Hindered rotor models with variable kinetic functions for accuratethermodynamic and kinetic predictions

Guillaume Reinisch,1,a� Jean-Marc Leyssale,1,b� and Gérard L. Vignoles2,c�

1CNRS, Laboratoire des Composites ThermoStructuraux, UMR 5801, 3 Allée de La Boétie,33600 Pessac, France2Université Bordeaux 1, Laboratoire des Composites ThermoStructuraux, UMR 5801,3 Allée de La Boétie, 33600 Pessac, France

�Received 6 July 2010; accepted 1 October 2010; published online 19 October 2010�

We present an extension of some popular hindered rotor �HR� models, namely, the one-dimensionalHR �1DHR� and the degenerated two-dimensional HR �d2DHR� models, allowing for a simple andaccurate treatment of internal rotations. This extension, based on the use of a variable kineticfunction in the Hamiltonian instead of a constant reduced moment of inertia, is extremely suitablein the case of rocking/wagging motions involved in dissociation or atom transfer reactions. Thevariable kinetic function is first introduced in the framework of a classical 1DHR model. Then, aneffective temperature and potential dependent constant is proposed in the cases of quantum 1DHRand classical d2DHR models. These methods are finally applied to the atom transfer reactionSiCl3+BCl3→SiCl4+BCl2. We show, for this particular case, that a proper accounting of internalrotations greatly improves the accuracy of thermodynamic and kinetic predictions. Moreover, ourresults confirm �i� that using a suitably defined kinetic function appears to be very adapted to suchproblems; �ii� that the separability assumption of independent rotations seems justified; and �iii� thata quantum mechanical treatment is not a substantial improvement with respect to a classical one.© 2010 American Institute of Physics. �doi:10.1063/1.3504614�

I. INTRODUCTION

Quantum chemical methods, as implemented in manycommercial software packages,1,2 allow nowadays for a rou-tine calculation of many properties of molecules. Amongthem, thermodynamic and kinetic data are crucial for theunderstanding and the prediction of chemical reactions. Areliable prediction of such properties requires tackling twomain issues. First, it is necessary to calculate accurate mo-lecular energies at least at some specific geometries definingthe minima �stable states� and saddle points �transition states�TS�� on the potential energy surface �PES�; ab initio meth-ods are known to be very adapted to this issue. A secondissue arises when one wishes to compute temperature depen-dent properties �entropy, heat capacity, reaction rate� forwhich densities of states �DOS� and/or partition functionsneed to be computed. This is usually achieved using thesimple harmonic-oscillator �HO� approximation. Indeed,most of the 3N-6 �or 3N-5 for linear molecules� internalvibrations are very well described under this approximation.However, it has been shown long ago that it fails when lowfrequency motions are involved.3,4

These problematic internal rotations are generally tor-sional movements �i.e., rotation of two molecular topsaround a single bond� as well as flexural �bending� motions.It has been shown that a specific treatment of torsional de-grees of freedom can greatly improve the accuracy of ther-

modynamic predictions.4–6 The one-dimensional hinderedrotor �1DHR� model introduced by Pitzer et al.,3,7 and de-scribed recently in a global frame by Pfaendtner et al.,5 is asimple and convenient way to compute the DOS associatedto internal rotations. This model considers individual anduncoupled rotations for which an effective 1D Hamiltonian isconstructed. The treatment applies to any molecule, whichcan be regarded as a rigid frame with attached symmetricaltops, or in other words, any molecule whose moments ofinertia for overall rotation do not depend on the value of theangle defining the internal rotation.3 This assumption leads tothe following effective Hamiltonian:

H1D = −�2

2Ired

�2

��2 + V��� , �1�

where Ired is the reduced moment of inertia and V��� is thehindering potential defining the variation of the potential en-ergy with the rotation angle �. V��� is usually computedusing ab initio calculations at discrete values of � and fittedwith an analytical form. Most of the time, relaxed geometryoptimizations are performed to partially account for the cou-pling with other modes. The calculation of the reduced mo-ment of inertia �introduced by Herschbach et al.8 and sum-marized by East and Radom6� takes into account thecoupling between internal and external rotations. Finally theeigenvalues ��i� of the Hamiltonian �Eq. �1�� are obtainedusing the one-dimensional Fourier grid Hamiltonian�FGH1D� algorithm9–11 and the partition function �Qrot

1D inEq. �2�� is evaluated through direct counting

a�Electronic mail: reinisch@lcts.u-bordeaux1.fr.b�Electronic mail: leyssale@lcts.u-bordeaux1.fr.c�Electronic mail: vignoles@lcts.u-bordeaux1.fr.

THE JOURNAL OF CHEMICAL PHYSICS 133, 154112 �2010�

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Qrot1D�T� =

1

�rot�

i

exp�− �i/kBT� , �2�

where �rot is the rotational symmetry number of the internalrotation and kB is Boltzmann’s constant.

One of the main drawbacks of the 1DHR model is that itdoes not take into account the coupling between internal ro-tations and other motions, although part of this coupling istaken into account through relaxed optimization. This maysacrifice accuracy in the case of highly coupled rotations12–14

and some authors have performed coupled n-DHRcalculations.13,15–17 However, such a treatment requires thecostly calculation of a multidimensional PES. Another ap-proach which takes into account the coupling with othermodes is the extended HR model of Vansteenkiste et al.16

Nevertheless, these authors have shown that in most casestorsional motions are correctly treated under the uncoupled1DHR model. The second limitation of 1DHR arises fromthe assumption of a rigid frame model with attached sym-metrical tops, leading to a constant kinetic function in theeffective 1D Hamiltonian �represented in Eq. �1� by the con-stant value Ired of the reduced moment of inertia�. If this isgenerally a good approximation for torsional motions �butnot always, as shown by Gang et al.12�, internal flexural mo-tions do not necessarily fall in that case. Rocking/waggingmotions frequently play an important role in dissociation re-actions and an accurate treatment of their evolution along theminimum energy path �MEP� is necessary to properly iden-tify the usually loose TS associated to these reactions. This isthe price to pay for a good prediction of dissociation reactionrates.18–20 Indeed, when considering the dissociation of twononlinear fragments, six internal transitory motions are evi-denced. The one associated to the reaction coordinate has notto be taken into account as it defines the dividing surface. Inthe variational reaction coordinate framework,21 the otherfive transitional motions correspond to internal rotations ofthe two fragments around two pivot points pA and pB at-tached, respectively, to fragments A and B. The conservedmodes are treated separately and an effective Hamiltonian isdefined as a function of the value of the reaction coordinate son the MEP and of pA and pB. Particularly, in this model,internal transitional modes are coupled with the three globalrotations to account for the Coriolis effect. The Hamiltonianat the dividing surface can thus be written as

H�s,pA,pB� = Hcons + HpA;pB

5D+rot, �3�

where Hcons is the part of the Hamiltonian associated to theconserved modes �conserved internal vibrations of the twofragments and global translations� and HpA;pB

5D+rot is the Hamil-tonian of the transitional modes. In general Hcons is takenunder the approximation of harmonic vibrations for A and Bmoieties. The identification of s, pA, and pB is achieved byminimizing the rate constant with respect to these variables.In that formalism the pivots are usually close to chemicalbond centers for rigid TS �resulting in dividing surfaces simi-lar to those used in transition state theory22,23 �TST�� whilethey tend to be located on the centers of masses of the twofragments for flexible TS �as postulated in the original flex-

ible TST �Refs. 24 and 25��. Because this multidimensionaltreatment needs a multidimensional �five-dimensional in thegeneral case� PES, H5D+rot is often separated into indepen-dent components. A usual approach, which has shown goodaccuracy in most cases, is to cast it in terms of an overallrotation �the effect is rigorously studied in the work ofHarms and Wyatt26�, a 1D torsional motion and two two-dimensional �2D� rotations.20,27–29 In this case, Eq. �3� is re-written as

H�s,pA,pB� = Hcons + Hrot + Htorsion + HA,pA

2D + HB,pB

2D , �4�

where HX,pX

2D is the bending movements of fragment X aroundthe pivot point pX.

Rocking/wagging motions can also be involved in someimportant classes of bimolecular reactions, like for instanceatom transfer reactions.30,31 In that case, the TS, generallywell located by a saddle point on the PES, involve forming/breaking chemical bonds, associated to three hindered rota-tions �one torsion and two bending motions�. As a conse-quence, the atom transferred from one fragment to the othernaturally acts as the pivot point. Then, a simple formulationfor the Hamiltonian is given by

H = Hcons + Hrot + Htorsion + H2D. �5�

The partition function associated to the 2D rocking/waggingHamiltonians �H2D� is computed using two main approaches:�i� an uncoupled model;32–34 and �ii� a coupled degeneratedmodel.29,35 In the first case, the most widely usedapproach30,34 is the one introduced by Benson20 and is basedon the HO approximation. The separability assumption thenallows to write the partition function as

Qrot2D�T� = � kBT

h�2 1

�w�r, �6�

where �w and �r are the corresponding frequencies of thewagging and rocking motions. However, this formulation di-verges when the frequencies �i tend to zero �free rotation� sothat it can only be applied at small separations. The analyti-cal variational TST study of the recombination reaction oftwo methyl radicals by Pacey34 suggests that using Eq. �6�induces an error on the reaction rate at low temperatures. Heproposes to better take into account the “real” potential en-ergy landscape associated to these motions. In other words,some kind of 1DHR treatment would be more appropriate.

The second approach for treating wagging/rocking mo-tions is to suppose a 2D degenerated symmetric tops modelas proposed by Pacey27 and Jordan et al.36 in the case of adegenerated rotor. Considering the triatomic system of Fig.1, an atom A rotating around the center of mass O of thediatomic B–B, the following Hamiltonian at a fixed value ofthe distance R between A and O �see the caption of Fig. 1 forthe definition of the other parameters� can be defined by:37

H2D = −�2

2I� 1

sin �

�

��sin �

�

��+

1

sin2 �

�2

��2� + V��� , �7�

where

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I =mAmBR2r2

r2mB + 2R2mA + �mAr2�/2�8�

is the moment of inertia obtained by Wilson’s G-matrixmethod,3,38 mA and mB being the masses of atoms A and B.The determination of the eigenvalues of Eq. �7� is performedwith V���=V0 sin2���, where V0 is a constant. In addition tothe ad hoc choice of a potential form, another importantrestriction of this approach is the choice of the center of massof the B2 fragment as the pivot point. Following the work ofPacey,27 Jordan et al.36 have presented a 2D classical Hamil-tonian for a degenerated rotor able to deal with arbitrarypotential forms

H2D =1

2I�p�

2 + p�2 /sin2 �� + V��� , �9�

where p� and p� are the conjugate angular momenta of theEuler angles � and � and I is the moment of inertia of anassumed symmetric top rotation. The related partition func-tion is then obtained using the DOS defined by

�2D�E� =42I

h2 �1 − cos��max�E��� , �10�

where �max�E� is the maximum angle allowed in the range�0:180°� at energy E. They supposed that the Hamiltonian ofEq. �9� can properly describe the 2D rocking motions occur-ring in a dissociation reaction. However, this approach isvery questionable with respect to the way the kinetic func-tion I is calculated. First, as in the work of Pacey,27 thekinetic function is defined by a constant moment of inertia.Second, while in the work of Pacey this constant is obtainedusing all geometric parameters of the system �see Eq. �8��, itis here assumed to be the moment of inertia of each fragmentaround its own pivot point. This formulation of I is clearlyunadapted because the moment of inertia of the rotor has todepend on the two fragments, as shown in Eq. �8�.

We propose in this paper a generalization of the 1D �Eq.�1�� and 2D �Eq. �9�� Hamiltonians in which the moment ofinertia is replaced by an angle-dependent kinetic function.This kind of function has been presented for multidimen-sional fully coupled studies4,15,21,39 but usually leads to com-plex expressions that are most of the time system-specific.Instead, we provide here a numerical, simple, and generalway to compute the kinetic function in the approximation of

a rigid rotor. The method takes into account all the relevantparameters: �i� fragments geometries, �ii� pivot points loca-tion, and �iii� direction of rotation. Then, different ap-proaches are presented in order to solve the Hamiltonianswhich can be �i� 1D classical, �ii� 1D quantum, or �iii� 2Ddegenerated classical. In particular, we propose a new way tocompute an effective potential and a temperature dependentmoment of inertia, allowing the direct use of some of theprocedures reported in previous works but in a more generalscheme. These methods are finally applied to study the atomtransfer reaction SiCl3+BCl3→SiCl4+BCl2, providing agood support to the discussion.

II. METHODS

A. Extending the unidimensional HR model

To describe our generalization of the HR models we firstconsider the general case of a 1D internal rotation as therelative rotation of two molecular fragments around a rota-tion axis. Invoking the Lagrange–Hamilton formalism, theHamiltonian of this movement is

H1D =p2

2f���+ V��� , �11�

where � is the rotation angle, p=−i��� /��2� �in the quantumformalism�, V��� is the rotational hindrance potential, andf��� is the kinetic function defined by

f��� = �i

mi� �Ri�����

�2

�12�

in which Ri and mi are, respectively, the position and mass ofatom i. Comparing Eqs. �1� and �11� we easily recognize thatthe 1DHR model is a specific situation of constant kineticfunction.

We present now a simple way to compute f��� within arigid-rotor approximation, assuming that the overall rotationand translation are not coupled to the internal motions. As aconsequence, both the linear and angular momenta are heldfixed to zero. The calculation of this function is illustratedFig. 2. The general case of an internal rotation is schematized

FIG. 1. Schematic representation of a triatomic system A–B2 where atom A�white disk� rotates around O, the center of mass of the diatomic B2 �blackdisks�. r is the bond length of B2, R is the distance between atom A and O,� and � are the Euler angles defining the rotation.

FIG. 2. Schematic representation of the kinetic function calculation. �a�Initial geometry representing the FF �black filled circle�, the pivot point p�black filled diamond�, the rotation axis v �perpendicular to the verticalplane�, and the MF �empty black circle�; �b� geometry obtained after arotation of � of the MF �black: current geometry, gray: initial geometry�;�c� geometry after cancellation of the global translation; �d� geometry aftercancellation of the global rotation.

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Fig. 2�a�. The molecule is divided into two counterparts: afixed fragment �FF� associated to a pivot point p and to arotation axis v around which rotates a mobile fragment �MF�.The relative atomic positions Ri

rel after a rotation of angle�, illustrated Fig. 2�b�, are given by

Rirel��0 + �� = PRi

0��0� , �13�

where P is a relative rotation operator Mrel�� ,p ,v� if atomi�MF and P=Id otherwise. In this equation, the Ri

0 are theinitial atomic coordinates corresponding to the rotation angle�0. Such a rotation always leads to nonzero linear and angu-lar momenta, an error that has to be corrected using both anoverall translation and an overall rotation. The overall trans-lation is a trivial center of mass correction and an interme-diate geometry �Fig. 2�c�� verifying Ptot=0 is obtainedthrough

RiPtot=0��0 + �� = Ri

rel��0 + �� − Grel + G0, �14�

where GX is the center of mass of geometry RiX.

As the overall rotation takes place in the same plane asthe internal rotation of Eq. �13�, the cancellation rotation axisis thus parallel to �v�. Also, it has to pass through the centerof mass G0=GPtot=0 in order for the latter to be conserved.The overall rotation angle �opt is thus the one able to cancelthe angular momentum with respect to the initial geometry

dJ��opt� = �i

�Mall��opt�RiPtot=0��0 + �� − Ri��0��

� �Ri��0� − G� = 0, �15�

where Mall is the overall rotation matrix �symbol�stands fora vector cross product�. �opt is numerically determined byminimizing dJ in �−� ;��, as pictured on Fig. 2�d�, andthe final atomic positions Ri

f after cancellation of linear andangular momenta are given by

Rif��0 + �� = Mall��opt�Ri

Ptot=0��0 + �� . �16�

The kinetic function at �0 can thus be evaluated using finitedifferences through

f��0� = �i

mi�Rif��0 + �� − Ri

0��0��

�2

�17�

and the kinetic function f��� is constructed step by step bysetting �Ri

0��0��k+1= �Rif��0+���k where k is the step num-

ber. This having no computational cost, as many steps asrequired can be achieved.

The simple procedure presented here can be compared tothe approach published some years ago by Katzer and Sax.40

These authors also used a numerical method, based on adiscrete set of geometries along the rotation path, to deter-mine the moment of inertia associated to a free pseudorota-tion. Our approach has the advantage of allowing the deter-mination of variable kinetic functions, often required to treathindered rotations. Also, it can be extended to non rigid ro-tation as is the case of the Katzer and Sax method.

B. Solving the Hamiltonian in classical and quantumformulations

At that point, we have at hand a much more accuratedefinition of the kinetic function than previously used in the1DHR model. The calculation of the classical DOS and par-tition function are realized through direct numerical integra-tion

��E� =1

h

0

2 −

��H�p,�� − E�d�dp , �18�

Q�T� =1

h

0

2 −

exp�− H�p,��/�kBT��d�dp . �19�

The classical calculation of the DOS is known to be suffi-ciently accurate to treat weakly hindered motions. However,a quantum resolution might be more suitable in other casesand especially at low temperatures. In order to easily solveEq. �11� using the FGH1D algorithm in the quantum formal-ism, a constant kinetic function is required. In that purpose,we define here an effective, temperature dependent, momentof inertia �Ieff� as

Ieff�T� = 0

2

f���P��,T�d� , �20�

where

P��,T� =exp�− V���/kBT�

Z�T��21�

captures the influence of the temperature on the occupancyof different values of � and so on different values of f���, inclose similarity with a Maxwell–Boltzmann averaging. In-deed, Z�T� is simply defined as

Z�T� = 0

2

exp�− V���/kBT�d� . �22�

Although some explicit dependence is lost during the defini-tion of this effective kinetic constant, we will show in Sec.III C 2 that this approximation is extremely reasonable bycomparing classical entropies obtained using either the realkinetic function f��� or the effective one Ieff.

Using the definition of Ieff�T� we can thus write a tem-perature effective 1DHR quantum Hamiltonian as

Heff1D�T� = −

�2

2Ieff�T��2

��2 + V��� . �23�

The determination of the temperature dependent eigenvalues��i

eff�T�� can thus be achieved using the FGH1D algorithm.9

This also allows us to obtain the temperature effective DOSgiven by

�eff1D�E,T� =

1

dE�rot�

i

exp�− �ieff�T�/kBT���E − �i

eff�T�� ,

�24�

where ��x�=1 if x� �−dE /2;dE /2� and ��x�=0 otherwise.The corresponding partition function is given by

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Qeff1D�T� =

1

�rot�

i

exp�− �ieff�T�/kBT� . �25�

In what follows, the generalized 1DHR model with an ex-plicit variable kinetic function in the classical formalism willbe noted “1DC,” the model with the effective constant in theclassical formalism “1DCe,” and the model with the effec-tive constant in the quantum formalism “1DQ.”

C. Extension to a degenerated 2D rotor

Similarly to the way we have extended the 1DHRmethod to the case of nonconstant kinetic functions, we natu-rally generalize the expression of the 2D degenerated hin-dered rotor model of Jordan et al.36 �Eq. �9�� as

Heff2D��,p�,p�,T� =

1

2Ieff�T��p�

2 + p�2 /sin2 �� + V��� . �26�

Extending the work of Jordan et al. to a hindrance potentialfunction V��� having possibly several accessible � domains,we obtain a general expression for the density of states

�eff2D�E,T� =

42Ieff�T��roth

2 �i=0

n

�cos��imin�E�� − cos��i

max�E��� ,

�27�

where �imin�E� and �i

max�E� are, respectively, the minimumand maximum values of � accessible around the ith mini-mum at a given energy E ��rot is the rotational symmetrynumber�. The partition function is thus given by a directintegration of the DOS over the energy

Qeff2D�T� =

0

�eff2D�E,T�dE . �28�

In analogy to the 1DC and 1DQ notations, we will refer tothis generalized degenerated 2D hindered rotor model as the“2DC” model.

III. APPLICATION TO THE REACTIONSiCl3+BCl3\SiCl4+BCl2

A. Computationnal details

All quantum chemical calculations are carried out usingthe GAUSSIAN 03 package.1 This includes geometry optimi-zation of reactants, products, and transition state, achieved atthe B3LYP/6–31g�d� level of theory. Vibrational frequenciesand zero point energies �ZPE� are also calculated using thismethod. The usual scaling factor of 0.96 is applied to theZPE. Single point energies �reactants, products, and TS� arerefined using the G3B3 �Ref. 41� method, known to provideaccurate electronic energies, generally within 2 kcal/mol.42

The hindering potentials V��� obtained from geometry opti-mizations �actually, TS optimizations as proposed byPfaendtner et al.5� at discrete values of the angle �, are cal-culated at the B3LYP/6–31g�d,p� level of theory. The result-ing energy profiles are fitted with polynomials for the wag-ging and rocking potentials, while the torsional motion isconsidered as a free rotation. Kinetic functions f��� are ob-tained for each internal motion using the method described in

Sec. II A, and fitted with analytical Fourier expressions. Thetemperature dependent moments of inertia Ieff�T� are directlycomputed through the numerical integration of Eq. �20�. TheFGH1D resolutions of Eq. �23� are achieved using 501 basisfunctions.

B. PES and TS

The energy barriers of the reaction are 22.3 kcal/mol inthe forward direction and 14.0 kcal/mol in the reverse direc-tion. The transition state, illustrated on Fig. 3, is character-ized by a chlorine atom �Cl�� partially bound to both thesilicon and boron atoms. Its main geometric parameters areB–Cl� and Cl�–Si distances of, respectively, 2.18 and 2.26

Å and a Cl�BSi angle of 25.4°.Among the 18 vibrational frequencies of these TS, 14

are relatively high �above 90 cm−1�. They correspond tostrong intramolecular bond stretching or angle bendingmodes. Another one, corresponding to the intrinsic reactioncoordinate, is imaginary �369i cm−1�. Finally, and as re-vealed by a normal mode visualization, three of them havelow frequencies and correspond to internal rotations of BCl2and SiCl3 around Cl�. The first one has a frequency of5.4 cm−1 and corresponds to the torsion around the dihedralangle � defined by ClB–B–Si–ClSi. During this motion thecentral chlorine atom Cl� moves close to the B–Si axis. Thesecond one is the wagging motion of the two molecularcounterparts around Cl� in the vertical plane of Fig. 3. It is

associated to a variation of the Cl�BSi angle �noted � in whatfollows� and has a frequency of 19.8 cm−1. Finally, the lastone is the rocking motion of the two counterparts around Cl�

in the horizontal plane of Fig. 3. Its frequency is 28.6 cm−1.In what follows, we first give an uncoupled 1D treatment

of these three internal rotations using the new 1DC, 1DCe,and 1DQ methods. To do so, we consider the reference ge-ometry illustrated Fig. 4�a�. This geometry is obtained by arotation of 25.4° of the � angle from the TS such that �=0.Indeed, as we consider the three rotations to be independentfrom each other, there should not be any coupling betweenthis wagging in the vertical plane and the rocking and torsionmodes. This geometry, with aligned Si–Cl�–B atoms, alsosimplifies considerably the study of these modes. Further-more, this reference TS �RTS� of Fig. 4�a� having slightlyweaker vibrational frequencies, and considering the very lowenergy difference with the actual TS ��0.4 kcal/mol�, its free

FIG. 3. Main geometric parameters of the transition state involved in thereaction SiCl3+BCl3→SiCl4+BCl2 as obtained from a B3Lyp/6-31G�d,p�optimization.

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energy should be lower than the one of the real TS whentemperature rises. It thus probably provides a better referencepoint to define the torsional and rocking motions under theassumption of independent modes. In this uncoupled 1Dtreatment, the wagging motion is thus defined by the varia-tion of � in the vertical �V� plane, the rocking by the varia-tion of � in the horizontal �H� plane and the torsion by thevariation of the previously introduced � angle. In order togive an account for the coupling between wagging and rock-ing modes, we also treat them in a second step using theproposed 2DC method. In that case, the two relevant rotationangles, defined in the Hamiltonian of Eq. �26�, are shownFig. 4�b�.

C. Treatment of the internal rotations

1. Torsional motion

The hindrance potential associated to the torsion is givenin Fig. 5. As can be seen on this figure, all energy barriers arelower than 0.004 kcal/mol. This motion can thus be consid-ered as a free rotation. In the 1DHR model, the kinetic func-tion is assumed to be a constant, identified as the reducedmoment of inertia defined in the work of Herschbach et al.8

All the Iij �i=2,3 ; j=1,2 ,3� give almost the same value of418.7 amu bohr2. The numerical method proposed inSec. II A gives a constant and similar value of the kineticfunction: 418.9 amu bohr2; whatever the value of � is. As aconsequence, the 1DHR and 1DQ treatments are identical.We plot on Fig. 6 the entropies and partition functions of thistorsional mode obtained within 1DQ and harmonic treat-ments. The torsional density of states and the partition func-tion in the 1DQ model have been computed according toEq. �25� with �rot=6. We observe a dramatic overestimation

of both the entropy and partition function when the HOapproximation is used. This is not surprising for such apoorly hindered motion.

2. Separable wagging and rocking motions

The potentials associated to the wagging and rockingmotions are presented on Fig. 7. They are obtained fromrelaxed TS optimizations, constraining the vertical �resp.horizontal� mirror symmetry, at discrete values of � �resp. ��,for, respectively, the wagging and rocking motions. As canbe seen on this figure, the rocking hindering potential has asingle minimum and seems rather compatible with a HOtreatment. On the opposite, the wagging motion exhibits twominima �around �30°� separated by a weak maximum at theRTS geometry ��=0�. For this particular motion, the har-monic approximation should definitely be overtaken.

We plot on Fig. 8 the numerical kinetic functions of theindependent wagging and rocking motions. During these cal-culations, only the internal degrees of freedom associated tothese motions, namely, the angles � and � for, respectively,the wagging and rocking motions are allowed to vary. Notethat some test calculations with other values of the B–Cl� orSi–Cl� distances have shown to have no effect on the kinetic

FIG. 4. Reference geometries used in the uncoupled �a� 1DC-1DQ and �b�2DC treatments of internal rotations. �a� The independent torsion, wagging,and rocking are, respectively, defined by angles �, �, and �; �b� the torsionis defined by the � angle and the coupled wagging and rocking are definedby the Euler angles � and �. �labels H and V stand, respectively, for thehorizontal and vertical planes�.

FIG. 5. Energy as a function of the torsional angle � as computed fromrelaxed TS optimizations at the B3LYP/6–31G�d,p� level of theory.

FIG. 6. Torsional entropies �black� and partition functions �gray� obtainedfrom 1DQ �straight line� and HO �dashed line� treatments.

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functions. As we can see, in contrast to the torsional motion,the variation observed on the kinetic functions with respectto the rotation angles are large �from 1 to �4�. Combiningkinetic and potential functions according to Eq. �20� allowsus to determine an effective, temperature dependent, momentof inertia for each of these two motions, shown on Fig. 9. Aswe can see on this figure, Ieff slightly increases from 445 to500 amu bohr2 for the wagging and from 560 to595 amu bohr2 for the rocking when the temperature goesfrom 0 to 1500 K. This is explained by the fact that whentemperature rises, configurations out of the energy minima,associated to higher kinetic functions, are becoming morepopulated.

We now show on Fig. 10 the entropies obtained for thetwo motions within the different 1D approximations: HO,1DQ, 1DCe, and 1DC. The corresponding partition functionsare also displayed as an inset to Fig. 10. We see immediatelythat all the 1DHR approaches show an almost perfect agree-ment whatever the temperature is. A similar agreement isobserved on the partition functions although some slight de-

viations can be seen between the different models. Neverthe-less, the excellent agreement between the exact classical�1DC� and effective classical �1DCe� entropies and partitionfunctions validates the use of an effective kinetic constant, atleast in the case presented here. Moreover, the good agree-ment between the effective classical �1DCe� and effectivequantum �1DQ� results shows that the classical approxima-tion performs well for this system; this might not be the casefor systems involving lower kinetic functions. We comparenow the 1DQ and HO treatments. Although the differencesobserved on the entropies given by these two models neverexceed 1 cal/mol/K in the range of 50–1500 K, some pointsneed to be noted. For the rocking motion, the two approachesare in excellent agreement at low temperatures �both for theentropy and the partition function�: this is not surprising be-cause of the almost harmonic shape of the potential close toits minimum. When temperature rises, the difference in en-tropy increases, and at 1400 K, the HO model overestimatesthe rocking entropy by around 0.6 cal/mol/K with respect tothe 1DQ model. Looking now at the wagging entropy, we see

FIG. 7. Energy as a function of the rotation angle for the flexural motions ascomputed from TS optimizations at the B3LYP/6–31G�d,p� level of theory.Full circles: wagging, DFT calculations; empty circles: rocking, DFT calcu-lations; straight line: wagging, polynomial fit; dashed line: rocking, polyno-mial fit.

FIG. 8. Kinetic function of the wagging �straight line� and rocking �dashedline� as a function of the rotation angle.

FIG. 9. Effective kinetic function of wagging �straight line� and rocking�dashed line� as a function of temperature.

FIG. 10. Entropy as a function of temperature for 1D separable wagging�black� and rocking �gray� motions. Squares: 1DQ model; triangles: 1DCemodel; straight lines: 1DC model; dashed lines: HO approximation. �Thecorresponding partition functions are displayed on the inset.�

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on Fig. 10 that at 50 K, the HO model shows values lower by1 cal/mol/K than the 1DQ model. However, the two entro-pies evolve rather differently with temperature, and at1400 K the situation is inverted: the entropy is around0.6 cal/mol/K lower for 1DQ than for HO. The differentevolutions with temperature observed for these two modelscan probably be explained with simple arguments. On onehand, the HO model considers a unique wagging potentialwell, which induces an underestimation of the entropy at lowtemperatures. On the other hand, at higher temperatures, theHO model is unable to deal with a complex potential formsuch as the wagging potential of Fig. 7. The fact that thewagging entropy predicted by the HO model is within1 cal/mol/K of the one obtained by the 1DQ model in theinvestigated temperature range is thus due to a compensationof errors.

3. Coupled rocking and wagging motions

In order to assess the separability of the wagging androcking motions, we now present a coupled approach basedon the 2DC model presented Sec. II C. In this case, themovement is described using the rotation angles � and � asdefined in Fig. 4�b�. Because this treatment �based on theHamiltonian of Eq. �26�� can only deal with a degenerated2D rotor defined by 1D potential V���� and kinetic f����functions, we have to choose a value of � at which thesefunctions are determined. In what follows, we apply this ap-proach to the two limiting cases of doubly degenerated wag-ging �noted 2DCw� and rocking �noted 2DCr� motions de-fined, respectively, by �=0 and �=90°. We plot on Fig. 11the entropies obtained with these two models �full symbols�and compare them to those obtained by taking twice theseparable wagging �1DQw� and rocking �1DQr� entropies.First, Fig. 11 shows that choosing a given motion to betreated with the 2DC formalism has a non-negligible influ-ence on the obtained entropy �and partition function�. In-deed, treating the internal rotations with the degenerated 2Dwagging �2DCw� leads to a substantially higher entropy than

what is obtained with a degenerated 2D rocking �2DCr�. Thedifference is �2 cal/mol/K at low temperatures and de-creases to �1 cal/mol/K at 1400 K. Second, the comparisonof these 2DC entropies with twice the 1DQ entropies of thecorresponding 1D motions allows us to assess the separabil-ity hypothesis of degenerated 2D wagging and rocking mo-tions. Indeed, as can be seen on Fig. 11, S2DCr

2�S1DQrfor

the whole temperature range, indicating that this degenerated2D rotor can be separated into two identical and independent1D motions. Although some agreement is observed at lowtemperatures, S2DCw

becomes increasingly lower than 2�S1DQw

when rising the temperature, indicating that such aseparation is less accurate for this particular motion. Similarconclusions are obviously drawn when we look at the corre-sponding partition functions in the inset of Fig. 11.

Finally, we present on Fig. 12 three estimations of thetotal entropy resulting from these two internal rotations:�i� the sum of the two individual entropies under the HOmodel; �ii� the sum of the two individual entropies under the1DQ model; and �iii� the average of the two 2DC entropies.We can see first that the independent 1DQ and the average ofthe two 2DC treatments give extremely close results with amaximum difference of �0.46 cal/mol/K at the highest in-vestigated temperature �1400 K�. We are not surprised to finda good agreement between these two models at low tempera-ture since the separability assumption should hold. To ourminds, there is no immediate answer about which of thesetwo approaches provides the best estimate of the entropy, asthey are based on distinct sets of assumptions. However, con-sidering that the rocking motion is nearly harmonic—see itspotential function �Fig. 7� and compare its 1DQ and HOentropies �Fig. 10� —it can probably be decoupled from thewagging, thus justifying a 1DQ treatment. Finally, we see onFig. 12 that the entropy obtained within the HO approxima-tion compares very well to the other one at low temperaturesand departs substantially from them when the temperatureincreases. If this discrepancy at high temperatures is not re-ally a surprise, the extremely good agreement at low tem-

FIG. 11. Entropy as a function of temperature obtained using the 2DCmethod. Full diamonds: wagging motion �2DCw�; Full triangles: rockingmotion �2DCr�. Also shown on this figure are twice the entropies obtainedusing the 1DQ approach. Empty squares: 2�S1DQw

; empty circles: 2�S1DQr

. �The corresponding partition functions are displayed on the inset.�

FIG. 12. Combined entropy of the two internal rotations �rocking and wag-ging� as a function of temperature. Straight line: sum of the two 1DHRentropies �S1DQw

+S1DQr�; dotted line: average of the two 2DHR entropies

� 12 �S2DCw

+S2DCr��; dashed line: sum of the two 1D HO entropies �SHOw

+SHOr�. �The corresponding partition functions are displayed on the inset.�

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perature requires a closer look. Looking at Fig. 10 we seethat it is only due to a lucky compensation of errors, the HOapproximation overestimating the rocking entropy and un-derestimating the wagging one by very similar amounts.

D. Rate constant

Combining now information gathered on the PES withthe partition functions determined previously, we can deter-mine the rate of the reaction within the formalism of thetransition state theory.43 We show on Fig. 13 the Arrheniusplots obtained with a full HO treatment of all vibrations andwith a specific treatment of the internal rotations: �i� an un-coupled 1DQ treatment of the three internal rotations; �ii� the1DQ treatment of the torsion and a 2DC treatment of thewagging motion �k2DCw

�; �iii� the 1DQ treatment of the tor-sion and a 2DC treatment of the rocking motion �k2DCr

�; and�iv� the arithmetic mean of the two 2DC rates �k2DCwr

�. Ascan be seen on this figure, the rate constants obtained withthe full HO model are much higher �by a factor 10� thanthose obtained by the other models. Among the latter, k2DCwand k2DCr

provide, respectively, the highest and lowest pre-dictions. Because of the underlying assumptions used, theyalso, respectively, represent an upper and lower bound of thereaction rate. The 1DQ treatment of the three rotational mo-tions provides an intermediate result, which is very close tothe average k2DCwr

. As already said, the 1DQ model probablyprovides the best estimation of this reaction rate. Based onthis result we recommend the Arrhenius parameters A=2.47�1013 cm3 /mol /s and Ea=26.4 kcal /mol to renderthe kinetics of this reaction.

IV. DISCUSSION

In this paper we have presented several methods allow-ing for the characterization, in a simple frame, of one and 2Dinternal rotations. Based on general expressions for hinderedrotations, we have proposed some extensions of the existingHamiltonians in order to take into account a variable kinetic

function. Doing so, we have extended the 1DHR and degen-erated 2DHR procedures to uncoupled 1D and degeneratedcoupled 2D wagging and rocking rotations for any pivotpoint. These new methods are thus particularly suitable tostudy dissociation and/or atom transfer reactions. The vari-able kinetic functions are calculated using an efficient andsimple numerical method in the rigid rotor approximation. Itproperly takes into account all the relevant parameters of thesystem: the geometry of the rotating fragments, the locationof the pivot point, and the direction of rotation. More spe-cifically, this variable kinetic function has been developedfor a classical treatment of 1D hindered rotors �the 1DCmodel�. In order to extend it to the cases of quantum 1Drotors �1DQ� and classical degenerated 2D rotors �2DC�, wehave then proposed the definition of an effective, tempera-ture dependent, kinetic function, obtained from a Maxwell–Boltzmann averaging of the real kinetic function.

As a case study, we have applied these methods to thetransition state of the atom transfer reaction SiCl3+BCl3→SiCl4+BCl2. Comparing the different models—full HOs,uncoupled classical 1D internal rotations, uncoupled quan-tum 1D internal rotations, and coupled degenerated classical2D internal rotations—we have concluded that �i� the har-monic approximation is insufficient to render the entropy ofthe system �leading to a ten times higher reaction rate�; �ii�1D quantum and classical treatments give almost equal re-sults; �iii� the two wagging/rocking modes can be consideredindependent; and �iv� the use of an effective constant �re-quired for 1D quantum or 2D classical treatments� is an ac-curate solution. Finally, this paper provides a theoretical es-timate of this reaction rate for the first time.

ACKNOWLEDGMENTS

We thank Dr Cedric Descamps �Snecma PropulsionSolide, LCTS� for useful discussions and comments aboutthis work. All the calculations presented here have been per-formed on the supercomputers of the Mésocencenter de Cal-cul Intensif en Aquitaine �MCIA�. Finally, Snecma Propul-sion Solide-SAFRAN Group is gratefully acknowledged byG.R. for financial support.

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