Determination of Arrhenius Parameters for Propagation in Free

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J. Phys. Chem. 1996, 100, 18997-19006

18997

Determination of Arrhenius Parameters for Propagation in Free-Radical Polymerizations: An Assessment of ab Initio Procedures Johan P. A. Heuts,1a-c Robert G. Gilbert,*,1b and Leo Radom*,1a Research School of Chemistry, Australian National UniVersity, Canberra, ACT 0200, Australia, and School of Chemistry, UniVersity of Sydney, Sydney, NSW 2006, Australia ReceiVed: May 15, 1996; In Final Form: September 30, 1996X

Ab initio calculations have been carried out for the addition of n-alkyl radicals to ethylene, with the aim of identifying levels of theory suitable for the reliable description of the mechanism and energetics of the propagation process in the free-radical polymerization of ethylene. For the calculation of absolute barriers, use of a method such as QCISD(T)/6-311G(d,p), together with MP2 basis set corrections to 6-311+G(3df,2p) and B3-LYP/6-31G(d) zero-point vibrational energy corrections, is recommended. For the calculation of Arrhenius frequency factors, the parameters required, i.e., geometries, fundamental frequencies, and rotational barriers, can be reliably obtained at much simpler levels of theory, such as UHF/6-31G(d) or even UHF/321G.

Introduction In free-radical polymerization, initiation and propagation occur via the addition of radicals to vinylic compounds. An understanding of the factors that govern the rate of these addition reactions is desirable for better process control and a more efficient development of new products. Although extensive and useful experimental information has been reported on radical addition reactions,2,3 many fundamental aspects are still not fully understood. Theoretical investigations may therefore be valuable in providing greater insight, and numerous theoretical studies have been reported in the literature.4,5 Several different theories of reaction dynamics may be employed to obtain information on the factors that govern the reaction rates, and in this study we have chosen the approach of transition state theory (TST).6 In TST, the expression for the rate coefficient k of a radical addition reaction is given by

k)

( )

E0 kBT Q exp h QalkeneQradical kBT †

(1)

where kB is Boltzmann’s constant, T is the temperature, h is Planck’s constant, E0 is the critical energy (i.e., the energy difference at 0 K between reactants and transition state), and Q†, Qalkene, and Qradical are the molecular partition functions of the transition state, alkene, and radical, respectively. The molecular properties required for the evaluation of the partition functions and the critical energy can be obtained by quantum chemical calculations.7 It has been found that a high level of theory is required for the calculation of reliable reaction barriers in these systems.5 On the other hand, it is likely that a lower level of theory might be sufficient for the calculation of Arrhenius frequency factors. The reason for this is that the frequency factor is determined by the molecular partition functions, which in turn depend on molecular properties such as geometry (moments of inertia), fundamental frequencies, and torsional potentials,6 all of which are generally known to be less sensitive to level of theory.7 Furthermore, because of the ratio of partition functions in eq 1, significant cancellation of errors may occur if the errors are systematic in nature. X

Abstract published in AdVance ACS Abstracts, November 1, 1996.

S0022-3654(96)01409-8 CCC: $12.00

The aim of the present study is to analyze in detail the theoretical predictions concerning the addition to ethylene of ethyl radical and to make use of our findings in assessing less detailed results for the addition to ethylene of n-alkyl radicals (up to n-heptyl) as models for the first few propagation steps in the free-radical polymerization of ethylene. Our study includes the calculation of geometries, energies, fundamental frequencies, and torsional potentials in order to establish suitable levels of theory for the reliable estimation of frequency factors and activation energies for radical addition reactions in general and propagation reactions in free-radical polymerizations in particular. Since the “chemical” implications of our calculations have been discussed in detail previously,8 we will only focus on the computational aspects in the present paper. Theoretical Procedures Standard ab initio molecular orbital7 and density functional theory (DFT)9 calculations were performed using the GAUSSIAN 92,10a GAUSSIAN 94,10b ACES II,10c and CADPAC10d programs. Calculations on open-shell systems such as radicals and the transition structures of radical addition reactions may be carried out in several different ways. In the first place, we can use either an unrestricted (UHF) or restricted open-shell (ROHF) Hartree-Fock procedure. UHF allows R and β orbitals to be different whereas ROHF restricts them to be the same. Both procedures have advantages and disadvantages. For example, while the ROHF wave function is a pure doublet (the spinsquared expectation value 〈S2〉 ) 0.75), the UHF wave function is contaminated by states of higher spin multiplicity (〈S2〉 > 0.75). However, the UHF wave function is more flexible and leads to the lower energy. We use both UHF- and ROHF-based procedures in this study. Electron correlation was taken into account in the present work by using Møller-Plesset perturbation theory and quadratic configuration interaction (QCI).11 Møller-Plesset theory was used with both UHF and ROHF starting points, resulting in UMP and several different formulations of restricted MP theory, e.g., ROMP,12a RMP,12b and ROHF-MBPT.12c UMP may show poor convergence if there is severe spin contamination.13 The results can sometimes be improved by using restricted procedures, or by projecting out the undesirable higher spin states, © 1996 American Chemical Society

18998 J. Phys. Chem., Vol. 100, No. 49, 1996

Heuts et al. TABLE 1: Calculated Geometrical Parametersa in the Transition Structures for the Anti Addition of n-Alkyl Radicals to Ethylene and Corresponding Reaction Barriers level

Figure 1. Schematic diagram showing key geometrical parameters in the transition structure for the addition of n-alkyl radicals to ethylene (X ) (CH2)nH, where n ) 0-5).

the latter leading to projected Møller-Plesset theory (PMP).14 Calculations were performed up to fourth-order Møller-Plesset perturbation theory (MP2, MP3, MP4). The QCI procedure was used only with a UHF wave function, resulting in UQCISD and UQCISD(T). Several different levels of theory were used in the calculation of geometries and vibrational frequencies. For the full set of radicals ranging from the ethyl radical to the n-heptyl radical and for the corresponding transition structures for the addition reaction to ethylene, geometries were optimized and vibrational frequencies determined at the UHF/3-21G and UHF/6-31G(d) levels of theory. The ethyl, n-propyl, and n-butyl radical additions were also studied with complete optimizations at UMP2/6-31G(d). Finally, for the addition of ethyl radical to ethylene, additional optimizations and frequency analyses were carried out at the ROHF/6-31G(d) and UQCISD/6-31G(d) levels of theory and with DFT calculations using the Becke (B)15 and Becke3 (B3)16 exchange functionals in combination with the Lee-Yang-Parr correlation functional (LYP)17 and the 6-31G(d) basis set. The energy levels of the hindered rotors, required in the frequency factor calculations, were calculated by solving the one-dimensional Schro¨dinger equation for a rigid rotor, using the finite element method extended to handle cyclic boundary problems.18 A basis set of free rotor functions, φm, given by

φm ) (1/x2π) exp(imθ) was used, where m is an integer between -200 and +200 and θ is the rotational angle. Rotational potentials were determined by fitting a three-term Fourier expansion through the stationary points of the potential. Calculations of the rotational constants required to obtain the partition functions of the internal rotations were performed with GEOM, a part of the UNIMOL software package.19 Results and Discussion Geometries. For our present purposes, it is desirable to establish relatively simple levels of theory that provide us with geometries that are suitable for the calculation of Arrhenius frequency factors and activation energies. These levels of theory do not necessarily need to be the same. Several key geometrical parameters in the optimized transition structures for the addition to ethylene of n-alkyl radicals, ranging from ethyl to n-heptyl, in fully extended structures with Cs symmetry, are listed in Table 1 (see also Figure 1). These are the C-C bond length being formed (r(C- -C)), the angle between the forming C-C bond and the R-C-C bond in the radical moiety (φ1), the angle between the forming C-C bond and the CdC bond in the ethylene moiety (φ2), and the square root of the product of the principal moments of inertia (xIaIbIc). Also

r(C- -C)b

φ1c

φ2d

xIaIbIce

∆E‡f

UHF/3-21G UHF/6-31G(d) UMP2/6-31G(d) ROHF/6-31G(d) UQCISD/6-31G(d) UB-LYP/6-31G(d) UB3-LYP/6-31G(d)

Ethyl + Ethylene TS 2.256 104.8 108.8 2.232 105.9 109.8 2.256 104.6 111.0 2.104 106.7 109.7 2.262 104.8 110.8 2.384 105.8 111.5 2.335 105.4 111.2

5.74 × 103 5.71 × 103 5.67 × 103 5.27 × 103 5.78 × 103 6.20 × 103 5.97 × 103

29.4 42.2 56.0 99.9 35.8 16.8 21.6

UHF/3-21G UHF/6-31G(d) UMP2/6-31G(d)

Propyl + Ethylene TS 2.259 105.6 108.5 2.234 106.4 109.7 2.262 105.0 110.8

1.15 × 104 1.14 × 104 1.13 × 104

29.1 41.9 53.2

UHF/3-21G UHF/6-31G(d) UMP2/6-31G(d)

Butyl + Ethylene TS 2.257 105.5 108.5 2.233 106.3 109.7 2.261 104.9 110.9

2.12 × 104 2.10 × 104 2.10 × 104

29.4 42.2 53.9

UHF/3-21G UHF/6-31G(d)

Pentyl + Ethylene TS 2.257 105.6 108.5 2.232 106.4 109.7

3.42 × 104 3.39 × 104

29.4 42.2

UHF/3-21G UHF/6-31G(d)

Hexyl + Ethylene TS 2.257 105.6 108.5 2.232 106.4 109.7

5.44 × 104 5.38 × 104

29.4 42.2

UHF/3-21G UHF/6-31G(d)

Heptyl + Ethylene TS 2.257 105.6 108.5 2.232 106.4 109.7

7.89 × 104 7.81 × 104

29.4 42.2

a See Figure 1. b Length (Å) of the forming C-C bond between the radical and ethylene. c Angle (deg) between the forming C-C bond and the R-C-C bond in the radical moiety. d Angle (deg) between the forming C-C bond and the CdC bond in ethylene. e Square root of the product of the principal moments of inertia (atomic units) (see eq 2). f Corresponding barrier to reaction (kJ mol-1), without zero-point or temperature corrections, calculated at the same level as the geometry optimization.

listed are the corresponding reaction barriers (∆E‡) at the various levels of theory. In addition to the fully extended anti structures, other stationary points on the potential energy surface corresponding to the rotation of the ethylene moiety about the forming C-C bond were also examined. We find that gauche addition of the radical to ethylene is in fact slightly favored over anti addition at all levels of theory except UHF/3-21G, with a largest difference of 1.6 kJ mol-1 at UMP2/6-31G(d). However, because our initial calculations on the anti and gauche additions indicated that the effect of level of theory is very similar for the two cases, we have chosen for simplicity to base most of our assessments on anti structures. We will deal explicitly with the gauche structures when we discuss barriers and make comparisons with experimental results. The changes in geometry with level of theory seen in Table 1 are consistent with results obtained in a more extensive study of the addition of methyl radical to ethylene.5 In general, the variation in the calculated geometric parameters at the different levels of theory is relatively small, with the exception of the DFT methods, which appear to overestimate the length of the forming C-C bond, and ROHF, which underestimates this length. The bond angles predicted by the various methods lie within a range of about 3°. Overall, it may be concluded that the geometries are all very similar. The choice of geometry will affect both the calculated reaction barrier and the frequency factor. Its effect on the calculated barrier can potentially have dramatic consequences, because a change in barrier as small as 5 kJ mol-1 will affect the rate coefficient by a factor of 10 at room temperature. This will be examined in a later section (see below).

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J. Phys. Chem., Vol. 100, No. 49, 1996 18999

Figure 2. Schematic representation of the six transitional modes in the transition state for the addition of an ethyl radical to ethylene and their respective harmonic frequencies (cm-1, UHF/6-31G(d)).

The choice of geometry will directly affect the frequency factor via the external rotational partition function, Qext,rot:20

Qext,rot )

( )x

xπ 8π2kT σ h2

3/2

IaIbIc

(2)

where σ is the overall symmetry number and Ia, Ib, and Ic are the principal moments of inertia. The geometric contribution is thus proportional to xIaIbIc. The values for this parameter obtained at different levels of theory are included in Table 1, and it can be seen that these results all lie within 10% of the UQCISD/6-31G(d) result. This indicates that even geometries obtained at a relatively simple level of theory, e.g., UHF/321G, may be used for the calculation of external rotational partition functions. Since our main interest lies in the calculation of Arrhenius parameters in free-radical polymerizations, which involve significantly larger molecules, we investigated the effect of extending the chain length of the radical. In order to simplify the problem, we have again constrained the chain to have a fully extended, Cs symmetry conformation and the addition to take place via an anti alignment in all cases. The results of these calculations are also shown in Table 1. We find that the geometric parameters in the transition state do not change significantly when the chain length is extended. The principal moments of inertia of course do vary since they are mass dependent. Fundamental Frequencies. Fundamental frequencies are important in the evaluation of both the Arrhenius frequency factor and the activation energy. Low frequencies have the greatest influence in determining the vibrational partition function of the molecules, whereas the higher frequencies have the greatest influence on the zero-point vibrational energy corrections. Hence, it is important to investigate the effect of different levels of theory on both ends of the vibrational spectrum. (i) Vibrational Partition Functions. Of particular importance in the calculation of the frequency factor are the so-called transitional modes in the transition structure of the reaction. These arise from the loss of three translational and three rotational degrees of freedom when the reactants combine to form the transition state.6 The six transitional modes for the

ethyl radical addition to ethylene are schematically represented in Figure 2. One of these transitional modes corresponds to the reaction coordinate and is characterized by an imaginary frequency. All normal modes will slightly change in going from reactants to transition state, but because of the ratio Q†/ (QalkeneQradical) in eq 1, the small changes in the nontransitional modes will approximately cancel. The contributions from the transitional modes, however, will not cancel against any internal modes in the reactants and will explicitly occur in eq 1. This means that the approximate sensitivity of the calculated frequency factor to level of theory of the vibrational frequencies may be seen in the contributions arising from the transitional modes. In order to investigate the sensitivity to level of theory of the calculated frequencies of the transitional modes, we performed normal mode analyses for the ethyl radical addition to ethylene with conventional ab initio methods up to UQCISD/ 6-31G(d) and the DFT procedures B-LYP/6-31G(d) and B3LYP/6-31G(d) (Table 2). The approximate effect of the different calculated frequencies on the frequency factor was examined by calculating the vibrational partition function Q arising from the appropriate transitional modes, given by20 6

[ ( )]

Q ) ∏ 1 - exp i)3

hνi

kBT

-1

(3)

where νi is the frequency of the ith transitional mode. Partition functions calculated with eq 3 at the several different levels of theory are also included in Table 2. Because calculated frequencies are known to show systematic differences when compared with experimental values,21-23 the partition functions have been calculated using both unscaled and scaled frequencies. Transitional modes 1 and 2 are omitted from these calculations, because they correspond to the reaction coordinate and an internal rotation, respectively. The reaction coordinate is lost as a degree of freedom in the transition state and is therefore omitted from the molecular partition function, while internal rotations are better described by a hindered rotor model (see below).6 Comparison of the UHF and UMP2 results shows that the unscaled frequencies are very similar and that the corresponding

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TABLE 2: Calculated Frequencies (cm-1) and Overall Vibrational Partition Functions of the Transitional Modes in the Transition Structure for Anti Addition of the Ethyl Radical to Ethylene at Different Levels of Theory level of theory mode

UHF/3-21G

UHF/6-31G(d)

UMP2/6-31G(d)

ROHF/6-31G(d)

UQCISD/6-31G(d)

B-LYP/6-31G(d)

B3-LYP/6-31G(d)

ν1 ν2 ν3 ν4 ν5 ν6 Qunscaleda Qscaleda,b

420i 57 177 276 413 620 3.29 3.25

483i 59 178 279 403 631 3.28 3.90

611i 61 172 267 407 595 3.45 3.19

2614i 75 207 316 460 709 2.64 3.08

551i 59 167 257 389 585 3.67 3.57

314i 49 141 239 348 521 4.61 4.11

384i 52 151 248 367 547 4.16 4.15

a Overall vibrational partition function at 323 K obtained from transitional modes 3-6. b In order to obtain Q scaled, frequencies were scaled by 1.0075 (HF/3-21G), 0.9061 (HF/6-31G(d)), 1.0485 (MP2/6-31G(d)), 1.0147 (QCISD/6-31G(d)), 1.0620 (B-LYP/6-31G(d)), and 1.0013 (B3-LYP/ 6-31G(d)), as recommended in ref 23.

partition functions are within 5% of one another, but the difference is increased to approximately 18% after scaling. This result may be associated with spin contamination in the UHF wave function, which will significantly affect UMP2 properties in particular. The unscaled UMP2/6-31G(d) results are quite close to the unscaled UQCISD/6-31G(d) results. After scaling, both UHF/6-31G(d) and UMP2/6-31G(d) partition functions deviate from the UQCISD/6-31G(d) partition function by approximately 10%. Remarkably (and perhaps fortuitously), UHF/3-21G does not seem to perform worse than either UHF/ 6-31G(d) or UMP2/6-31G(d). A comparison between the unscaled UHF and ROHF results immediately reveals one of the major shortcomings of the restricted Hartree-Fock procedure, i.e., the incorrect description of bond-breaking and bond-formation processes, as indicated by the very high imaginary frequency. Furthermore, it can be seen that all the frequencies are overestimated by ROHF, leading to partition functions that are significantly too low. The results after scaling seem to be better, but considering the fact that restricted Hartree-Fock theory does not describe bond-breaking processes adequately, this result might be fortuitous, and we do not recommend the ROHF procedure for general use in the study of radical addition reactions. The unscaled DFT frequencies are all smaller than the corresponding frequencies obtained with the conventional molecular orbital procedures. This result is not really surprising because, as we have seen from the geometric parameters listed in Table 1, the transition structures obtained with the DFT methods are not as tight as the transition structures obtained with the conventional ab initio methods, thus leading to “floppier” modes. Scaling the frequencies lowers the partition function obtained with the B-LYP/6-31G(d) and B3-LYP/631G(d) procedures to values which are closer to (but still approximately 16% higher than) the scaled partition function obtained at the UQCISD/6-31G(d) level of theory. On the basis of the above analysis, we recommend the use of scaled UHF/6-31G(d) frequencies as a cost-effective alternative to UQCISD/6-31G(d) for the low-frequency transitional modes. If a somewhat greater uncertainty in the results is acceptable, the cheaper UHF/3-21G level of theory might be used. Since we are interested in propagation reactions in free-radical polymerizations, the effect of extending the chain length of the radical on the calculated frequencies of the transitional modes was investigated. The unscaled UHF/6-31G(d) frequencies of the transitional modes are listed in Table 3 for the addition to ethylene of n-alkyl radicals ranging from ethyl to n-pentyl radical. (The results for the n-hexyl and n-heptyl radical additions being omitted because it is very difficult to identify the individual transitional modes in these two species.) We can

TABLE 3: Calculated Frequencies (cm-1, UHF/6-31G(d)) of the Transitional Modes in the Transition Structures for Anti Addition of n-Alkyl Radical Additions to Ethylene calculated frequency mode

ethyl

propyl

n-butyl

n-pentyl

ν1 ν2 ν3 ν4 ν5 ν6

483i 59 178 278 404 631

459i 54 131 267 401 633

458i 46 100 247 401 633

459i 44 82 249 401 633

see from this table that some frequencies hardly change at all (ν1, ν5, and ν6) and some change a little (ν2 and ν4), while ν3 changes substantially. The decrease in frequencies may be attributed to the greater mass of the longer chains and is largest for the motions which directly involve the larger mass, notably ν3 (see Figure 2). (ii) Zero-Point Vibrational Energies. The calculation of reaction barriers involves the calculation of the energy difference between reactants and transition state, including zero-point vibrational energy (ZPVE) contributions. As noted above, the ZPVE is dominated by the high-frequency vibrations. Because of spin contamination in the transition states, it would be desirable to calculate ZPVE’s using methods such as QCISD or QCISD(T).24 However, it is not currently feasible to use these methods for the calculation of frequencies in the relatively large systems involved in polymerization processes because of the computational resources that are required. Hence, our task is to find a simpler level of theory that will yield suffciently reliable ZPVE’s. In order to do this, we have calculated the ZPVE’s at the UHF/6-31G(d), ROHF/6-31G(d), and MP2/631G(d) levels of theory, the DFT procedures B-LYP/6-31G(d) and B3-LYP/6-31G(d), because they have been shown to yield very good results in frequency calculations,23,25 and UQCISD/ 6-31G(d) as the reference calculation. The calculated ZPVE’s and ∆ZPVE’s (i.e., the ZPVE contributions to the reaction barrier) are listed in Table 4. It can be seen that there are large differences between the results obtained at the several different levels of theory, the difference between the scaled ∆ZPVE’s at the UHF/6-31G(d) and UMP2/ 6-31G(d) levels being as large as 4.7 kJ mol-1. This is mainly caused by the large difference in the calculated ZPVE’s of the transition structures. A detailed study of the fundamental frequencies in the reactants and transition structures at the two levels of theory shows that the major contributions to this difference originate from differences in the following modes (the values in parentheses being the differences in contributions to ∆ZPVE calculated at the UMP2/6-31G(d) and UHF/6-31G(d) levels of theory): 500-1000 cm-1 modes in the transition

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TABLE 4: Effect of Level of Theory on the Scaled Calculated Zero-Point Vibrational Energiesa in the Anti Addition of Ethyl Radical to Ethylene ZPVEc b

level

ethene

ethyl radical

TS

ZPVEd

UHF UMP2 ROHF B-LYP B3-LYP UQCISD

0.050 03 0.050 32 0.050 03 0.050 39 0.050 23 0.050 28

0.057 87 0.059 43 0.058 29 0.058 75 0.058 49 0.059 11

0.110 03 0.113 67 0.111 54 0.112 13 0.111 71 0.112 65

5.6 10.3 8.4 7.9 7.9 8.6

a

Scale factors of 0.9207 for HF/6-31G(d), 0.9670 for MP2/6-31G(d), 1.0126 for B-LYP/6-31G(d), 0.9806 for B3-LYP/6-31G(d), and 0.9776 for UQCISD/6-31G(d) were used for the calculated ZPVE’s, as recommended in ref 23. b Calculations were performed with a 6-31G(d) basis set and the correlated calculations used the frozen-core approximation. c Calculated zero-point vibrational energies in hartrees. d ∆ZPVE ) ZPVE -1 TS - ZPVEethyl radical - ZPVEethylene (kJ mol ).

structures (+2.3 kJ mol-1) and ethylene (+0.9 kJ mol-1) and 1000-1600 cm-1 modes in the transition structures (+1.4 kJ mol-1). The ROHF/6-31G(d) result is very close to the UQCISD/631G(d) result. A closer examination of the fundamental frequencies in the transition state shows that the high-frequency modes are virtually the same (i.e., significantly too low) for UHF/6-31G(d) and ROHF/6-31G(d) but that all the frequencies in the lower and middle range of the spectrum are significantly higher (i.e., up to 70 cm-1 per mode) for ROHF/6-31G(d). Hence, the good agreement between the ROHF/6-31G(d) and UQCISD/6-31G(d) results is caused by a fortuitous cancellation of an overestimation in the low frequencies and an underestimation in the higher frequencies with ROHF and we do not recommend the ROHF procedure for general use. Both DFT procedures yield ∆ZPVE’s of approximately 8 kJ mol-1 and agree very well with the UQCISD/6-31G(d) result. This is perhaps not surprising, because (scaled) DFT procedures, especially B3-LYP, have been found to perform very well in predicting vibrational frequencies.23,25 On the basis of these results, we recommend the B3-LYP/ 6-31G(d) level of theory for the calculation of zero-point vibrational energies. We note that the scaled ∆ZPVE value for the favored gauche addition at this level of theory is 8.3 kJ mol-1. Torsional Potentials. The low-frequency modes are the most important contributors to the frequency factors, as can be seen from the functional form of the vibrational partition function given by eq 3. Several of these modes actually correspond to hindered internal rotations,26,27 and three of these modes are important in the present systems, namely, the rotation-inversion of the CH2 group in the radicals, the rotation of the C2H4 moiety in the transition state about the forming C-C bond (τ1 in Figure 3 or the transitional mode ν2 in Figure 2), and the rotation in the transition state of the CH2 group of the radical moiety, now dragging along the ethylene moiety and hence resulting in the rotation of a C3H6 moiety (τ2 in Figure 3).8 In the remainder of this paper, these three torsions will be referred to as methylene, ethylene, and propylene rotations, respectively. These internal rotations are better described by a hindered rotor model than by a harmonic oscillator model.26,27 In order to solve the one-dimensional Schro¨dinger equation for this motion, the potential for the particular rotation is required, and we approach this problem by obtaining optimized geometries for the stationary points along the potential and fitting a three-term Fourier expansion to the energies at these points. Full optimizations and frequency analyses of the three rotations were performed at the UHF/3-21G, UHF/6-31G(d),

Figure 3. Schematic representation of the transition structure for the anti addition of an ethyl radical to ethylene, showing the two lowfrequency modes which are replaced by hindered rotations: τ1, the ethylene rotation, and τ2, the propylene rotation.

Figure 4. Schematic representation of the CH2 rotation-inversion process in the ethyl radical (i.e., X ) H) and propyl radical (i.e., X ) CH3) involving the sequential movement of the two hydrogen atoms at the radical center. The dihedral angle θ defines the torsional angles listed in Table 5.

UMP2/6-31G(d), B-LYP/6-31G(d) and B3-LYP/6-31G(d) levels of theory. Single-point energies for the UHF/6-31G(d) and UMP2/6-31G(d) stationary points were also determined at the UQCISD(T)/6-31G(d) level of theory. We find that the DFT procedures significantly underestimate the rotational barriers (e.g., the relative energies of the eclipsed conformation with respect to the staggered conformation in the propylene rotation are found to be 5.51 and 5.74 kJ mol-1 for B-LYP/6-31G(d) and B3-LYP/6-31G(d), respectively, as compared with 6.82, 7.73, and 6.84 kJ mol-1 for UHF/3-21G, UHF/6-31G(d), and UMP2/6-31G(d), respectively). This might have been expected from the loose transition structures, and hence these results are not discussed explicitly here. As the ethyl and propyl radicals have been studied in great detail previously,28,29 we will not discuss their respective methylene rotations, but we note that the methylene rotation actually corresponds to a rotation-inversion process. This process, which we treat as a hindered rotation for simplicity, is schematically shown in Figure 4, and the results of the calculations are listed in Table 5. The ethylene rotations in the transition states of both the propyl and ethyl radical additions to ethylene yield very similar results (Table 6). This finding might have been anticipated, since the additional methyl group in the propyl moiety is not encountered by the ethylene moiety during the rotation. A very different situation occurs for the propylene torsion in the transition states of both reactions (Table 7). In the case of the ethyl radical addition to ethylene, the propylene group eclipses three times with a hydrogen atom, whereas in the transition state for the reaction of the propyl radical, the propylene eclipses twice with a hydrogen atom and once with a methyl group. The latter leads to an increase in the relative energy by about 10 kJ mol-1, as seen in Figure 5 and Table 7. As can be seen from Tables 5-7, there is a very good fit of the Fourier series to the calculated points. The UQCISD(T) barriers are generally slightly lower than the corresponding UMP2 barriers, with the UHF barriers being larger again. The UMP2 potentials, in general, do not deviate significantly from the UQCISD(T) potentials, which suggests that single-point

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TABLE 5: Rotational Potentials for the Methylene Rotation in Ethyl and Propyl Radicals at Several Levels of Theory from direct calculations level UHF/3-21G

UHF/6-31G(d)

UMP2/6-31G(d)

UHF/3-21G

UHF/6-31G(d)

UMP2/6-31G(d)

θ

a

Eb

QCIc

0.0 30.0 60.0 90.0 0.0 30.0 60.0 90.0 0.0 30.0 60.0 90.0

Ethyl Radical 0.00 0.39 0.00 0.39 0.00 0.00 0.71 0.51 0.00 0.00 0.71 0.51 0.00 0.00 0.66 0.50 0.00 0.00 0.66 0.50

0.0 29.6 58.5 90.1 0.0 27.3 62.4 90.0 0.0 25.7 61.7 90.0

Propyl Radical 0.00 0.50 0.00 0.60 0.76 0.74 1.29 0.95 0.00 0.00 0.74 0.53 0.84 0.87 1.21 1.05 0.00 0.00 0.74 0.62

Fourier fit Vid

θe

TABLE 6: Rotational Potentials for the Ethylene Rotation in the Ethyl + Ethylene TS and the Propyl + Ethylene TS at Several Levels of Theory from direct calculations

Ef level

0 0 0.39 0 0 0.71 0 0 0.66

0.07 -0.07 0.53 -0.85 -0.16 0.83 -0.84 -0.28 0.74

0.0 30.0 60.0 90.0 0.0 30.0 60.0 90.0 0.0 30.0 60.0 90.0

0.00 0.39 0.00 0.39 0.00 0.71 0.00 0.71 0.00 0.71 0.00 0.71

0.0 29.7 59.0 90.0 0.0 26.2 61.6 90.0 0.0 24.6 60.9 90.0

0.00 0.50 0.00 0.60 0.76 1.29 0.00 0.74 0.84 1.21 0.00 0.74

a Torsional angles of stationary points, as defined in Figure 4 (deg). Relative energies, without ZPVE (kJ mol-1). c Relative energies (kJ mol-1) at the UQCISD(T)/6-31G(d) level of theory (single points).d Vi (kJ mol-1) as defined by V(θ) ) 0.5{V2(1 - cos 2θ) + V4(1 - cos 4θ) + V6(1 - cos 6θ)}, listed in this order from top to bottom. e Torsional angles of stationary points calculated with the Fourier series (deg). f Relative energies of stationary points calculated with the Fourier series (kJ mol-1). b

energy calculations with methods more highly correlated than UMP2 are not required for the rotational potentials. The effect of the different rotational potentials on the calculated partition functions of the ethylene and propylene rotations, while maintaining the moment of inertia constant, was studied by calculating the partition functions, using the UMP2/ 6-31G(d) optimized transition structures. It can be seen (Table 8) that the partition functions obtained with different rotational potentials lie within about 25% of one another. Hence, if uncertainties of this magnitude are acceptable, rotational potentials may even be obtained at the UHF/3-21G level of theory, allowing larger systems to be studied. Frequency Factors. Frequency factors were calculated by numerical differentiation of eq 1. Assuming a complete separability of the Hamiltonian for the vibrational and rotational modes, the methylene, ethylene, and propylene torsional modes were described as hindered rotations and included in the rotational rather than the vibrational partition functions. Frequency factors for the anti addition of ethyl radical to ethylene, calculated with parameters obtained at the UHF/3-21G, UHF/ 6-31G(d), UMP2/6-31G(d), B-LYP/6-31G(d), B3-LYP/631G(d), and UQCISD/6-31G(d) (with a UMP2/6-31G(d) rotational potential) levels of theory, are listed in Table 9. It can be seen that the values obtained using the DFT procedures are significantly too high, which may be attributed to the lower low-frequency modes and the lower rotational barriers. The results obtained at the other levels of theory lie within about 25% of the UQCISD/6-31G(d) result, with a maximum value of 1.80 × 108 dm3 mol-1 s-1 for UHF/6-31G(d) and a lowest value of 1.13 × 108 dm3 mol-1 s-1 for UMP2/6-31G(d).31 The good result obtained with UHF/3-21G is important because it

UHF/3-21G

UHF/6-31G(d)

UMP2/6-31G(d)

UHF/3-21G

UHF/6-31G(d)

UMP2/6-31G(d)

a

θ

b

E

QCI

c

Ethyl + Ethylene TS 0.0 3.72 59.3 0.16 118.6 3.28 180.0 0.00 0.0 3.83 3.45 58.6 0.00 0.00 119.9 3.63 4.18 180.0 0.19 1.11 0.0 3.63 3.21 55.0 0.00 0.00 121.4 4.86 4.20 180.0 1.56 1.23

Fourier fit Vid

θe

Ef

-0.39 0 3.72 -0.19 61.3 0.15 -3.33 120.0 3.28 180.0 0.00 0.00 0.0 3.83 -0.27 60.8 -0.01 -3.64 120.8 3.63 180.0 0.19 1.86 0.0 3.63 -0.25 58.2 -0.03 -3.93 123.3 4.87 180.0 1.56

Propyl + Ethylene TS 0.0 3.36 -0.17 58.8 0.00 -0.28 119.0 3.02 -3.12 180.0 0.07 0.0 3.81 3.28 -0.01 59.0 0.00 0.00 -0.44 120.2 3.47 3.90 -3.49 180.0 0.31 1.34 0.0 3.50 3.05 2.03 54.9 0.00 0.00 -0.50 122.1 4.68 3.90 -3.66 180.0 1.87 1.53

0.0 3.36 61.3 -0.01 120.7 3.03 180.0 0.07 0.0 3.81 61.4 -0.01 121.4 3.47 180.0 0.31 0.0 3.50 58.6 -0.04 124.6 4.70 180.0 1.87

a Torsional angles of stationary points, defined as the CCCC dihedral angle (deg). b Relative energies, without ZPVE (kJ mol-1). c Relative energies (kJ mol-1) at the UQCISD(T)/6-31G(d) level of theory (single points). d Vi (kJ mol-1) as defined by V(θ) ) 0.5{V1(1 - cos θ) + V2(1 - cos 2θ) + V3(1 - cos 3θ)}, listed in this order from top to bottom. e Torsional angles of stationary points calculated with the Fourier series (deg). f Relative energies of stationary points calculated with the Fourier series (kJ mol-1).

suggests that this relatively simple level of theory can be used for the reliable estimation of frequency factors, thus allowing larger systems to be investigated.27 The UHF/3-21G, UHF/6-31G(d), and UMP2/6-31G(d) results for the anti addition of propyl radical to ethylene are also listed in Table 9. When comparing these results with those obtained for the anti addition of the ethyl radical, a similar observation is made: the results obtained for the propyl radical addition at the three levels of theory lie within a 25% range of the average value, with UHF/6-31G(d) yielding the highest value for the frequency factor. From a consideration of the performances of the various procedures examined for all the separate contributions to the frequency factor, we recommend the UHF/6-31G(d) level of theory as a cost-effective procedure for general use. The frequency factor for the gauche addition (i.e., the energetically favored pathway) calculated at UHF/6-31G(d) has a value of 1.64 × 108 dm3 mol-1 s-1, which is in reasonable agreement with the experimental value of 2.0 × 108 dm3 mol-1 s-1.30 Reaction Barriers. We begin by investigating the sensitivity of the calculated barriers to choice of geometry (Table 10). We have used UHF/6-31G(d), UMP2/6-31G(d), and UQCISD/631G(d) geometries in our studies, with single-point calculations at a variety of levels of theory using 6-31G(d) and 6-311G(d,p) basis sets. It can be seen that the barriers obtained for the UHF/ 6-31G(d) geometries are in general smaller than the barriers obtained for the UMP2/6-31G(d) geometries. Examination of the QCISD and QCISD(T) barriers shows that the barriers obtained employing UQCISD/6-31G(d) geometries lie between those obtained using UHF/6-31G(d) and UMP2/6-31G(d)

Propagation in Free-Radical Polymerizations

J. Phys. Chem., Vol. 100, No. 49, 1996 19003

TABLE 7: Rotational Potentials for the Propylene Rotation in the Ethyl + Ethylene TS and the Propyl + Ethylene TS at Several Levels of Theory from direct calculations level UHF/3-21G

UHF/6-31G(d)

UMP2/6-31G(d)

UHF/3-21G

UHF/6-31G(d)

UMP2/6-31G(d)

a

θ

b

E

QCI

c

Fourier fit Vid

θe

level of theory 0.0 60.0 120.0 180.0 0.0 60.0 120.0 180.0 0.0 60.0 120.0 180.0

6.82 0.00 6.82 0.00 7.73 0.00 7.73 0.00 6.84 0.00 6.84 0.00

Propyl + Ethylene TS 0.0 19.16 74.2 2.51 120.7 7.92 180.0 0.00 0.0 19.21 16.65 70.2 2.46 1.51 120.8 8.27 7.03 180.0 0.00 0.00 0.0 17.73 16.35 69.9 1.40 1.14 120.8 7.04 6.78 180.0 0.00 0.00

0.0 71.2 120.6 180.0 0.0 70.6 120.6 180.0 0.0 71.2 122.0 180.0

19.16 2.45 7.92 0.00 19.21 2.46 8.27 0.00 17.73 1.39 7.05 0.00

-9.45 -5.14 -9.76 -8.62 -5.65 -9.11

partition function

Ef

Ethyl + Ethylene TS 0.0 6.82 0.00 60.2 0.00 0.00 119.8 6.82 -6.82 180.0 0.00 0.0 7.73 6.80 0.00 60.3 0.00 0.00 0.00 119.8 7.73 6.80 -7.73 180.0 0.00 0.00 0.0 6.84 6.57 0.00 60.3 0.00 0.00 0.00 119.7 6.84 6.57 -6.84 180.0 0.00 0.00 -9.69 -5.30 -9.47

TABLE 8: Calculated Partition Functions at 323 K of the Ethylene and Propylene Rotationsa with Rotational Potentials Obtained at the UHF/3-21G, UHF/6-31G(d), and UMP2/6-31G(d) Levels of Theory

a Torsional angles of stationary points, as defined in Figure 5 (deg). Relative energies, without ZPVE (kJ mol-1). c Relative energies (kJ mol-1) at the UQCISD(T)/6-31G(d) level of theory (single points).d Vi (kJ mol-1) as defined by V(θ) ) 0.5{V1(1 - cos θ) + V2(1 - cos 2θ) + V3(1 - cos 3θ)}, listed in this order from top to bottom. e Torsional angles of stationary points calculated with the Fourier series (deg). f Relative energies of stationary points calculated with the Fourier series (kJ mol-1).

geometries. The variation in barrier with different choices of geometry is approximately (2 kJ mol-1. The results in Table 10 show that the calculated barrier is very sensitive to level of theory, and they are consistent with results obtained previously in studies of the addition of methyl radical to ethylene.5 Next we examine in more detail the effect of basis set and level of theory on the calculated barrier. We have considered basis sets up to 6-311+G(3df,2p) at the UMP2, PMP2, and RMP2 levels of theory and up to 6-311G(df,p) at the UQCISD level of theory for UHF/6-31G(d), UMP2/6-31G(d), and UQ-

propylene rotation

UHF/3-21G UHF/6-31G(d) UMP2/6-31G(d)

Ethyl + Ethylene TS 7.047b 6.980b 5.927b

6.203c 5.887c 6.195c

UHF/3-21G UHF/6-31G(d) UMP2/6-31G(d)

Propyl + Ethylene TS 8.437d 7.936d 6.702d

3.845e 3.780e 4.706e

a UMP2/6-31G(d) geometries from the anti addition were used in all cases. b I ) 7.3194 amu bohr2. c I ) 10.3364 amu bohr2. d I ) 9.4991 amu bohr2. e I ) 14.5520 amu bohr2.

TABLE 9: Calculated Frequency Factors (dm3 mol-1 s-1) and Temperature Corrections to the Reaction Barriers, ∆Eact (kJ mol-1), at 323 K for the Anti Additions of the Ethyl and Propyl Radicals to Ethylene at Several Different Levels of Theorya level of theory

frequency factor

UHF/3-21G UHF/6-31G(d) UMP2/6-31G(d)

∆Eact

Ethyl Radical Addition 1.43 × 108 1.80 × 108 1.13 × 108 1.46 × 108 2.37 × 108 2.33 × 108

1.29 1.67 1.05 1.01 1.58 1.67

Propyl Radical Addition 0.73 × 108 1.00 × 108 0.68 × 108

2.37 3.29 2.67

UHF/3-21G UHF/6-31G(d) UMP2/6-31G(d) UQCISD/6-31G(d) B-LYP/6-31G(d)b B3-LYP/6-31G(d)c

b

Figure 5. Rotational potentials for the propylene rotation (τ2 in Figure 3) for the additions of ethyl radical (s) and propyl radical (- - -) to ethylene. A torsional angle of 0° corresponds to the situation in which the forming C-C bond eclipses a C-H bond (ethyl radical) or C-C bond (propyl radical).

ethylene rotation

a All parameters required in the frequency factor calculations were obtained at the levels of theory indicated, except for the rotational potential used in the UQCISD/6-31G(d) frequency factor, which was taken to be the UMP2/6-31G(d) potential. The frequencies used in these calculations were scaled with the appropriate scale factors for low-frequency modes, as recommended in ref 23. b Potential used for ethylene torsion: V(θ) ) 0.5{-0.28(1 - cos 2θ) - 2.45(1 - cos 3θ)}, potential used for propylene torsion: V(θ) ) 0.5 {5.52(1 - cos 3θ)}. c Potential used for ethylene torsion: V(θ) ) 0.5{0.36(1 - cos θ) 0.17(1 - cos 2θ) - 2.79(1 - cos 3θ)}, potential used for propylene torsion: V(θ) ) 0.5{5.74(1 - cos 3θ)}

CISD/6-31G(d) geometries. Only the results for UQCISD/631G(d) geometries are listed in Table 11, but we note that similar results are generally obtained with the other geometries. It can be seen that in general the barrier for gauche addition is approximately 2 kJ mol-1 lower than the barrier for anti addition at each of the levels of theory examined. Furthermore, it can be seen that for both the gauche and anti additions, the barrier does not significantly change at any of the levels of theory beyond 6-311G(d,p) all the way to 6-311+G(3df,2p). The results in Table 11 allow us to examine in more detail the previously suggested basis set additivity approximation:5 ∆E‡[UQCISD(T)/6-311+G(3df,2p)] ≈ ∆E‡[UQCISD(T)/6-311G(d,p)] + ∆E‡[RMP2/6-311+ G(3df,2p)] - ∆E‡[RMP2/6-311G(d,p)] In particular, we can compare the basis set corrections obtained using RMP2 with corresponding UMP2 and PMP2 values. Table 12 displays the barriers for the gauche addition at the UQCISD(T)/6-311G(d,p) level of theory, basis set corrections from 6-311G(d,p) to 6-311+G(3df,2p) with the three different MP2 procedures, and the final (additive) UQCISD(T)/6-311+G(3df,2p) barriers.

19004 J. Phys. Chem., Vol. 100, No. 49, 1996

Heuts et al.

TABLE 10: Effect of Level of Theory on the Calculated Barrier to Reaction for the Anti Addition of Ethyl Radical to Ethylene (kJ mol-1), Using 6-31G(d) and 6-311G(d,p) Basis Sets and UHF/6-31G(d), UMP2/6-31G(d), and UQCISD/6-31G(d) Geometries 6-31G(d) level

UHF/6-31G(d)

UMP2/6-31G(d)

UHF UMP2 UMP3 UMP4 PUHF PMP2 PMP3 PMP4 ROHF RMP2 RMP3 RMP4 UB-LYP UB3-LYP UQCISD UQCISD(T)

42.2 56.7 55.5 50.8 0.5 18.7 25.0 20.3 93.5 36.3 46.3 35.1 10.7 18.2 34.2 29.5

43.6 56.1 55.2 52.0 7.2 22.9 29.0 25.8 78.2 36.4 43.9 37.3 17.7 21.8 37.2 33.7

6-311G(d,p) UQCISD/6-31G(d)

35.8 31.9

TABLE 11: Basis Set Effect on the Calculated Reaction Bariers for Anti and Gauche Additions at the UMP2, PMP2, RMP2, UQCISD, and UQCISD(T) Levels of Theorya calculated reaction barrierb basis set

UMP2 PMP2c RMP2 UQCISD UQCISD(T)

6-31G(d) 6-311G(d,p) 6-311G(df,p) 6-311+G(2df,p) 6-311+G(3df,2p)

57.4 53.6 53.8 54.1 53.4

Anti Addition 19.4 36.1 16.5 31.2 16.6 30.7 17.6 30.7 17.0 29.8

6-31G(d) 6-311G(d,p) 6-311G(df,p) 6-311+G(2df,p) 6-311+G(3df,2p)

55.6 51.4 51.5 52.2 52.0

Gauche Addition 18.0 34.1 14.6 28.7 14.7 28.1 16.0 28.2 15.8 27.8

35.8 32.6 32.9

32.0 27.7

34.4 30.9 31.1

30.4 25.8

a Single-point energy results for UQCISD/6-31G(d) optimized geometries. b Reaction barriers without zero-point vibrational energy and temperature corrections (kJ mol-1). c Only the first higher spin state was annihilated in these calculations.

TABLE 12: Calculated Reaction Barriers and Basis Set Corrections for the Gauche Addition of Ethyl Radical to Ethylene Using UHF/6-31G(d), UMP2/6-31G(d), and UQCISD/6-31G(d) Optimized Geometries geometry

∆E‡(QCI)a ∆(UMP2)b ∆(PMP2)c ∆(RMP2)d ∆E‡(QCI) + ∆(UMP2) ∆E‡(QCI) + ∆(PMP2) ∆E‡(QCI) + ∆(RMP2)

UHF/ 6-31G(d)

UMP2/ 6-31G(d)

UQCISD/ 6-31G(d)

23.5 +2.0 +2.5 -0.1 25.5 26.0 23.4

27.6 -0.5 +0.1 -1.5 27.1 27.7 26.1

25.8 +0.6 +1.1 -0.9 26.3 26.9 24.9

a ∆E‡(QCI) is the barrier (kJ mol-1) at the UQCISD(T)/6-311G(d,p) level of theory, without zero-point vibrational energy and temperature corrections. b ∆(UMP2) ) ∆E‡(UMP2/6-311+G(3df,2p)) - ∆E‡(UMP2/ 6-311G(d,p)) (kJ mol-1). c ∆(PMP2) ) ∆E‡(PMP2/6-311+G(3df,2p)) - ∆E‡(PMP2/6-311G(d,p)) (kJ mol-1). d ∆(RMP2) ) ∆E‡(RMP2/6311+G(3df,2p)) - ∆E‡(RMP2/6-311G(d,p)) (kJ mol-1).

We can see that the most positive corrections are obtained with the PMP2 procedure and the most negative with RMP2. Similar trends are observed for all three choices of geometries. In addition, the basis set corrections are most positive for the UHF/6-31G(d) geometries and most negative for the UMP2/631G(d) geometries. Examination of the UQCISD(T)/6-311G(d,p) barriers (∆E‡(QCI)) shows that the barrier obtained

UHF/6-31G(d)

UMP2/6-31G(d)

49.0 53.1 51.9 45.9 8.2 15.9 22.3 16.2 98.2

49.6 52.0 51.6 47.1 14.2 19.7 26.2 21.7 82.8

18.1 25.0 31.1 25.3

23.3 26.8 34.0 29.5

UQCISD/6-31G(d)

32.6 27.7

employing UMP2/6-31G(d) geometries is approximately 4 kJ mol-1 higher than the barrier obtained employing UHF/6-31G(d) geometries, with the barrier obtained with UQCISD/6-31G(d) geometries lying in between. The opposite dependence on choice of geometry observed for ∆E‡(QCI) on the one hand and the basis set correction at the MP2 level of theory (∆MP2) on the other leads to much smaller differences in the calculated barriers for different choices of geometries, after basis set correction. Taking into account all the values for the (additive) UQCISD(T)/6-311+G(3df,2p) barriers listed in Table 12, our best estimate for the reaction barrier is 25.6 kJ mol-1. In this specific case of the addition of alkyl radicals to ethylene, the UMP2 procedure seems to perform best in the basis set correction, as indicated by the good agreement with UQCISD(T) and UQCISD in the trend observed for the smaller basis sets (Table 11). For general use, however, we recommend RMP2 for the basis set correction, because of the possible adverse consequences of varying spin contamination with UMP2. We note also that the RMP2 procedure has been shown to correlate better than PMP2 with UQCISD(T).5 The barrier obtained with this generally recommended procedure has a value of 24.8 kJ mol-1, which is slightly lower than our best estimate in this specific case. In the light of the observation that the calculated barriers at the (additive) QCISD(T)/6-311+G(3df,2p) level of theory only show a relatively small dependence on the choice of geometry, and the previous recommendation of UHF/6-31G(d) (in preference to UMP2/6-31G(d)) in the case of large spin contamination,5 we recommend the use of UHF/6-31G(d) optimized geometries for general use. The vibrationless barrier (∆E‡) obtained in this manner is the basis for our further calculation of the activation energy, which is given by

Eact ) E0 - R

∂ ln(Q†/QalkeneQradical) ∂T-1

+ RT

(4)

Together with the zero-point energy correction (∆ZPVE), ∆E‡ gives the critical energy E0. The last two terms in eq 4, to which we will refer collectively as ∆Eact(T) and which show the variation in Eact with temperature, are obtained automatically with the frequency factor calculations and are included in Table 9. It can be seen that ∆Eact values at 323 K range from 1.01 to 1.67 kJ mol-1. As in the case of frequency factors, we recommend the use of UHF/6-31G(d) for calculating ∆Eact(T).

Propagation in Free-Radical Polymerizations Calculation of ∆Eact at 323 K at the UHF/6-31G(d) level of theory for the favored gauche addition yields a value of 1.53 kJ mol-1. In a previous section, we recommended the use of the B3LYP/6-31G(d) procedure for calculating ∆ZPVE, and taking this into account, eq 4 can now be written as

Eact ) ∆E‡(QCI) + ∆(MP2) + ∆ZPVE(B3-LYP/6-31G(d)) + ∆Eact(T) (5) Taking into account all possible combinations of ∆E‡(QCI) + ∆(MP2) (see Table 12) and ∆Eact (see Table 9), our best estimate for the activation energy at 323 K for the reaction of ethyl radical with ethylene comes out as 35.1 kJ mol-1. Use of the generally recommended procedures of this study (i.e., UHF/6-31G(d) geometries, UHF/6-31G(d) values of ∆Eact(T), B3-LYP/6-31G(d) ∆ZPVE’s, and ∆E‡(QCI) + ∆(RMP2) barriers) gives an activation energy at 323 K of 33.3 kJ mol-1. Both theoretical estimates are in reasonable agreement with the reported experimental value of 30.5 kJ mol-1 for the addition of ethyl radical to ethylene30 (and may be compared with an experimental value of 34.3 kJ mol-1 (at atmospheric pressure) for the highpressure polymerization of ethylene32). Because the proposed additivity scheme involves a calculation at the QCISD(T)/6-311G(d,p) level of theory, which will be computationally very demanding for large systems, it would be useful to have a scheme for approximating the basis set effect in going from QCISD(T)/6-31G(d) to QCISD(T)/6-311G(d,p). From the available data (ref 5 and Table 11), it can be seen that UMP2 performs somewhat better than RMP2 and QCISD. As mentioned before, however, when there is greater spin contamination than in the case of an alkyl radical addition to ethylene, the good correlation between UMP2 and QCISD(T) may not exist, which decreases the attraction of UMP2 for general application. RMP2 and UQCISD are then the preferred options for the proposed basis set correction, with the UQCISD procedure probably being the more robust, but also the more expensive. A similar basis set correction has been proposed in the G2(MP2,SVP) approximation to G2 theory,33 where it was shown that the results obtained were only marginally worse than those obtained with G2(MP2), but the reduced computational expense significantly extended the range of systems that can be examined. Concluding Remarks In the present study we have attempted to define levels of theory suitable for the reliable determination of Arrhenius parameters for the propagation reaction in free-radical polymerization. Our study confirms previous recommendations5 with respect to the calculation of reliable reaction barriers for radical addition reactions: good results are likely to be obtained through calculations at the QCISD(T)/6-311G(d,p) level of theory, with MP2 basis set correction to 6-311+G(3df,2p). We find that simpler levels of theory are sufficient for the calculation of the Arrhenius frequency factor to a reasonable accuracy. The parameters required to calculate the frequency factor in the transition state theory formulation, namely, geometries, fundamental frequencies, and torsional potentials, are much less sensitive to level of theory than are the barriers. In summary, we recommend the following procedures for general use in the determination of Arrhenius parameters for the propagation reaction in free-radical polymerization: (a) UHF/6-31G(d) for the optimization of geometries to be used in barrier and frequency factor calculations; (b) UHF/6-31G(d) for

J. Phys. Chem., Vol. 100, No. 49, 1996 19005 the calculation of rotational barriers and low-frequency vibrational modes, to be used in the calculation of frequency factors and temperature corrections to the reaction barrier; (c) B3-LYP/ 6-31G(d) for the calculation of zero-point vibrational energies; (d) UQCISD(T)/6-311G(d,p) for the calculation of reaction barriers together with an RMP2 estimate of the basis set corrections from 6-311G(d,p) to 6-311+G(3df,2p); (e) UHF/ 3-21G for the calculation of frequency factors and RMP2 for basis set corrections from 6-31G(d) to 6-311+G(3df,2p) in calculations on larger systems if somewhat larger uncertainties are acceptable. Acknowledgment. We gratefully acknowledge helpful discussions with Dr. Meredith Jordan, Dr. Anthony Scott, Dr. Richard Wong, and Professor H. Bernhard Schlegel, the provision of the one-dimensional Schro¨dinger equation code by Dr. George Bacskay, the provision of an Overseas Postgraduate Research Scholarship for J.P.A.H., support by the Australian Research Council, and a generous allocation of time on the Fujitsu VP-2200 supercomputer of the Australian National University Supercomputer Facility. References and Notes (1) (a) Australian National University. (b) University of Sydney. (c) Present address: School of Chemical Engineering and Industrial Chemistry, University of New South Wales, Sydney, NSW 2052, Australia. (2) For recent experimental studies on radical addition reactions, see: (a) Citterio, A.; Sebastiano, R.; Marion, R.; Santi, R. J. Org. Chem. 1991,56, 5328. (b) He´berger, K.; Walbiner, M.; Fischer, H. Angew. Chem., Int. Ed. Engl. 1992, 31, 635. (c) Avila, D. V.; Ingold, K. U.; Lusztyk, J.; Dolbier, W. R.; Pan, H.-Q. J. Am. Chem. Soc. 1993, 115, 1577. (d) He´berger, K.; Fischer, H. Int. J. Chem. Kinet. 1993, 25, 249. (e) He´berger, K.; Fischer, H. Int. J. Chem. Kinet. 1993, 25, 913. (f) Wu, J. Q.; Beranek, I.; Fischer, H. HelV. Chim. Acta 1995, 78, 194. (g) He´berger, K.; Lopata, A. J. Chem. Soc., Perkin Trans. 2 1995, 91. (g) Zytowski, T.; Fischer, H. J. Am.Chem. Soc. 1996, 118, 437. (h) Avila, D. V.; Ingold, K. U.; Lusztyk, J.; Dolbier, W. R.; Pan, H.-Q. J. Org. Chem. 1996, 61, 2027. (3) For recent experimental studies on propagation rate coefficients, see: (a) Davis, T. P.; O’Driscoll, K. F.; Piton, M. C.; Winnik, M. A. Macromolecules 1990, 23, 2113. (b) Davis, T. P.; O’Driscoll, K. F.; Piton, M. C.; Winnik, M. A. Polym. Int. 1991, 24, 65. (c) Deibert, S.; Bandermann, F.; Schweer, J.; Sarnecki, J. Makromol. Chem., Rapid Commun. 1992, 13, 351. (d) Pascal, P.; Winnik, M. A.; Napper, D. H.; Gilbert, R. G. Makromol. Chem., Rapid Commun. 1993, 14, 213. (e) Hutchinson, R. A.; Aronson, M. T.; Richards, J. R. Macromolecules 1993, 26, 6410. (f) Schoonbrood, H. A. S.; Van den Reijen, B.; De Kock, J. B. L.; Manders, B. G.; Van Herk, A. M.; German, A. L. Macromol. Rapid Commun. 1995, 16, 119. (g) Bergert, U.; Buback, M.; Heyne, J. Macromol. Rapid Commun. 1995, 16, 275. (h) Bergert, U.; Beuermann, S.; Buback, M.; Kurz, C. H.; Russell, G. T.; Schmaltz, C. Macromol. Rapid Commun. 1995, 16, 425. (i) Shipp, D. A.; Smith, T. A.; Solomon, D.; Moad, G. Macromol. Rapid Commun. 1995, 16, 837. (j) Buback, M.; Gilbert, R. G.; Hutchinson, R. A.; Klumperman, B.; Kuchta, F.-D.; Manders, B. G.; O’Driscoll, K. F.; Russell, G. T.; Schweer, J. Macromol. Chem. Phys. 1995, 196, 3267. (k) Lyons, R. A.; Hutovic, J.; Piton, M. C.; Christie, D. I.; Clay, P. A.; Manders, B. G.; Kable, S. H.; Gilbert, R. G. Macromolecules 1996, 29, 1918. (4) For recent theoretical studies on the addition of substituted alkyl radicals to alkenes, see: (a) Houk, K. N.; Paddon-Row, M. N.; Spellmeyer, D. C.; Rondan, N. G.; Nagase, S. J. Org. Chem. 1986, 51, 2874. (b) Arnaud, R.; Subra, R.; Barone, V.; Lelj, F.; Olivella, S.; Sole, A.; Russo, N. J. Chem. Soc., Perkin Trans. 2 1986, 1517. (c) Clark, T. J. Chem. Soc., Chem. Commun. 1986, 1774. (d) Fueno, T.; Kamachi, M. Macromolecules 1988, 21, 908. (e) Gonzales, C.; Sosa, C.; Schlegel, H. B. J. Phys. Chem. 1989, 93, 2435. (f) Gonzales, C.; Sosa, C.; Schlegel, H. B. J. Phys. Chem. 1989, 93, 8388. (g) Arnaud, R. New J. Chem. 1989, 13, 543. (h) Zipse, H.; He, J.; Houk, K. N.; Giese, B. J. J. Am. Chem. Soc. 1991, 113, 4324. (i) Arnaud, R.; Vidal, S. New J. Chem. 1992, 16, 471. (j) Tozer, D. J.; Andrews, J. S.; Amos, R. D.; Handy, N. C. Chem. Phys. Lett. 1992, 199, 229. (k) Schmidt, C.; Warken, M.; Handy, N. C. Chem. Phys. Lett. 1993, 211, 272. (l) Wong, M. W.; Pross, A.; Radom, L. J. Am. Chem. Soc. 1993, 115, 11050. (m) Wong, M. W.; Pross, A.; Radom, L. Isr. J. Chem. 1993, 33, 415. (n) Davis, T. P.; Rogers, S. C. Macromol. Theory Simul. 1994, 3, 905. (o) Wong, M. W.; Pross, A.; Radom, L. J. Am. Chem. Soc. 1994, 116, 6284. (p) Wong, M. W.; Pross, A.; Radom, L. J. Am. Chem. Soc. 1994, 116, 11938. (q) Espinosa-Garcı´a, J.; Corchado, J. C.; Leroy, G. J. Phys. Chem. 1995, 99, 13926. (r) Barone, V.; Orlandini, L. Chem. Phys. Lett. 1995, 246, 45.

19006 J. Phys. Chem., Vol. 100, No. 49, 1996 (5) See, for example: Wong, M. W.; Radom, L. J. Phys. Chem. 1995, 99, 8582. (6) For reviews, see, for example: (a) Pechukas, P. In Dynamics of Molecular Collisions, Part B; Miller, W. H., Ed.; Plenum Press: New York, 1976. (b) Laidler, K. J.; King, M. C. J. Phys. Chem. 1983, 87, 2657. (c) Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983, 87, 2664. (d) Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and Recombination Reactions; Blackwell: Oxford, 1990. (7) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (8) Heuts, J. P. A.; Gilbert, R. G.; Radom, L. Macromolecules 1995, 28, 8771. (9) See, for example: Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules, Oxford University Press: New York, 1989. (10) (a) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Foresman, K. B.; Johnson, B. G.; Schlegel, H. B.; Robb, M. A.; Repogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzales, C.; Matin, R. L.; Fox, D. J.; DeFrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. GAUSSIAN 92, Revision D.3; Gaussian Inc.: Pittsburgh, PA, 1992. (b) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Repogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; DeFrees, D. J.; Baker, J. ; Stewart, J. P.; Head-Gordon, M.; Gonzalez, C.; Pople, J. A. GAUSSIAN 94, Revision B.3; Gaussian, Inc.: Pittsburgh, PA, 1995. (c) Stanton, J. F.; Gauss, J.; Watts, J. D.; Lauderdale, W. J.; Bartlett, R. J. ACES II, Version 0.2; Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, 1992. (d) Amos, R. D.; Alberts, I. L.; Andrews, J. S.; Colwell, S. M.; Handy, N. C; Jayatilaka, D.; Knowles, P. J.; Kohayashi, R.; Koga, N.; Laidig, K. E.; Maslen, P. E.; Murray, C. W.; Rice, J. E.; Sanz, J.; Simandiras, E. D.; Stone, A. J.; Su, M. D. CADPAC5, Department of Chemistry, University of Cambridge, Cambridge, England, 1992. (11) Pople, J. A.; Head-Gordon, M.; Raghavachari, K. J. Chem. Phys. 1987, 87, 5968. (12) (a) Amos, R. D.; Andrews, J. S.; Handy, N. C.; Knowles, P. J. Chem. Phys. Lett. 1991, 185, 256. (b) Knowles, P. J.; Andrews, J. S.; Amos, R. D.; Handy, N. C.; Pople, J. A. Chem. Phys. Lett. 1991, 186, 130. (c) Lauderdale, W. J.; Stanton, J. F.; Gauss, J.; Watts, J. D.; Bartlett, R. J. Chem. Phys. Lett. 1991, 187, 21. (13) See, for example: (a) Knowles, P. J.; Somasundran, K.; Handy, N. C.; Hirao, K. Chem. Phys. Lett. 1985, 113, 8. (b) Handy, N. C.; Knowles, P. J.; Somasundram, K. Theor. Chim. Acta 1985, 68, 87. (c) Gill, P. M. W.; Radom, L. Chem. Phys. Lett. 1986, 132, 16. (d) Nobes, R. H.; Pople, J. A.; Radom, L.; Handy, N. C.; Knowles, P. J. Chem. Phys. Lett. 1987, 138, 481. (e) Lepetit, M. B.; Pe´lissier, M.; Malrieu, J. P. J. Chem. Phys. 1988, 89, 998. (f) Gill, P. M. W.; Pople, J. A.; Radom, L.; Nobes, R. H. J. Chem. Phys. 1988, 89, 7307. (g) Jensen, F. Chem. Phys. Lett. 1990, 169, 519. (h) Nobes, R. H.; Moncrieff, D.; Wong, M. W.; Radom, L.; Gill, P. M. W.; Pople, J. A. Chem. Phys. Lett. 1991, 182, 216. (14) (a) Schlegel, H. B. J. Chem. Phys. 1986, 84, 4530. (b) Schlegel, H. B. J. Phys. Chem. 1988, 92, 3075. (15) Becke, A. D. Phys. ReV. A 1988, 38, 3098. (16) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (17) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (18) (a) Nordholm, S.; Bacskay, G. B. Chem. Phys. Lett. 1976, 42, 253. (b) Bacskay, G. B. Unpublished work. (19) Gilbert, R. G.; Jordan, M. J. T.; Smith, S. C.; Pitt, I. G.; Greenhill, P. G. UNIMOL Program suite, available from the authors: School of Chemistry, University of Sydney, NSW 2006 Australia, or by electronic

Heuts et al. transfer from [email protected]. (20) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976. (21) (a) Pople, J. A.; Schlegel, H. B.; Krishnan, R.; DeFrees, D. J.; Binkley, J. S.; Frisch, M. J.; Whiteside, R. A.; Hout, R. F.; Hehre, W. J. Int. J. Quantum Chem. Symp. 1981, 15, 269. (22) Pople, J. A.; Scott, A. P.; Wong, M. W.; Radom, L. Isr. J. Chem. 1993, 33, 415. (23) (a) Scott, A. P.; Radom, L. J. Phys. Chem., submitted. (b) The standard scale factors for theoretical frequencies in ref 22 were evaluated using a least-squares procedure and are thus most appropriate for high frequencies. They are useful, for example, for evaluating zero-point vibrational energies. The most important frequencies for the evaluation of vibrational partition functions and related thermodynamic quantities (e.g., enthalpy and entropy), however, are the low frequencies. With this in mind, the vibrational data in ref 22 were reexamined (ref 23a) leading to (i) scale factors for low frequencies of 1.0075 (HF/3-21G), 0.9061 (HF/6-31G(d)), 1.0485 (MP2/6-31G(d)), 1.0620 (B-LYP/6-31G(d)), 1.0013 (B3-LYP/631G(d)), and 1.0147 (QCISD/6-31G(d)) and (ii) scale factors for zero-point vibrational energies of 0.9207 (HF/3-21G), 0.9135 (HF/6-31G(d)), 0.9670 (MP2/6-31G(d)), 1.0126 (B-LYP/6-31G(d)), 0.9806 (B3-LYP/6-31G(d)), and 0.9776 (QCISD/6-31G(d)). We adopt these values in the present work. (24) (a) Stanton, J. F. J. Chem. Phys. 1994, 101, 371. (b) Chen, W.; Schlegel, H. B. J. Chem Phys. 1994, 101, 5957. (25) (a) Johnson, B. G.; Gill, P. M. W.; Pople, J. A. J. Chem. Phys. 1993, 98, 5612. (b) Bauschlicher, C. W.; Partridge, H. J. Chem. Phys. 1995, 103, 1788. (c) Mebel, A. M.; Morokuma, K.; Lin, M. C. J. Chem. Phys. 1995, 103, 7414. (d) Rauhut, G.; Pulay, P. J. Chem. Phys. 1995, 103, 1788. (26) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955. (27) See, for example: (a) Pitzer, K. S.; Gwinn, D. W. J. Chem. Phys. 1942, 10, 428. (b) Lewis, G. N.; Randall, M. Thermodynamics, 2nd ed. (revised by K. S. Pitzer and L. Brewer); McGraw-Hill: New York, 1961. (c) Troe, J. J. Chem. Phys. 1977, 66, 4758. (d) Truhlar, D. G. J. Comput. Chem. 1991, 12, 266. (28) For the ethyl radical, see: (a) Pacansky, J.; Gardini, G. P.; Bargon, J. J. Am. Chem. Soc. 1976, 98, 2665. (b) Pacansky, J.; Dupuis, M. J. Chem. Phys. 1978, 68, 4276. (c) Pacansky, J.; Dupuis, M. J. Am. Chem. Soc. 1982, 104, 415. (d) Claxton, T. A.; Graham, A. M. J. Chem. Soc., Faraday Trans. 2 1987, 83, 2307. (e) Chettur, G.; Snelson, A. J. Phys. Chem. 1987, 91, 3488. (d) Pacansky, J.; Koch, W.; Miller, M. D. J. Am. Chem. Soc. 1991, 113, 317. (f) Quelch, G. E.; Gallo, M. M.; Schaefer, H. F. J. Am. Chem. Soc. 1992, 114, 8239. (g) Sears, T. J.; Johnson, P. M.; Jin, P.; Oatis, S. J. Chem. Phys. 1996, 104, 781. (29) For the propyl radical, see: (a) Pacansky, J.; Dupuis, M. J. Chem. Phys. 1979, 71, 2095. (b) Claxton, T. A.; Graham. A. M. J. Chem. Soc., Faraday Trans. 2 1988, 84, 121. (c) Bernardi, F.; Fossey, J. J. Mol. Struct. (THEOCHEM) 1988, 180, 79. (d) Pacansky, J.; Waltman, R. J.; Barnes, L. A. J. Phys. Chem. 1993, 97, 10694. (30) Kerr, J. A.; Parsonage, M. J. EValuated Kinetic Data on Gas Phase Reactions; Butterworths: London, 1972. (31) The frequency factor results given in this paper differ slightly from those given in ref 8, because the scale factors for the frequencies differ slightly from the values used previously. A sensitivity analysis of the scale factor dependence of the frequency factor has been included in ref 8. (32) Schweer, J. Ph.D. Thesis, Go¨ttingen University, Germany, 1988. (33) Curtiss, L. A.; Redfern, P. C.; Smith, B. J.; Radom, L. J. Chem. Phys. 1996, 104, 5148.

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