Quantum Dynamics of the Abstraction Reaction of H withCyclopropaneXiao Shan* and David C. Clary*

Physical and Theoretical Chemistry Laboratory, Department of Chemistry, University of Oxford, South Parks Road,Oxford OX1 3QZ, U.K.

ABSTRACT: The dynamics of the abstraction reaction of H atoms with the cyclopropanemolecule is studied using quantum mechanical scattering theory. The quantum scatteringcalculations are performed in hyperspherical coordinates with a two-dimensional (2D)potential energy surface. The ab initio energy calculations are carried out with CCSD(T)-F12a/cc-pVTZ-F12 level of theory with the geometry and frequency calculations at theMP2/cc-pVTZ level. The contribution to the potential energy surface from the spectatormodes is included as the projected zero-point energy correction to the ab initio energy. The2D surface is fitted with a 29-parameter double Morse potential. An R-matrix propagationscheme is carried out to solve the close-coupled equations. The adiabatic energy barrierand reaction enthalpy are compared with high level computational calculations as well asexperimental data. The calculated reaction rate constants shows very good agreementwhen compared with the experimental data, especially at lower temperature highlightingthe importance of quantum tunnelling. The reaction probabilities are also presented anddiscussed. The special features of performing quantum dynamics calculation on the chemical reaction of a cyclic molecule arediscussed.

1. INTRODUCTIONIn a reduced-dimensionality (RD) quantum scattering calcu-lation,1−6 only a subset of the internal degrees of freedom(DoFs) are treated explicitly. Such methods provide oppor-tunities for dynamical studies of the chemical reactions ofpolyatomic (typically over 6 atoms) systems, and some of theearly works can be found in refs 7−12. In the past decade, ourgroup has applied the RD method to various chemical reactionsinvolving H atom abstractions and exchange processes. Thesereactions include H + CH4 → H2 + CH3,

13−15 H + C2H6 →H2 + C2H5,

16−18 H + CH3OH → H2 + CH2OH/CH3O,17

H + C3H8 → H2 + n-C3H7/i-C3H7,17,19 H + CH3NH2 → H2 +

CH2NH2/CH3NH,20 Cl + CH4/CHD3 ↔ HCl + CH3/

CD3,21−23 CH3 + CH4 → CH4 + CH3,

24 H + C4H10 → H2 +n-C4H9/i-C4H9,

25 and H + HCF3 ↔ H2 + CF3.26 In our RD

calculations, normally two internal DoFs are treated explicitlyin the quantum dynamics calculations and potential surface:the chemical bonds that are formed and broken in a reaction.The contribution of the rest of the DoFs, the spectator modes,to the reaction dynamics is accounted as zero-point energies(ZPEs) in the construction of the two-dimensional (2D) potentialenergy surface (PES). In this case, the 2D PES has the correctbarrier height and energetics of reactants and products. Inaddition, other modes can be included in the quantum dynamicscalculations if of particular interest.18

In most of the earlier works, the 2D PESs were fitted with2D potential functions, in particular a 25-parameter double-Morse function16 and two 29-parameter double-Morsefunctions.13−15,17−24 The two 29-parameter functions differ bythe function used to define the position of the transition stateof a reaction on the surface.24 The latest improvement to our

method focuses on the construction of the 2D PES, inparticular reducing the number of necessary ab initio quantumchemistry calculations. The so-called (1 + 1)D methods utilizesthe minimum energy path (MEP) of a reaction and approxi-mate the rest of the PES of such reaction with a harmonicfunction27 or a one-dimensional Morse function.25,26 The approachwe use has some similarities to other methods, such as the reactionpath Hamiltonian28 and reaction surface Hamiltonian.29

All of the RD methods are largely dependent on the accuracyof the PES for a reaction, which in turn depends on the abinitio quantum chemistry calculations. In principle, the highestlevel of theory with complete basis set would yield the mostaccurate results. However, it would be computationally tooexpensive to use such a method to construct even a full PES fora chemical reaction such as the title reaction

+ ‐ → + ‐H cyc C H H cyc C H3 6 2 3 5 (R1)

Von Horsten et al.27 tested a series of ab initio methods forthe single energy calculations of each grid point on the PES forthe H abstraction reactions from noncyclic alkane molecules.We follow this work and make comparisons of two commonlyused high level methods in the present study to further testtheir performances in different systems.The H abstraction from cyclopropane (cyc-C3H6) is of parti-

cular interest because of the unusually strong C−H bond30−32

and because of its relevance in combustion processes.33

Received: August 28, 2014Revised: September 30, 2014Published: October 1, 2014

Article

pubs.acs.org/JPCA

© 2014 American Chemical Society 10134 dx.doi.org/10.1021/jp5087174 | J. Phys. Chem. A 2014, 118, 10134−10143

An earlier study shows the bond dissociation energy, being445 kJ/mol,34 which is similar to the C−H bond strength inmethane. In addition, the product of the abstraction reaction,cyclopropyl (cyc-C3H5), can undergo a ring opening isomer-ization to form the thermodynamically more stable allyl radical(CH2CHCH2).

35 This procedure is often used by experimen-talists36−40 to produce the allyl radical. In such experi-ments, cyc-C3H5 are first generated by the H abstractionreaction from cyc-C3H6 using a radical or atoms. Recently, theH + cyc-C3H6 → H2 + C3H5 reaction has been studied experi-mentally41,42 and theoretically43 to investigate the mechanismbehind the production of rovibrationally hot H2 molecule in thereaction. Two coexisting reaction mechanisms were found:43

R1 and H-addition/ring opening.In addition to the recent theoretical studies on the energetics

of R1 in ref 43, the reaction rate constants of R1 have beenanalyzed using methods that are based on conventional transi-tion state theory (TST).44,45 The present work performs RDquantum calculations on R1 to investigate its kinetics anddynamics in more detail. In particular the cumulative andproduct-state- and reactant-state-dependent reaction probabil-ities are reported. In addition, the contribution of the quantumtunnelling effect to the overall reaction rate constant is analyzedand compared with experiment. This is the first time to ourknowledge that a quantum scattering calculation has beenapplied to a chemical reaction involving a cyclic molecule, whichalso introduces some new features such as the treatment of thecenter-of-mass (CoM) of the cyc-C3H5 fragment.The remainder of the article is organized as follows. In

section 2, we discuss the 2D PES of R1, in particular the abinitio calculations for the stationary points on the surface andthe fitting to the ab initio data with a 29-parameter double-Morse potential function. The theoretical background includingkey steps of the R-matrix propagation method46 to solve the2D time-independent nuclear motion Schrodinger equationand the calculations of the reaction rate constants in quantumtheory and transition state theory are in section 3. The resultsof our scattering calculations are presented in section 4 withdiscussions. Our main conclusions are in section 5.

2. AB INITIO POTENTIAL ENERGY SURFACEThis section is divided into two parts. In the first, we discuss theenergies of the stationary points on the PES of R1. The fittingof the surface is reported in the second part. All the ab initiocalculations were carried out using MOLPRO package.47 Forthe geometry optimizations on a grid of fixed lengths for thebonds being broken and formed and frequency calculations,we used the second order Møller−Plesset perturbation theory(MP2) with a correlation consistent polarized valence triple-ζDunning basis set48 (cc-pVTZ). A single-point energy calcula-tion was carried out, for each optimized stationary structure, atcoupled cluster level including single, double, and perturbativetriple excitations [CCSD(T)] with the augmented cc-pVTZbasis set (aug-cc-pVTZ). We also employed the explicitlycorrelated coupled cluster method, CCSD(T)-F12a/cc-pVTZ-F12 method,49,50 for the single point energy calculations inthe current study, where cc-pVTZ-F12 stands for the triple-ζcorrelation consistent F12 MOLPRO basis set of Peterson et al.51

For simplicity reasons, we shall for the rest of this article use“CCSD(T)” and “F12a” for the CCSD(T)/aug-cc-pVTZ andCCSD(T)-F12a/cc-pVTZ-F12 methods, respectively.2.1. Optimized Geometry and Reaction Energetics.

We show respectively in Figure 1a−c the molecular geometries

of the transition state (TS), reactants, and products of the R1.The important bond lengths and angles are highlighted inTable 1. It can be seen that our optimized geometry at MP2level is in a close agreement with Yu et al.43 at CCSD/cc-pVDZlevel of theory. In particular when comparing to Yu et al.,43 forthe TS structure, our method only underestimates the bondlength of H−Ha and Ha−C by ∼3.7% and ∼0.070%, respectively,and the bond angle H−Ha−C by ∼0.95%. Note that in this workwe refer to the abstracted H atom as Ha for clarity.In Figure 1d, we illustrate the van der Waals (vdW) complex

that is formed by the reacting particles, where the H atom isdirectly above the center of the C3 ring of cyc-C3H6. On the 2DPES, it can be found in the reactant channel and is a dominantfeature. The key geometry coordinates of this complex are alsoshown in Table 1. The distance between H and the center of theC3 ring is 3.6204 Å. The equivalent H−Ha distance is 3.0731 Å.This structure is a minimum on the 2D PES. When the bondlengths H−Ha and C−Ha are frozen at 3.0731 and 1.0785 Å,respectively, the vdW complex structure would be obtained evenif one starts the partial optimization from a collinear geometryfor the three atoms. On the 2D PES, the area surrounding thispoint shows similar bending behavior of the three atoms. At agreater H−Ha distance, the vdW interaction is not as strong, andhence this structure is no longer favored energetically. A nearcollinear geometry would be obtained from a partial optimizationstarting from a collinear geometry. These results directly affectedour choice of coordinate system to run the scattering calculationwith. We shall discuss this point in detail in the next section.Table 2 shows the reaction energetics. Once again we compare

our results to ref 43. where the energy is obtained using anextrapolated coupled-cluster/complete basis set (CBS) methodwith their CCSD/cc-pVDZ geometry, and the extrapolationwas based on the cc-pVDZ, cc-pVTZ, and cc-pVQZ basis setsand CCSD and MP2 level of theory. Our energy calculationsfor the stationary points were done, as mentioned before, atboth CCSD(T) and F12a levels of theory. It can be seen thatboth methods (13.61 and 13.47 kcal/mol for the CCSD(T)and the F12a methods, respectively) show good agreement inestimating the reaction barrier height comparing to the CBS data(13.03 kcal/mol).Both methods show that the reaction is endothermic, which

is also calculated by Yu et al.43 Our results for the endothermi-city are again in good agreement with the CBS data. However,both our results and the CBS data somewhat overestimate thereaction enthalpy when comparing to the experimental value52−56

of 2.06 kcal/mol. The barrier height for the R1 is much largerthan that for the H abstraction reaction from noncyclic C3H8by a H atom from either the primary or the secondary C atoms(10.56 and 7.76 kcal/mol, respectively).27 Also for C3H8, the Habstractions by H atom are both exothermic reactions. In fact thereaction energetics here is more comparable to the benchmarkH + CH4 → H2 + CH3 reaction,

13−15,27 for which the barrierand enthalpy are 14.2 and 0.6 kcal/mol. The vdW well depth forR1 is predicted to be 0.21 and 0.26 kcal/mol by the CCSD(T)and F12a methods, respectively. Note that they both overestimatethe binding energy when comparing with the CBS result of0.06 kcal/mol. This is mainly due to the ZPE contribution, whichdepends on the frequency calculation done at MP2 level oftheory. In comparison, the F12a method shows better perform-ance than the CCSD(T) method in predicting the adiabaticreaction barrier and the reaction enthalpy. In addition, it is alsoapproximately 2−3 times faster than the CCSD(T) method.

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The single-point energy calculations for the grid points on thePES in this study were therefore done at the F12a level of theory.We report the harmonic frequencies of the TS and cyc-C3H6

in Table 3. This table also includes the vibrational frequenciesof the TS after the rectilinear projection.57 We used therectilinear projection method to project out the contributionof the two active vibrational modes, the remaining vibrationalfrequencies were then used in the calculation of the ZPEs,which we add to our single-point calculation for the grid pointsand used in fitting the 2D PES. Note that the ZPEs used tocalculate energies in Table 2 are calculated with the frequencies

before projection, and their values are also reported in Table 3.It can be seen that the projection procedure has significant impacton the frequencies of only two vibrational modes, in particular,from 1165.87 and 871.08 cm−1 to 1171.22 and 859.88 cm−1. Bothmodes correspond to the bending of H−Ha−C from collinearconfiguration.

2.2. Potential Energy Surface. In our quantum scatteringcalculation we utilize the hyperspherical coordinates, ρ and δ.They are defined to this reaction as

μρ δ

μρ δ= =

mR

mr[ cos( )] , [ sin( )]1 2 2 2 2 2

(1)

with

=+

+m

m m mm m

2 (3 5 )3 71H C H

C H (2)

Table 1. Key Geometry Coordinates of the Stationary Pointson the 2D PES for the H + cyc-C3H6 → H2 + cyc-C3H5Reactiona

bond lengths andangles

MP2/cc-pVTZ

CCSD/cc-pVDZ43

TS C−Ha 1.4375 1.4385H−Ha 0.85084 0.8832H−Ha−C 174.32 176.0

cyc-C3H6 C−H 1.0785 1.0955cyc-C3H5 C−Hb 1.0758 1.0938vdW Complex H-CoM of C3 ring 3.6204 4.2734

C−Ha 1.0785 1.0953aAll bond lengths are in Å, and the bond angles are in degrees. Notethat the positions of specified H-atoms, Ha and Hb, can be found inFigure 1.

Table 2. Reaction Energetics of the H + cyc-C3H6 → H2 +cyc-C3H5 Reaction

a

MP2/cc-pVTZ

CCSD(T)/aug-cc-pVTZ

CCSD(T)-F12a/cc-pVTZ-F12 ref 43

ΔVa‡ 19.14 13.61 13.47 13.03

ΔrH 9.54 3.27 4.18 4.03ΔEvdW −0.28 −0.21 −0.26 −0.06

aAll the energies are in kcal/mol. The ZPE corrections were includedfor all the computational results. The geometry optimization and thefrequency calculations that lead to the ZPE for our data werecalculated at the MP2/cc-pVTZ level of theory.

Figure 1. Optimized structure of the stationary points on the 2D PES for R1. The H atom being abstracted in the reaction is marked as “Ha” in theTS structure. The remaining H atom on the C atom after the abstraction is marked as “Hb” in the product cyc-C3H5 structure. In the vdW complexstructure, one of the H atoms in CH2 on the same side as the incoming H atom to the C3 ring is marked as “Ha”.

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=mm22

H(3)

=+

+m

m m mm m

(3 5 )3 63

H C H

C H (4)

and

μ = m m m1 2 33 (5)

where, in eq 1, R and r are the Jacobi coordinates. Normally(R, r) are defined as the distance between the centers of mass(CoMs) of the H−Ha and cyc-C3H5 fragments and the bonddistance of H−Ha, respectively.To construct the 2D PES, whether using a fitting functions

or the recent development of the (1 + 1)D method, the startingpoint is the minimum energy path (MEP). In Figure 2a, weshow the MEP of the R1 in hyperspherical coordinatesconverted from the normal Jacobi coordinates. The blackdots, apart from the TS (marked on the graph), are obtaineddirectly from the results of the intrinsic reaction path (IRC)calculation in the MOLPRO program;47 the black curve is thecontinuation of the ab initio points. The TS is marked as thelarge red dot on the graph. Points representing the asymptoticregion, i.e., large ρ values, are included in the graph as blue dots.It can be seen that this reaction has a late TS, and hence, onewould expect that the vibrationally ground state dominates theproduct. We shall discuss this point in more detail in section 4.A more interesting feature of the MEP can be found in

the reactant channel. In particular in the 5.1 ≤ ρ ≤ 5.9 au and0.42 ≤ δ ≤ 0.61 rad region, the ρ-value of the MEP curvedecreases as δ increases. This is in contrast to the normallyexpected behavior of the curve, which is shown on the graphas the dashed black curve. In the last section, we reported anenergetically favored vdW complex structure in the reactantchannel of this reaction. Here we show the position of thecomplex on the MEP as the brown dot in Figure 2a, and its co-ordinate is (ρ, δ) = (5.119 au, 0.6027 rad). In fact for the MEPpoints close to the vdW complex, a noncollinear H−Ha−Cconfiguration is observed in the calculation with the bond anglesuggesting that the incoming H atom tends to sit above the C3ring. The presence of this vdW complex in the reactant channeldenies the usage of (1 + 1)D reduced dimensionality methods.In the (1 + 1)D calculations,25−27 the geometry of the X−H−Y

triatomic fragment in the reacting complex is assumed to becollinear or near collinear along the MEP. For the MEP aroundthe vdW complex in R1, assuming a collinear configuration of

Table 3. Harmonic Frequencies (in cm−1) and ZPEs (in kcal/mol) at MP2/cc-pVTZ Levela

frequencies ZPE

cyc-C3H6 3298.19 3279.63 3279.57 3197.11 3188.28 3188.24 51.731533.89 1484.89 1484.88 1233.93 1221.02 1121.011169.17 1087.00 1056.07 1055.94 909.52 909.42867.94 745.85 745.80

cyc-C3H5 3287.19 3255.54 3249.74 3170.36 3163.35 1513.09 42.911479.63 1283.32 1191.64 1136.66 1110.77 1077.941015.57 957.30 882.27 802.34 795.37 624.21

TS 1342.30 (i,a) 3272.30 3261.80 3257.22 3177.59 3176.99 50.492255.22 (a) 1520.78 1486.41 1267.22 1202.62 1165.871124.50 1121.60 1074.40 1033.94 1023.56 951.98886.45 871.08 815.53 760.92 313.03 274.70

TS (projected) 3272.21 3261.80 3255.40 3177.59 3176.57 1521.56 47.251486.41 1269.36 1202.62 1171.22 1123.81 1121.601074.40 1033.79 1023.56 951.98 886.52 859.88815.53 760.42 313.03 272.73

aThe imaginary frequency at TS is marked with “i” in parentheses. The active modes of TS are marked with “a” in parentheses.

Figure 2. Plots of the MEP of R1 projected onto the hypersphericalcoordinate (ρ, δ)-plane with ρ and δ values converted from the normalJacobi coordinate (a) and the alternative Jacobi coordinate (b). Theblack dots and curves are the ab initio MEP and its continuation,respectively. The blue dots are the ab initio data in the asymptoticregions. The red and brown dots are the position of the TS and thevdW complex, respectively. The dashed black curve (a) is the normallyexpected route of the MEP in (ρ, δ)-plane.

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the X−H−Y fragment or freezing the bond angle at 180° in theab initio calculations would clearly introduce large errors in thePES. Therefore, in order to study the dynamics of the reactionwith RD methods, we have to use a fitting function. Previously,a 29-parameter double-Morse potential function13−15,17−24 hasbeen successfully applied to several H-abstraction reactions byour group, and more detailed discussions of this function can befound in ref 24. It can be written as

ρ δ ρ ρ δ ρ

ρ ρ δ ρ

= − + −

+ − − − −

+

V f g h c

f g h c

c

( , ) ( )({1 exp[ ( ) ( )]} )

( )({1 exp[ ( ) ( )]} )1 1 1

214

2 2 22

28

29 (6)

with

ρ ρ ρ= + −f c c c( ) exp( )c1 1 2 4

3

ρ ρ ρ= + +g c c c( )1 5 6 72

ρ ρ ρ ρ ρ= + + − + +h c c c c c c( ) log( )1 8 9 10 11 12 132

ρ ρ ρ= + −f c c c( ) exp( )c2 15 16 18

17

ρ ρ ρ= + +g c c c( )2 19 20 212

ρ ρ ρ ρ ρ= + + + + +h c c c c c c( ) log( )2 22 23 242

25 26 272

where ci (i = 1, 2, 3, ..., 29) are parameters. In some of theearlier works, this fitting potential function has been appliedto reactions with vdW wells in both reactant and productchannels.13,22 However, it should be noted that in those cases,the vdW structures all had a near collinear or collinearconfiguration for the X−H−Y fragment of a reacting complex.The function is fitted to the ab initio data via a least-squaresprocedure.58 We measure the convergence of the fitting pro-cedure with the sum of the squared residue (RSS).14,21,24

Typically, a RSS value on the order of 10−4 would mean anadequate potential fit.24 However, even with our best fitted set ofparameter values, the RSS obtained is as large as 0.029. Therefore,it is clear that the 29-parameter double-Morse potential functionis not appropriate to fit the ab initio PES data in the hyper-spherical coordinates.In the previous works of our group,16−20 we used an

alternative definition of the Jacobi coordinates, especially forreactions involving alkanes with more than one C atom.16−20

In this alternative system, r is the same as the normal Jacobicoordinates, while R is defined as

= +R rm

r2CH

HHHa a (7)

Effectively, this alternative definition is assuming the CoMof cyc-C3H5 fragment on the C atom, from which the H atomis abstracted in the reaction. This system has been appliedto H-abstraction reactions from noncyclic alkanes up topropane.16−19 We plot in Figure 2b the MEP of the R1 inhyperspherical coordinates converted from the alternativeJacobi coordinates. The black dots are the ab initio data,while the black curve is the continuation of the data. The vdWcoordinate is once again included as the brown dot, with thecoordinates being (ρ, δ) = (5.646 au, 0.5397 rad). It can beseen that the MEP has a normal behavior and that the vdW ispresented along the MEP. In this case, the structure of thepotential energy well in the reactant channel is maintained on

the surface, but we do not face the complication that it is lyingoutside the normally expected route of the MEP as shown inthe last coordinate system. Our double-Morse potential fittingfunction can handle this with ease. The potential yields an RSSvalue of 2.8 × 10−4, and a total of 163 ab initio grid points wereused in the fit. The values of the parameters are reported inTable 4, and a contour plot of the fitted potential function isshown in Figure 3, with the positions of the TS and the vdWcomplex indicated on the plot.

3. SCATTERING THEORYWith the PES constructed as described in Section 2, weproceed to solve the nuclear motion Schrodinger equation. Inthe hyperspherical coordinate system; the Hamiltonian can bewritten as46

μ ρ ρ δ ρ ρρ δ = − ∂

∂+ ∂

∂− −

+

⎪ ⎪

⎪ ⎪⎧⎨⎩

⎫⎬⎭

HJ

V1

21 3

4( , )

2

2 2

2

2 2

2

2

(8)

where J is the total angular momentum operator, and V(ρ,δ)is the potential energy, which is calculated using eq 6 withthe parameter values given in Table 4. For J = 0, we have theHamiltonian

μ ρ ρ δ ρρ δ = − ∂

∂+ ∂

∂− +

⎧⎨⎩⎫⎬⎭H V

12

1 34

( , )0

2

2 2

2

2 2(9)

We then carry out the time-independent scattering simulationusing the R-matrix propagation scheme, which was developed by

Table 4. Values of the Parameters Defining the CCSD(T)-F12a/cc-pVTZ-F12 2D PES

c1 0.1792 c9 0.5050 c17 3.6143 c25 8644.2894c2 −27.3273 c10 −0.1084 c18 1.8439 c26 1991.9187c3 1.4881 c11 −44.6845 c19 3.6370 c27 −58.1164c4 1.9875 c12 26.2979 c20 2.5737 c28 1.1323c5 5.5220 c13 0.1459 c21 −0.0863 c29 −117.6630c6 −0.3538 c14 2.4558 c22 −11.9089c7 0.1360 c15 0.1718 c23 0.0579c8 1.0460 c16 −0.5894 c24 −0.0012

Figure 3. Contour plot of the fitted PES, produced with eq 6, and theparameter values are given in Table 4. The position of the TS and thevdW complex are marked on the plot with arrows.

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Stechel et al.46 We only outline the key steps of the scheme andthe application of the approximate boundary conditions. Moredetailed description of the R-matrix method and its applicationto the J-shifting model can be found in earlier studies.14,15,24,59

The hyperspherical coordinate space is first divided into evenwidth sectors in ρ; for each sector the δ-dependent Hamiltonianis given by

μρ δρ δ = − ∂

∂+δH V

12

( , )i2

2

2 i(10)

We then expand the sector-dependent wave function for thequantum state k as a function of ρ as

∑ρ δ ρ ρ ρ φ δ ρΨ =′

′ ′f( , ; ) ( ; ) ( ; )kk

N

k k ki i i(11)

where ϕk′(δ;ρi) is the δ-dependent wave function obtained bydiagonalizing eq 11 using a discrete variable representation witha particle-in-a-box basis.60 The size of the contracted basis, N ineq 11, is set to be larger than the number of open channels inthe asymptotic region.46 The expression of the wave functionin eq 11 is applied to eq 8, the problem is reduced to solve theso-called close-coupled equations

ρρ ρ ρ ρ ρ+ =f W f

dd

( , ) ( ) ( , ) 02

2 i i i(12)

where W is a diagonal matrix. It is given by

ρ μ ε ρμρ μρ

= − − −+⎛

⎝⎜⎜

⎞⎠⎟⎟W E

J J( ) 2 ( )

38

( 1)2kk ki i

i2

i2

(13)

In J = 0 case, it can be reduced to

ρ μ ε ρμρ

= − −⎛⎝⎜⎜

⎞⎠⎟⎟W E( ) 2 ( )

38kk ki i

i2

(14)

To solve eq 12, the R-matrix is now propagated through allsectors from the classical forbidden region at a small ρ toasymptotic ρ.46,61 In the asymptotic region, we apply theapproximate boundary conditions in hyperspherical coordi-nates.21,62 The scattering matrix (S-matrix) is extracted from theR-matrix, for sector ρi,

ρ ρ ρ ρ ρ ρ ρ= ′ − ′ −−S R O O R I I( ) ( ( ) ( ) ( )) ( ( ) ( ) ( ))i i i i1

i i i(15)

where

λ λ ρ λ λ ρ= − =− −I Oexp( i ) and exp(i )kk k k kk k k1/2

i1/2

i (16)

with

λ ρ ρ= W( ) ( )k kki i (17)

Once the S-matrix is calculated, one can then obtain otherphysical properties. In the present study, we report the state-to-state and cumulative reaction probability with respect to thetotal energy, E. When J = 0, they are given by

= | |→=

→P E S E( ) ( )i fJ

i f0 2

(18)

and

∑= | |=→P E S E( ) ( )J

i fi fcum

0

,

2

(19)

respectively. The subscripts “i” and “f ” denote, respectively, theinitial and final states of the reactants and products. It should benoted that in the ideal case, Pkk′

J = 0(E) should be evaluated in thefinal ρ-sector only. However, since we apply the approximateboundary conditions directly in the hyperspherical coordinates,oscillatory behavior can be observed as a function of sector ρi.

63

To remove the problem, we calculate the S-matrix over anumber (Snum) of sectors in the asymptotic region, and averageover the probability matrix elements to produce Pkk′

J = 0.64,65

The reaction rate constant, kscatt(T), is then calculated usingthe cumulative reaction probability with the J-shifting model.3,66

kscatt(T) is given by

∫π

= −‡

‐

∞=

⎛⎝⎜

⎞⎠⎟k T

Q T

Q T Q TP E

Ek T

E( )( )

2 ( ) ( )( ) exp d

mJ

scatttot

( )

totH

totcyc C H 0

cum0

B3 6

(20)

where QtotH (T) and Qtot

cyc‑C3H6(T) are the total partition functionsof H and cyc-C3H6, while Qtot

‡(m)(T) is the partition function ofthe TS. Note that for the vibrational partition function of theTS only the frequencies of vibrational spectator modes after theprojection procedure are included in the function, and hencehere we have m = 3N − 6 − 2.In this study, in order to analyze the contribution from any

quantum effects, the classical transition state theory (TST) rateconstant is also calculated. It is given by

π=

−Δ‡ ′

‐

‡⎛⎝⎜⎜

⎞⎠⎟⎟k T

Q T

Q T Q Tk T

Vk T

( )( )

2 ( ) ( )exp

m

TSTtot

( )

totH

totcyc C H B

a

B3 6

(21)

where ΔVa‡ is the adiabatic vibrational barrier height; the values

were reported in Table 2. For the TS partition function, all thevibrational modes except the one that corresponds to thetransition state vector are included in the vibrational partitionfunction calculation, and m′ = 3N − 6 − 1. Note that the R1involves multiple H abstraction sites, it is accounted for in botheqs 20 and 21 by the symmetry number in the rotationalpartition function.

4. SCATTERING CALCULATION RESULTS ANDDISCUSSION

In order to test the convergence of the reaction probabilities forR1, we have conducted a series of test calculations varying thescattering parameters. The resulting values of the parametersare reported in Table 5. Note that in this study we follow the

experience of previous works, fix Snum to the final 15% of Nρ. Inthe energy range (0 to 1.5 eV) that we are interested in, thereare only seven open channels. In our calculations, the numberof contracted basis function required is 10.We present in Figure 4 the plots of cumulative reaction

probability for R1 versus total energy as the black curves.

Table 5. Values of the Parameters Used in the ScatteringCalculationsa

parameter final value parameter final value

ρmin 3.2 au Nδ 200ρmax 12 au Emax 1.5 eVNρ 750 Einc 0.001 eVSnum 110 N 10

aDistances are in atomic units. See Ref 24. for a definition of theparameters.

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The product-state- and reactant-state-dependent contributionsare shown in Figure 4a,b, respectively. The adiabatic energybarrier, ΔVa

‡, is ∼0.584 eV. It is marked on both graphs as theblack dashed line. It can be seen that the reaction probabilitycurve rises before this value, and from our calculation, thereaction starts at total energy of ∼0.443 eV. This result indicatesa quantum tunnelling effect is contributing to the reactionmechanism. Another picture of the tunnelling contribution interms of magnitudes at low temperature can be found whencomparing the rate constants of the quantum calculation with theclassical TST results, which will be discussed in more detail laterin this section.In Figure 4a, the red, blue, and brown curves are the

probabilities for the reaction producing the H2 product in theground, first, and second excited vibrational states. It can beseen that the reaction is dominated by the ground state H2,which was predicted from our discussion before on the reactionhaving a late TS. The v = 1 product is not found until the totalenergy is typically above ∼0.85 eV. The v = 2 H2 channel opensat very high energy; its contribution to the overall reaction rateconstant is very small.The red, blue, and brown curves in Figure 4b show,

respectively, the reactant-state-dependent reaction probabilitiesfrom (v = 0), (v = 1), and (v = 2) states. One interesting featureis that the reactants in the first vibrational excited state hasa comparable or even higher (at lower energy) contributionto the ground state reactants. Both states are involved in the

quantum tunnelling effect, their reaction probabilities both risebefore the adiabatic barrier. It should be noted that althoughfrom the figure it seems the (v = 0) and (v = 1) reactant-state-dependent probabilities have the same threshold, our calcula-tion suggests that the two channels open at a total energy of∼0.373 and ∼0.435 eV, respectively. At ∼0.849 eV the reactantchannel associated with the second excited vibrational stateopens.We show our calculation of the rate constants for reaction R1

in Figure 5a,b in the temperature (T) ranges of 200 to 2000 K

and 333 to 1000 K, respectively. On both figures, the solidcurves are our quantum scattering calculation results. The dashedcurves are the classical TST rate constants. It can be seen that thetwo curves converge at high T, over 1000 K. We can also see thatour quantum rate constant curve bends away from the classicalTST curve at low T. At around 500 K, the rate constant of thequantum calculation being 8.342 × 10−16 cm3 molecule−1 s−1

is ∼5 times greater than the classical TST result of 1.604 ×10−16 cm3 molecule−1 s−1; at around 300 K, it becomes∼20 times greater. This clearly shows the strong contributionof the quantum tunnelling effect to the reaction, especially atlower temperature. This tunnelling effect was also observed inthe reaction probability calculation of Figure 4. To confirm such

Figure 4. Plots of the cumulative reaction probability versus totalenergy and the product-state-dependent reaction probabilities (a) andthe reactant-state-dependent reaction probabilities (b).

Figure 5. Comparison of the reaction rate constants of R1 from thepresent study to previous experimental and theoretical studies in thetemperature ranges of 200 to 2000 K (a) and 333 to 1000 K (b).

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effect exists in the reaction, we compare our calculated reactionrate constants to the experimental data.Two experimental studies of R1 were conducted by Marshall

and co-workers at different T ranges: 358 to 550 K30 and 628 to779 K.31 The fitted Arrhenius functions to the experimentaldata were reported, respectively, as

= ± − ±

− −

−

k

RT

log( /cm mol s )

14.21 0.13 (11.7 0.26 kcal mol /2.3 )

3 1 1

1

and

= ± − ±

− −

−

k

RT

log( /cm mol s )

13.6 1.0 (11.6 3.11 kcal mol /2.3 )

3 1 1

1

We show them in Figure 5a,b as the red and blue curves, res-pectively. The experimental error estimates are also presented.Comparing to the results in the higher T range (blue curves),both the quantum and classical TST curves show good agree-ment to the experiment. For the lower T range data, the classicalTST method is clearly underestimating the rate constant. Ourquantum calculation, however, shows again very good agree-ment with the experiment. This provides clear evidence thatquantum tunnelling is contributing to the overall reaction. Therate constant of a previous theoretical study44 is also includedin Figure 5a,b as the brown curves. This study was based onthe TST framework, and the values of the parameters in theArrhenius equation were chosen to fit the lower T experimentaldata. As a result of the fitting nature, on the graphs it has betteragreement to the red curves than our quantum calculationresults, but worse agreement to the blue curves. To have clearercomparison with the experiments, the values of the reaction rateconstants from our quantum and classical TST calculations aswell as the experimental data are presented in Table 6. We cansee that in both low and high T cases, the quantum results are invery good agreement to the experiments.

5. CONCLUSIONSWe applied a reduced dimensionality quantum scatteringcalculation on the chemical reaction of H abstraction by anH atom from the cyclopropane molecule. The latest availableab initio quantum chemistry methods were used to constructthe potential energy surface. The cyclic hydrocarbon produced

some special aspects for the calculation and resulting fittingprocedures, which are described in detail.In the quantum scattering calculation, the R-matrix propagation

scheme was applied to the fitted 2D PES. We examined thecumulative reaction probability as well as the product-state- andreactant-state-dependent reaction probabilities. The state-depend-ent probability results confirmed our finding in the ab initiocalculations that the reaction has a late transition state. We alsofound that a quantum tunnelling effect has a significant contribu-tion to the reaction at lower total energies. The reaction rateconstants computed from our quantum scattering calculationand classical TST were compared with experimental data at twodifferent temperature ranges. Our quantum results show verygood agreement to the experiments in both cases, while theclassical TST underestimated the rate constants at lower T due tothe lack of quantum tunnelling.This is the first time that quantum dynamical methods have

been applied to chemical reaction involving cyclic alkanemolecules. The success of the study suggests the approach canbe extended to larger cyclic systems in future studies.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: (X.S.) xiao.shan@chem.ox.ac.uk.*E-mail: (D.C.C.) david.clary@chem.ox.ac.uk.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was supported by the EPSRC Programme Grant No.EP/G00224X/1 and Levenhulme Trust Project Grant No.RPG-2013-321.

■ REFERENCES(1) Bowman, J. M. Reduced Dimensionality Theories of QuantumReactive Scattering. Adv. Chem. Phys. 1985, 61, 115−167.(2) Bowman, J. M. Reduced Dimensionality Theory of QuantumReactive Scattering. J. Phys. Chem. 1991, 95, 4960−4968.(3) Bowman, J. M. Overview of Reduced Dimensionality QuantumApproaches to Reactive Scattering. Theor. Chem. Acc. 2002, 108, 125−133.(4) Clary, D. C. Quantum Theory of Chemical Reaction Dynamics.Science 1998, 279, 1879−1882.(5) Clary, D. C. Quantum Dynamics of Chemical Reactions. Science2008, 321, 789−791.(6) Szichman, H.; Baer, M.; Nakamura, H. Four MathematicalDimensional Quantum Mechanical Studies of Tetra-Atom Systems:State-to-state J = 0 Probabilities for the H2 + OH → H2O + HReaction. J. Chem. Phys. 1997, 107, 3521−3527.(7) Nyman, G.; Clary, D. C. Quantum Scattering Calculations on theCH4 + OH → CH3 + H2O Reaction. J. Chem. Phys. 1994, 101, 5756−5771.(8) Nyman, G.; Clary, D. C.; Levine, R. D. Potential Energy SurfaceEffects on Differential Cross Sections for Polyatomic Reactions. Chem.Phys. 1995, 191, 223−233.(9) Takayanagi, T. Reduced Dimensionality Calculations ofQuantum Reactive Scattering for the H + CH4 → H2 + CH3Reaction. J. Chem. Phys. 1996, 104, 2237−2242.(10) Nyman, G. Quantum Scattering Calculations on the NH3 + OH→ NH2 + H2O Reaction. J. Chem. Phys. 1996, 104, 6154−6167.(11) Garrec, J.-L. L.; Rowe, B. R.; Queffelec, J. L.; Mitchell, J. B. A.;Clary, D. C. Temperature Dependence of the Rate Constant for theCl− + CH3Br Reaction Down to 23 K. J. Chem. Phys. 1997, 107,1021−1024.

Table 6. Comparison of Rate Constants for the H +cyc-C3H6 → H2 + cyc-C3H5 Reaction in cm3 molecule−1 s−1;Powers of 10 Are in Parentheses

temperature (K) kTST kscatt exptl

360 6.517(−19) 8.083(−18) 2.054(−17)a

390 2.912(−18) 2.830(−17) 7.247(−17)a

420 1.059(−17) 8.379(−17) 2.135(−16)a

450 3.267(−17) 2.167(−16) 5.447(−16)a

480 8.812(−17) 5.019(−16) 1.236(−15)a

510 2.128(−16) 1.061(−15) 2.547(−15)a

540 4.684(−16) 2.076(−15) 4.844(−15)a

630 3.267(−15) 1.091(−14) 6.231(−15)b

660 5.593(−15) 1.730(−14) 9.495(−15)b

690 9.169(−15) 2.644(−14) 1.395(−14)b

720 1.447(−14) 3.914(−14) 1.985(−14)b

750 2.209(−14) 5.632(−14) 2.746(−14)baFrom ref 30. bFrom ref 31.

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp5087174 | J. Phys. Chem. A 2014, 118, 10134−1014310141

(12) Clary, D. C.; Palma, J. Quantum Dynamics of the WaldenInversion Reaction Cl− + CH3Cl→ CH3Cl + Cl−. J. Chem. Phys. 1997,106, 575−583.(13) Kerkeni, B.; Clary, D. C. Ab Initio Rate Constants fromHyperspherical Quantum Scattering: Application to H + CH4 → H2 +CH3. J. Chem. Phys. 2004, 120, 2308−2318.(14) Banks, S. T.; Clary, D. C. Chemical Reaction Surface VibrationalFrequencies Evaluated in Curvilinear Internal Coordinates: Applica-tion to H + CH4 ⇌ H2 + CH3. J. Chem. Phys. 2009, 130 (024106), 1−12.(15) Banks, S. T.; Tauternmann, C. S.; Remmert, S. M.; Clary, D. C.An Improved Treatment of Spectator Mode Vibrations in ReducedDimensional Quantum Dynamics: Application to the HydrogenAbstraction Reactions μ + CH4, H + CH4, D + CH4, and CH3 +CH4. J. Chem. Phys. 2009, 131 (044111), 1−12.(16) Kerkeni, B.; Clary, D. C. Ab Initio Rate Constants fromHyperspherical Quantum Scattering: Application to H + C2H6 and H+ CH3OH. J. Chem. Phys. 2004, 121, 6809−6821.(17) Kerkeni, B.; Clary, D. C. Quantum Reactive Scattering of H +Hydrocarbon Reactions. Phys. Chem. Chem. Phys. 2006, 8, 917−925.(18) Kerkeni, B.; Clary, D. C. The Effect of The Torsional andStretching Vibrations of C2H6 on the H + C2H6 → H2 + C2H5

Reaction. J. Chem. Phys. 2005, 123 (064305), 1−8.(19) Kerkeni, B.; Clary, D. C. Quantum Dynamics and Kinetics ofthe Abstraction Reactions by H Atoms of Primary and SecondaryHydrogens in C3H8. Mol. Phys. 2005, 103, 1745−1755.(20) Kerkeni, B.; Clary, D. C. Quantum Scattering Study of theAbstraction Reaction of H Atoms From CH3NH2. Chem. Phys. Lett.2007, 438, 1−7.(21) Banks, S. T.; Clary, D. C. Reduced Dimensionality QuantumDynamics of Cl + CH4 → HCl + CH3 on an Ab Initio Potential. Phys.Chem. Chem. Phys. 2007, 9, 933−943.(22) Remmert, S. M.; Banks, S. T.; Harvey, J. N.; Orr-Ewing, A. J.;Clary, D. C. Reduced Dimensionality Spin-Orbit Dynamics of CH3 +HCl ⇌ CH4 + Cl on Ab Initio Surfaces. J. Chem. Phys. 2011, 134(204311), 1−15.(23) Shan, X.; Remmert, S. M.; Clary, D. C.; Zhang, B.; Liu, K.Crossed-beam and Reduced Dimensionality Studies of the State-to-State Integral Cross Sections of the Cl + HCD3(v) → HCl(v′) + CD3

Reaction. Chem. Phys. Lett. 2013, 587, 88−92.(24) Remmert, S. M.; Banks, S. T.; Clary, D. C. ReducedDimensionality Quantum Dynamics of CH3 + CH4 → CH4 + CH3:Symmetric Hydrogen Exchange on an Ab Initio Potential. J. Phys.Chem. A 2009, 113, 4255−4264.(25) Shan, X.; Clary, D. C. Quantum Effects in the AbstractionReaction by H Atoms of Primary and Secondary Hydrogens in n-C4H10: A Test of a New Potential Energy Surface ConstructionMethod. Phys. Chem. Chem. Phys. 2013, 15, 1222−1231.(26) Shan, X.; Clary, D. C. A Reduced Dimensionality QuantumMechanical Study of the H + HCF3 ↔ H2 + CF3 Reaction. Phys. Chem.Chem. Phys. 2013, 15, 18530−18538.(27) von Horsten, H. F.; Banks, S. T.; Clary, D. C. An Efficient Routeto Thermal Rate Constants in Reduced Dimensional QuantumScattering Simulations: Applications to the Abstraction of Hydrogenfrom Alkanes. J. Chem. Phys. 2011, 135 (094311), 1−14.(28) Miller, W. H.; Handy, N. C.; Adams, J. E. Reaction PathHamiltonian for Polyatomic Molecules. J. Chem. Phys. 1980, 72, 99−112.(29) Carrington, T., Jr.; Miller, W. H. Reaction Surface Hamiltonianfor the Dynamics of Reactions in Polyatomic Systems. J. Chem. Phys.1984, 81, 3942−3950.(30) Marshall, R. M.; Purnell, J. H.; Sheppard, A. Reaction ofHydrogen Atoms with Cyclopropane in the Temperature Range 358−550K. J. Chem. Soc., Faraday Trans. 2 1986, 82, 929−935.(31) Marshall, R. M.; Purnell, J. H.; Satchell, P. W. Reaction ofHydrogen Atoms with Cyclopropane in the Temperature Range from628 to 779 K. J. Chem. Soc., Faraday Trans. 1 1984, 80, 2395−2403.(32) Galano, A.; Alvarez-Idaboy, J. R.; Vivier-Bunge, A. Non-AlkaneBehavior of Cyclopropane and Its Derivatives: Characterization of

Unconventional Hydrogen Bond Interactions. Theor. Chem. Acc. 2007,118, 597−606.(33) Dobe, S.; Turanyi, T.; Iogansen, A. A.; Berces, T. RateConstants of the Reaction of OH Radicals with Cyclopropane andCyclobutane. Int. J. Chem. Kinet. 1992, 24, 191−198.(34) Baghal-Vayjooee, M. H.; Benson, S. W. Kinetic andThermochemistry of the Reaction Cl + Cyclopropane ⇌ HCl +Cyclopropyl. Heat of Formation of the Cyclopropyl Radical. J. Am.Chem. Soc. 1979, 101, 2838−2840.(35) Arnold, P. A.; Carpenter, B. K. Computational Studies on theRing Openings of Cyclopropyl Radical and Cyclopropyl Cation. Chem.Phys. Lett. 2000, 328, 90−96.(36) Greig, G.; Thynne, J. C. J. Reactions of Cyclic Alkyl Radicals.Part 2. Photolysis of Cyclopropane Carboxaldehyde. Trans. FaradaySoc. 1967, 63, 1369−1374.(37) Parker, J. H. Hydrogen Fluoride Vibrational EnergyDistributions for the Reactions of Fluorine Atoms with Cyclanes.Int. J. Chem. Kinet. 1975, 7, 433−441.(38) Hudgens, J. W.; Dulcey, C. S. Observation of the 3s 2A1Rydberg States of Allyl and 2-Methylallyl Radicals with MultiphotonIonization Spectroscopy. J. Phys. Chem. 1985, 89, 1505−1509.(39) Arnold, P. A.; Cosofret, B. R.; Dylewski, S. M.; Houston, P. L.;Carpenter, B. K. Evidence of a Double Surface Crossing betweenOpen- and Closed-Shell Surfaces in the Photodissociation ofCyclopropyl Iodide. J. Phys. Chem. A 2001, 105, 1693−1701.(40) Oh, C. H.; Shin, S. K.; Kim, H. L.; Park, C. R. Photodissociationof Cyclopropyl Cyanide at 193 nm. Chem. Phys. Lett. 2002, 363, 404−408.(41) Shuman, N. S.; Srivastava, A.; Picconatto, C. A.; Danese, D. S.;Ray, P. C.; Valentini, J. J. Evidence of a Surprising Channeling of Ring-Opening Energy to the H2 Product in the H + c-C3H6 → H2 + C3H5

Abstraction Reaction. J. Phys. Chem. A 2003, 107, 8380−8382.(42) Teslja, A.; Valentini, J. J. State-to-state Reaction Dynamics: ASelective Review. J. Chem. Phys. 2006, 125 (132304), 1−22.(43) Yu, H. G.; Muckerman, J. T. Exploring the Multiple ReactionPathways for the H + cyc-C3H6 Reaction. J. Phys. Chem. A 2004, 108,10844−10849.(44) Cohen, N. The Use of Transition-State Theory to ExtrapolateRate Coefficients for Reactions of H Atoms with Alkanes. Int. J. Chem.Kinet. 1991, 23, 683−700.(45) Truong, T. N. Reaction Class Transition State Theory:Hydrogen Abstraction Reactions by Hydrogen Atoms as Test Cases.J. Chem. Phys. 2000, 113, 4957−4964.(46) Stechel, E. B.; Walker, R. B.; Light, J. C. R-Matrix Solution ofCoupled Equations for Inelastic Scattering. J. Chem. Phys. 1978, 69,3518−3531.(47) Werner, H.-J.; Knowles, P. J.; Amos, R. D.; Bernhardsson, A.;Berning, A.; et al. MOLPRO, a package of ab initio programs, version2010.2; see http://www.molpro.net.(48) Dunning, T. H., Jr. Gaussian Basis Sets for Use in CorrelatedMolecular Calculations. I. The Atoms Boron Through Neon andHydrogen. J. Chem. Phys. 1989, 90, 1007−1023.(49) Adler, T. B.; Knizia, G.; Werner, H.-J. A Simple and EfficientCCSD(T)-F12 Approximation. J. Chem. Phys. 2007, 127 (221106), 1−4.(50) Knizia, G.; Adler, T. B.; Werner, H.-J. Simplified CCSD(T)-F12Methods: Theory and Benchmarks. J. Chem. Phys. 2009, 130(054104), 1−20.(51) Peterson, K. A.; Adler, T. B.; Werner, H.-J. SystematicallyConvergent Basis Sets for Explicitly Correlated Wavefunctions: TheAtoms H, He, B-Ne and Al-Ar. J. Chem. Phys. 2008, 128 (084102), 1−12.(52) NIST Chemistry WebBook, NIST Standard Reference Data-base; http://webbook.nist.gov.(53) DeFrees, D. J.; McIver, R. T., Jr.; Hehre, W. J. Heats ofFormation of Gaseous Free Radicals via Ion Cyclotron DoubleResonance Spectroscopy. J. Am. Chem. Soc. 1980, 102, 3334−3338.(54) Tsang, W. Heats of Formation of Organic Free Radicals byKinetic Methods. In Energetics of Organic Free Radicals; Martinho

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp5087174 | J. Phys. Chem. A 2014, 118, 10134−1014310142

Simoes, J. A., Greenberg, A., Liebman, J. F., Eds.; Blackie Academicand Professional: London, 1996; pp 22−58.(55) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Pople, J. A.Assessment of Gaussian-2 and Density Functional Theories for theComputation of Enthalpies of Formation. J. Chem. Phys. 1997, 106,1063−1079.(56) Chase, M. W. NIST-JANAF Thermochemical Tables, 4th ed. J.Phys. Chem. Ref. Data, Monograph 9 1998, 1−1951.(57) Lu, D.-H.; Truhlar, D. G. Bond-Distance and Bond-AngleConstraints in Reaction-Path Dynamics Calculations. J. Chem. Phys.1993, 99, 2723−2738.(58) Matlab, version 7.1.0.183 (R14) Service Pack 3; TheMathWorks, Inc.: Natick, MA, 2007.(59) von Horsten, H. F.; Clary, D. C. Reactive Resonances in the F +CHD3 Reaction: a Quantum Dynamics Study. Phys. Chem. Chem. Phys.2011, 13, 4340−4356.(60) Muckerman, J. T. Some Useful Discrete Variable Representa-tions for Problems in Time-Dependent and Time-IndependentQuantum Mechanics. Chem. Phys. Lett. 1990, 173, 200−205.(61) Light, J. C. The R-Matrix Method. In The Theory of ChemicalReaction Dynamics; Clary, D. C., Ed.; D. Reidel Publishing Company:Dordrecht, The Netherlands, 1986; pp 215−234.(62) Brooks, A.; Clary, D. C. Application of HypersphericalCoordinates to Four-Atom Reaction Scattering: H2 + CN → H +HCN. J. Chem. Phys. 1990, 92, 4178−4190.(63) Nyman, G.; Yu, H. G. Quantum Theory of BimolecularChemical Reactions. Rep. Prog. Phys. 2000, 63, 1001−1059.(64) Yu, H. G.; Nyman, G. Three-Dimensional Quantum ScatteringCalculations on the Cl + CH4 → HCl + CH3. Phys. Chem. Chem. Phys.1999, 1, 1181−1190.(65) Yu, H. G. A Five-Dimensional Quantum Scattering Model forthe Type AB + XCD3 ⇌ ABX + CD3 Reaction in HypersphericalCoordinates: Application to OH + CH4 ⇌ H2O + CH3. J. Chem. Phys.2001, 114, 2967.(66) Bowman, J. M. Reaction and Molecular Dynamics; Springer: NewYork, 2000.

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp5087174 | J. Phys. Chem. A 2014, 118, 10134−1014310143