Computational and Theoretical Chemistry 978 (2011) 88–97

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Computational and Theoretical Chemistry

journal homepage: www.elsevier .com/locate /comptc

DFT characterization of the first step of methyl acrylate polymerization:Performance of modern functionals in the complete basis limit

Ilhan Yavuz a,1, Gökçen Alev A. Çiftçioglu b,1

a Department of Physics, Marmara University, 34722 Istanbul, Turkeyb Department of Chemical Engineering, Marmara University, 34722 Istanbul, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 August 2011Received in revised form 27 September 2011Accepted 28 September 2011Available online 8 October 2011

Keywords:Density functional theoryFree-radical polymerizationFocal point analysisHindered rotor approximationMethyl acrylate

2210-271X/$ - see front matter � 2011 Elsevier B.V.doi:10.1016/j.comptc.2011.09.043

E-mail addresses: ilhan.yavuz@marmara.edu.tr (I. Yedu.tr (Gökçen Alev A. Çiftçioglu)

1 Co-authors.

We obtain values of the reaction barrier for the reaction of methyl acrylate CH2@CHCOOCH3 (MA) withthe radical CH3CHCOOCH3 (HMA�) by density functional theory (DFT) using a variety of functionals andbasis sets. Structures for the reactants and the transition state are optimized in B3LYP/cc-pVTZ. Weextrapolate energies for these structures to the complete basis set (CBS) limit for each of the functionalsB3LYP, PBE, TPSS, BMK, HSE2PBE, mPW1PW91, B97-1, wB97-XD, and M06-2X. The extrapolation followsthe energies obtained by the basis sets cc-pVnZ with n = 2, 3, and 4. The estimate of the barrier height issensitive to the basis and the choice of functional.

In order to recover the rate constant for the radical addition we require partition functions as well asthe barrier height. To obtain the partition functions for internal rotation in MA, the radical HMA�, and thetransition state for their addition HMAMA�(TS), we trace one-dimensional torsional potentials in B3LYP/cc-pVTZ. Using this data we employ a range of approximations to the partition function ranging from theharmonic oscillator limit, interpolation schemes linking the harmonic oscillator and free rotor limits, andsemi-classical expressions. Comparison with the partition functions obtained by direct sum of Boltzmannfactors with energy eigenvalues obtained by solution of the Schrödinger equations (total eigenvalue sumor TES) for the one-dimensional torsional potentials show that Mielke and Truhlar’s TDPPI-HS approxi-mation is very accurate.

Estimates of activation energies and rate constants for the addition reaction based on the modern func-tionals wB97-XD and M06-2X in the CBS limit and the TES partition functions reproduce the best exper-imental measurement.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Free radical addition has generally been described within theabsolute rate theory [1], which represents the rate according to

kðTÞ ¼ kBTh

Q z

Q MQ Re�DEz

0=RT ð1Þ

The reaction barrier DEz0 and partition functions for the acti-vated complex Q�, the monomer QM, and the radical QR are requiredto evaluate this expression. Quantum chemical methods can giveall the information necessary for evaluation of the rate constantunder the assumptions that: (1) energy is separable into termsdealing with rotation, vibration, and electronic contributions; (2)vibrations are separable into normal modes of motion; and (3) nu-clear motion takes place on a single potential surface defined bythe electronic energy of a single state.

All rights reserved.

avuz), gciftcioglu@marmara.

The model for radical polymerization is that (1) an initiator I2

is decomposed by heat or light to radicals I�. (2) The radical addsto a monomer M to form the first of a sequence of larger radicalsIM1

�. (3) Propagation extends the chain length according toM + IMj

�? IMj+1�. These addition reactions are called ADj in the

work we describe below. Finally (4) termination ends the growthof the polymer, either by a coupling reaction or by dispropor-tionation.

Most attention has been given to the first few propagationsteps. For methyl acrylate the minimal model of the first propaga-tion step AD1 involves the species shown in Fig. 1, in which MA isthe monomer, HMA� represents the radical and the transition stategeometry is represented by HMAMA�(TS).

The initiation step has not been overlooked. Dossi et al. [2]studied the addition of initiating radicals methyl (Me�), phenyl(Ph�), benzoyl (BzO�), tert-butoxy (tBuO�), and 2-cyanoprop-2-yl(CNP�, derived from 2,20-azoisobutyronitrile, called AIBN) to themonomers methyl acrylate (MA), methyl methacrylate (MMA),acrylonitrile, and styrene. They find reasonable agreement withexperimental enthalpies and activation energies for the addition

(a) MA (b) HMA•

(c) HMAMA• (TS)

Fig. 1. Structures for species in the addition reaction AD1: (a) MA, (b) HMA� and (c)HMAMA� (TS). Torsions are M1 = ethenyl-ester; M2 = carbonyl-methoxy; M3 = meth-oxy methyl; R1 methoxy-methyl; R2 = carbonyl-methoxy; R3 = C2H4 radical-ester;R4 = Methyl-CH radical site. The transition state has corresponding torsions,D1 M M3; D2 M M2; D3 M M1; D4 M R3; D5 M R2; D6 M R1; D7 M R4; D8 is thetorsion about the bond forming in the transition state. We found and confirmedB3LYP/6-31G(d) coordinates reported by Coote [11], and computed correspondingcoordinates in B3LYP/cc-pVTZ. Notice that the ester groups are stacked in theoptimized structure; the carbonyls are reversed in orientation in this case, butalternatively the carbonyls could be aligned. The reversed form shown is favored byabout 3 kJ mol�1.

I. Yavuz, G.A.A. Çiftçioglu / Computational and Theoretical Chemistry 978 (2011) 88–97 89

reactions of methyl radical and the monomers within the modelB3LYP/6-31G(d) (see Table 1 for a guide to density functionals men-tioned in this report). A substantial improvement is effected byenhancing the basis to 6-311+G(d,p). Harmonic approximationswere used for partition functions for all vibrations. Computed acti-vation energies for the reactions of the several initiator radicalsand monomers range from about 40 kJ mol�1 for CNP� to about4 kJ mol�1 or Ph�; log(kI) ranges from 3 to 11. The activation energiesfor this reaction vary more weakly with the choice of monomer; formethyl radical, log(kI) ranges from 9 to 10. For the first propagationstep computations predict a range of log(kP) from 4.4 (acrylonitrile)

Table 1Density functionals mentioned in this work.

Abbreviation References

B3LYP Becke [24] and Lee et al. [25]B1B95 Zhao et al. [26] and Becke [27]MPWB1K Zhao and Truhlar [28]BMK Boese and Martin [29]BB1K Zhao et al. [30]MPW1K Zhao and Truhlar [25]MPW1B95 Zhao and Truhlar [25]M06-2X Zhao and Truhlar [31]wB97-XD Chai and Head-Gordon [32]HSE2PBE Heyd and Scuseria [33]MPW1PW91 Zhao and Truhlar [25]B97-1 Becke [34]TPSS Tao et al. [35]PBE Perdew et al. [36]

to 5.5 (MMA). Corresponding experimental values are 4.7 and 6.3 sothe model seems to underestimate rates somewhat.

Perhaps the most through study specific to the MA system is thework of Yu et al. [3]. These authors pursued the following steps forAD1, AD2, and AD3:

1. Optimization of the structures of the monomer, radical, andtheir transition state by UB3LYP/6-31G(d).

2. Exploration of the effect of basis set choice with single pointcalculations with UB3LYP, using Pople basis sets ranging from6-31G(d) to 6-311G(3df,2p).

3. Exploration of the effect of functional choice with the 6-31G(d,p) basis, using B1B95 and MPWB1K, the latter parame-trized to treat barriers.

4. Treatment of low frequency internal motions either as har-monic oscillations or as hindered internal rotation.

These authors explored chain length effects and solvent effectsas well, which we do not discuss here. They observed that B3LYPbarrier heights for the addition of HMA� radical with monomerMA (corrected for zero-point vibrational energy) varied with basis,from 22.7 kJ mol�1 with 6-31G(d) to 28.0 kJ mol�1 with 6-311+G(d,p). The barrier heights for MPWB1K/6-31G(d,p) andB1B95/6-31G(d,p) were 18.3 and 20.4 kJ mol�1 respectively.

Results of modeling may be compared with the Arrheniusparameters from pulsed laser studies by Beuermann and Buback[4], A = 1.66 � 104 L mol�1 s�1 and Ea = 17.7 kJ mol�1. The activa-tion energy, recovered from regression of ln(kP) vs 1/T, dependsnot only on the reaction energy barrier but also on the treatmentof the low-frequency internal motions and the associated vibra-tional partition functions. When these were described in the har-monic approximation, B3LYP values increased from 30 to35 kJ mol�1, the MPWB1 K value rose to 25.5 kJ mol�1, and theB1B95 value became 27.9 kJ mol�1. Treating the motions as hin-dered rotations required tracing the potentials and evaluating par-tition functions for the torsional states by summing Boltzmannfactors referring to the computed eigenvalues. This correction in-creased barrier heights by 0.9 kJ mol�1 in reaction AD1. Effectsare more significant for later propagation steps AD2 and AD3,and in those cases tend to reduce barrier heights.

Degirmenci et al. [5] pursued the following steps:

1. Optimization of the structures of the monomer, radical, andtheir transition state by B3LYP/6-31+G(d);

2. Exploration of the effect of functional choice by single point cal-culations with the 6-311+G(3df,2p) basis, using BMK, MPW1 K,BB1 K, MPWB1 K, and MPW1B95;

3. Treatment of low frequency internal motions as harmonicoscillations.

These authors added methyl radical to MA to produce the reac-tant CH3MA� for their AD1 propagation step CH3MA�+MA ?CH3MAMA� (see Fig. 4 in their report). Computed values of log(kP)at 303.15 K show that use of the MPW1 functional improves rateconstant estimates substantially over use of the B3LYP functional.Regression of log(kP) computed with the model MPWB1K/6-311+G(3df,2p)//B3LYP/6-31+G(d) vs. 1/T over the range 250–350 Kproduced estimates of Arrhenius parameters A and Ea of

Table 2In(kP) at 303.15 K for reaction AD1 computed with various DFT functionals and the6-311+G(2df,2p) basis set (see Ref. [5]).

Exp [5] B3LYP BMK MPW1K BB1K MPWB1K MPW1B95

9.60 1.45 3.13 3.80 2.86 4.70 5.46

1 There are many ways of classifying functionals, all of which are confounded atleast in part by the proliferation of constructions. A clear discussion is provided by[31].

90 I. Yavuz, G.A.A. Çiftçioglu / Computational and Theoretical Chemistry 978 (2011) 88–97

3.28 � 103 L mol�1 s�1 and 24.3 kJ mol�1, respectively. These areagain to be compared with experimental values of A = 1.66 � 107

L mol�1 s�1 and Ea = 17.7 kJ mol�1. The values of log(kP) at 303.15 Kfor various choices of functional are shown in Table 2.

Izgorodina and Coote [6] have examined the free radical propa-gation reactions for a range of species, (notably acrylonitrile) andhave also studied the accuracy of a variety of approximations tothe torsional partition functions [7]. Particularly accurate calcula-tions are obtained in the G3(MP2)-RAD adaptation of thermochem-ical methods [8]. Izgorodina and coworkers used G3(MP2)-RAD toevaluate the reliability of DFT methods for thermochemistry ofradical reactions [9]. In the case of the AD1 reaction at issue in thisdiscussion, the following sequence:

1. Optimization of the structures of the monomer, radical, andtheir transition state by B3LYP/6-31G(d).

2. Use of the G3(MP2)-RAD scheme for single-point energyestimates.

3. Treatment of low frequency internal motions as harmonic oscil-lations, with scaling of the B3LYP/6-31G(d) frequencies;

produced a reaction rate of 2 � 104 L mol�1 s�1 in remarkableagreement with experiment [10]. Oddly, the ZPE-corrected barrierG3(MP2)-RAD seems very low, 4.8 kJ mol�1, while the ZPE-cor-rected barrier in B3LYP/6-31G(d) is 20.2 kJ mol�1 according toGaussian logs supplied in Supplementary data [11].

While all the reports mentioned above made considerable con-tributions to understanding of the initiation and early stages ofpropagation in the I(MA)j�–MA radical polymerization, the roughly60-fold discrepancy between the best model of the rate constantfor AD1 and the experimental value suggests that further refine-ment in the model is required. In this work we take three stepsin this direction: the first is to describe the basis set dependenceof the reaction barrier in the systematic fashion suggested by thefocal point analysis (FPA) [12], and the second is to employ somemore recently developed density functionals. The third is to incor-porate a variety of treatments of the internal rotation partitionfunctions. To anticipate the course of our work,

1. We optimize the structures of the monomer, radical, and theirtransition state by B3LYP/cc-pVTZ, a substantially larger basisthat has been employed heretofore;

2. We pursue the complete basis limit by single point energies atthe structures defined in step 1, using basis sets cc-pVnZ withn = 2, 3, and 4; these results are extrapolated to the completebasis limit;

3. We employ a variety of density functionals to evaluate reactionbarriers, including the familiar hybrid B3LYP, pure functionalsPBE and TPSS, extensively parametrized methods includingBMK and M06-2X, and methods incorporating long-range anddispersion corrections such as B97X-D;

4. We survey a variety of treatments of low frequency internalmotions beyond the harmonic approximation, including thedirect sum of Boltzmann factors for energy levels of the tor-sional potentials defined in B3LYP/cc-pVTZ.

2. Basis-set extrapolation of the reaction barrier for addition ofHMA to MA

We estimate the reaction barrier for addition of MA with aHMA� radical produced by addition of an H atom to MA, for severalchoices of functional and basis set.

The geometries of MA, HMA� and HMAMA�(TS) are optimized atthe B3LYP/cc-pVTZ level. The basis, a member of the family of Dun-ning correlation-consistent sets [13] cc-pVnZ with n = 2, 3,4, . . ., exceeds in flexibility any previous choice of basis for

structural determination. The vibrational frequencies and the zeropoint vibrational energy (ZPE) of each species are calculated forthese structures in the same model. Table 3 summarizes the re-sults, including a structural comparison with the B3LYP/6-31G(d)model. Differences in bond lengths are usually minor (less than0.01 Å), but the length of the forming bond is found to be about0.04 Å shorter in the B3LYP/cc-pVTZ calculations.

The simplest initiator, H atom, defines the simplest radical reac-tant as shown in Fig. 1. We substituted methyl radical for hydrogenatom as a first model of larger initiators. As Table 3 shows, themethyl substitution has almost no effect on the reaction barrier.The difference is found to be roughly 0.04 kJ mol�1.

We retain these B3LYP/cc-pVTZ structures and compute single-point reaction energy barriers for a variety of functional choices,and for reference, restricted open-shell MP2 values. We followthe changes in computed barriers as the basis set is systematicallyimproved. Results are collected in Table 4. The CCn notation refersto Dunning’s correlation-consistent series of basis sets cc-pVnZ.The smallest of these basis sets, CCD, contains 3 s functions, 2 pfunctions, and a single d function for first-row atoms C and O,and 2 s functions and one p function for H. These are given thenotation 3s2p1d and 2s1p, respectively. At each stage of improve-ment, when n increases to n + 1, the basis is enhanced with anadditional function of each angular momentum, and an additionalset with angular momentum l = n + 1; explicitly, CCT has 4s3p2d1ffor C and O, and 3s2pd for H. The CCD basis with 233 members forthe transition state is comparable in quality with the 6-31+G(d) ba-sis used for structure determination by Degirmenci et al. [5]. OurCCT choice for this task has 542 members. The 6-311+G(3df,2p) ba-sis used for single point energy estimates in previous work con-tains 585 members, while our CCQ basis has 1050 members.Extrapolation by the formula

EðXÞ ¼ ECBS þ A=X3 ð2Þ

(for which E(X) is the energy or property calculated with the basiscc-pVXZ) typically makes a further correction over the CCQ esti-mates of reaction barriers of less than 1 kJ mol�1. We evaluatedthe constants in the formula from our energy values in CCT andCCQ. The values [CC5] in Table 4 are estimates derived from Eq. (2).

Modern density functionals fall into two broad classes, whichwe might call constrained and parameterized.1 Constrained func-tionals are designed to conform with known properties of the den-sity functionals, while parameterized functionals incorporatequantities that can be chosen to enhance performance in any of sev-eral specific contexts. In either case, the functional is judged by a sta-tistical figure of merit reflecting its fidelity to experimental data.Density functionals may be categorized by their location on a hierar-chy called Jacob’s ladder [14]. The local density approximation calledLDA which refers only to local values of the density occupies the firstrung. Gradient-corrected (GGA) functionals the second; more rigor-ous treatment of the orbital kinetic energy defines the third level,also called meta-GGA. Hyper-GGA functionals include exact ex-change information, and are non-local. Finally, use of the virtualorbital space defines the highest rung, called RPA after the randomphase approximation for electronic excitations. The LDA approxima-tion is the common starting point. We have chosen PBE [15] as a rep-resentative of the GGA class, and TPSS [16] as a representative of themeta-GGA class.

Parametrized functionals are often hybrid in composition; thatis, they incorporate some amount of ‘‘exact’’ (non-local, Hartree–Fock) exchange. The extent of admixture is chosen empirically.B3LYP is the most familiar example. The Truhlar research group

Table 3ZPE corrected energy barrier of the MA polymerization. Geometry of MA, RMA� and RMAHMA�(TS) (R = H, Me) are optimized at B3LYP/cc-pVTZ level. Cf. data from Ref. [11].

HMA�+MA MA HMA� HMAMA�(TS) Barrier (kJ mol�1)

Energy (a.u.) �306.588612 �307.1684124 �613.7471406 25.95Gibbs Free Energy. (a.u.) �306.523912 �307.096106 �613.589043 81.32ZPE (kJ mol�1) 249.5875 273.7586 528.1322 4.79ZPE corrected Energy barrier. (kJ mol�1) 30.74

MeMA�+MA MA MeMA� MeMAMA�(TS) Barrier (kJ mol�1)

Energy (a.u.) �306.588612 �346.4956558 �653.074543 25.53Gibbs Free Energy. (a.u.) �306.523912 �346.397141 �652.889678 82.37ZPE (kJ/mol) 249.5875 349.226 603.7744 4.96ZPE corrected Energy barrier. kJ mol�1 30.70

Coote [11] MA HMA� HMAMA�(TS) Barrier (kJ mol�1)

B3LYP/6-31G(d) �306.4677530 �307.0484408 �613.5105425 14.84G3(MP2)-RAD �306.02164 �306.58567 �612.60547 auThermal corrections 19.9 21.8 40.1 14.2 kJ mol�1

HMAMA�(TS) structure B3LYP/6-31G(d) B3LYP/cc-pVTZ

C1AC2 1.3659 1.3620C2AC3 1.4713 1.4673C3AO5 1.3590 1.3513C3AO4 1.2202 1.2128O5AC6 1.4343 1.4341C1� � �C13 2.2986 2.2592C13AC15 1.4942 1.4888C16AO17 1.2206 1.2132C16AO18 1.3552 1.3516O18AC19 1.4374 1.4369C2@C1� � �C13AO16 60.40 63.75

ZPE and thermal corrections use scaled B3LYP/6-31(d) frequencies.Bond distances in Angstroms, The torsion angle in degrees.TS forming bond distances are shown in bold.

Table 4Extrapolation of ZPE corrected values of the energy barrier for various functional choices in the cc-pVnZ basis set sequence (units are in kJ mol�1).

B3LYP PBE TPSS BMK HSE2PBE mPW1PW91 B97-1 wB97-XD M06-2X ROMP2

CCD 21.39 0.67 11.09 19.26 10.53 16.98 10.01 3.41 6.67 5.54CCT 30.74 8.63 18.59 23.84 17.41 23.92 17.61 10.21 10.93 4.37CCQ 32.75 10.48 20.28 27.27 18.93 25.77 19.36 12.01 12.55 5.74[CC5] 33.65 11.20 20.98 26.89 19.58 26.32 20.05 12.55 12.72 6.23CBS 34.49 11.92 21.65 27.43 20.20 26.96 20.74 13.18 13.14 6.74

CC5 values are estimated by the extrapolation formula. The extrapolation formula parameters are defined by CCT and CCQ energy values.

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has developed a series of hybrid MPW functionals [17]: theMPW1K variant would be placed on the (hybrid) GGA rung, whileMPW B1K and MPW1B95 are on the (hybrid) meta-GGA level. TheM06-2X represents the hybrid hyper-GGA class. We have addedthe hybrid-meta BMK to our test set, along with the B97-1 disper-sion-corrected empirical functional and its range-corrected off-spring wB97X-D. Since we might imagine using a DFT methodadapted to extended systems for description of larger acrylate olig-omers, we have also included the HSE functional incorporatinglong-range screening of exchange, especially developed for con-densed phases.

Table 4 shows that the computed energy barriers are sensitiveboth to the functional and the basis. Among the density functionalchoices, B3LYP predicts the largest barrier, while wB97-XD andM06-2X give the smallest. The ROMP2 method predicts still smal-ler barriers. As the basis is systematically improved, the computedbarrier rises. Between the CCT basis and the extrapolated limit, wefind an increase in the barrier of about 2–3 kJ mol�1. In the case ofthe smaller barriers, this modest effect is significant.

A referee has suggested that tracing the spin density over thereaction path could provide some insight into the performanceof functionals. Our broad expectation is that since the transitionstate is early in the reaction (the newly forming Carbon� � �Carbonbond is quite long at the transition state, 2.25 Å), the spin willreside primarily on the HMA� fragment at the transition statestructure. We find that this is generally true. The spin popula-tions at carbons R (the nominal radical site in HMA�), T (thenominal radical site in the Adduct HMAMA�) and S (the site ofattack) comprise about 80% of the system’s total unit spin. Spindensities of the most active sites around the forming bond in ad-duct HMAMA� as labeled R, T and S, respectively, in the preced-ing discussion are shown in Table 5. The hybrid functionals thatproduce useful barriers retain more than 60% of a spin on theradical reactant HMA� in the transition state. B3LYP (which esti-mates a high barrier) transfers more spin to MA, while ROMP2transfers much less. Pure density functionals transfer still lessthan B3LYP but predict lower barriers. While we see hints of apattern, the picture is not very clear.

Table 5The spin populations at carbons R (the nominal radical site in HMA�), T (the nominal radical site in the Adduct HMAMA�) and S (the site of attack), calculated by various choices inthe cc-pVQZ basis set.

Barrier Large Small Accurate

Functional B3LYP TPSS MPW1PM91 ROMP2 PBE MO62X wB97XD

Spin R 0.600 0.572 0.627 0.733 0.542 0.649 0.640Spin T �0.175 �0.157 �0.210 0.011 �0.114 �0.200 �0.205Spin S 0.387 0.376 0.414 0.093 0.362 +0.374 0.383Total spin at RTS 0.812 0.791 0.831 0.837 0.790 0.823 0.818

92 I. Yavuz, G.A.A. Çiftçioglu / Computational and Theoretical Chemistry 978 (2011) 88–97

The Truhlar functionals have been developed with barrierheights expressly in mind, so we expect good performance fromMPW and M06-2X long-range corrections which give properasymptotic behavior of the exchange improve barrier heights[18] so we expect good results from wB97-XD. Broadly, the pres-ence of a substantial portion of exact (i.e., HF) exchange, either glo-bal or as a long-range feature of the functional, is necessary to goodperformance.

It is therefore not so outlandish that B3LYP overestimatesbarrier heights so seriously in this reaction type – it contains18% exact exchange, substantially less than MO6-2X (54%) orBMK (42%). The MPW1B95 and MPWB1K variants contain 31%and 44% respectively. It is more puzzling that ROMP2 so seriouslyunderestimates the reaction barrier, since HF exchange is fullypresent and one expects that the dynamical correlation wouldbe reasonably well comprehended by the many body perturbationcorrection, we can conjecture that the seriously stretched C� � �Cbond (ca 2.25–2.30 Å) in the radical addition transition state is asevere test of the second order perturbation theory. Small denom-inators in the perturbation correction term might overestimatestabilization.

The energy barrier is not immediately comparable with theArrhenius parameter Ea, the activation energy. To estimate thisquantity we need to evaluate the rate constant as a function oftemperature. This requires accurate estimates of the partitionfunctions for the reactants and transition state, which in turn re-quires a description of the energy terms for translation, overall(bodily) rotation, and internal motions. The internal motions inmany cases are well described as harmonic vibrations, but largeamplitude modes (internal rotations) require more detailedtreatment.

3. Torsional Motions for MA, HMA, and HMAMA(TS)

Here we characterize internal rotations for reactants and prod-ucts in the addition reaction called AD1, of MA with the radical de-rived by adding a H atom to the primary carbon of the alkenefragment R = HMA�, and the transition state for the addition

Table 6Guide to torsions in MA, HMA�, and the radical addition transition state, HMAMA�(TS).

Label Brief description Correspondence of torsionsin reactants to torsions inthe transition state

M1 C2 (ethenyl) vs ester D2 = M1M2 Methoxy vs carbonyl D3 = M2M3 Methoxy methyl D1 = M3R1 Methoxy methyl D4 = R1R2 Methoxy vs carbonyl D5 = R2R3 C2 (ethyl) vs ester D6 = R3R4 Methyl vs radical center D2 = M1D8 Torsion around the forming bond D8

D = HMAMA�(TS). Fig. 1 defines the torsion angles for MA (M1,M2, and M3); the radical (R1, R2, R3, and R4); and the transitionstate for the addition (D1. . .D8).

Table 6 summarizes the types and relations of the torsions.More explicit description of the potential for each torsional modeappears below.

4. Computation of the torsional potentials

In common with many investigations, we use a single modelchemistry to generate internal rotation potentials as well as thegeometries at extreme points, i.e. equilibrium and transition statestructures. The 6-31G(d) basis was used by Yu et al. [3] as well asLin et al. [7], while the 6-31+G(d) basis was chosen by Degirmen-ci et al. [5] Like these authors we used the B3LYP implementationin GAUSSIAN 03 [19] and also GAUSSIAN 09 [20] and exploredpotentials for MA, HMA� and HMAMA�(TS) in 6-31G(d) and cc-pVTZ basis sets. Each torsion is traced in 30� steps, in a ‘‘relaxed’’scan so that a geometry optimization is conducted for each choiceof angle. A rigid scan, in contrast, would enforce an assumptionthat individual torsions are exactly independent, and that eachreduced moment of inertia is independent of torsion angle. Therelaxed scan generally yields smaller values of the barrier heightsthan the rigid scan provides. Yu et al. [3] established that the 30�step is sufficient to define reliable Fourier coefficients for avariety of internal motions. The internal rotation potential isrepresented in Fourier series (see Eq. (3)).

VðsÞ ¼Xn

k¼0

½akð1� cosðksÞÞ þ bkðsinðksÞÞ� ð3Þ

We obtained ak and bk coefficients by least-squares regression onthe discrete set of computed V(s) values. We concluded that 10terms in the Fourier series is sufficient to represent the torsionalpotentials accurately. Values for the coefficients are to be found inthe Supplementary material, along with figures displaying the po-tential for each torsion. We characterize the motions as type A(methyl rotations), type B (ethenyl or ethyl rotations relative tothe COOMe ester, type C (methoxy rotation relative to a substitutedcarbonyl) and type D (the unique rotation around the forming bondin the transition state).

5. Torsion type A: methyl rotations

The potentials for methyl rotations have familiar threefold sym-metry. D1 is a methoxy methyl twist with a modest barrier, likeM3, the rotation of the methoxy methyl group in the monomer.D4 is another methoxy methyl twist with a modest barrier whichresembles the methoxy torsion R1. The low barrier height of lessthan 5 kJ mol�1 (note vertical scale) suggests that these are essen-tially free rotors. The potential is well described by a single cosineterm. R4 and D7 would seem to be closely analogous internal rota-tions, each well described by a single cosine term and for which thePitzer–Gwinn partition functions would serve. The barriers for R4

Fig. 2. In the transition state the distance of the forming bond is almost unchangedduring the D 8 torsion.

I. Yavuz, G.A.A. Çiftçioglu / Computational and Theoretical Chemistry 978 (2011) 88–97 93

and D7 are much higher than the barriers for D1 and M3: the prox-imity to the reactive center and the forming bond has an impact.Use of the larger basis cc-pVTZ basis set does not noticeably alterthe potential for these rotations. Graphs illustrating this remarkare to be found in Supplementary material.

6. Torsion type B: ethenyl and ethyl (C2 fragment) rotationrelative to ester

In the monomer, the rotation M1 of the ethylene fragment rel-ative to the ester fragment has two minima very close in energyand a relatively high barrier, ca. 30 kJ mol�1. Torsion R3, like M1,is a motion of the C2 fragment relative to the ester fragment. Thetwo minima are not far different in energy (ca. 5 kJ mol�1) butthe barrier in this case (the C2 fragment being CH3CH� rather thanCH2CH) is much higher, more than 45 kJ mol�1 in contrast to M1’s30 kJ mol�1. D2 is the torsion in the transition state that resemblesM1, while D6 is the torsion in the transition state that correspondsto R3. However, in each case the CC fragments in D2 and D6 are en-gaged in the bond forming in the transition state. The second min-imum in D6 is elevated but the barrier is slightly reduced. D3 is likeM2 an asymmetric potential with two distinct minima and highbarriers. The second minimum lies high, so the potential is approx-imately the superposition of two harmonic wells.

In all but D6, use of the larger basis cc-pVTZ basis set does notnoticeably alter the potential for these rotations. Even in this casethe general shape of the curve and the magnitudes of barriersagree.

7. Torsion type C: methoxy group rotation relative to the XCOfragment

The rotation M2 of the methoxy group relative to the ethenyl-carbonyl fragment is strongly opposed, with a barrier height ofmore than 50 kJ mol�1, and asymmetric; the 180� form is almost40 kJ mol�1 higher than the 0� form. Torsion R2, like M2, is a twistof the methoxy group vs. the substituted carbonyl. These motionseach have a high barrier (almost 50 kJ mol�1) and a strong prefer-ence (more than 30 kJ mol�1) for the zero-degree form shown inFig. 1. D3 and D5 are the correlates of M2 and R2, respectively. Eachpotential is asymmetric with two distinct minima and high barri-ers. Use of the larger basis cc-pVTZ basis set does not noticeably al-ter the potential for these rotations.

8. Torsion type D: rotation around the forming CC bond

Torsion D8 is the twist around the forming bond; in the scan asusual we allowed all degrees of freedom to be optimized, apartfrom the torsion D8. It was not necessary to constrain the formingbond’s distance; its value hardly changes during the torsion, pre-sumably because the gradient of the electronic energy is alwaysvery close to zero during the torsion. This is displayed in Fig. 2.

The B3LYP calculation in the larger cc-pVTZ basis reveals twosignificant barriers to rotation around the forming bond (seeFig. S4.1 in Supplementary material). In contrast, B3LYP in 6-31G(d) describes the torsion as almost perfectly unhindered. Thisis a consequence of the prediction by B3LYP/6-31G(d) of a greaterlength (2.2986 Å) of the forming bond in the transition state than isfound in B3LYP/cc-pVTZ.

9. Recovery of torsional energy eigenvalues from torsionalpotentials

Internal rotations are torsional motions about an axis RAB of onefragment of a molecule A, with moment of inertia IA, relative to

another fragment B with moment of inertia IB. The Hamiltonianfor internal rotation

H ¼ � h2

2Ir

d2

ds2 þ VðsÞ ð4Þ

refers to an angle s about a torsional axis RAB and a reduced mo-ment of inertia, Ir

1Ir¼ 1

IAþ 1

IBð5Þ

To define the terms in the Schrödinger equation for internal motion,we need the reduced moment of inertia for each relative rotation.For torsions within a molecule XY of a fragment X relative to theremainder of the molecule Y an axis of rotation must be chosen inorder to define a reduced moment of inertia. The scheme calledIð2;3Þr [21], used by many researchers in this field [3,6,7,22], definesthe axis of internal rotation as extending from the center of massof fragment X to the center of mass of the remainder Y.

We used the Fourier Grid Hamiltonian (FGH) method [23] withfine 2p/1000 radian mesh to obtain the eigenvalues of each tor-sional potential. The partition function is defined by the set ofeigenvalues for this Hamiltonian.

QHR ¼1rX

n

e�bEn ð6Þ

In Eq. (6), r is the symmetry number, En is the eigenvalue of the ntheigenstate, and b = 1/kBT (kB: Boltzmann constant).

10. Approximate partition functions for internal motion

It is hardly necessary to develop approximation to the simplermotions (center of mass translation, body rotation, and harmonicmotions with small displacements). However low frequency mo-tions, deriving from large-amplitude modes that can be describedeither as soft and often severely anharmonic vibrations referringto a single minimum energy structure or as more or less hinderedinternal rotations linking several minima. The most rigorous eval-uation of the partition function employs energy eigenvalues com-puted as described above. The result of this ‘‘Total EigenvalueSummation’’ (TES) serves as a reference for comparison of theapproximations to partition functions.

In the limit of very low barriers to internal rotation, the motioncan be described as free rotation, V(s) = 0, and the partition func-tion takes on a simple form:

Table 7Classes of approximations to partition functions for hindered rotors.

Type Methods Assumptions Requirements Note

I Truhlar T91 [37] N-fold symmetric periodic potential Geometries, frequencies, Analytic interpolationMulti-conformational harmonicoscillator (MC-HO [38])

Unsymmetric potentials Geometries, frequencies,minima

Reduces to harmonic oscillator case forsymmetric potentials

Chuang–Truhlar CT-Cx [39] Unsymmetric potentials Geometries, frequencies,minima

Extension of T91 to multiple minima

Pitzer–Gwinn TPG [39] N-fold symmetric periodic potential Geometries, frequencies One-dimensional numerical tables;analytic approximations

II Pitzer–Gwinn RPG [40] V fit to N-fold symmetric periodic potential Geometries, frequencies,minima, barriers

Classical integral evaluated analytically

Segmented Variants V fit to a cosine form for each minimum andits left or right barriers

Geometries, frequencies,minima, barriers

Classical integral evaluated analyticallyfor each segment

III Wigner–Kramers WK [41] General potential Geometries, Full one-dimensional potential

Semi-classical integral requiresnumerical integration

Pitzer–Gwinn (rev, Hui-Yun) MPG[42]

General potential Geometries, full one-dimensional potential

Semi-classical integral requiresnumerical integration

Truhlar–Mielke TDPPI-HS [43] General potential Geometries, full one-dimensional potential

Revised semi-classical integral requiresnumerical integration

IV Eigenvalue Sum TES General potential Geometries, full one-dimensional potential

Solution of Schrödinger equation

Table 8FHR (see Eq. (10)) values for various hindered rotor methods.

Temp. (K) Methods

MC-HO CT-Cx Class. WK TDPPI-HS TES

248.15 1.182 1.195 2.444 2.359 2.509 2.476253.15 1.197 1.212 2.454 2.377 2.516 2.484273.15 1.249 1.272 2.489 2.440 2.543 2.512298.15 1.301 1.332 2.524 2.503 2.570 2.542305.15 1.314 1.347 2.533 2.518 2.577 2.549323.15 1.344 1.380 2.553 2.553 2.593 2.566

94 I. Yavuz, G.A.A. Çiftçioglu / Computational and Theoretical Chemistry 978 (2011) 88–97

Q FR ¼1r�h

ffiffiffiffiffiffiffiffiffiffi2pIr

b

sð7Þ

At the other extreme, very high barriers make the internalrotational motion essentially harmonic about each of theminimum energy conformations, V =

PjAj(s � sj)2. The partition

function for this limit is closely related to the simplest harmonicoscillator form

Q HO ¼e�bhx=2

1� e�bhx ð8Þ

It is possible that there are high barriers between minima, butthat the minima have distinct energies. Then the harmonic oscilla-tor partition function must incorporate all minimum energyarrangements,

Q MC-HO ¼X

j

e�½ujþ�hxj=2�b

1� e�b�hxjð9Þ

The sum in Eq. (9) extends over only distinct energy minima withrelative energies uj; this incorporates any torsional symmetry.

Intermediate cases require special treatment, and have beenthoroughly discussed. Here we summarize some important cases(Table 7). Further details are to be found in Supplementarymaterial.

For methyl rotations (Type A) the Pitzer–Gwinn partition func-tion for a threefold hindered rotor would be accurate. A classical orhigh-temperature limit could be a suitable approximation to theirpartition functions. The torsions of type B, i.e. M1, R3, D2 and D6follow an asymmetric potential with two distinct minima and highbarriers. The partition function for this kind of motion can be esti-mated by the multi-conformer harmonic oscillator. For D3 the sec-ond minimum is so high that the partition function for this motionwould be well approximated by a single harmonic oscillator form.Rotations of type C include M2 and R2 and their correlates D3 andD5. In each case the second minimum lies high so the partitionfunction for this motion would be well approximated by a singleharmonic oscillator form. The partition function for type D motion,the torsion D8 around the forming bond, would be the most diffi-cult to approximate. In the Supplementary material we provide athorough discussion of errors in the approximate treatment of par-tition functions for the internal modes with the TES value as thereference point. In brief, the TDPPI-HS of Mielke and Truhlar is con-sistently accurate.

11. Impact of internal rotation on the AD1 reaction rate

The accuracy of the treatment of any one torsion is finally not soimportant as the accuracy of the ratio of the transition statepartition function to the reactants’ partition functions. Here weconstruct this overall ratio over a range of temperatures, and dis-cuss the factors for each kind of internal rotation: the low-barriermethyl torsions, the high barrier cis–trans torsions linking con-formers of comparable energy, and the high-barrier torsions link-ing minima of very different energies. We wish not only toestablish what treatments of hindered rotation are serviceablebut also under what circumstances a harmonic approximation ispragmatically justified.

The rate is dependent on ratios of partition functions, and itmay be that although the harmonic approximation may not pro-vide a faithful description of individual partition functions, the crit-ical expressions appearing in the absolute rate theory may yet beadequately approximated by the harmonic expression. We con-struct values of

FHR ¼ lnðQ zHR=Q M;HRQR;HRÞðQ zHO=Q M;HOQR;HOÞ

!ð10Þ

to help us judge whether this might be the case. This quantity is anadditive correction to the Arrhenius form of the rate constant:

lnðkAD1Þ ¼ lnðAHOÞ þ FHR � Ea=RT ð11Þ

Table 8 contains values for FHR with several approximations for thepartition functions for internal rotation, each divided by the corre-sponding harmonic oscillator value.

The table shows that the Chuang–Truhlar interpolation effects alesser correction than the Mielke–Truhlar TDPPI-HS method, andthe modified Pitzer–Gwinn method provides values slightly

Table 9Activation energies Ea (kJ mol�1) of MA polymerization for different levels of theory at the CBS limit and various hindered rotor schemes.

B3LYP PBE TPSS BMK HSE2PBE mPW1PW91 B97-1 wB97-XD M06-2X MP2

HO 38.78 16.22 25.95 31.73 24.49 31.26 25.04 17.47 17.44 11.04MCHO 40.17 17.60 27.33 33.11 25.88 32.64 26.42 18.86 18.83 12.42CT-Cx 40.36 17.79 27.53 33.30 26.07 32.83 26.61 19.05 19.02 12.61Classical 39.73 17.16 26.90 32.68 25.44 32.20 25.98 18.42 18.39 11.99WK 40.43 17.87 27.60 33.38 26.14 32.90 26.69 19.12 19.09 12.69TDPPI-HS 39.53 16.96 26.69 32.47 25.24 32.00 25.78 18.22 18.18 11.78TSE 39.62 17.05 26.78 32.56 25.33 32.09 25.87 18.31 18.27 11.87PLP-SEC exp. [4] 17.7 ± 0.7 kJ mol�1

HO: harmonic oscillator. MCHO: multi-conformation harmonic oscillator. CT-Cx: Chuang–Truhlar. WK: Wigner–Kirkwood. TDPPI-HS: Truhlar–Mileke. TSE: total eigenvaluesum (best reference).

I. Yavuz, G.A.A. Çiftçioglu / Computational and Theoretical Chemistry 978 (2011) 88–97 95

greater still. The correction afforded by these approximations is notenormous, but may make a difference of a factor of 5–10 in theoverall rate constant relative to the value obtained within the har-monic approximation.

12. Estimated Arrhenius parameters and rate constants for AD1

With the partition functions in hand, it is possible to evaluatethe Arrhenius parameters by regression of ln(kP) vs. 1/T. Tables 9and 10 incorporate a variety of choices of partition functions forinternal rotations and choices of density functional. In each casethe reaction barrier is the value obtained by basis set extrapolationfor the specific functional choice, but the torsional potentialswhich define the partition functions are always those estimatedby B3LYP/cc-pVTZ.

The best estimates of the activation energy are provided byTruhlar’s M06-2X functional and Head-Gordon’s WB97-XD.

Table 10KpðTÞ(in L mol�1 s�1, at 25 �C) of MA polymerization for different levels of theory at the C

B3LYP PBE TPSS BMK

HO 8.46E�01 1.68E+03 3.31E+01 3.22E+00MCHO 3.11E+00 6.16E+03 1.22E+02 1.18E+01CT-Cx 3.21E+00 6.36E+03 1.25E+02 1.22E+01Classical 1.06E+01 2.09E+04 4.13E+02 4.01E+01WK 1.03E+01 2.05E+04 4.04E+02 3.93E+01TDPPI-HS 1.11E+01 2.19E+04 4.33E+02 4.20E+01TSE 1.07E+01 2.13E+04 4.20E+02 4.08E+01PLP-SEC exp. [4] 1.31E+04 L mol�1 s�1

Table 11aActivation energies for AD1 reaction compared with previous studies [3,5] and PLP-SEC ex

HO

MPWB195/6-11G(3df,2p)//B3LYP/6-31G(d) 16.82M06-2X/CBS//B3LYP/cc-pVTZ 17.44wB97-XD/CBS//B3LYP/cc-pVTZ 17.47Degirmenci et. al. [5] 24.26Yu et. al. [3] 25.50 (AD1) 26.90 (AD3)PLP-SEC experiment [4] 17.7 ± 0.7 kJ mol�1

Table 11bPropagation rate constants at 298.15 K for AD1 reaction compared with previous studies

HO

MPWB195/6-11G(3df,2p)//B3LYP/6-31G(d) 987M06-2X/CBS//B3LYP/cc-pVTZ 1030wB97-XD/CBS//B3LYP/cc-pVTZ 1010Degirmenci et al. [5] 184Yu et al. [3] 333(AD1) 612(AD3)PLP-SEC experiment [4] 13100 L mol�1 s�1

ROMP2 underestimates the activation energy, while both puredensity functionals and especially the widely used hybrid func-tional B3LYP overestimate the activation energy. The choice of par-tition function for internal motion does not make a large impact(errors in the partition functions for reactants tending to cancel er-rors in the partition functions for the transition state), spanning anarrow range of about 1 kJ mol�1.

The rate constant for AD1 reflects the over-estimate of activa-tion energies by B3LYP and other density functionals, and the un-der-estimate of activation energies by ROMP2 in this system. Againthe M06-2X and wB97-XD functionals perform well, providing aremarkably accurate match between their predicted kAD1 and theexperimental report. One should perhaps take care in this compar-ison, since we have computed the rate constant for the first addi-tion (AD1) while the experimental measurement is not specificto that first step (later steps seem to be a little faster).

In Tables 11a and 11b, we collect values of kP and Ea from liter-ature sources, including our own calculations with the MPWB195

BS limit and various hindered rotor schemes.

HSE2PBE mPW1PW91 B97-1 wB97-XD M06-2X MP2

5.94E+01 3.89E+00 4.78E+01 1.01E+03 1.03E+03 1.36E+042.18E+02 1.43E+01 1.76E+02 3.71E+03 3.77E+03 4.98E+042.25E+02 1.47E+01 1.81E+02 3.82E+03 3.89E+03 5.14E+047.42E+02 4.85E+01 5.97E+02 1.26E+04 1.28E+04 1.69E+057.26E+02 4.75E+01 5.84E+02 1.23E+04 1.25E+04 1.66E+057.77E+02 5.08E+01 6.25E+02 1.32E+04 1.34E+04 1.77E+057.55E+02 4.94E+01 6.07E+02 1.28E+04 1.30E+04 1.72E+05

periment [4]. Activation energies are in units of kJ mol�1.

CT-Cx WK TDPPI-HS TSE

16.89 15.56 16.22 16.0619.02 19.09 18.18 18.2719.05 19.12 18.22 18.31– – – –– – – 26.40 (AD1) 21.50 (AD3)

[3,5] and PLP-SEC experiment [4]. Rate constants are in units of L mol�1 s�1.

CT-Cx WK TDPPI-HS TSE

2298 16,307 15,413 14,3673890 12,500 13,400 13,0003820 12,300 13,200 12,800– – – –– – – 1717(AD1) 10320(AD3)

Fig. 4. Ln(k) (above) and Arrhenius plots for the rate constant for the first step ofpolymerization (AD1) as estimated by the M06-2X/cc-PVXZ sequence. In the lowerplot the dashed red line represents the result of the PLP-SEC experiment [4]. Thiscomputation almost exactly reproduces the experimental line.

Fig. 3. Basis set dependence (dependence on the cardinal number) of the activationenergy Ea (in kJ mol�1) results for the first step AD1 in MA polymerization withwB97-XD, M06-2X and ROMP2 methods. The horizontal dashed line represents thePLP-SEC experiment [4].

96 I. Yavuz, G.A.A. Çiftçioglu / Computational and Theoretical Chemistry 978 (2011) 88–97

functional (similar to the MPWB1K recommended by Yu et al. [3])This was the best performer that we found in early study, in whichwe used only the 6-31G(d) basis.

We want to emphasize the impact of basis set extrapolation.The impact of the basis has been observed in a number of investi-gations; although structures seem satisfactory with DFT of almostany kind in a modest basis, energy calculations are reported to bemuch improved by use of the 6-311(3df,2p) basis. Fig. 3 shows thatthe activation energy as estimated by the density functional meth-ods wB97-XD and M06-2X is sensitive to the quality of the basis.Oddly the ROMP2 value is relatively unresponsive to the basisset flexibility. The impact of the improvement in basis on the rateconstant is substantial, as is shown in Fig. 4.

13. Conclusions and recommendations

Accurate values of reaction barriers are critical to successfulmodeling of reaction rates, while the treatment of the partitionfunctions associated with hindered rotations is also an importantaspect of the representation of rates. On the first issue, we con-firm that B3LYP in modest basis sets defines reliable structuresand vibrational frequencies. B3LYP’s overestimate of reactionenergy barriers for free radical additions makes it unsuitable forquantitative work, though it is useful for the description of theimpact on reaction parameters owing to substituent or otherincremental effects. For estimates of the energy barrier, werecommend that modern density functionals, especially tailoredfor the task, such as MPWB195, M06-2X, or wB97X-D should beused.

The effect of internal rotation can be significant as well. Herewe have seen that for free radical addition of systems with anumber of internal rotors, the harmonic approximation canunderestimate the effect by as much as a factor of three. When-ever possible we recommend that the partition function for hin-dered internal rotational modes be evaluated as a sum ofBoltzmann factors referring to the eigenvalues of the torsionalpotentials (TES). The TDPPI-HS method is an accurate approxi-mation to partition functions for hindered rotors but its scopeof application may be limited by the expense of constructing tor-sional potentials for large molecules; this is also the case for theWK and MPG methods, which also perform well. We can per-haps hope for a degree of transferability of torsional contribu-tions to an overall partition function from analogies in smallsystems to larger systems. For example, one may wish to gener-ate the torsional partition functions for the product of the AD1reaction. The only new motion is the torsion around the bondconnecting HMA� with MA to form HMAMA� (P-AD1). Once thepartition function for this motion is characterized one has allthe necessary analogs for the AD2 reactants, transition state,and product.

We have shown that reaction barriers are sensitive to the basis;enhancing the basis tends to increase the barrier in the AD1 reac-tion of MA polymerization. Of course use of an extensive basis be-comes infeasible for large systems. To judge from our limitedresults, there is some possibility that the value at the basis set limitis systematically about 20% greater than the value obtained at thecc-pVTZ level.

Acknowledgements

The authors wish to acknowledge generous financial and com-putational support by the Türkpetrol Foundation, Marmara Univer-sity’s Scientific Research Department (BAPKO) and Department ofChemical Engineering, the Chemistry Department and the officeof the provost of the University of Virginia, and finally, TR-GRIDof ULAKBIM. Guidance and encouragement by Prof. Dr. Zikri Altun(Marmara University) and advice by Prof. Carl Trindle (Universityof Virginia) were essential to our work.

I. Yavuz, G.A.A. Çiftçioglu / Computational and Theoretical Chemistry 978 (2011) 88–97 97

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.comptc.2011.09.043.

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