.08(4) 4000 8.25(4) ' Accurate quantal state-selected rate constants are availabe only for a , b See the corresponding footnotes t o Table 11. Baer, Halavee, and Persky (ref 15). n = 0.

the n = 1 state has an unphysical loop a t the symmetric stretch line. This occurs because the natural collision coordinates that are used are not single-valued in this region. However, this distortion of the Marcus-Coltrin tunneling path (MCP) for n = 1 has a negligible effect upon the calculation of tunneling probabilities because this portion of the Marcus-Coltrin tunneling path is used in the tunneling calculations only for very small translational energies. For classically allowed regions the MCP coincides

with the minimum energy path (MEP), which is well behaved. Because of the local minimum in the adiabatic curve for n = 1at s = 0, the region near s = 0 is a classically allowed region except a t very small translational energies. The corner cutting of the Marcus-Coltrin tunneling path has the effect of making the adiabatic barriers thinner for the MCP than for the MEP and therefore the MCPVA method gives larger reaction probabilities than the MEPVA method. The state-selected reaction probabilities

+

TABLE XI: State-Selected Rate Constants (Units of 'cm molecule-' s - ' ) for the Collinear Reaction C1 t HD -+ C1H Using t h e Extended LEE'S Potential Energy Surface of Baera-c n = l n=O

--

T, K 200 300 400 600

1000 1500 2400 4000 ai

kA(n,T) 4.05(-1) 2.10( 1 ) 1.58(2) 1.26(3) 7.28(3) 1.89(4) 4.1 9( 4) 7.86( 4)

1089


Generalized Transition Stak Theory

kMEPVA( n , T )

kMCPVA(n,T)

9.27(- 1) 3.04( 1) 1.88(2 ) 1.28( 3) 6.7 3( 3) 1.70(4) 3.75(4) 7.13(4)

See the Corresponding footnote t o Table 11. Baer, Halavee, and Persky (ref 15).

1.26 3.67( 1) 2.15(2) 1.39( 3) 7.04( 3) 1.75(4) 3.82( 4) 7.19(4)

k ( n, T ) d

-

kMEPVA(n,T) kMCPVA(n,T)

kA(n,T)

1.72(3) 5.44( 3) 1.02(4) 2.02(4) 3.84(4) 5.71(4) 8.37(4) 1.19(5)

1.68( 3) 5.43(3) 1.02(4) 2.03(4) 3.85(4) 5.7 3( 4) 8.39( 4) 1.19( 5 )

3.16 6.76(1) 3.32(2)

D

1.73(3) 5.46( 3) 1.02(4) 2.02(4) 3.84(4) 5.71(4) 8.37( 4) 1.19(5)

Accurate quantal state-selected rate constants are available only for

n = 0.

TABLE XII: State-Selected Rate Constants (Units of cm molecule-' s - l ) for the Collinear Reaction I t H, -+ IH t H Using t h e Potential Energy Surface of Raff et al.a-c -~ n-0 n=l T, K 200 300 400 600

1000 1500 2400 4000

-kA(n, -T )

( n ,T )

k

4.71(- 32) 5.30(- 20) 5.87(- 1 4 ) 6.89(- 8 ) 5.42(- 3) 1.64 1.29(2) 2.62( 3)

4.35(6.03(7.71(1.11(-

33) 21)

15) 8)

1.11(- 3) 4.07(- 1) 3.94(1) 9.86(2)

k(n,TId

1.18(- 22) 1.35(-16) 1.66(- 10) 1.40(-5) 4.46(- 3)

kA(n,T ) 6.08(- 26) 6.28(- 1 6 ) 6.67(- 11) 7.50(- 6) 9.04(- 2) 1.07(1) 4.17(2) 5.30( 3 )

kMEPVA(n,T) 6.27(-27) 7.92(- 1 7 ) 9.66(- 1 2 ) 1.32(-6) 2.02(-2) 2.88 1.37(2) 2.14( 3 )

k(n,TId 8.12(- 1 2 ) 5.85(-8) 4.48(-4) 6.28(- 1) 2.49( 1)

a , b See the corresponding footnotes to Table 11. N o tunneling occurs for the adiabatic barriers therefore the MarcusColtrin path is identical with the minimum energy path and the MCPVA method reduces to the MEPVA method. Gray, Truhlar, Clemens, Duff, Chapman, Morrell, and Hayes (ref 20).

TABLE XIII: State-Selected Rate Constants (Units of cm molecule-' s - ' ) for the Collinear Reaction I Using t h e RMC Potential Energy

n= 0 T, K 200 300 400 600

1000 1500 2400 4000

kAin,T)

kMEPVA(n,T)

4.58(- 32) 5.20(- 20) 5.78(- 1 4 ) 6.82(-8) 5.39(- 3) 1.63 1.29(2) 2.62( 3)

4.22(- 33) 5.89(- 21) 7.56(- 15)

1.09(-8)

1.10(-3) 4.04(- 1 ) 3.93(1) 9.8 6( 2 )

+

H,

-

IH t H

n = l k(n,TId

k

T)

5.98(- 26)

1.50(- 20) 1.79(- 1 4 ) 2.35(-8) 2.14(- 3) 7.14(- 1)

6.24(- 1 6 ) 6.61(-11) 7.46(- 6) 9.01(- 2) 1.07(1) 4.17(2) 5.29(3)

kMEPVA ( n , T )

5.58(-27) 7.36(-17) 9.18(-12) 1.28(- 6) 2.00(-2) 2.86 1.37(2) 2.14( 3)

k(n,TId 1.45(- 1 1 ) 1.05(-7) 7.83(- 4) 1.05 4.14(1)

See the corresponding footnotes to Table 11. N o tunneling occurs for the adiabatic barriers therefore the Marcus-Coltrin path is identical with the minimum energy path and the MCPVA method reduces t o the MEPVA method. Gray, Truhlar, Clemens, Duff, Chapman, Morrell, and Hayes (ref 20).

--

predicted by these approximation methods are compared probabilities at about E = 20.1 kcal/mol, but it is obscured with the accurate quantal calculations of D i e ~ t l e rWu, ,~~ in P(n=l,Erei) by the rapid rise of the 1 1 reaction Johnson, and L e ~ i n eDuff , ~ ~and one of the p r ~ b a b i l i t y . ~ ~The , ~ ' fact that the 1 0 probability is and Top and Baer36in the upper right part of Figure 1. Our comparable to the 1 1 is another indication that calculations indicate that the MCPVA method gives very the vibrationally adiabatic model breaks down for the accurate results for the n = 0 state. Marcus and Coltrin7 excited state. Nevertheless it is interesting that the peaks drew the same conclusion, although they calculated the occur at E = 20.4 kcal/mol, in very in both PVA(n=l,Erel) probabilities seimiclassically rather than quantum megood agreement with the accurate resonance position. The chanically, and they did not introduce any approximations predicted Breit-Wigner widths r (full widths at halflike the Morse approximation for the outer turning points. maxima) are 0.04 (MEPVA) and 0.07 kcal/mol (MCPVA), The corner cutting for the MCP is even greater in the n both much less than the accurate quantal r = 0.51 = 1 state than iin the n = 0 state; however, In PMCPVA-kcal/mol. This kind of model for resonance positions has (n=l,Erel)does nlot differ from In PmPVA(n=l,Erel) as much been applied previously to the I + H23sand H H2 reas the corresponding quantities do for n = 0. This deaction~.~~ creased sensitivity to method is due to the local minimum The good agreement of the MCPVA reaction probain the adiabatic curve for n = 1which decreases the range bilities for n = 0 with the accurate quantal ones is also of distances over which the adiabatic potential curves of reflected in the low-energy part of the P(E) plot (lower the two methods differ. This minimum also gives rise to right of Figure 1). The MEPVA results are too low and the prominent resonance in PVA(n=l,Erel). The predicted the results of the adiabatic theory with classical treatment PVA(n=l,EreJ do not agree well with the accurate P(n= of the reaction coordinate (step function) provide only a l,EreJ,which does not seem to show the resonance. The crude estimate, underestimating P ( E ) at low E and resonance does occur in the 1 0 and 0 -* 0 reaction overestimating P(E) for E > 12 kcal/mol. Because the

-

+

-

1090

The Journal of Physical Chemistry, Vol. 83,

No. 8, 1979


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Figure 1. Potential energy contours, reaction paths, and state-specific and average reaction probabilities for H -4- H2 H2 H for the PK2 potential energy surface. Figures 1, 3, 5-12, 14, and 16 are explained in the second paragraph of section V. At the upper left are potential energy contours representing energies 4.5, 9.0, 13.5, and 18.0 kcal/mol above the bottom of the reactant potential curve. Adiabatic barriers used in the Marcus-Cobin method are plotted in the lower left for relative translational energies approximately equal to those which give the maximum contribution to the state-selected ratel constants at 300 K. These energies are 4.70 and 2.06 kcal/mol for the n = 0 and 1 states, respectively, corresponding to total energies of 11.OO and 20.37 kcal/mol. Various reaction probabilities are plotted at the right. At the upper right the solid curves are state-selected reaction probabilities PMEPVA(n,€) for n = 0 and 1 and the dashed curves are state-selected reaction probabilities PMCPVA(n,€) for n = 0 and 1. Also at the upper right, symbols represent accurate quantal calculations: squares are the state-selected reaction probabilities of Diestler (ref 33) for his low energy point 'for n = 0 and all but the bst three of his published hgh energy points for n = 1; triangles are a selected set of stateselected reaction probabilities published by Wu, Johnson, and Levine (ref 34), the octagons are the complete set of state-selected reaction probabilities published by Duff and Truhlar (ref 35), and the diamonds are a selected set of state-selected reaction probabilities of Top and Baer (ref 36). At the lower right, the, piecewise constant solid curve is the quantized average reaction probability with classical treatment of reaction coordinate motion; in accordance with eq 59 it assumes successively the values 011, 111, 112, and 212 (Le., 0.0, 1.0, 0.5, and 1.0) in this energy range. The other solid curve is PMEPVA(€) defined by eq 61, and the dashed curve is PMCPVA(€), also defined by eq 61. The symbols in the P ( € ) plot are accurate quantal calculations: the squares are the results published by Diestler for all his energies above the threshold for the first excited state, the triangles are again a selected set of the results of Wu, Johnson, and Levine, the octagons are the same results of Duff and Truhlar, and the diamonds are the same results of Top and Baer, except the three lowest energy points are omitted.

latter two methods are poor in the threshold region, the corresponding slate-selected rate constants for n = 0 shown in Table I1 are much lower than the accurate quantal ones of Schatz et al.40 The MCPVA rate constants, however, are in much better agreement, underestimating the state-selected rate constant for n = 0 by only 25% or less in the temperature range 200-1000 K. Next consider the canonical rate constants in Table XIV. Figure 2 shows that the maxima of the free-energy curves occur at the saddle point for all temperatures up to 1500 K; in fact, h'(T) and hCVT(T)are identical in the temperature range 200-1500 K, but differ at 2400 K. Table XIV shows that microcanonical variational transition state

theory and the unified statistical theory with classical transmission coefficients differ significantly from conventional transition state theory and canonical variational theory only at 1000 and 1500 K. At 1000 K kMvT(T) and kus(T) are 4 and 6% lower than k'(T), respectively, and at 1500 K they are 10 and 14% lower. Therefore generalized transition state effects are small in this system for the temperatures studied. Tunneling is very important in this system. This is evident from examining the four rate constants for which no tunneling correction factor is included: hi(T), hcvT(T), hPVT(T),and hus(T). At 200 K all four underestimate the rate by a factor of 46 and at 1000 K they are too low by

1092


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Flgure 5. Same as Figure 1 except for the D f D2 D2 D reaction for the TK potential energy surface. Potential energy contours in the upper left represents energies 5, 10, 15, and 20 kcal/mol above the bottom of the reactant potential curve. Adiabatic barriers for the Marcus-Coltrin path are plotted in the lower left at relative translational energies approximately equal to those which give the maximum contributionto the statsselected rate constants at 300 K. These energies are 6.20 and 2.75 kcal/mol for the n = 0 and 1 states, respectively, corresponding to total energies of 10.67 and 15.88 kcal/mol. The accurate quantal reaction probabilities shown below the n = 1 threshold are from Truhlar, Kuppermann, and Adams (ref 14 and 44).

compared with the accurate quantal ones.14144 The MCPVA method again gives dramatic improvement over the MEPVA method. Although tunneling is important for this system at low temperatures, at higher temperatures the state-selected rate constants for the three vibrationally adiabatic methods shown in Table IV become comparable. In comparison to the accurate state-selected rate constants of Truhlar, Kuppermann, and ad am^,^^,^^ hA(n=O,T=lOOOK) and kMCPvA(n=O,T=lOOOK) are only 12 and 9% too low, and K) is 1% too high. A t 1500 K these kMCPvA(n=O,T=lOOO errors are 7,5, and 1%again, respectively. Including lower temperatures, the MCPVA method gives the overall most accurate state-selected rate constants; it is 49% too high at 200 K, 12% high at 300 K, and from 3 to 1%too high from 400 to 1500 K. Table XVI compares the approximate canonical rate constants with accurate quantal ones+14,46 We first note that all four rate constants without tunneling corrections, k * ( n ,kcw(7'), k"w(T), and kus(r), severely underestimate the accurate quantal rate constants at low temperatures. However, at T I 1000 K, all four are accurate within 15%.

The lower temperature results can be improved by quantal transmission coefficients; for example, the Wigner and CVE methods decrease the error in hi(7') from a factor of 6.5 at 200 K to a factor of 2.2 or 2.0. At 300 K, the error in ht(7') is reduced by these methods from a factor of 2.3 to 12% (Wigner) or 27% (CVE). For T 1 1000 K, the quantal transmission coefficients do not improve hi( T ) or hCVT(7'). However adjusting the location of the dividing surface to minimize recrossing effects yields kpw( T ) which is improved by a quantal transmission coefficient. Thus or k(T) at the good agreement of k'(7') with kflVT/CVE(7') 1000 K arises from cancellation of two different kinds of errors: recrossing effects and neglect of tunneling. As for the previous two systems the MCPVA method of correcting kwvT(T) gives the best results over the entire temperature range. It is interesting that, unlike the results for H + H2, the MCPVA method overestimates the accurate quantal results at all temperatures. The largest error is for 200 K where it is 49%. The error decreases to 2% or less at T 2 600 K. The quantally unified statistical model with MCPVA tunneling deviates from hMCPvA(T) by 3% at 1000 K and 8% at 1500 K. A t even



I

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Figure 6. Same as Figure 1 except for H D2 HD D for the TK potential energy surface. Potential energy contours in the upper left represent energies 4.5, 9.0, 13.5, and 18 kcal/mol above the bottom of the reactant potential curve. Adiabatic barriers for the Marcus-Coltrin method are plotted at the! lower left for relative translational energies approximately equal to those which give the maximum contribution to the state-selected rate constants at 300 K. These energies are 6.89 and 5.13 kcal/mol for the n = 0 and 1 states, respectively, corresponding to total energies of 11.36 and 18.26 kcal/mol. The accurate quantal state-selected reaction probabilities for the n = 0 state (diamonds) and n = 1 state (octagons), and average reaction probability (squares) are a complete set of results for energies between the lowest and highest energies shown. These are from the calculations of Truhlar, Kuppermann, and Adams (ref 14 and 44) and Adams, Smith, and Hayes (ref 45).

higher temperatures the results would be expected to differ more and the quantally unified statistical model is expected to give the best estimate of the thermal rate constant at those temperatures. D. H + D zand D + H 2 ( T K Surface). The results for the H D, reaction on the T K surface are summarized in Figure 6 arid Tables V and XVII, and the results for D H2 on the 'l'K surface are summarized in Figure 7 and Tables VI and XVIII. As in the previous systems the MCPVA state-selected reaction probabilities for n = 0 are in excellent agreement with the accurate quantal res u l t ~ The . ~worst ~ ~ agreement ~ ~ ~ ~of ~the four isotopic cases is for H + D,, and the method is noticeably less accurate for the two cases with D2 reactant than for the two cases with H, reactant. The MEPVA method underestimates P(n=O,Erel)at low energy for all four isotopes. When the MCPVA state-selected rate constants are compared to the accurate quantal ones14,46for n = 0 the results for H t D2are 60 and 6% too high at 200 and 300 K, respectively, less than 5% too low for 400 K IT I 1000 K, and 2% too high at 1500 K. For D + H2,the MCPVA

+

+

results are within 24% of the accurate results over the entire 200-1000 K temperature range. The MEPVA method and the adiabatic theory with classical treatment of the reaction-coordinate motion both severely underestimate h(n=O,T)at low temperatures. For H + Dz,errors are by factors of 4 and 9, respectively, at 200 K, less than 24% at 600 K, less than 14% at 1000 K, and less than 4% at 1500 K. For D + H2, errors are by factors of 11and 39, respectively, a t 200 K and 18% (MEPVA) and 27% (classical transmission coefficient) at 1000 K. For H D2, as for D + D2, tunneling becomes much less important at higher temperatures than for the H H2 systems. Tables XVII and XVIII contain the comparison of the approximate canonical rate constants with the accurate quantal r e s ~ l t s . The ~ ~ ?effects ~ ~ of varying the location of the dividing surface are large for both systems and are more pronounced for D H2 than for the other three T), hydrogenic systems studied. At 200 K hcvT(T ) , and kus(T) for D + H2 are factors of 2.8 lower than h f ( T ) . As the temperature is increased, this effect is less pronounced: at 1500 K hCVT(T),hWVT(T),and kus(T) for D

+

+

+

1096

The Journal of Physical Chemistry, Vo/. 83, No. 8, 1979


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Figure 7. Same! as Figure 1 except for D -I-H2 DH H for the TK potential energy surface. Potential energy contours at the upper left represent energies 5, 10, 15, and 20 kcal/mol above the bottom of the reactant potential curve. Adiabatic curves used in the Marcus-Coltrin method are plotted at the lower left for relative translational energies of 5.30 and 0.15 kcal/mol for the n = 0 and 1 states, respectively, corresponding to total energies of 11.59 and 18.46 kcal/mol. For the ground state this energy is approximately the one which contributes the most to the state-selected thermal rate cortstant at 300 K. For the excited state PMCPVA(n=l,E,) exp(-E,,/&) with T = 300 K has two local maxima, one within 0.15 kcallmol of the threshold energy and one at E,,, = 0.96 kcai/mol. The accurate quantal reaction probabilities shown below the n = 1 threshold are from the calculations of Truhlar, Kuppermann, and Adams (ref 14 and 44).

+

H, are only 26, 28, and 33% lower than k i ( T ) , respectively. As in the E1 Hz and D + Dz systems the four methods with no tunneling corrections severely underestimate the rate constant at low temperatures for both H Dz and D H,. Addition of the Wigner tunneling correction to conventional transition state theory offers improvement; at 200 K, for H Dz an error of a factor of 7.2 is decreased to 2.0, and for D H2 a factor-of-13.9 error is decreased to 2.7. A t 300 K k t ( T ) is too low by factors of 2.1 and 2.6 for H + D, and D + H2,respectively, whereas kw(T) is too high by 1 and 10%- The CVE tunneling correction also improves the accuralcy at low temperatures when used with k i ( T ) ; however, because of the large effects of the variational adjustment of the dividing surfaces, especially for D + H,, this3 result is suspect. For D + H2 kiiCvE(T) underestimates k(Z3 by only 25% at 200 K but overestimates h ( T ) by as much as a factor of 2.3 for 300 K 5 T I1500 K. Using the CVE correction with the variational methods provides much better results at all but the lowest temperature; for D H2, kCVTIcVE(T) and kpVTICVE( T )

+

+

+

+

+

+

underestimate k(T) by a factor of 3 at 200 K but for T i n the range 300-1500 K both give errors less than 17%. The MCPVA method gives the best overall agreement for both systems. For H + D2, kMCPVA(T) is too large by 57 YO,but it is within 6% of the accurate k(T) for 300 K IT I1000 K. For D H2, kMCPVA(T)overestimates h ( T )by less than 24% * Notice that the effect of moving the dividing surface is larger for D Hz and H + D2 than for H + Hz and D Dz. This is due to the fact that the derivative of the transverse stretching frequency with respect to s is zero at the conventional transition state for the symmetric systems but not for the asymmetric systems. For this reason we would generally expect larger effects of varying the dividing surface in asymmetric collinear systems as compared to symmetric collinear systems if the potential energy surfaces are similar to the TK surface. E. Cl H2. The results for C1 + Hz are summarized in Figure 8 and Tables VI1 and XIX. The LEPS surface used for the C1 + H2 system and its isotopic variants has a small classical endoergicity of 3.01 kcal/mol. The

+

+

+

+

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IH -k H reaction for the RMC potential energy surface. Potential energy contours are Figure 16. Same! as Figure 1 except for the I H2 at the upper left represent energies 10, 20, 30, 40, and 50 kcallmol above the bottom of the reactant potential curve. For this system no tunneling occurs for the adiabatic barriers and the MCPVA method reduces to the MEPVA method. Therefore the paths of outer turning points are absent from the contour plot and the adiabatic barriers used in the Marcus-Coltrin method are absent from the V,(n,s) plots. A selected set of the accurate quantal state-selected reaction probabilities of Gray, Truhlar, Clemens, Duff, Chapman, Morreil, and Hayes are indicated on the P @,E,,) plot as diamonds ( n = 0) and octagons ( n = 1). A selected set of the state-selected probabilities are used to compute the accurate quantal average reaction probability shown as diamonds in the P(€)plot.

effect is attributed to the neglect of quantum mechanical tunneling. Thirdly, we mention that there is no completely consistent way to include quantal effects in transition state theory. The (other methods considered in this article attempt to correct the first two deficiencies but there is no way to completely alleviate the third shortcoming. Classical transition state theory provides an upper bound on the exact classical rate constant. This leads naturally to the classical variational methods which give the lowest upper bound to the exact classical rate constant. Although quantized transition state theory does not provide a rigorous upper bound to the accurate quantal rate constants, the present article shows that the variational transition state theories are still useful for high temperatures where conventional transition state theory predicts rates that are too large. The canonical variational theory involves a fixed dividing surface for each temperature whereas the microcanonical variational theory allows the dividing surface to be a function of energy. In both cases the dividing surface is chosen to minimize classical recrossings as much as possible since transition state theory assumes no classical recrossing of the dividing surface. The

unified statistical theory goes one step beyond the microcanonical variational theory and is an attempt to estimate how much recrossing occurs even at the dividing surface with the least amount of classical recrossing. For the hydrogenic systems, the rate constant k f ( T ) of conventional transition state theory is in error by less than 23% at 1500 K and the variational theories and the unified statistical theory offer some improvement. However for C1 Hz, Dz, and Tz the differences are much more important; a t 1500 K, the error in conventional transition state theory is 35-50% but it is decreased to less than 15% by the rate constants k”(T) and kus(Tj of microcanonical variational transition state theory and the unified statistical theory. Canonical variational transition state theory gives EL result kCVT(T) of intermediate accuracy between conventional and microcanonical variational transition state theory. The results are even more encouraging for the I + H2 system where h f ( T ) is too large by factors of 2.4 and 2.1 for two different potential surfaces at 1500 K, but kCvT(T),kpvT(T), and kus(T) are all accurate within 18% on one surface and 3% on the other. The improvement is even more dramatic at lower temperatures.

+

1110


6.C. Garrett and D. G. Truhlar

kw(n

gives better overall agreement for each system. For the I + H2 systems, k”cvE(T) gives good agreement at low temperatures, however, our results indicate that this is due to cancellation of errors. Even for the other cases, the CVE tunneling correction is inconsistent with the vibrationally 1 3 33 adiabatic derivation of transition state theory. zc The remaining approximate methods that we consider 1 sj3 c- r / I E include not only quantal effects on the reaction-coordinate / -5 33motion but also effects of classically recrossing the con3c t 7 5:: ventional transition state. The simplest theories that s - 4: incorporate both these effects are the canonical and microcanonical variational transition state theories with C W uc 523 t quantal reaction-coordinate correction, yielding - Cr 0 hcvTjcvE(T) and hrvTICvE(T). Again the CVE quantal -=z 43 correction is not completely consistent with underlying I dynamical assumptions of these theories but it does take account of the potential energy along the whole reaction I 1s coordinate and allow for both tunneling and nonclassical I 3 reflection. Since k*”’(T) gives more accurate results than -. 3 c, 3 kCvT(T) at high temperatures we summarize the results S :3C-R2 T). At low temperatures recrossing only for h~vT/CvE( Figure 17. Same as Figure 2 except for the I 4- H, IH + H system effects generally become small and hpVTICvE( T ) gives for the RMC potential energy surface. approximately the same agreement with k ( T ) as seen for As compared to conventional transition state theory the ht/cvE(T). Notable exceptions to this are the H + D2, D variational theories require knowledge of the potential + H2, C1+ Hz, C1+ HD, and I + H2 reactions. For H + surface in the whole valley from reactants to products and D2, D + Hz, C1+ H2,and C1+ HD, the variational theory not just the potential along the valley floor but also the is an improvement over conventional transition state valley width (local force constants). They also require theory and in fact gives overall better results than kw(T) more computational effort, especially kWw( T )and kus(T) for each system. which require numerical Boltzmann averaging. A more consistent method of including both tunneling The simplest method to approximate the tunneling and recrossing effects in transition state theory is the effects at low temperature is to use the Wigner tunneling adiabatic theory of reactions with a quantal treatment of correction. To employ this method involves knowing only the reaction coordinate motion. This provides a way to one more piece of information beyond that needed for add quantal corrections to the reaction-coordinate motion conventional transition state theory; it is necessary to know to krw( T ) because for classical reaction-coordinate motion the imaginary frequency for the unbound normal mode at the adiabatic theory is identical with microcanonical the saddle point. This method gives much better accuracy variational theory. For the quantal adiabatic calculations, than conventional transition state theory at low temone needs to know no more information about the potential peratures for many of the systems studied here. However surface than what was required to do the microcanonical for all five of the hydrogenic systems studied, hw(Dis still variational calculation; however, in the quantal calculations too low by as much as a factor of 5 at 200 K. The results it is necessary to compute reaction probabilities for the are much more encouraging for the C1 + H2 and isotopone-mathematical-dimensional adiabatic barriers. Two ically varying systems. For all systems except C1 + HD, methods are used corresponding to two different tunneling kw(T) is in error by less than 30% at 200 K. For the C1 paths: the minimum energy path (MEP) and the MarHD system and two I H2 systems, ht(T) is either very cus-Coltrin path (MCP). We first consider the minimum close to k(T) or too big, therefore kw(T) is no improvement energy path vibrational adiabatic method, which yields for these reactions. Although the empirical success of the kMEPVA(T).The results of this theory are consistently too Wigner tunneling correction is noteworthy for practical low at all temperatures for the systems studied. It does work, the assumptions of its derivation are not generally provide an improvement over conventional transition stat,e justified. theory for most cases, and at 1500 K its predictions are Another way to incorporate quantal effects on reacaccurate within 11% for all cases except for the I + H2 tion-coordinate motion in conventional transition state system. theory is to use accurate quantal transmission probabilities The Marcus-Coltrin path vibrationally adiabatic mefor the classical potential energy barrier along the minithod, yielding hMCPvA( T ) , gives dramatic improvement over mum-energy reaction path; this is the conservation of the MEPVA method. For all five hydrogenic systems it vibrational energy (CVE) model. To employ this method gives the best overall agreement of the methods studied. entails knowing the potential energy as a function of However, for the C1+ H2and isotopically varying systems, distance along the minimum energy path, not just in the kMCPvA( T ) is sometimes much too small at low temperavicinity of the saddle point but everywhere. It also requires tures. For C1 + Hz, Dz, T2,and DH kw(T) gives better the additional work of calculating the transmission agreement with the accurate h(T) for 200 K 5 T 5 400 K coefficients for this one-dimensional potential curve. At than kMCPvA(T)does. However, excluding the I + H2 low temperatures the results of applying this correction system, the MCPVA method is the most consistently to conventional transition state theory to yield kj/cvE(T) accurate; it is within 67% of the accurate k(T) at all are not as accurate as hw(T). For all the hydrogenic temperatures for all other systems. For the I + H2 resystems except D H2 kJ’cvE(T) is too high at 200 and action, the adiabatic approximation breaks down and both 300 K. The largest error is a factor of 14 for H + H2 for the MEPVA and MCPVA methods predict too much the PK2 potential energy surface. For C1 + D2, Tz,and nonclassical reflection. However the microcanonical DH, kljcvE(T) gives fairly good agreement with k(T) for variational theory with classical reaction-coordinate motion 200 K 5 T I 4 0 0 K, however, in this temperature range, v

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Generalized Transition State Theory. Quantum ... - ACS Publications

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