MEP–MPE Potential Energy Surface forthe Cl � CH4 3 HCl � CH3 Reaction

ERNESTO GARCIA, CARLOS SANCHEZ, AURELIO RODRIGUEZ,ANTONIO LAGANADepartamento de Quımica Fısica, Universidad del Paıs Vasco, Vitoria, Spain

Received 20 June 2004; accepted 8 November 2004Published online 19 October 2005 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.20808

ABSTRACT: A potential energy functional based on a many-process expansionminimum energy path approach has been fitted to the ab initio potential energy valuesof the Cl � CH4 system. The functional form adopted, the parameters of the fittedsurface, and the variation of the cone of acceptance as a function of relevant parametersare discussed. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem 106: 623–630, 2006

Key words: potential energy surface; rate coefficients; Cl � CH4; reactive scattering

1. Introduction

A key step of reactive molecular dynamics isthe fitting of the calculated values of the po-

tential energy surface (PES) using a suitable func-tional form [1]. Along with this problem goes thechoice of an appropriate set of coordinates to over-come the fact that physical coordinates (such as theinternuclear distances in which most of the cur-rently used potential energy functionals are ex-pressed) cannot be used to formulate the interaction

of neutral molecules over the whole range, neitheras direct powers nor as inverse powers.

To overcome this difficulty, it has been sug-gested the use of bond order (BO) coordinates, de-fined for the generic �� atom–atom pair as n�� �exp[����(r�� � re��)], with r�� the internuclear dis-tance, re�� the corresponding equilibrium value and��� a parameter related to the bond strength of the�� diatom [2, 3]. As a matter of fact, using BOcoordinates, the PES of some diatomic and tri-atomic systems have been formulated as pure poly-nomials [2–8].

BO coordinates are suitable for this purpose,since they have built in the metrics of molecularbonds. They become, in fact, large at short dis-tances, one at the atom–atom equilibrium distanceand zero at infinity. To generalize the BO formula-tion of the PES to large systems, it has been foundalso convenient to adopt a many-process expansion(MPE) [9–12]. In the MPE scheme given the set of

Correspondence to: A. Lagana; e-mail: lag@unipg.itA. Lagana is currently at the Dipartimento di Chimica, Uni-

versita di Perugia, Perugia 06123, Italy.Contract grant sponsor: MCyT.Contract grant sponsor: ASI.Contract grant sponsor: MIUR.Contract grant sponsor: COST.Contract grant sponsor: European Community.Contract grant number: HPRN-CT-1999-00007.

International Journal of Quantum Chemistry, Vol 106, 623–630 (2006)© 2005 Wiley Periodicals, Inc.

atoms ���� . . . , one can formulate the PES as asum of all the possible processes � connecting reac-tants to products:

V����· · · � ��

W��s��V��t��, (1)

where s� is the evolution coordinate (or reactioncoordinate) of process � driving the transformationof the system from reactants to products while t� isthe set of coordinates describing the local deforma-tion of the system. In Eq. (1), V�(t�) is the � processpotential describing the cross section of the reactionchannel at each point of the evolution coordinate,while W�(s�) is a weight function that properly av-erages the contributions coming from different pro-cesses. In this way, the permutational symmetry ofthe system is fully taken into account.

The evolution coordinate s� of process � can besuitably taken to be an angle. This is the case of thehyperspherical BO (HYBO) coordinates [13, 14]. Inthe specific case of the fixed angle atom diatomreactive process � � �� 3 �� � � (with � theexchanged atom and �� the angle formed by the tworelated BO variables n�� and n��), the reaction coor-dinate is taken to be the angle � defined as [10, 12]:

� � arctan�n��

n���. (2)

The angle � can be seen as the rotation (fromreactants to products) angle of a diatomic-like BO(ROBO) potential given as a function of the hyper-radius � (that can be seen as a collective coordinateof the system). The hyperradius � is defined as

� � �n��2 � n��

2 . (3)

For the generic atom diatom reaction (i.e., in afull three-dimensional treatment), one has to con-sider the three possible reactive processes � � ��3�� � �, � � �� 3 �� � �, and � � �� 3 �� � �(each of which refers to a different � process) andtake into account that the value of �� varies duringthe reactive process. Accordingly, the weight of Eq.(1) has the normalized form

W����� �w�����

¥�� w�������. (4)

When the weight w� depends on �� in a way thatprivileges the process closer to collinearity, the re-

sulting functional form is called the largest anglegeneralization of ROBO (LAGROBO). The LA-GROBO functional has been successfully used to fitthe PES of several three-atom [10, 12, 15] and four-atom [16–18] systems.

To generalize the treatment to polyatomic reac-tions with more than four atoms, one needs toabandon the idea of including all processes and tofollow each of them from the beginning to the endwith a specific continuity variable. One can insteadconsider only those parts of the processes that areassociated with one or more low-energy minimumenergy paths (MEPs) relevant to the key mecha-nisms of the overall process [19, 20].

Based on this idea, we have developed a newminimum energy path–many process expansion(MEP–MPE) potential energy surface for the Cl �CH4 that is important for modeling the strato-spheric ozone depletion and the hydrocarbonflames. Accordingly, this article is organized as fol-lows: in Section 2, the MEP–MPE functional is de-scribed; in Section 3, the BO and MEP parametersfor the Cl � CH4 process are given; in Section 4, thevalue of the parameters of the process potential arereported; and in Section 5, the properties of theMEP–MPE PES discussed and the relationship ofthe temperature dependence of the thermal ratecoefficient on the cone acceptance.

2. MEP–MPE Functional Form

According to the already mentioned basic idea offounding the PES on a subset of MPE segmentsselected using a MEP energetic criterion, the MEP–MPE functional representation of the interaction isobtained by limiting the sum of Eq. (1) to the N localpaths connecting the geometries of the MEP (orMEPs) associated with the stationary points of thesurface relevant to the investigated processes. Theneach V� process potential is formulated as a sum oftwo contributions: a contribution V�,mep associatedwith the MEP closer to the considered arrangementand a contribution V�,mbe introducing the correc-tions associated with the deviation of the geometryconsidered from the nearest MEP configuration:

V� � V�,mep � V�,mbe. (5)

The V�,mbe contribution is further articulated intotwo (2), three (3), and four (4) body components as

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624 VOL. 106, NO. 3

in the popular many-body expansion (MBE) ap-proach [21]:

V�,mbe � �i

Vi�,mbe�2� � �

i

Vi�,mbe�3� � �

i

Vi�,mbe�4� , (6)

where the index i runs over all the two, three, andfour subsets of atoms. The two-body terms read

Vi�,mbe�2� � Di�,mbe

�2� �ni � ni�,mep�2 (7)

and account for the difference between the relatedBO coordinate ni and its reference value ni�,meptaken at the corresponding point on the nearestminimum energy path. It is worth noting that thecorrection of Eq. (7) is anharmonic in the physicalspace in spite of its harmonic-like formulation inthe BO space. The three-body terms read

Vi�,mbe�3� � Di�,mbe

�3� ��i� � �i�,mep�2 (8)

and account for the difference between the relatedactual bending angle �i� and that of the correspond-ing geometry on the nearest minimum energy path�i�,mep. The four-body terms read:

Vi�,mbe�4� � Di�,mbe

�4� � i� � i�,mep�2 (9)

and account for the difference between the relatedactual torsion angle i� and that of the correspond-ing geometry on the nearest minimum energy path i�,mep. Finally the weight function w� is defined:

w� � �j

exp��b�nj � nj,mep�2�, (10)

where the sum extends over all the j pairs of atomsof the system and the b parameter determines thewidth of the Gaussian.

3. BO and MEP Parameters at theCl � CH4 Stationary Points

To apply the MEP–MPE functional to the Cl � CH43HCl � CH3 reaction, one first has to define the BOparameters of the related asymptotic pairs of atoms.These are 1.645, 1.275, 1.120, and 0.741 Å for theequilibrium internuclear distances and 1.910, 1.868,1.957, and 1.944 Å�1 for the � parameter of the ClOC,ClOH, COH, and HOH pairs, respectively.

After defining the BO space one maps the calcu-lated ab initio electronic energy values (high level

ab initio calculations of the potential energy valuesof the ClCH4 reaction have been reported in theliterature [22–28]) onto it and carries out a detailedanalysis of the topology of the surface.

Only one process � (the one with the lowest MEP)having three stationary points X was considered inour study of the ClCH4 surface. The first stationarypoint (XAR) is that of the isolated Cl and CH4 reac-tants. The second stationary point (XAS) is that asso-ciated with the saddle to reaction. In this stationarypoint, the COH bond of CH4 facing the Cl atom isstretched to bind Cl in a collinear-like COHOClbond, while the remaining (three) COH bonds aresymmetrically arranged around the COHOCl axis.The third stationary point (XAP) is that of the isolatedHCl and CH3 products. Accordingly, the evolutioncoordinate � [see Eq. (1)] was partitioned into twosegments. The first segment goes from the reactants tothe saddle (� � R–S), while the second goes from thesaddle to the products (� � S–P).

An illustration of the partition of the MEP for theCl � CH43 HCl � CH3 reaction is given in Figure1. In Figure 1, isoenergetic contours of the MEP–MPE PES presented in this study at the collinear valueof the ClOHOC angle are plotted as a function of the

FIGURE 1. Isoenergetic contour maps of the MPE–MEP PES for the Cl � CH4 3 HCl � CH3 reaction inthe C3v symmetry plotted as a function of the ClOHand HOC bond order coordinates. The stationarypoints R (reactants), S (saddle) and P (products) aremarked by a star, while the minimum energy path po-tential (Vmep) is evidenced as a solid line.

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 625

n(HOC) and n(ClOH) BO coordinates. In this case,the origin of the coordinates corresponds to the dis-sociation in the Cl, H, and CH3 fragments. The equi-librium configuration of the products is located atn(HOC) � 0 (x-axis of Fig. 1) and n(ClOH) � 1(corresponding to equilibrium distance of the HClmolecule). The equilibrium configuration of reactantsis placed on the y-axis of Figure 1 [n(ClOH) � 0] at avalue of n(HOC) larger than 1, since the equilibriumlength of the COH bond in CH4 is shorter than that ofthe isolated CH diatom taken for the construction ofthe BO variable. It is worth noting that the minimumenergy path (indicated in Fig. 1 as the solid curve lineconnecting R to P) is close to the arc centered on theorigin and connecting reactants and products equilib-rium geometries. In Figure 1, the two R–S and S–Psegments in which the minimum energy path is par-titioned are delimited by stars.

For the relevant atomic (i) subsets, the ni�,mep,�i�,mep and i�,mep were determined as follows. Atthe R stationary point, the equilibrium geometry ofCH4 is given by four CH bonds of length 1.087 Åhaving tetrahedral symmetry as from spectroscopicdata. The S stationary point is associated with thetransition state whose geometry is given by the abinitio data of Truong et al. [22]. In this geometry, theCl, C, and the reactive hydrogen lie on the sameline. The relevant internuclear distances arer(H1Cl) � 1.431 Å and r(CH1) � 1.388 Å (with H1the reactive hydrogen abstracted by the Cl atomfrom the methane molecule). The remaining CHbonds have a length of 1.388 Å and a C3v symmetryaround the COH1OCl axis (three planar angles of101.2°). The P stationary point is that of the HCl �CH3 products with the two molecules at their equi-librium geometry. This means that the HCl inter-nuclear distance is set at 1.275 Å and the CH inter-nuclear distances of CH3 are set at 1.080 Å. Theequilibrium geometry of CH3 is taken to be planar(�(HCH) � 120°) as from spectroscopic data. Therelated energetics is: the energy zero is set at thereactant asymptote; the transition state has an en-ergy of 7.9 kcal/mol (as from ab initio calculations[22]); the product asymptote has an energy of 6.7kcal/mol (as from spectroscopic data).

4. Parameters of the ProcessPotential

A crucial part of the fitting procedure consists ofdetermining the contributions for the relevantatomic subsets (i) [see Eqs. (7)–(9)] to the process

potentials V� (for sake of simplicity here after theprocess label � is dropped from our formalism,since we consider only one process). AccordinglyVmep (the potential energy associated with the min-imum energy path) and Vmbe (the related energycorrection term) are evaluated out of the abovedefined three stationary points. To this end, theVmep values are interpolated using a third degreepolynomial from the ab initio values at XAR, XAS,and XAP stationary points. Continuity conditionsfor the potential energy and its derivatives are im-posed at the extrema. A similar procedure is used toevaluate the corrective contributions of Vmbe. In thiscase, the relevant Di,mbe

(k)X parameters of Eqs. (7)–(9)are calculated using again a third-degree polyno-mial to interpolate the values obtained at the sta-tionary points.

The Di,mbe(2)X parameters are, in general, given by

the value of the dissociation energy of the isolateddiatoms (78.3, 106.4, 78.2, 109.5 kcal/mol for i �ClOC, ClOH, COH, and HOH, respectively). Forthe reactant geometry, the Di,mbe

(2)R value for COHhas been scaled by a factor of 0.93. For the saddlegeometry, the Di,mbe

(2)S common value for ClOH1 andCOH1 has been set at 60.0 kcal/mol to take intoaccount the relaxation of these bonds with respectto that of the related ClOH (for the HCl product)and COH (for the CH4 reactant) bonds. The com-mon value is due to the assumption that the twobonds have the same strength, since they break andform simultaneously. No changes were introducedfor the products.

The three-body correction includes only a fewimportant contributions. In fact, for the reactantgeometry only the contribution due to theHOCOH bending motion of the CH4 molecule isconsidered (the related Di,mbe

(3)R parameter is set at 34kcal/mol), while for the product geometry only theHOCOH bending motion of the CH3 molecule isconsidered (the related Di,mbe

(3)P parameter is set at 14kcal/mol, implying that the CH3 molecule is flop-pier than the CH4 one). In both cases, because of thesymmetry of the methane molecule and the methylradical all bending motions are dealt in the sameway. At the saddle geometry, a distinction betweenthe bending motions of H1OCOH and HOCOH ismade. In practice, the bending motions centered onthe C atom involving the reacting H1 atom areconsidered more energetic (35 kcal/mol) than thosenot involving it (30 kcal/mol). The value of theDi,mbe

(3)S parameter for H1OCOH is slightly higherthan that of the pure CH4 reactant molecule butsignificantly higher than that of the pure CH3 prod-

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626 VOL. 106, NO. 3

uct molecule. For the stationary point associatedwith the transition state particular attention hasbeen paid to the description of the bending motionof ClOH1OC. In this case, a rather small value hasbeen adopted for Di,mbe

(3)S , since the associated angleis large (180°). Clearly, this parameter affects thereactivity of the PES, since it determines the cone ofacceptance of the chlorine atom by the methanemolecule. As a matter of fact, a null value of Di,mbe

(3)S

implies that the corresponding term of the three-body correction Vi,mbe

(3) is zero for all values of theClOH1OC angle. On the contrary, when Di,mbe

(3)S

differs from zero, Vi,mbe(3) has a nonvalue in the case

that the geometry of the system coincides with thereference configuration of the (collinear) transitionstate and becomes increasingly more positive (i.e.,the potential becomes more repulsive) as the angleClOH1OC increasingly deviates from collinearity.Accordingly, when Di,mbe

(3)S is zero, even for geome-tries sufficiently far from collinearity, the potentialenergy is small enough to allow an abstraction ofthe hydrogen atom. As Di,mbe

(3)S increases, the intervalof energetically favorable angles of attack decreasesbecause large deviations from collinearity make thepotential quite repulsive. As we shall comment inmore detail later, Di,mbe

(3)S is also the parameter weplayed with to allow the calculated rate coefficientmatch the measured data.

The four body Di,mbe(4)X corrections are, in general,

neglected. Exception is made for the ClOH1OCOHtorsion of the product geometry for which a smallcontribution of 0.4 kcal/mol is considered to accountfor the loss of planarity of the CH3 molecule.

The last parameter needed to construct the MPE–MEP is the b parameter of the Gaussian that definesthe weight function of Eq. (10). For the title system,a value of 460.5 is adopted in order to reduce theweight to 10�8 for displacements of 0.2 from theMEP value of the BO coordinate.

5. Properties of the MEP–MPE PES

A sketch of the MEP–MPE PES built in this wayis given in this section by plotting in Figure 2 itsisoenergetic contours for the saddle geometry. Inthe plot, the potential energy is calculated as afunction of the x- and y-projections of the Cl atomposition vector around the CH4 molecule in its tran-sition state geometry. This means that the asymp-totic energy is 22 kcal/mol larger. In Figure 2, thesaddle to reaction (7.9 kcal/mol) corresponds to theminimum placed at x � �2.80 Å and y � 0 (i.e.,

along the collinear approach). As apparent fromFigure 2, the cone of acceptance (defined by theangle centered at the coordinates origin and cover-ing the regions of the surface with a potential en-ergy of 21 kcal/mol�1) for the MEP–MPE surface(with DClOH1OC,mbe

(3)S � 0) is, as expected, fairly wide(120°). The dynamical full-dimensional quasi-classical calculations performed on the MEP–MPEPES indicate that the cone of acceptance is too large.As a matter of fact, the room temperature rate co-efficient was estimated to be [29] 2.07 � 0.18 �10�13 cm3 molecule�1 s�1, a value twice as large asthe IUPAC recommended one that is 1.0 cm3 mol-ecule�1 s�1 [30].

To scale the calculated room temperature ratecoefficient to the experimental data, the DClOH1OC,mbe

(3)S

parameter was varied. At DClOH1OC,mbe(3)S � 1.84 kcal/

mol a room temperature rate coefficient of (1.01 �0.04) � 10�13 cm3 molecule�1 s�1 was obtained [30]that is in good agreement with the IUPAC recom-mended value. The optimized MEP–MPE surfacewell reproduce also the experimental dependenceof the rate coefficient on the temperature. The de-crease of the reactivity of the optimized PES with

FIGURE 2. Isoenergetic contour maps of the MPE–MEP PES plotted as a function of the x- and y-Carte-sian coordinates of the Cl atom around the CH4 frag-ment at its transition state geometry. Contours arespaced by 1 kcal/mol�1. The zero of energy is fixed atthe asymptotic reactants’ Cl, C and two hydrogen at-oms are forced to lie on the some z � 0 plane equilib-rium potential minimum.

MEP–MPE PES FOR Cl � CH4 3 HCl � CH3

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 627

respect to the unoptimized one can be understoodin terms of a reduction of the cone of acceptance forthe Cl atom approaching the CH4 target molecule atthe transition state geometry. In fact, as apparentfrom Figure 3, where the isoenergetic contours ofthe optimized MEP–MPE surface are plotted (as inFig. 2), the cone of acceptance is narrower (82°).

To better characterize the optimized MEP–MPEPES the energetic and the geometric properties ofits stationary points have been calculated. Table Iquotes the endoergicity ( V) and the potential bar-rier to reaction ( V‡), as well as the geometry andthe harmonic frequencies of the reactants, productsand saddle point. These values are compared withthose of the analytical PES of Corchado et al. [27](hereafter labeled as CTE PES). In Table I, dataobtained from several ab initio calculations are alsoindicated. Truong et al. [22] calculated the geome-tries and the energies at MPSAC2/MC-311G(2d,d,p)level, whereas the frequencies were calculated atMP2/MC-311G(2d,d,p) level. Those values wereused by Roberto-Neto et al. [25] to optimize theparameters of some semiempirical functions to as-semble the PES. Dobbs and Dixon [23] calculatedthe geometries and the frequencies at the MP2(FU)/TZ�2P level and the energies at QCISD(T)/cc-pVQZ(no g) level. Duncan and Truong [24] calcu-lated the geometries and frequencies at the QCISD/6-311G(d,p) level and the energies at the

PMP4(SDTQ)/6-311�G(2df,2pd) level at theBH&HLYP/6-311G(d,p) geometries. Yu and Ny-man [26] calculated all properties at MP2-SAC/6-311G(2df,2pd) level. Corchado et al. [27] calculatedthe geometries and frequencies at the MP2/pTZlevel and the energies at the CCSD(T)/IB level.Troya et al. [28] calculated the geometries and fre-quencies at the UMP2/6-311G(2df,2pd) level and theenergies at the PUMP4/6-311G(2df,2pd) level.

The endoergicity of both CTE and MEP–MPEanalytical PESs is slightly different, although bothvalues fall inside the range of the ab initio values.Similarly, the potential barrier to reaction of bothanalytical PESs lies inside the ab initio values and,in this case, heights are very much the same (7.7 vs.7.9 kcal/mol).

The equilibrium geometries of the CH4 reactant,HCl and CH3 product molecules derived from theoptimized MEP–MPE PES agree with the ab initiovalues calculated using different levels of theory.The same is not true in the case of the analyticalCTE PES, for which the equilibrium distance of theCH bond in both CH4 and CH3 molecules is largerthan the ab initio values aside from the fact thatthey are set to be equal. The discrepancies betweenthe values of the harmonic frequencies obtainedfrom the analytical PESs and those obtained fromthe ab initio calculations appear to be much larger.As a matter of fact, the harmonic frequencies pre-dicted by the optimized MEP–MPE PES underesti-mate the ab initio values in 7%. Moreover, the lowerfrequencies �3 and �4 for the CH3 radical are, re-spectively, underestimated of about 30% and over-estimate of about 40%, with respect to the ab initiovalues. A similar discrepancy was also found forthe CTE PES.

The saddle-point geometry calculated by the var-ious ab initio studies are quite similar. Moreover,the geometry of the saddle point of the optimizedMEP–MPE PES falls inside the range of the ab initioestimates. On the contrary, the geometry of the CTEsaddle significantly differs from the ab initio oneand, in particular, from that used in the fittingprocedure [27]. In fact, the ClOH1 distance at thesaddle is 0.1 Å shorter, whereas the COH dis-tance is 0.02 Å larger.

As to the harmonic frequencies at the saddlepoint, the higher frequencies (�1 � �6) of the opti-mized MEP–MPE PES are slightly lower than thoseobtained from the ab initio calculations (the corre-sponding harmonic frequencies of the CTE PES areeven smaller). The discrepancies of the lower fre-quencies (�7 � �11) are larger, with both (�7, �8)

FIGURE 3. As Figure 2, isoenergetic contour maps ofthe optimized MEP–MPE PES.

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628 VOL. 106, NO. 3

overestimated and (�9, �10, �11) underestimated.The ab initio imaginary frequency at the saddlepoint, despite the wide range of the calculated val-ues, is clearly underestimated by the analyticalPESs and, in particular, by the optimized MEP–MPE one. It is worth to remind the strong depen-dence of the imaginary frequency on the level oftheory used in the ab initio calculations. For in-stance, Dobbs and Dixon [23] obtained an imagi-nary frequency of 1262i cm�1 when using a MP2/TZ�2P method, whereas this value decreasedsignificantly (360i cm�1) when using a B3LYP/aug-

cc-pVDZ method, in spite of the fact that the geom-etry and the other frequencies are similar.

6. Conclusions

A recently proposed new fitting procedure basedon BO coordinates has been proposed for reactivesystems having four or more atoms.

The peculiarity of the suggested functional formis that it moves from a rotating bond order formu-lation of the reactive channels by selecting the most

TABLE I ______________________________________________________________________________________________Relative potential energies (in kcal/mol) for the products (�V) and the saddle point (�V‡), with respect to theequilibrium state of the reactants, geometry (in Å and degrees), and harmonic frequencies (in cm�1) of thestationary points for the optimized MEP–MPE PES.*

Ab initio calculations Analytical PESs

Ref. [22] Ref. [23] Ref. [24] Ref. [26] Ref. [27] Ref. [28] CTE [27] MPE

V 6.8 6.98 6.38 5.98 6.0 6.43 6.1 6.7 V‡ 7.9 8.90 7.87 6.87 7.6 6.87 7.9 7.9CH4(Td)

r(CH) 1.091 1.083 1.093 1.091 1.085 — 1.094 1.087�1 3218 3212 3166 3212 — — 3037 3017�2 3080 3076 3046 3067 — — 2871 2944�3 1582 1591 1573 1563 — — 1501 1481�4 1364 1369 1367 1346 — — 1339 1325

HClr(HCl) 1.276 1.276 1.276 1.276 1.273 — 1.274 1.275� 3048 3063 3045 3030 — — 2993 2989

CH3(D3h)r(CH) 1.080 1.072 1.093 1.080 1.074 — 1.094 1.080�1 3369 3369 3309 3361 — — 3165 3163�2 3176 3178 3127 3166 — — 2994 3028�3 1452 1441 1436 1435 — — 1240 1018�4 426 460 433 437 — — 580 601

ClHCH3 saddle pointr(ClH1) 1.431 1.452 1.443 1.432 1.451 1.444 1.356 1.432r(CH1) 1.388 1.375 1.387 1.359 1.359 1.389 1.380 —r(CH) 1.086 1.078 1.088 1.087 1.080 1.079 1.098 1.086�(H1CH) 101.2 101.2 101.6 101.2 — 101.4 — 101.2�(ClH1C) 180 180 180 180 180 180 180 180�1, �2 3295 3305 3259 3302 3303 3312 3039 3133�3 3118 3132 3100 3125 3131 3140 2910 3021�4, �5 1441 1448 1449 1436 1457 1449 1372 1432�6 1227 1213 1207 1229 1223 1227 1126 1238�7, �8 874 958 939 926 923 982 782 1118�9 572 511 505 537 519 524 732 457�10, �11 324 378 355 387 337 396 312 215�12 949i 1262i 1228i 915i 1136i 1143i 760i 247i

* Values from ab initio calculations and the analytical CTE PES are also shown.

MEP–MPE PES FOR Cl � CH4 3 HCl � CH3

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 629

favorable minimum energy paths to reaction. Thenthe selected MEP approaches are piecewise approx-imated by simple analytical functions. The totalpotential function is then assembled by taking aweighed combination of the potential channels as-sociated with the various processes and by correct-ing the MEP contributions for the displacement ofthe actual geometry from the nearest MEP geome-try.

The resulting parametrization of the potential isintimately connected with the key features of thePES. Accordingly, a calibration of the surface fea-tures can be carried out by optimizing the param-eters to the reproduction of some kinetics and dy-namics properties of the reaction.

This is indeed what has been done in this studyof the Cl � CH4 system, for which the appropriatereactive process has been chosen, a suitable set ofbond order parameters has been selected, and agood reproduction of the ab initio and experimentalcharacteristics of the system has been obtained.Since the resulting MEP–MPE surface showed notto be particularly well reproducing the thermal ratecoefficient especially as the temperature lowers, afine-tuning of the surface was performed by vary-ing the parameter governing the amplitude of thecone of acceptance. Once this was achieved, theagreement between calculated and measured ratecoefficients become highly satisfactory.

ACKNOWLEDGMENTS

Aurelio Rodrıguez acknowledges the financialsupport provided through the European Commu-nity’s Human Potential Programme under contractHPRN-CT-1999-00007 [Reaction Dynamics].

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