Potential energy surface, thermal, and state-selected rate coefficients, and kinetic isotope effects for Cl+CH4→HCl+CH3

Potential energy surface, thermal, and state-selected rate coefficients, and kinetic

isotope effects for Cl+CH 4 →HCl+CH 3

J. C. Corchado, D. G. Truhlar, and J. Espinosa-Garcı ́a

Citation: The Journal of Chemical Physics 112, 9375 (2000); doi: 10.1063/1.481602

View online: http://dx.doi.org/10.1063/1.481602

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JOURNAL OF CHEMICAL PHYSICS

VOLUME 112, NUMBER 21

1 JUNE 2000

Potential energy surface, thermal, and state-selected rate coefﬁcients,

and kinetic isotope effects for Cl¿CH₄\HCl¿CH₃

J. C. Corchado

´

Departamento de Quımica Fısica, Facultad de Ciencias, Universidad de Extremadura,

06071 Badajoz, Spain

D. G. Truhlar

Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis,

Minnesota 55455-0431

´

J. Espinosa-Garcıa

´

Departamento de Quımica Fısica, Facultad de Ciencias, Universidad de Extremadura,

06071 Badajoz, Spain

͑Received 25 October 1999; accepted 6 March 2000͒

A new potential energy surface is reported for the gas-phase reaction ClϩCH₄→HClϩCH₃. It is

based on the analytical function of Jordan and Gilbert for the analog reaction HϩCH₄→H₂ϩCH₃,

and it is calibrated by using the experimental thermal rate coefﬁcients and kinetic isotope effects.

The forward and reverse thermal rate coefﬁcients were calculated using variational transition state

theory with semiclassical transmission coefﬁcients over a wide temperature range, 200–2500 K.

This surface is also used to analyze dynamical features, such as reaction-path curvature, the

coupling between the reaction coordinate and vibrational modes, and the effect of vibrational

excitation on the rate coefﬁcients. We ﬁnd that excitation of C–H stretching modes and Cl–H

stretching modes enhances the rate of both the forward and the reverse reactions, and excitation of

the lowest frequency bending mode in the CH₄reactant also enhances the rate coefﬁcient for the

forward reaction. However, the vibrational excitation of the CH₃umbrella mode ͑lowest frequency

mode in products͒ slows the reaction at temperatures below 1000 K, while above 1000 K it also

I. INTRODUCTION

ratio between two rate coefﬁcients, with the one correspond-

ing to the reaction involving the lighter isotopolog in the

numerator.

Chlorine atom chemistry is of great importance over a

wide temperature range. At low temperatures ͑200–300 K͒,

Cl plays an active role as ozone destroyer in the atmosphere,

and at high temperatures ͑1000–2500 K͒ it is important for

combustion. In recent years the reaction with methane,

ClϩCH₄→HClϩCH₃, has been widely studied as a mecha-

nism to eliminate chlorine atoms from the atmosphere by

converting them into inactive HCl. This reaction is also pro-

totypical for halogen–hydrocarbon combustion. Experimen-

tally, this reaction has been the subject of a number of stud-

ies using various experimental techniques over a range of

temperatures.^1–22The rate coefﬁcients determined by various

experimental groups agree well with each other at room tem-

perature, with a recommended value of 1.0ϫ10^Ϫ13cm³

molecule^Ϫ1s^Ϫ1͑Refs. 12, 14͒. In addition to rate constants,

some experimental studies^{1,13,16,17,20}provide kinetic isotope

effects ͑KIEs͒, which are ratios of rate coefﬁcients for isoto-

pologs or isotopomers and which are a sensitive test of the

shape of the potential energy surface ͑PES͒, especially the

barrier width and vibrational force constants near the dy-

namic bottleneck. The ¹²C/¹³C KIE is 1.066Ϯ0.002 ͓Refs.

16͑a͒ and 16͑b͔͒ at 298 K, the CH₄/CD₄KIE has been de-

termined to be 15Ϯ4 at 295–304 K ͑Refs. 1, 13, 20, 22͒,

decreasing to 5 at 400 K ͑Ref. 1͒, and the CH₄/CH₃D KIE is

1.6 at 250 K ͓Ref. 16͑c͔͒, decreasing to 1.4–1.5 at 295–298

K ͓Refs. 13, 16͑c͒, 22͔. Note that the KIE is deﬁned as the

Ab initio calculations^23–28have predicted potential en-

ergy barrier heights, V^Þ, ranging from 8.0 to 9.9 kcal/mol,

and vibrationally adiabatic ground-state barrier heights,

G

⌬V_a͑i.e., zero-point-inclusive barrier heights͒, ranging

from 3.6 to 5.5 kcal/mol. For a more complete study one

needs to know more than just the properties of the saddle

point. One example of a more complete study is the mapping

of the reaction valley near all state-selected dynamical bottle-

necks. Furthermore, the title reaction is an interesting system

because it represents a heavy-light-heavy mass combination

͑H–L–H͒, which, a priori, is a good candidate for large

‘‘corner cutting’’ tunneling effects in which reaction-path

curvature leads to internal centrifugal forces that tend to

make the system leave the minimum energy path ͑MEP͒,²⁹

and tunneling paths on the concave side of the MEP are

favored. It is possible that tunneling is enhanced when the

system leaves the valley described by a quadratic expansion

around the MEP to ‘‘cut the corner’’ of the potential energy

surface to a large extent. To study these effects or to rule

them out requires a description of the PES over a wider

region than is required to only describe the zero-point vibra-

tional valley around the MEP.²⁹Rate coefﬁcients for the title

reaction have been calculated by several groups.^{24–27,30,31}

Chen et al.²⁴calculated rate coefﬁcients for the reverse reac-

0021-9606/2000/112(21)/9375/15/$17.00

9375

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

Corchado, Truhlar, and Espinosa-Garcıa

tion based on RRKM theory³²with a tunneling correction

evaluated using the simple Wigner factor,³³and they ob-

tained excellent agreement with experimental values. How-

ever, since the Wigner factor ignores variational effects on

the location of the effective barrier for tunneling and since it

includes only the lowest-order nonzero term in a power se-

ries in ប, this agreement may be fortuitous. Gonzalez-Lafont

et al.²⁵used ab initio information along the reaction path to

calculate rate coefﬁcients using canonical variational transi-

tion state theory ͑CVT͒ ͑Refs. 34,35͒ with small-curvature

semiclassical adiabatic ground-state ͑SCSAG͒ ͑Ref. 35͒ tun-

neling contributions; their calculations agree with the experi-

mental values within a factor of 5.5 at 300 K. Dobbs and

Dixon²⁶used conventional transition state theory to calculate

the rate coefﬁcients; they obtained values lower than experi-

ment at low temperature but in good agreement with experi-

ment for TϾ300 K. Duncan and Truong²⁷calculated rate

coefﬁcients using CVT and centrifugal-dominant SCSAG

tunneling ͑SCT͒,³⁶based on ab initio information on the re-

action path. Their rate coefﬁcients are signiﬁcantly lower

than the experimental values by factors increasing from 2.2

to 11.2 as the temperature decreases from 500 to 200 K. In

1996, two of the present authors reported an analytical PES

for this reaction³⁰based on the CH₅PES of Joseph et al.,³⁷

and they carried out rate coefﬁcient calculations using CVT

with large-curvature tunneling³⁸͑LCT͒ transmission coefﬁ-

cients including information about the corner cutting region

of the potential energy surface. Although the rate coefﬁcients

were in good agreement with the experimental data, the

CH₄/CD₄KIEs were too high compared to the experimental

ones. More recently, one of the authors and co-workers³¹

reported dual-level direct dynamics calculations of rate coef-

ﬁcients and deuterium and carbon-13 KIEs by CVT with

microcanonical optimized multidimensional tunneling

͑␮OMT͒ ͑Ref. 38͒ and obtained rate coefﬁcients and KIEs in

good agreement with experimental data, although the lower-

level surface has a ‘‘deep’’ well in reactant valley.

In the present paper, to correct the deﬁciency of the ear-

lier PES, we report the construction of a new analytical PES

for the title reaction based on the PES for the HϩCH₄reac-

tion by Jordan and Gilbert,³⁹which has the advantage that is

symmetric with respect to any permutation of the four CH₄

hydrogens. The previously reported surface³⁰treats the hy-

drogen being abstracted differently than the remaining hy-

drogen atoms. ͑As pointed out by Jordan and Gilbert,³⁹the

asymmetric hydrogen treatment is an inconvenient feature

when the analytical surface is used for trajectory calcula-

tions.͒ We then use this PES to perform variational transition

state theory calculations of thermal and vibrational-state-

selected rate coefﬁcients and KIEs. For the thermal rate co-

efﬁcients and KIEs we include multidimensional tunneling

effects, by both the SCT ͑Ref. 36͒ and LCT ͑Ref. 38͒ meth-

ods.

For this reason and because of the difﬁculty of converging

higher-order correlation effects, we believe that none of the

unextrapolated theoretical calculations^23–28is entirely reli-

able. The ﬁrst attempt to extrapolate the barrier height was

carried out by Truong et al.²³who used the scaling all

correlation-energy ͑SAC͒ ͑Ref. 40͒ method to extrapolate

second-order Mo”ller–Plesset perturbation theory ͑MP2͒ to

the complete basis set and full conﬁguration interaction

level. Their calculation is, by construction, in good agree-

ment with the experimental heat of reaction, and it yields a

barrier height of 7.9 kcal/mol. Recently, one of us⁴¹has pro-

posed a new method, the ‘‘inﬁnite-basis’’ ͑IB͒ method, for

extrapolation to a complete one-electron basis set for corre-

lated electronic structure calculations. This method is based

on the extrapolation of energies obtained by using

correlation-consistent polarized double- and triple-zeta basis

sets,⁴²cc-pVDZ and cc-pVTZ ͑henceforth abbreviated pDZ

and pTZ͒. The extrapolated energies are more accurate than

even larger basis sets.^41,43

The total energy for the basis set limit is written as^41,43

E^t_ϱ^otalϭE^H_ϱ^FϩE^c_ϱ^orr

,

͑1͒

3^␣

3^␣Ϫ2^␣

2^␣

3^␣Ϫ2^␣

3^␤

3^␤Ϫ2^␤

E^t_ϱ^otal

ϭ

E^H₃^F

Ϫ

E^H₂^F

ϩ

E^c₃^orr

2^␤

3^␤Ϫ2^␤

Ϫ

E^c₂^orr

,

͑2͒

where ‘‘corr’’ is correlation energy, ‘‘HF’’ is Hartree–Fock

energy, ␣ and ␤ are parameters, and the subscripts 2 and 3

denote use of the pDZ and pTZ basis sets. The values of ␣

and ␤ that are employed in this paper where taken from Ref.

43, where they were ﬁtted in order to reproduce the experi-

mental atomization energies of a series of test molecules. We

will apply the IB method at two levels of electron correla-

tion, namely, MP2 ͑Mo”ller–Plesset second-order perturba-

tion theory with correlation of only the valence electrons͒,

and CCSD͑T͒ ͑singles and doubles coupled-cluster approach

including a quasiperturbative estimate of connected triple ex-

citations, again correlating only valence electrons͒.

In the present work we have optimized all the stationary

point geometries at the MP2/pTZ level and we carry out

single-point calculations with the IB method. The geometric

parameters of all the stationary points are listed in Table I,

together with other ab initio calculations and experimental

data, when available, for comparison. The optimum geom-

etry predicted by all the calculation levels listed in Table I

have a linear C–HЈ–Cl angle and a saddle point with C_3v

symmetry.

The MP2/pTZ reactant and product geometries agree

with the experimental data, and the MP2/pTZ transition

structure ͑TS͒ agrees with other ab initio calculations ͑within

a 0.01 Å range͒. Among the four high-level ab initio calcu-

lations compared in Table I, we note that the DFT calcula-

tions of Ref. 27 stand out for having a longer C–HЈ bond

than the other three calculations. The situation is more com-

plicated with respect to the barrier height because there is a

signiﬁcant basis set effect on the classical barrier height. This

behavior agrees with the more general experience of many

II. METHODS AND CALCULATION DETAILS

A. Estimation of the barrier height

Dobbs and Dixon²⁶noted that the classical barrier height

is very sensitive to extension of the one-electron basis set.

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

ClϩCH₄reaction

TABLE I. Ab initio and SAC properties for reactant, product, and saddle point.

MP2/pTZ^a,b

TS

Parameter CH H

H Cl

Ј

Cl

TTBGS^cDD^d

DT^eMP2/pTZ^aCCSD͑T͒/IB^fBest^g

Ј

CH₃

3

Geometry^h

R͑CH͒

R͑CH ͒

1.085 1.074

1.085

1.086

1.388

1.431

1.078 1.077

1.375 1.443

1.452 1.431

1.080

1.359

1.451

Ј

R͑ClH ͒

1.273

Ј

Energyⁱ

⌬E

6.7

1.4

7.9

3.7

6.8^j

1.9

8.9

4.9

8.9

3.6

9.9

5.5

6.2

6.0

1.4

7.6

3.7

6.8

2.2

8.4

4.5

⌬H₀

V^Þ

0.9^k

9.0

⌬V^G_a^Þ

4.8^k

Spin

l

S²

0.762

0.759

0.784

͗

͘

^aThis work: optimized at the MP2/pTZ level.

^bReactants and products.

^cFrom Ref. 23. MP2-SAC/MC-311G͑2d,d,p͒ energies and geometries with MP2/MC-311G͑2d,d,p͒ frequencies.

^dFrom Ref. 26. MP2/TZϩ2P level.

^eFrom Ref. 27. DFT/6-311G͑d,p͒ level.

^fThis work: CCSD͑T͒/IB//MP2/pTZ with parameters from Ref. 43. See Table II.

^gSame as f and including ϩ0.8 kcal/mol from the spin–orbit interaction.

^hH is the hydrogen abstracted. Experimental values ͑Ref. 44͒: CH (T_d): 1.091 Å, CH₃(D_3h): 1.079 Å, ClH

Ј

4

͑C_ϱv): 1.275 Å.

ⁱ⌬E: zero-point-exclusive energy of reaction, also called classical energy of reaction; ⌬H₀: zero-point-

inclusive energy of reaction, equal to enthalpy of reaction at 0 K; V^Þ: zero-point-exclusive barrier height, also

called classical barrier height; ⌬V^G_a^Þ: vibrationally adiabatic ground-state barrier height evaluated at the

saddle point, equal to the conventional transition state theory approximation to the enthalpy of activation at

0 K.

^jSingle-point CCSD͑T͒/cc-pVQZ energy calculations from Ref. 26.

^kZero-point energy at the MP2/MC-6311G͑2d,d,p͒ level from Ref. 23.

^lCalculated at the MP2/pTZ level.

workers⁴⁵that the truncation of the one-electron basis set is a

major source of error in most ab initio calculations of mo-

lecular energies. Therefore, we have calculated the energy

changes ͑reaction and activation͒ based on the IB basis-set

extrapolation method^41,43to obtain the basis-set limit for the

energy at this level. Tables I and II give the IB values for the

energy of the product and the transition structure relative to

reactants, together other theoretical and experimental values

for comparison. Note that this is the ﬁrst time that the IB

method has been used to calculate barrier heights. The IB

barrier height is in excellent agreement with the previous²³

SAC calculation, being only 0.3 kcal/mol lower.

have the same energy. Including the spin–orbit effect would

lower the energy of the lower state of the Cl atom to one-

third of ⌬E below its nonrelativistic energy ͑0.8 kcal/mol͒.

As usual, we assume that at the saddle point the spin–orbit

coupling is essentially fully quenched and the nonrelativistic

treatment will give a good approximation to the correct en-

ergy. As discussed previously,³¹the effect of the lowering in

the reactants energy due to the spin–orbit coupling can be

simulated in our nonrelativistic calculations by adding 0.8

kcal/mol to the barrier height and reaction enthalpy. Note

that the quenching of the spin–orbit interaction energy of Cl

has been usually ignored by other workers, and indeed the

uncertainty in the usual ab initio calculations does not allow

one to eliminate other sources of error of this magnitude.

Nevertheless, the spin–orbit effect is systematic, and it

should be included. With this contribution added, the MP2/

pTZ enthalpy of reaction at 0 K is 2.2 kcal/mol, only 1.2

kcal/mol higher than the experimental value⁴⁴of 1.0 kcal/

mol.

The chlorine atom has two low-lying electronic states

with a separation of ⌬Eϭ882 cm^Ϫ1͑2.5 kcal/mol͒ ͑Ref. 44͒

due to spin–orbit coupling. In nonrelativistic calculations,

such as the present ab initio calculations, both states would

TABLE II. Single-point energy calculations ͑kcal/mol͒.

⌬E

V^Þ

B. Calibration of the analytical PES

MP2/pTZ

MP2/IB͑orig.͒

MP2/IB^b

6.2

4.6

4.3

7.3

6.2

6.0

6.8

9.0

7.0

6.6

9.6

8.1

7.6

8.4

a

The potential energy surface for the ClϩCH₄

→HClϩCH₃reaction is similar to that for the well-studied

HϩCH₄→H₂ϩCH₃reaction. Both are hydrogen abstraction

reactions from methane to yield the CH₃radical, and there is

a slow change in geometry of the CH₃group from pyramidal

to planar along the reaction path. With this similarity in

mind, we have modiﬁed several parameters of the

HϩCH₄→H₂ϩCH₃surface of Jordan and Gilbert³⁹in order

CCSD͑T͒/pTZ

CCSD͑T͒/IB͑orig.͒

CCSD͑T͒/IB^b

a

CCSD͑T͒/IB^b,c

^aExtrapolated to inﬁnite basis with original parameters of Ref. 41.

^bExtrapolated to inﬁnite basis with improved parameters of Ref. 43.

^cIncludes spin–orbit contributions of ϩ0.8 kcal/mol.

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

Corchado, Truhlar, and Espinosa-Garcıa

TABLE III. Fitted parameters for the analytical PES.

to get an analytical PES for the title reaction. The new sur-

face consists of four London–Eyring–Polanyi⁴⁶͑LEP͒ func-

tions augmented by bending terms as in the earlier CH₅

surface³⁷on which it is based. This potential is written in

terms of stretching ͑str͒, out-of-plane ͑op͒ bending, and va-

lence ͑val͒ bending terms and has the form

Parameter^a

Value

R⁰_CH

1.0939 Å

¹D_CH

³D_CH

a_CH

112.341 kcal/mol

38.680 kcal/mol

1.7030 Å^Ϫ1

0.1350 Å^Ϫ1

6.6140 Å^Ϫ1

b_CH

c_CH

R⁰_CCl

¹D_CCl

³D_CCl

VϭV_strϩV_opϩV_val

,

͑3͒

1.7811 Å

where V_stris the stretching term given by

87.149 kcal/mol

23.583 kcal/mol

0.8650 Å^Ϫ1

4

V ϭ

str

V

R

,R_CCl,R_ClH͒,

CH

i

͑4͒

͑

͚

3

␣

CCl

iϭ1

R⁰

1.2745 Å

ClH

¹D_ClH

³D_ClH

106.266 kcal/mol

36.664 kcal/mol

1.8700 Å^Ϫ1

and where V₃represents the LEP functional form used by

Joseph et al.,³⁷V_opis the out-of-plane bending term, corre-

lating with the out-of-plane motion of the methyl radical, and

V_valis a harmonic term for valence bends. The latter terms

are designed to be reasonable both for the methane molecule

and the methyl radical. The mathematical expressions of the

bending terms are given by Jordan and Gilbert,³⁹and they

are not repeated here. It is however critical to note that V₃

involves three singlet parameters (¹D_XϪY, ␣_XϪY, and

␣

ClH

0.413 952 Å^Ϫ2

2.099 96 Å^Ϫ2

2.865 95 Å^Ϫ2

a₁

a₂

a₃

^aFor a complete deﬁnition of the parameters and the analytical form of the

PES, see Ref. 39.

R⁰_XϪY) and ﬁve triplet parameters (³D_X–Y, ␤_X–Y, c_X–Y

,

saddle point the length of the bond that is broken ͑C–HЈ͒

increases by 27%, and the length of the bond that is formed

͑Cl–HЈ͒ is only 6% larger than at products. This indicates

that the reaction of the Cl atom with methane proceeds via a

‘‘late’’ transition state, i.e., it is a productlike transition state.

The saddle point has one imaginary frequency. The absolute

value of the imaginary part of this frequency is lower in our

calculations than the ab initio values, although it is well

known that sometimes ab initio calculations overestimate

this value.⁴⁵Moreover, as noted above, our objective was to

reproduce experimental KIEs, and the value of this imagi-

nary frequency actually rose during this adjustment. The

combined effect of potential energy and zero-point energy at

the saddle point is embodied in, ⌬H^Þ₀, the conventional

transition-state enthalpy of activation at 0 K, and this agrees

within 1.5 kcal/mol with the ab initio calculations and with

the IB calculations performed in this work.

a_X–Y, and R_X–Y) for each bond X–Y.³⁷

The calibration process used in this work has several

steps. In the ﬁrst step, we changed the parameters (R_C⁰_H

*

,

0

1

R_CCl, R_ClH, D_CH, D_CCl, D_ClH, ␣_CH, ␣_ClH) of the PES

related to the geometric, energetic, and vibrational properties

of the reactants and products, so that the geometries, endot-

hermicity, and vibrational frequencies agreed reasonably

with the available experimental values. In the second step,

we reﬁt some parameters (³D_CH³D_CCl³D_ClH, ␣_CCl) in

, ,

order to reproduce the characteristics of the ab initio calcu-

lated saddle point, in particular, the geometry, barrier height,

and the vibrational frequencies. During this ﬁtting, we exer-

cised special care to insure that no artiﬁcially deep wells

were introduced in the reactant or product valley. Finally,

since a main objective in this work is to analyze the dynami-

cal details of the rate process ͑importance of tunneling, tem-

perature dependence, excited-state reactivity, etc.͒ and to cal-

culate possible sources of the KIEs, as the third step of our

recalibration, we reﬁt some of the parameters of the analyti-

cal PES, in order to reproduce more accurately the experi-

mental rate coefﬁcients at 300 K and improve the agreement

with experimental ¹³C and deuterium ͑CD₄vs CH₄) KIEs at

300 K. The ﬁnal parameters that are different from the

original³⁹HϩCH₄parameters are given in Table III.

The results of the ﬁnal ﬁt are listed in Table IV for

reactants and products and in Table V for the saddle point. In

general, the experimental^44,47reactant and product properties

are well reproduced, with the most signiﬁcant difference be-

ing 0.02 Å for the C–H bond length in CH₃. With respect to

the vibrational frequencies, small differences were obtained,

although these small errors partially cancel between reactants

and products, yielding an enthalpy of reaction at 0 K of 0.9

kcal/mol, in very good agreement with the experimental

value of 1.0 kcal/mol.⁴⁴Comparison of the properties of the

saddle point of this surface with ab initio calculations shows

that this surface reproduces, in general, the trends and values

although the new surface has a shorter Cl–HЈ bond. At the

C. Dynamical calculations

With the new surface calibrated as described in the pre-

vious section, the reaction path was calculated starting from

the saddle point geometry and going downhill to both reac-

TABLE IV. Reactant and product properties^acalculated using the analytical

surface.

System

R͑C–H͒

R͑Cl–H͒

Frequencies

b

CH₄

1.094

3037 ͑t͒, 2871,

1501 ͑e͒, 1339 ͑t͒

3165 ͑e͒, 2994,

1240 ͑e͒, 580

2993

c

CH₃

1.094

ClH^d

1.274

^aDistances in Å, frequencies in cm^Ϫ1, energies in kcal/mol.

^bExperimental values ͑Ref. 44͒: CH₄(T_d): 1.091 Å, 3019͑t͒, 2917, 1534͑e͒,

1306͑t͒.

^cExperimental values ͑Ref. 44͒: CH₃(D_3h): 1.079 Å, 3184͑e͒, 3002,

1383͑e͒, 580.

^dExperimental values: ClH͑C_ϱv): 1.275 Å ͑Ref. 44͒, 2991 cm^Ϫ1͑Ref. 47͒.

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9379

J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

TABLE V. Saddle point properties^acalculated using the analytical surface.

Electronic structure calculations

ClϩCH₄reaction

Parameter m

R^Þ͑CH͒

SPES^b

APES^c

TTBGS^d

DD^e

DT^f

Best^g

1.098

1.389

1.356

1.096

1.377

1.386

1.086

1.388

1.431

1.078

1.375

1.452

1.077

1.443

1.431

1.080

1.359

1.451

R^Þ(CH )

Ј

R^Þ͑ClH )

Ј

⌬E

6.1

0.9

7.7

3.1

6.1

0.9

8.1

4.4

7.5^h

2.2^h

8.8^h

4.4^h

7.6^h,i

2.7^h

9.7^h

5.7^h

9.7^h

4.4^h

6.8^h

2.2^h

8.4^h

4.5^h

⌬H₀

⌬E^Þ

10.7^h

6.3^h

⌬H^Þ(0 K)

Frequencies

1,2

3

3039 ͑e͒

2910

3022 ͑e͒

2960

3295 ͑e͒

3118

3305 ͑e͒

3132

3328 ͑e͒

3162

3303 ͑e͒

3131

4,5

6

1372 ͑e͒

1126

1419 ͑e͒

1190

1449 ͑e͒

1227

1448 ͑e͒

1213

1468 ͑e͒

1211

1457 ͑e͒

1223

7,8

9

782 ͑e͒

732

1102 ͑e͒

543

874 ͑e͒

572

958 ͑e͒

511

920 ͑e͒

541

923 ͑e͒

519

10,11

F

312 ͑e͒

760i

344 ͑e͒

1196i

324 ͑e͒

949i

378 ͑e͒

1262i

385 ͑e͒

996i

337 ͑e͒

1136i

^aDistances in Å, frequencies in cm^Ϫ1, energies in kcal/mol. H is the hydrogen abstracted. Symmetry C

.

Ј

3v

^bSymmetric PES. This work.

^cAsymmetric PES, from Ref. 30.

^dFrom Ref. 23. MP2-SAC/MC-311G͑2d,d,p͒ energies and geometries with MP2/MC-311G͑2d,d,p͒ frequencies.

^eFrom Ref. 26. MP2/TZϩ2P level.

^fFrom Ref. 27. DFT/6-311G͑d,p͒ level.

^gCCSD͑T͒/IB//MP2/pTZ from present work with scaled MP2/MC-311G͑2d,d,p͒ frequencies for the saddle

point ͑scale factor 0.96͒ and experimental frequencies for the reactants and products.

^hWe added the ϩ0.8 kcal/mol contribution from the spin–orbit interaction to all energies in this table to provide

a consistent comparison.

ⁱSingle-point CCSD͑T͒/cc-pVQZ energy calculations from Ref. 26.

tants and products in mass-weighted Cartesian coordinates,

using Euler’s single-step method⁴⁸with step-size of 0.0005

amu^1/2a₀, where 1 a₀ϵ1 bohrϵ0.529 Å. The Hessian ma-

trix was evaluated every 30 points along this reaction-path.

Along this minimum energy path ͑MEP͒, the reaction coor-

dinate, s, is deﬁned as the signed distance from the saddle

point, with sϾ0 referring to the product side. In the rest of

paper, the units of s are a₀, and all calculations are carried

out in mass-scaled coordinates with a reduced mass ␮ equal

to 1 amu. Thus, distances through the mass-scaled coordi-

nates in a₀are equivalent to distances through mass-

weighted coordinates in amu^1/2a₀. We calculate the

reaction-path between sϭϪ3.0 a₀and sϭϩ3.0 a₀; all the

rate coefﬁcients are well converged with respect to the gra-

dient step-size, distance between Hessian matrices ͑0.015

a₀), and extent of reaction-path calculated. Along the MEP a

generalized normal-mode analysis was performed using a re-

dundant curvilinear projection operator⁴⁹formalism. With

this information, we calculated the ground-state vibrationally

adiabatic potential curve

B_mm(s), measuring the coupling between generalized nor-

Ј

mal modes m and mЈ induced by motion of the system along

the reaction path. The B_mF(s) coupling terms are the com-

ponents of the reaction path curvature, ␬(s), deﬁned as

FϪ1

1/2

2

␬ s͒ϭ

͑

B

s͒

͔

͑6͒

͓

͑

ͩ

ͪ

͚

mF

kϭ1

and they control the nonadiabatic ﬂow of energy between

these modes and the reaction coordinate.⁵¹These coupling

terms are used in the calculation of transmission coefﬁcients

that include the effects of the reaction path curvature. They

also allow us to give a qualitative explanation of the effects

of vibrational excitation of reactants and/or products.

Finally, the energies, vibrational frequencies, geom-

etries, and gradients along the MEP were used to estimate

rate coefﬁcients and kinetic isotope effects ͑KIEs͒ by using

variational transition state theory ͑VTST͒ with semiclassical

multidimensional tunneling. We calculated thermal rates us-

ing canonical variational theory^34,35͑CVT͒, which locates

the dividing surface between reactants and products at a

G

V^G_as͒ϭV

s͒ϩ␧ s͒,

͑5͒

͑

MEP

int

where V_MEP(s) is the classical energy along the MEP with its

zero of energy at the reactants (sϭϪϱ), and ⑀^G_int(s) is the

zero-point energy at s from the generalized normal-mode vi-

brations orthogonal to the reaction coordinate. We also cal-

culated the coupling terms,⁴⁹B_mF(s), measuring the cou-

pling between normal mode m and the motion along the

reaction coordinate, mode F, and the Coriolis-type terms,⁵⁰

point s^CVT(T) along the reaction path that minimizes the

*

generalized TST rate coefﬁcients, k^GT(T,s), for a given tem-

perature T. Thermodynamically, this is equivalent to locating

the transition state at the maximum ⌬G^GT,0͓T,s^CVT(T)͔, of

*

the standard-state free energy of activation proﬁle

⌬G^GT,0(T,s).^34,35Thus, the thermal rate coefﬁcient will be

given by

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´

Corchado, Truhlar, and Espinosa-Garcıa

9380

J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

k_BT

1

ប␻

m

k^CVTT͒ϭ␴

K⁰exp Ϫ⌬G^GT,0T,s^CVT͒/k T , ͑7͒

Q_mn,T͒ϭexp Ϫ ϩn

,

͑8͒

͑

͓

͑

͔

͑

ͫ

ͩ

ͪ

ͬ

B

h

*

2

k_BT

where n is the vibrational state of mode m. Substituting Eq.

͑8͒ for the usual canonical partition function of mode n

yields a state-selected free energy of activation proﬁle called

⌬G^GT,0(n,T,s). This adiabatic model assumes that vibra-

tional modes preserve their character along the reaction path.

This approximation is most likely to be fulﬁlled for excited-

state reactions involving excitation of high-frequency modes

in reactants and in thermoneutral or exothermic reactions

where the dynamical bottleneck is located early along the

reaction path. The adiabatic approximation may be poor for

excited-state reactions with vibrational excitation of the low-

frequency modes.⁵⁸

Since the assumption of vibrational adiabaticity all along

the reaction path may not hold for this reaction, we also used

the partial-reaction-path adiabaticity approximation ͑PRP͒.⁶⁰

In this method, we assume that the adiabaticity of the state-

selected high-frequency mode transverse to the reaction path

is conserved until a point where the generalized normal

modes become strongly coupled. We identify this point with

the ﬁrst point on the way from reactants to products where

the reaction-path curvature peaks. Beyond that point, the en-

ergy in the state-selected high-frequency mode is transferred

freely to all the other motions, and, as a result, that particular

mode can ﬁnd itself in its ground state. The operational way

of implementing this method is the following: We start by

locating the ﬁrst point, s_ϩ, where the reaction-path curva-

ture shows a peak. The region before that point will be taken

as adiabatic, and the free energy of activation along that

portion of the reaction path will be taken as the state-selected

free energy of activation, ⌬G^GT,0(n,T,s). After that point,

the free energy of activation will be taken as the one for the

ground state, ⌬G^GT,0(nϭ0,T,s). Thus, the rate coefﬁcient

can be written as⁶⁰

with k_Bbeing Boltzmann’s constant, h being Planck’s con-

stant, ␴ being the symmetry factor ͑the number of equivalent

reaction paths, which were assumed to be 4 and 2 for the

forward and reverse nonsubstituted reactions, respectively͒,

and K⁰being the reciprocal of the standard-state concentra-

tion, taken as 1 molecule cm^Ϫ3

.

In the present work, we used the general polyatomic rate

coefﬁcient code POLYRATE.⁵²We correctly⁵³included the

²P_1/2excited state of Cl ͑with excitation energy⁴⁴882 cm^Ϫ1

)

in the reactant electronic partition function. The rotational

partition functions were calculated classically, and vibrations

are treated as quantum mechanical separable harmonic oscil-

lators, with the generalized normal-modes deﬁned in curvi-

linear coordinates.^49,54The chosen curvilinear coordinates

were all the possible bond lengths and bond angles. The

advantage of curvilinear coordinates^49,54͑nonlinear functions

of Cartesian coordinates͒ is that all the vibrational frequen-

cies were real over the whole of the reaction path, while with

rectilinear coordinates^50,55͑linear functions of Cartesian co-

ordinates͒, we found that the lowest bending frequencies had

unphysical imaginary values over a wide range of the reac-

tion coordinate. The appearance of imaginary frequencies

along the reaction path is a consequence of the inadequacy of

the Cartesian coordinates for describing vibrational motions,

and this behavior has also been found in other hydrogen

abstraction reactions.^49,54,56,57

The vibrational-state-selected rate coefﬁcients were cal-

culated using a vibrationally adiabatic version of generalized

transition state theory.^58,59The adiabatic expression differs

from the form of the thermal rate only in the vibrational

partition function for the state-selected mode, which is re-

placed by

max ⌬G^GT,0n,T,s͒/k T

͑

B

Ϫϱрsрs

k_BT

h

ϩ

k^PRP-CVTn,T͒ϭ␴

K⁰exp Ϫmax

.

͑9͒

͑

max ⌬G^GT,0nϭ0,T,s͒/k T

ͫ

ͬ

ͭ

ͮ

͑

B

s

рsрϩϱ

ϩ

Note that, as discussed in previous work,⁶⁰the rates calcu-

lated using this method will probably underestimate the rates

of the vibrationally excited reactions and the true value will

lie between the fully adiabatic and PRP rate coefﬁcients.

Finally, we added the tunneling contributions. The

ClϩCH₄reaction has a heavy–light–heavy mass combina-

tion, and, therefore, a large curvature tunneling ͑LCT͒ calcu-

lation should be performed.^29,61We used the microcanonical

optimized multidimensional tunneling (␮OMT͒ approach³⁸

in which, at each total energy, the larger of the SCT and LCT

tunneling probabilities is accepted as the best estimate. In the

LCT calculations we allowed the system to reach all the

energetically accessible vibrational excited states of the

mode that receives the transferred atom.⁶¹All transmission

coefﬁcients also include the classical adiabatic ground-state

͑CAG͒ transmission coefﬁcient⁶²that adjusts the quantal cor-

rection for the difference between V^G_aat its maximum and at

the CVT transition state.

III. RESULTS AND DISCUSSION

A. Reaction path and curvature terms

Figure 1 shows the classical potential energy, V_MEP, the

ground state vibrationally adiabatic potential energy relative

to reactants, ⌬V^G_a, and the change of the local zero-point

energy (⌬ZPE͒, as functions of s over the range from

sϭϪ2.0 to sϭϩ2.0 a₀, i.e., from a geometry with a C–H

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

ClϩCH₄reaction

FIG. 2. Generalized normal-mode vibrational frequencies as a function of

reaction coordinate s. Doubly degenerate modes are dashed curves, and

nondegenerate modes are solid curves. These vibrational frequencies were

calculated by a curvilinear vibrational analysis and were used to generate the

⌬ZPE curve of Fig. 1.

FIG. 1. Classical potential energy curve, V_MEP, zero-point energy, ⌬ZPE,

and vibrationally adiabatic potential energy curve, ⌬V^G_a, as a function of s

for the unsubstituted reaction. All quantities are with respect to the reactants.

distance of 1.090 Å and a Cl–H distance of 1.871 Å to a

geometry with a C–H distance of 1.805 Å and a Cl–H dis-

tance of 1.274 Å. Note that ⌬V^G_aand ⌬ZPE are deﬁned as

the difference between V_a^Gat s or ZPE at s and their values

for reactants.

a minimum near sϭ0 with a reasonable steep rise immedi-

ately thereafter ͑Fig. 1͒, and, therefore, variational effects

may be expected for this reaction. Large variational effects

are the expected behavior for a nearly thermoneutral reaction

with a heavy–light–heavy mass combination.^29͑a͒,35The rea-

son for this is that the dip in the reactive mode frequency is

most pronounced in such cases.³⁴

Figure 1 shows that the maximum of ⌬V^G_ais shifted with

respect to the saddle point ͑sϭ0͒; in particular it occurs at

sϭϩ0.29 bohr. Therefore, large variational effects may be

expected for this reaction ͑a ‘‘variational effect’’ is a differ-

ence between a prediction of variational transition state

theory and a prediction of conventional transition state

theory͒.

Along the MEP, the coupling terms, B_mF(s), between

the reaction coordinate and the orthogonal bound modes con-

trol the nonadiabatic ﬂow of energy between these modes

and the reaction coordinate.^{50,51,63–70}Figure 3 shows the cur-

vature (␬) of the reaction path as a function of s. There are

two sharp peaks, one on the reactant side and one on the

product side of the saddle point, as usual for most thermo-

neutral reactions. The ﬁrst is due to strong coupling of the

reaction path to the CH₄reactive stretch ͑which is mϭ6 at

this peak͒ which has B_mFϭ15.6 a^Ϫ₀¹at the maximum of ␬ at

sϭϪ0.53 a₀and to the umbrella bend mode ͑which is mϭ7

at this location͒ which contributes to a lesser extent (B_mF

ϭ5.6 a^Ϫ₀¹at sϭϪ0.48 a₀). Therefore, a priori, excitation

of these modes might be expected to enhance the rate. The

second peak is in the exit channel (sϭϩ0.02 a₀), and it is

lower than the ﬁrst. It is due to the coupling of the reaction

path to the Cl–H stretch ͑which is mϭ6 at sϭϩ0.05 a₀),

To analyze the origin of the variational shift in the maxi-

mum of ⌬V^G_a, in Fig. 2 we show the variation of the gener-

alized normal mode frequencies along the reaction path. The

lowest curve in Fig. 2 tends to zero in the reactant limit ͑s

ϭϱ), i.e., it is a transitional mode, and in that limit there are

nine frequencies corresponding to CH₄. The lowest-

frequency C–H stretching mode numbered mϭ4 transforms

adiabatically into the stretching frequency of the bond that

breaks during the reaction; thus it is called the reactive mode,

and it correlates adiabatically with the mode numbered mϭ9

at the saddle point, where the modes are numbered in order

of decreasing frequency and degenerate modes are counted

twice. However, in the rest of this paper we will label the

modes by the way that they correlate adiabatically. Thus the

mode that we call the reactive mode at the saddle point is

mϭ6. At about sϭϪ0.5 a₀and again at sϭϩ0.2 a₀, the

reactive mode exhibits an avoided crossing with a methyl

umbrella mode ͑mϭ7, which correlates adiabatically to mϭ9

at the saddle point͒. The transitional modes have a maximum

close to the minimum of the reactive and umbrella modes in

the saddle point region, although their frequencies increase

less than the other ones decrease. As a result, the ZPE shows

which has B_mFϭ3.3^Ϫ_a¹and to the CH₃bend ͑which is mϭ9

0

at this location͒, which has B_mFϭ4.2 a^Ϫ₀¹. This might be an

indication that these product modes could appear vibra-

tionally excited. However, if we examine ⌬V^G_acurve ͑Fig. 1͒

we note that the energy difference between the maximum of

the ⌬V^G_aat sϭ0.29 and the products is only 2.89 kcal/mol

and therefore, there is not enough available energy to excite

one quanta the Cl–H stretch mode (2993 cm^Ϫ1

Ϸ8.5 kcal/mol) if the system passes through the transition

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´

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

Corchado, Truhlar, and Espinosa-Garcıa

FIG. 4. Coriolis coupling term, B_mm, between the reactive mode and the

Ј

umbrella mode of CH₃as a function of reaction coordinate s.

FIG. 3. Reaction path curvature, ␬, as a function of reaction coordinate s.

IV. RATE COEFFICIENTS FOR THE ISOTOPICALLY

UNSUBSTITUTED REACTION

state region in the ground vibrational state, although there is

enough energy to excite one quanta the CH₃umbrella mode

(550 cm^Ϫ1Ϸ1.6 kcal/mol͒. Therefore, for thermal reactions

the Cl–H stretch mode will appear vibrationally unexcited,

while the CH₃umbrella mode can appear vibrationally ex-

cited. This situation would be typical of a reaction with a

‘‘late’’ transition state where the reagent vibrational energy

is more effective than the translational energy for enhancing

the reaction rate, while the products are formed with small

vibrational excitation.^64,68–70The present analytical PES

does have a ‘‘late’’ transition state. These qualitative conclu-

sions on the population of the vibrational states agree with

the experimentally observed populations of Zare et al.^15,17,21

and with the results of the theoretical study of Duncan and

Truong.²⁷

In order to shed more light on the dynamical features of

this reaction, we also calculated the coupling between the

vibrational modes, B_mm͑Fig. 4͒. Although several modes

seem to be coupled, one particular B_mmterm is especially

large; the one that measures the coupling between the reac-

tive mode and the CH₃umbrella mode. These are the two

modes that are responsible for the peaks in ␬(s) in the two

avoided crossing zones as seen in Fig. 2 and as discussed

above.

In canonical variational transition state theory the loca-

tion of the generalized transition state dividing surface is

optimized along the reaction path for each temperature in

order to minimize the rate coefﬁcients.^34,35The value of the

reaction coordinate, s, where the canonical variational tran-

sition state occurs is denoted as s^CVT(T). In the present case,

*

bottleneck properties of the thermally averaged reaction,

based on this CVT approach, show signiﬁcant dependence

on such optimization, and the value of s^CVT(T) ranges from

*

ϩ0.29 a₀at 0 K to Ϫ0.57 a₀at 2500 K. Comparison of

columns 2 and 3 of Table VI shows the variational effect on

the thermal rate coefﬁcients, decreasing from a factor of 5.2

at 200 K to a factor of 1.4 at 1000 K, then increasing again to

a factor of 3.1 at 2500 K.

The shifting of the transition state as we increase the

temperature occurs suddenly. For temperatures below 967 K

the variational transition state is located in the product chan-

TABLE VI. Rate coefﬁcients without tunneling ͑k͒ and transmission coef-

ﬁcients (␬) for the CH₄ϩCl reaction.

a

T͑K͒

k͑TST͒

k͑CVT͒

k͑CUS͒

␬͑SCT͒ ␬(␮OMT͒

␬͑ZCT͒

b

200 6.2͑Ϫ15͒ 1.2͑Ϫ15͒ 1.1͑Ϫ15͒

250 3.1͑Ϫ14͒ 8.6͑Ϫ15͒ 7.9͑Ϫ15͒

300 9.5͑Ϫ14͒ 3.4͑Ϫ14͒ 3.0͑Ϫ14͒

400 4.4͑Ϫ13͒ 2.1͑Ϫ13͒ 1.7͑Ϫ13͒

600 2.8͑Ϫ12͒ 1.8͑Ϫ12͒ 1.2͑Ϫ12͒

800 8.8͑Ϫ12͒ 6.2͑Ϫ12͒ 3.9͑Ϫ12͒

1000 2.0͑Ϫ11͒ 1.4͑Ϫ11͒ 8.3͑Ϫ12͒

1500 7.7͑Ϫ11͒ 3.7͑Ϫ11͒ 2.7͑Ϫ11͒

2000 1.8͑Ϫ10͒ 6.9͑Ϫ11͒ 5.6͑Ϫ11͒

2500 3.3͑Ϫ10͒ 1.1͑Ϫ10͒ 9.2͑Ϫ11͒

1.80

1.45

1.29

1.15

1.06

1.04

1.02

1.01

1.00

2.30

1.69

1.43

1.22

1.09

1.05

1.03

1.01

1.00

12.8

5.48

3.33

1.98

1.35

1.18

1.11

1.05

1.03

1.02

The coupling of the vibrational modes and the reaction

coordinate has an important effect on tunneling.^29,35,71Our

analysis of this coupling shows that the reaction-path curva-

ture peaks at a point close to the variational transition state,

which is the region where tunneling is most likely to take

place. This makes it necessary to take into account the reac-

tion path curvature in order to correctly describe the tunnel-

ing effect.

^aIn all cases ␬͑LCT͒ϭ␬(␮OMT͒ to at least three signiﬁcant ﬁgures.

^b6.2͑Ϫ15͒ stands for 6.2ϫ10^Ϫ15, in cm³molecule^Ϫ1s^Ϫ1

.

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

TABLE VII. Forward rate coefﬁcients for the ClϩCH₄→HClϩCH₃reaction.

ClϩCH₄reaction

Theory

Experiment

Ref. 11 Ref. 12

T͑K͒ CVT/␮OMT CUS/␮OMT

RCT^a

APES^b

Ref. 4

Ref. 21

c

200

250

300

400

500

600

800

1000

1500

2000

2500

1.5͑Ϫ14͒

4.7͑Ϫ14͒

1.1͑Ϫ14͒

4.1͑Ϫ13͒

1.1͑Ϫ12͒

2.3͑Ϫ12͒

7.0͑Ϫ12͒

8.5͑Ϫ12͒

2.6͑Ϫ11͒

5.2͑Ϫ11͒

8.6͑Ϫ11͒

1.4͑Ϫ14͒

4.3͑Ϫ14͒

9.9͑Ϫ14͒

3.4͑Ϫ13͒

8.1͑Ϫ13͒

1.6͑Ϫ12͒

4.3͑Ϫ12͒

4.9͑Ϫ12͒

1.9͑Ϫ11͒

4.2͑Ϫ11͒

7.3͑Ϫ11͒

5.6͑Ϫ15͒ 1.5͑Ϫ14͒ 1.1͑Ϫ14͒

2.4͑Ϫ14͒ 4.2͑Ϫ14͒ 4.1͑Ϫ14͒

6.9͑Ϫ14͒ 8.8͑Ϫ14͒ 1.0͑Ϫ13͒ 9.4͑Ϫ14͒ 1.0͑Ϫ13͒ 9.3͑Ϫ14͒

3.1͑Ϫ13͒ 2.7͑Ϫ13͒ 3.5͑Ϫ13͒ 3.4͑Ϫ13͒ 3.1͑Ϫ13͒ 3.0͑Ϫ13͒

8.6͑Ϫ13͒ 6.4͑Ϫ13͒ 8.8͑Ϫ13͒ 7.5͑Ϫ13͒ 8.2͑Ϫ13͒ 6.5͑Ϫ13͒

1.9͑Ϫ12͒ 1.3͑Ϫ12͒

6.2͑Ϫ12͒ 3.7͑Ϫ12͒

1.4͑Ϫ11͒ 8.2͑Ϫ12͒

5.6͑Ϫ11͒ 3.1͑Ϫ11͒

1.3͑Ϫ10͒ 7.2͑Ϫ11͒

2.4͑Ϫ10͒ 1.3͑Ϫ10͒

1.1͑Ϫ14͒

4.3͑Ϫ14͒

1.3͑Ϫ12͒

3.0͑Ϫ12͒

a

˜

Roberto-Neto, Coitino, and Truhlar ͑Ref. 31͒. CVT/␮OMT calculations based on a dual-level direct dynamic

calculation.

^bOur earlier asymmetric PES ͑Ref. 30͒. CVT/LCT calculations.

^c1.5͑Ϫ14͒ stands for 1.5ϫ10^Ϫ14, in cm³molecule^Ϫ1s^Ϫ1

.

nel, while for temperatures over 967 K it is located on the

reactant side of the reaction path. The reason is that

⌬G^GT,0(T,s) presents a double-peaked shape, as does the

⌬V^G_acurve plotted in Fig. 1. For temperatures below 967 K,

the higher ⌬G^GT,0͑T,s͒ peak is found on the product side

s^CVT(966 K͒ϭ0.14 , while at temperatures higher than 967

expression used to describe the surface ͑asymmetric vs sym-

metric͒ but rather to the different criteria used to calibrate it.

In our ﬁrst paper, the calibration was based on the stationary

point properties ͑geometries, frequencies, and energies͒,

while in the present paper, the calibration is designed to re-

produce the rate coefﬁcients and the KIEs. With these new

criteria, the tunneling effect is considered in the calibration.

Table IX lists phenomenological activation energies

computed as local two-point slopes of Arrhenius plots for the

forward and reverse reactions, in order to provide the most

appropriate comparison to experiment. Our theoretical re-

sults show excellent agreement with experimental values in

various temperature ranges up to 500 K.

Our activation energies at 298 K ͑from slopes of Arrhen-

ius plots from 293 K to 303 K͒ are 2.63 and 1.07 kcal/mol

for the forward and reverse reactions, respectively. Using

these theoretical activation energies a value of 1.56 kcal/mol

is obtained for the enthalpy of reaction at 298 K, in agree-

ment with the experimental value from enthalpies of forma-

tion, ⌬H²_r^{98 K}ϭ1.65 kcal/mol ͑Ref. 44͒. This result is encour-

aging, although of course all transition state properties cancel

out, as does the barrier shape. This comparison conﬁrms that

we have a reasonable ⌬H⁰₂₉₈by means of the well known

relation⁷⁵between the overall enthalpy of reaction and the

difference of the forward and reverse enthalpy of activation.

͓

͔

*

K, ⌬G^GT,0(T,s) has its higher peak on the reactant side

s^CVT(968 K͒ϭϪ0.57 a₀. At 967 K both peaks are equally

͓

͔

*

high. This sudden change in the location of the transition

state leads to a discontinuity in the activation energy ͑and in

an Arrhenius plot͒ around 967 K. We will avoid this problem

by using the canonical uniﬁed statistical theory ͑CUS͒.^35,72

This approach takes account of both maxima ͑global and

local maxima͒ in ⌬G^GT,0(T,s) by using a canonical version

of the branching analyses of Hirschfelder and Wigner and

Miller.⁷³Thus the CUS rate coefﬁcient has a continuous tem-

perature derivatives even where the variational transition

state undergoes a sudden switch.

Tables VII and VIII list the CUS/␮OMT rate coefﬁ-

cients, along with experimental^{4,11,12,21,74}and other

theoretical^30,31rate coefﬁcients for the temperature range

200–2500 K for both the forward and reverse reactions, re-

spectively. In both cases, with the new parametrization of the

surface, the calculated rates agree better than the previously

reported surface³⁰with the experimental data in the tempera-

ture range that has been studied experimentally. Since the

experimental data were used in the parametrization, this rep-

resents simply a check on the consistency of the parametri-

zation. It is encouraging though that the new results are more

consistent with direct dynamics calculations³¹than were the

earlier results based on the previous analytic surface. For the

forward reaction, with our earlier asymmetric PES ͑Ref. 30͒

we obtained a transmission coefﬁcient of 106 and a CVT rate

coefﬁcient of 1.5ϫ10^Ϫ16cm³molecule^Ϫ1s^Ϫ1at 200 K,

while in this work we obtain values of 12.8 and 1.2

ϫ10^Ϫ15cm³molecule^Ϫ1s^Ϫ1, respectively; the latter value is

in better agreement with previous direct dynamics ␮OMT

transmission coefﬁcient calculations,³¹which range from 4.3

to 7.4 at 200 K. This discrepancy between the present result

and the earlier one is not due to the form of the analytical

A. Kinetic isotope effects

The ¹²C/¹³C KIEs are listed in Table X in the tempera-

ture range 200–2500 K. Our results slightly overestimate

these KIEs, especially at low temperatures, as compared to

previous experimental¹⁶and theoretical³¹values. To analyze

this behavior in more detail, in the same Table X we also

give a factor analysis. This factorization of the KIEs involves

writing them as a product, where each factor in the product

arises from the ratio, for the two isotopologs or isotopomers,

of a particular subset of partition coefﬁcients or of some

other factor in the overall rate expression. For the sake of

clarity, the factorization of partition functions is based on

conventional transition state theory, and the variational and

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´

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

Corchado, Truhlar, and Espinosa-Garcıa

TABLE VIII. Reverse rate coefﬁcients for the ClϩCH₄→HClϩCH₃reaction.

Theory

Experiment

T͑K͒

CVT/␮OMT

CUS/␮OMT

RCT^a

APES^b

Ref. 11

Ref. 74

c

200

250

300

400

500

600

800

1000

1500

2000

2500

3.8͑Ϫ14͒

3.6͑Ϫ14͒

5.4͑Ϫ14͒

7.5͑Ϫ14͒

1.3͑Ϫ13͒

1.9͑Ϫ13͒

2.6͑Ϫ12͒

4.4͑Ϫ13͒

3.8͑Ϫ13͒

1.1͑Ϫ12͒

2.1͑Ϫ12͒

3.6͑Ϫ12͒

3.3͑Ϫ14͒

5.1͑Ϫ14͒

7.0͑Ϫ14͒

1.1͑Ϫ13͒

1.7͑Ϫ13͒

2.3͑Ϫ13͒

4.5͑Ϫ13͒

6.5͑Ϫ13͒

1.8͑Ϫ12͒

3.6͑Ϫ12͒

6.2͑Ϫ12͒

2.3͑Ϫ14͒

3.0͑Ϫ14͒

3.8͑Ϫ14͒

5.6͑Ϫ14͒

8.1͑Ϫ14͒

1.1͑Ϫ13͒

2.1͑Ϫ13͒

3.6͑Ϫ13͒

9.9͑Ϫ13͒

2.1͑Ϫ12͒

3.8͑Ϫ12͒

5.8͑Ϫ14͒

8.4͑Ϫ14͒

1.5͑Ϫ13͒

2.5͑Ϫ13͒

3.7͑Ϫ12͒

7.1͑Ϫ13͒

6.6͑Ϫ13͒

1.4͑Ϫ12͒

2.6͑Ϫ12͒

4.2͑Ϫ12͒

4.8͑Ϫ14͒

8.6͑Ϫ14͒

1.2͑Ϫ13͒

6.6͑Ϫ14͒

1.6͑Ϫ13͒

2.8͑Ϫ13͒

a

˜

Roberto-Neto, Coitino, and Truhlar ͑Ref. 31͒. CVT/␮OMT calculations based on a dual-level direct dynamic

calculation.

^bCVT/LCT results obtained using the earlier asymmetric PES from Ref. 30.

^c3.8͑Ϫ14͒ stands for 3.8ϫ10^Ϫ14, in cm³molecule^Ϫ1s^Ϫ1

.

tunneling effects are introduced in additional factors. This

factorization can be useful for understanding the chemical

nature of the KIEs. Thus, we write^37,76

The CH₄/CD₄KIEs are listed in Table XI in the same

temperature range. The theoretical KIEs are in good agree-

ment with the available experimental data. Note that the new

surface corrects a major deﬁciency of our previous surface,³⁰

which was the poor description of the deuterium KIEs. For

example, at 300 K, the KIE was 41.8 with our previous sur-

face, but now is 14.0, i.e., a factor of 3 lower, in closer

agreement with the experimental values ͑ϳ11–18͒. To ex-

plain this reduction, we performed another factor analysis.

This is also given in Table XI. We ﬁnd that the reduction is

k

¹²CH₄͒

¹³CH₄͒

͑

Þ

KIEϭ

where ␩

ϭ␩ ␩ ␩

␩

␩ ␩ ,

tun var CUS

͑10͒

trans

rot vib

is the ratio of the relative translational partition

trans

Þ

functions, ␩_rotis from rotational partition functions as evalu-

ated at the saddle point, ␩ is from vibration ͑including

Þ

vib

zero-point energy͒ at the saddle point, ␩

is the ratio

tun

due, mainly, to variational effects, ␩_varϭKIE^CVT/KIE^TST

,

of tunneling factors

␬

^{␮ OMT}(¹²CH₄)/␬^␮OMT(¹³CH ) , ␩

͓ ͔

4 var

ϭKIE^CVT/KIE^TST, i.e., the ratio of KIEs calculated using

i.e., the ratio of KIEs calculated using variational and con-

ventional transition state theory, which at 300 K decreases

from 0.91 ͑Ref. 30͒ to 0.52 for the new surface. Table XI

also allows us to explain the origin of the experimentally

variational and conventional transition state theories, and

␩

_CUSϭKIE^CUS/KIE^CVT, i.e., the ratio of KIEs calculated us-

ing the CVT and CUS methods. Table X indicates that for

temperatures where experimental values are available, the

single most important factor in the KIEs is the tunneling

factor.

TABLE X. ¹²C/¹³C kinetic isotope effects for the forward reaction and

factor analysis.

KIE

SPES factors^a

TABLE IX. Activation energies ͑kcal/mol͒.

Þ

vib

d

T͑K͒

SPES^b

RCT^c

Expt.

␩

tun

var

CUS

Experiment

200

250

300

400

500

600

800

1000

1500

2000

2500

1.216

1.173

1.141

1.099

1.074

1.060

1.043

1.051

1.038

1.034

1.031

1.094

1.070

1.057

1.042

1.034

1.030

1.025

1.022

1.020

1.019

1.02

1.01

1.00

1.03

1.02

1.01

1.00

0.99

1.01

1.13

1.10

1.08

1.06

1.04

1.03

1.02

1.01

1.00

1.069^e

T͑K͒

CVT/␮OMT

CUS/␮OMT

Ref. 4

Ref. 11

1.066^e,f

Forward reaction

200–300

300–500

500–600

2.38

3.38

4.46

5.67

5.74

6.59

8.36

2.29

3.14

4.04

5.01

7.19

7.97

9.55

2.6Ϯ0.4

3.5Ϯ0.5

3.1Ϯ0.3

600–967

968–1000

1000–1500

1500–2000

Þ

a

Reverse reaction

␩

_transand ␩_rotare independent of temperature, and their values are 1.06 and

200–300

300–500

500–600

0.96

1.61

2.38

3.47

3.58

4.65

7.12

0.87

1.37

1.96

2.80

5.02

6.03

8.32

0.96, respectively.

1.4^a

^bThis work, CUS/␮OMT results for SPES analytic surface.

^cThis work: CVT/␮OMT using the AM1-SRP4͓MP2͔-IC direct dynamics

˜

600–967

level of Roberto-Neto, Coitino, and Truhlar ͑Ref. 31͒.

968–1000

1000–1500

1500–2000

^dThis factor includes the CAG transmission coefﬁcient ͑see Ref. 62͒ as well

as the Boltzmann factor of the potential energy difference at the two iso-

topic variational transition states.

^eSaueressig et al. ͓Ref. 16͑a͔͒.

^a296–495 K.

^fCrowley et al. ͓Ref. 16͑b͔͒.

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

ClϩCH₄reaction

TABLE XI. CH₄/CD₄kinetic isotope effects for the forward reaction and

factor analysis.

TABLE XIII. CH₄/CH₂D₂kinetic isotope effects for the forward reaction.

SPES^a

KIE

SPES factors^a

T͑K͒

CVT/␮OMT

CUS/␮OMT

Expt.^b

Þ

vib

T͑K͒ SPES^bRCT^cAPES^d

Expt.

e

var

␩

tun

CUS

200

250

298

300

400

500

600

800

1000

1500

2000

2500

4.61

3.63

3.01

2.98

2.24

1.85

1.63

1.38

0.95

1.23

1.17

1.14

4.56

3.56

2.93

2.91

2.16

1.76

1.53

1.29

0.87

1.19

1.12

1.10

200

250

295

298

300

304

400

500

600

87.7

29.3

14.9

14.3

14.0

13.3

5.6

3.3

2.4

1.6

0.8

18.9

11.4

8.0

7.9

7.8

7.6

4.6

3.3

2.6

1.9

1.6

1.3

1.2

267.8

92.0

22.4

10.3

6.4

6.2

6.1

0.37 0.96 5.20

0.45 0.94 3.15

0.51 0.92 2.35

0.52 0.92 2.30

0.52 0.92 2.28

0.53 0.92 2.23

0.63 0.89 1.60

1.43

44.2 16.4^f

42.3 18.5,^g12.2^h

41.8

38.9 10.9ⁱ

5.9

3.1

13.4

7.2

4.7

2.9

2.3

1.8

1.6

1.5

5.2ⁱ

1.95 0.69 0.86 1.35

1.48 0.74 0.85 1.23

1.05 0.78 0.85 1.12

0.87 0.44 0.89 1.08

0.67 1.00 0.92 1.03

0.64 0.99 0.92 1.02

0.60 0.98 0.92 1.01

800

1000

1500

2000

2500

1.4

1.3

1.2

^aThis work.

^bMatsumi et al. ͑Ref. 19͒.

Þ

a

␩

_transand ␩_rotare independent of temperature, and their values are 1.25 and

1.69, respectively.

Cl ϩ CH₃D→DCl ϩ CH₃,

Cl ϩ CH₃D→HCl ϩ CH₂D,

^bThis work, CUS/␮OMT results for the SPES analytic surface.

^cThis work, CVT/␮OMT using the AM1-SRP4͓MP2͔-IC direct dynamics

˜

method of Roberto-Neto, Coitino, and Truhlar ͑Ref 31͒.

^dCVT/LCT results obtained using the earlier asymmetric PES from Ref. 30.

^eThis factor includes the CAG transmission coefﬁcient ͑see Ref. 62͒ as well

as the Boltzmann factor of the potential energy difference at the two iso-

topic variational transition states.

the ﬁrst with symmetry factors 1 and 2, for the forward and

reverse reactions, respectively, and the second with symme-

try factors of 3 and 2, respectively. The total rate coefﬁcient

corresponds to the sum of these two partial reactions. Table

XII shows that our results agree better with the results ob-

tained by Roberto-Neto et al.³¹for their direct dynamics im-

plicit surface at high temperature than at low temperature.

Our values are larger than the experimental measurements.

Finally, we calculated the CH₄/CH₂D₂KIEs ͑Table

XIII͒, which consists of two reactions,

^fWallington and Hurley ͑Ref. 13͒.

^gBoone et al. ͑Ref. 22͒. This value is based on k(CH₄)ϭ1.0

ϫ10^Ϫ13cm³molecule^Ϫ1s^Ϫ1

.

^hMatsumi et al. ͑Ref. 19͒, at ‘‘room temperature.’’

ⁱChiltz et al. ͑Ref. 1͒.

measured KIEs; in particular we ﬁnd that the differences in

the vibrational frequencies introduced by the isotopic substi-

tution.

We also have calculated the CH₄/CH₃D KIEs ͑Table

XII͒. For the ClϩCH₃D, two reactions must be considered,

Cl ϩ CH₂D₂→DCl ϩ CH₂D,

Cl ϩ CH₂D₂→HCl ϩ CHD₂,

both with symmetry factors of 2 for the forward and reverse

reactions. The agreement with the only experimental

measurement¹⁹is poor, with our CUS/␮OMT value being

almost twice the experimental value.

TABLE XII. CH₄/CH₃D kinetic isotope effects for the forward reaction.

SPES^a

The nonsmooth behavior of the KIEs near 1000 K is due

to the transition state switch discussed in Sec. III B ͑which

occurs at 967 K for the perprotio case͒. Even without con-

sidering temperatures close to this switch, the KIEs from the

various theoretical treatments differ by disturbingly large

amounts. Clearly the KIEs are very sensitive to the potential

energy surface and the assumptions of the dynamical treat-

ment.

T͑K͒

CVT/␮OMT

CUS/␮OMT

RCT^b

Expt.

200

250

295

296

298

300

400

500

600

2.63

2.22

1.97

1.96

1.95

1.94

1.61

1.43

1.32

1.20

0.99

1.12

1.09

1.07

2.61

2.20

1.94

1.93

1.92

1.58

1.40

1.29

1.16

0.95

1.09

1.06

1.05

1.49

1.41

1.36

1.35

1.27

1.22

1.18

1.12

1.10

1.05

1.03

1.02

1.58^c

1.36^d

1.51^c

1.54^e

B. Vibrational-state selected rate coefﬁcients

800

The effect of umbrella motions on reactions of the form

XϩCH₄→XHϩCH₃͓where XϭO(³P) or Cl͔ has recently

been studied by approximate quantum mechanical scattering

calculations.^77,78For example, it was found that the reverse

reaction, CH₃ϩHCl→CH₄ϩCl, is impeded by excitation of

the CH₃umbrella mode, in agreement with the state-selected

variational transition state theory results by Duncan and

Truong.²⁷However, Duncan and Truong also predicted that

the umbrella mode enhances the forward reaction, and

stretch excitation enhances the reaction in both directions.

1000

1500

2000

2500

^aThis work.

^bThis work, CVT/␮OMT using the AM1-SRP4͓MP2͔-IC direct dynamics

˜

method of Roberto-Neto, Coitino, and Truhlar ͑Ref. 31͒.

^cSaueressig et al. ͓Ref. 16͑c͔͒.

^dWallington and Hurley ͑Ref. 13͒.

^eBoone et al. ͑Ref. 22͒. This KIE is based on k(CH₄)ϭ1.0ϫ10^Ϫ13cm³

molecule^Ϫ1s^Ϫ1

.

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

Corchado, Truhlar, and Espinosa-Garcıa

Nyman et al.⁷⁹predicted considerably larger rate coefﬁcients

for stretch-excited CH₄than for ground-state CH₄.

The above analysis of the B_mFcoupling terms showed

that excitation of the reactants ͑C–H symmetric stretching

and the lowest frequency mode in CH₄) may enhance the

forward reaction rates. In a similar fashion, it is possible that

the excitation of some vibrational modes of the products ͑for

example, the Cl–H stretching and CH₃umbrella modes͒ en-

hances the reverse reaction. In the present section we will

present VTST calculations ͑including multidimensional tun-

neling contributions͒ on vibrational state-selected rate coef-

ﬁcients in order to shed more light on this subject.

We will start analyzing the vibrationally adiabatic poten-

tial energy curves of the ground state and two excited states

͓Figs. 5͑a͒ and 5͑b͔͒. In particular we consider the reactive

mode and the CH₃umbrella mode. We use the convention of

numbering the modes in order of decreasing frequency, and

we correlate the modes adiabatically. Thus, in this section,

the reactive mode is the one that starts as mϭ4, becomes

mϭ6 near the saddle point, and ends as mϭ4 ͑see Fig. 2͒.

Thus it transforms from a CH₄symmetric stretch to an HCl

stretch. The CH₃umbrella mode starts as mϭ7, becomes

mϭ9 near the saddle point, and then becomes mϭ7 again.

To be sure that there is no misunderstanding, we note explic-

itly that in spectroscopic notation ͑where one gives the low-

est numbers to totally symmetric modes͒, the ‘‘reactive

mode’’ transforms from ␯ of CH₄to ␯ of HCl, and the

2

‘‘umbrella mode’’ transforms from ␯ of CH₄to ␯ of CH₃.

3

The excitation of the reactive mode by one quantum lowers

the vibrationally adiabatic barrier to 2.17 kcal/mol and the

reverse one to 0.94 kcal/mol as compared to 3.78 kcal/mol^Ϫ1

and 2.89 kcal/mol for the forward and reverse ground-state

barriers. Thus, vibrational excitation of this mode is expected

to enhance both the forward and reverse reactions. This ex-

citation shifts the maximum of the vibrationally adiabatic

barrier from sϭϩ0.29 a₀for the ground state to s

ϭϩ0.96 a₀for the excited-state. Thus the variational transi-

tion state is in the exit channel. The local maximum of the

vibrationally adiabatic potential curve in the entrance chan-

nel is called a supernumerary transition state,⁸⁰and its analog

for the ClϩH₂reaction has recently been discussed in

detail.⁸¹

Excitation of the CH₃umbrella mode by one quantum

lowers the vibrationally adiabatic barrier height with respect

to reactants from 3.78 kcal/mol ͑at sϭ0.25 a₀) to 2.52 kcal/

mol ͑at sϭ0.29 a₀), but it raises the vibrationally adiabatic

barrier height with respect to products to 3.80 kcal/mol from

2.89 kcal/mol. These results agree well with the theoretical

estimation of Duncan and Truong²⁷for the reverse reaction.

If this mode were to remain adiabatic from each asymptote

to the respective dynamical bottleneck, we would expect that

the excitation of this mode in methyl accelerates the forward

reaction but retards the reverse reaction.

FIG. 5. Vibrationally adiabatic ground-state potential energy curve ͑dotted

line͒, vibrationally adiabatic curve for exciting by one quantum the mode

that transforms into the ClH stretch mode ͑mode mϭ6 at sϭ0 a₀; solid line͒,

and vibrationally adiabatic curve for exciting by one quantum the mode that

transforms adiabatically into the CH₃umbrella mode ͑mode mϭ9 at

sϭ0 a₀; dashed line͒, each as a function of reaction coordinate s. Each

curve in ͑a͒ is relative to the value for that state at reactants, while in ͑b͒

they are relative to the value for that state at products.

state. This situation does not allow a more quantitative

analysis for the forward reaction with a fully adiabatic ap-

proximation; instead, we will use the PRP ͑Ref. 60͒ adiabatic

approximation.

However, comparison of the B_mmCoriolis coupling

Ј

terms to the state-selected vibrationally adiabatic potential

curves indicates that the energy can ﬂow between modes in

the entrance valley and, therefore, the symmetric stretch and

the umbrella modes do not preserve their character along the

reaction path all the way from reactants to the transition

The value of s_ϩfor the forward reaction is taken as s

ϭϪ0.53 a₀,while for the reverse reaction it is taken as s_ϩ

ϭ0.02 a₀; these are the values where the curvature of the

reaction path shows its ﬁrst peak in each direction ͑see

Sec. III A͒.

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J. Chem. Phys., Vol. 112, No. 21, 1 June 2000

ClϩCH₄reaction

TABLE XIV. Ratio between excited-state and thermal rate coefﬁcients for

the reactive mode excitation in the forward and reverse reactions.

TABLE XV. Ratio between excited-state and thermal rate coefﬁcients for

the excitation of the mϭ9 mode for the forward and reverse reactions.

CH₄ϩCl→CH₃ϩHCl

CH₃ϩHCl→CH₄ϩCl

T͑K͒

CVT

CVT/SCT

Ref. 79

CVT

CVT/SCT

T͑K͒

CVT

CVT/SCT

CVT

CVT/SCT

200

250

300

400

600

406.0

119.0

52.0

17.2

5.2

489.0

138.0

58.0

18.9

5.6

2.8

3.4

2.2

1.75

1.52

43.0

20.0

12.0

6.0

2.8

1.7

180.0

73.0

40.0

19.4

9.3

6.2

4.8

4.6

4.3

97.0

48.0

30.0

15.6

7.6

5.0

6.6

4.8

4.0

200

250

300

400

600

56.8

24.6

13.8

6.37

2.65

1.58

1.17

1.08

1.03

1.01

235.0

68.5

29.9

10.4

3.46

1.89

2.36

1.69

1.44

1.31

0.10

0.16

0.22

0.34

0.54

0.69

0.84

1.61

2.31

2.90

0.16

0.21

0.27

0.38

0.57

0.72

1.52

2.36

3.06

3.63

800

2.7

800

1000

1500

2000

2500

1.79

1.44

1.28

1.19

1000

1500

2000

2500

4.0

3.5

We calculated the PRP rate coefﬁcients for the reaction

with the reactive mode excited for both the forward and re-

verse reaction. The ratios of these results to the ground-state

rate coefﬁcients are given in Table XIV. Each ratio is ob-

tained from rate coefﬁcients calculated at the same level of

theory, i.e., CVT or CVT/SCT. These ratios measure the

enhancement of the reaction rate when the reactive C–H

stretching mode is excited for the forward reaction and the

enhancement when the Cl–H stretching mode is excited in

the reverse reaction. Unfortunately, a theory for performing

this kind of calculation with the LCT tunneling approxima-

tion has not been developed yet, and therefore we will carry

out our analysis based on CVT and CVT/SCT rate coefﬁ-

cients. Although LCT tunneling dominates in the ground-

state calculations, the analysis of the CVT and CVT/SCT

results will be valuable for carrying out at least a qualitative

analysis.

ences are due to the differences in the dynamical methods,

and not to the reduced dimensionality.

A similar set of ratios was calculated for the case of

excitation of the umbrella mode, and they are listed in Table

XV. In this case, the vibrational excitation of the umbrella

mode in CH₄causes an acceleration of the forward reaction

rate that ranges between 235 and 1.3 at 200 K and 2500 K,

respectively. The reverse reaction, however, has a different

behavior. Thus, for temperatures below 800 K, the thermal

reverse rate coefﬁcient is larger when either the umbrella

motion of CH₃or the stretch of HCl is excited. Excitation of

the CH₃umbrella mode reduces the reverse rate coefﬁcient

by as much as a factor of 6.2 at 200 K. However, for tem-

peratures above 800 K, excitation of the umbrella vibrational

mode speeds up the reverse reaction, and the factor is as

large as 3.6 ͑at 2500 K͒.

It is also interesting to note that tunneling has more ef-

fect on the rate coefﬁcient ratios for the umbrella-mode case

than for the stretch-mode case.

Once again, the PRP bottlenecks were found to be in the

reaction-path regions where the vibrational mode is in its

excited state.

The ﬁrst conclusion we can draw from these results is

that excitation of the reactive mode increases the rate coef-

ﬁcient signiﬁcantly for both the forward and reverse reac-

tions. It is encouraging that the CVT and CVT/SCT ratios

are reasonably close ͑within a factor of 2͒ for the entire tem-

perature range.

According to the CVT/SCT values, the forward rates are

enhanced by a factor ranging from 489 at 200 K to 58 at 300

K and to 1.52 at 2500 K. The reverse rates show a lower

enhancement at temperatures below 600 K ͑97 at 200 K and

30 at 300 K͒, but the ratio between excited-state and thermal

rates is larger than for the forward reaction for temperatures

above 600 K ͑3.5 at 2500 K͒.

It is interesting to point out that the dynamical bottle-

neck was always located before the s_ϩvalue for the forward

reaction, and after it for the reverse reaction ͑i.e., the dy-

namical bottleneck occurs in the region where nϭ1͒.

The enhancements we calculated for the forward reac-

tion are compared to those of Nyman et al.⁷⁹in Table XIV.

Nyman et al. employed reduced-dimensionality scattering

calculations, treating CH₄as a pseudo-atom. We might ex-

pect that this reduced-dimensionality method is less accurate

than our calculations because they do not take account of all

the degrees of freedom. Duncan and Truong also obtained

larger enhancements than the reduced-dimensionality

calculations.²⁷However, one cannot rule out that the differ-

C. Comparison with the earlier asymmetric PES

The largest differences between the earlier PES

30

͑asymmetric͒ and this new PES ͑symmetric͒ are due to the

different calibration criteria used in the two cases. While in

the earlier PES the criterion was to reproduce the properties

of experimental measurements and electronic structure cal-

culations of the stationary points ͑geometry, frequency and

changes of energy, reaction and activation͒, in this new PES

the criterion is to reproduce the experimental rate coefﬁcients

and kinetic isotope effects. The most important effect of the

calibration is that the shape of the adiabatic curve is greatly

modiﬁed. We note especially that it has a wider barrier,

which is correlated with the lowering of the absolute value of

the imaginary frequency, from 1196i to 760i cm^Ϫ1, at the

saddle point. This behavior leads a reduction of the

CH₄/CD₄KIEs from 41.8 to 14.0 at 300 K, and the latter

value is in good agreement with the experimental values.

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´

9388

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Corchado, Truhlar, and Espinosa-Garcıa

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V. CONCLUSIONS

In this work we have constructed a new potential energy

surface for the ClϩCH₄→HClϩCH₃reaction based on the

symmetric analytical potential energy function for the

HϩCH₄→H₂ϩCH₃reaction. The new potential energy sur-

face is semiempirical and is based on using the experimental

rate coefﬁcients and kinetic isotope effects as calibration cri-

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‘‘late’’ transition state ͑productlike transition state͒.

The forward and reverse rate coefﬁcients were calcu-

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CVT/␮OMT method. With this new reparametrization, the

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25

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ACKNOWLEDGMENTS

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vestigacion Cientıﬁca y Tecnica del Ministerio de Educacion

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Potential energy surface, thermal, and state-selected rate coefficients, and kinetic isotope effects for Cl+CH4→HCl+CH3

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