The Dynamic Equilibrium Between (AlOMe)n Cages and - American

Article pubs.acs.org/Macromolecules

The Dynamic Equilibrium Between (AlOMe)n Cages and (AlOMe)n·(AlMe3)m Nanotubes in Methylaluminoxane (MAO): A FirstPrinciples Investigation Zackary Falls, Nina Tymińska, and Eva Zurek* Department of Chemistry University at Buffalo, State University of New York Buffalo, New York 14260−3000, United States S Supporting Information *

ABSTRACT: Species likely to be present in methylaluminoxane (MAO) are studied via dispersion-corrected DFT, which we show is able to accurately predict thermochemical parameters for the dimerization of trimethylaluminum (TMA). Both cage-like, (AlOMe)n,c, and TMA-bound nanotubes, (AlOMe)n,t·(AlMe3)m, are found to be important components of MAO. The most stable structures have aluminum/oxygen atoms in environments whose average hybridization approaches sp3/sp2. The (AlOMe)n,t·(AlMe3)m isomers with the lowest free energies possess Al−μ-Me−Al bonds. At 298 K a novel Td-(AlOMe)16,c oligomer is one of the most stable structures among the six stoichiometries with the lowest free energies: (AlOMe)20,c·(AlMe3)2, Td-(AlOMe)16,c, (AlOMe)18,c, (AlOMe)20,c·(AlMe3), (AlOMe)10,t·(AlMe3)4, and (AlOMe)20,c. As the temperature rises, the abundance of (AlOMe)n,t·(AlMe3)m decreases, and that of (AlOMe)n,c increases. Because the former are expected to be precursors for the active species in polymerization, this may in part be the reason why the cocatalytic activity of MAO decreases at higher temperatures.

1. INTRODUCTION In the past three decades, the serendipitous discovery1 of methylaluminoxane (MAO) and its dramatic effect2 as a cocatalyst in olefin polymerization has revolutionized industrial polyolefin production.3 Sinn and Kaminsky were the first to realize that adding water to systems such as Cp2ZrMe2/TMA (trimethylaluminum) caused them to become highly active as ethylene polymerization catalysts.1 The partial hydrolysis of TMA, which is found as the dimer in solution, is thought to result in the formation of MAO, which behaves as an activator in the polymerization process. MAO abstracts a methyl or alkyl group from the precatalyst, allowing the olefin to coordinate to the metal center followed by a subsequent insertion into the metal−alkyl bond via a Cossée−Arlman-type mechanism4−6 (slight modifications may be made as a result of ion-pairing between the anion and the catalyst7,8). The interest in these single-site catalysts stems from their high stereoselectivity, and the narrow molecular weight distribution of the polymer produced.9−14 MAO and its modified variants have been been used extensively as co-catalysts with a variety of postmetallocene catalysts.15 For example, highly branched lowdensity polyethylene was synthesized upon MAO activation.16 Typically, a large excess of MAO is necessary to achieve high © XXXX American Chemical Society

catalytic activities, but recently a complex was synthesized that displayed the highest activity when the Al:Ti ratio was 8.17 Despite tremendous efforts MAO has thwarted experimental characterization, and is sometimes referred to as a “black-box”. Prior to Atwood’s structural determination of the (Al7O6Me16)− anion,18 it was believed that aluminoxanes consisted of linear or cyclic structures with three-coordinate aluminum and two-coordinate oxygen atoms.19,20 Evidence for four-coordinate aluminum atoms were put forward by Barron and co-workers who were able to synthesize and characterize a series of cages, [(tBu)Al(μ3-O)]n, n = 6, 7, 8, 9, 12.21,22 The methyl analogues of two of these cage-structures, (AlOMe)6,c and (AlOMe)12,c, are illustrated in Figure 1. It was postulated that a dynamic equilibrium between a plethora of cage-like structures ensues in MAO, whereas the bulkier tert-butyl substituents in Barron’s studies hindered such an interconversion. A number of analogous [(tBu)Ga(μ3-S)]n cages were prepared as well, and it has been shown that the cubane, n = 4, can form larger oligomers, n = 6−8, under thermolysis in Received: September 11, 2014 Revised: October 22, 2014

A

dx.doi.org/10.1021/ma501892v | Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

Article

edges were found to be favored. For example, the bond belonging to two square faces (s−s) colored green in (AlOMe)6,c, see Figure 1b, is strained, whereas the bond comprising a square and a hexagonal face (s−h) highlighted in Figure 1(a) is less strained. The Gibbs free energies were employed to calculate a Boltzmann distribution of (AlOMe)n with varying n, and in a wide temperature range a “sodalite”-like (AlOMe)12,c cage, shown in Figure 1c, was found to have the highest abundance. Curiously, this structural motif arises in a number of diverse systems ranging from (MgO)12 and (ZnO)12 clusters58 to a CaH6 solid that may be superconducting under pressure.59 Because the aforementioned strained s−s edges are destabilizing, it was assumed that the predominant MAO cages would be “spherical” since the topology of “long” polyhedra necessitated a greater amount of LLA sites. These strained edges were also found to be the sites most reactive with TMA45 or the metallocene.49 Perhaps unsurprisingly the incorporation of TMA into the spherical cages was found to be unfavorable, and the optimal C:Al ratio was predicted to be close to unity. A deficiency in the computational method employed in refs 7, 35, 45, and 49 was that neither generalized gradient approximation (GGA) type functionals nor hybrid functionals such as B3LYP can reproduce the dimerization energy of TMA. At the time the work was performed, however, correlated wave function based methods would have been prohibitively expensive, and functionals incorporating dispersion as well as metaGGA functionals were not available. Recent studies by Boudene and co-workers of cage structures using metaGGAtype functionals have illustrated that an (AlOMe)18 oligomer may be the predominant species in solution at roomtemperature, and identified geometric descriptors that can be employed to calculate thermodynamic parameters of arbitrary cages.60 Linnolahti and co-workers performed periodic calculations at the Hartree−Fock and B3LYP levels of theory on various armchair, zigzag, and chiral nanotubes.41 On the basis of a comparison of the energies of some of these systems with the spherical (AlOMe)12,c structure it was suggested that aluminoxanes may be nanotubular. Interestingly, the finite (2,2) nanotubes32 correspond to oligomers that were previously precluded from further study,35 since they were found to be less stable than the spherical alternatives. However, Linnolahti et al. showed via MP2 calculations that the nanotubular (AlOMe)12,t which had been capped at the ends with two TMA dimers (Al16O12Me24) had a lower Gibbs free energy as compared to the sodalite-like (AlOMe)12,c and two free TMA dimers.32 In Figure 1d, we illustrate the nanotubular (AlOMe)12,t, with one of the four strained bonds comprising two square faces highlighted. The authors also pointed out that the Al16O12Me24 structure may be an important component of MAO, because it can account for the experimentally determined C:Al:O ratio and average molecular weight. Within this contribution we focus on studying computationally the dynamic equilibrium between spherical MAO cages (with the formula (AlOMe)n,c, n = 6−20) and free TMA, as well as finite (2,2) MAO nanotubes that have been capped at the ends with TMA (with the formula (AlOMe)n,t·(AlMe3)m, n = 6−20 and even, m = 1−4). The stability of the structures is shown to be linked to the hybridization of the aluminum and oxygen atoms, with the former preferring sp3 and the latter sp2 hybridization. Methyl groups bridging two aluminum atoms are present in species with the formula (AlOMe)n,t·(AlMe3)m. These interactions are highly stabilizing, and are reminiscent of

Figure 1. (a, b) The hexameric cage, (AlOMe)6,c. Bonds belonging to (a) one square and one hexagonal face (s−h) and (b) two square faces (s−s), are highlighted in green. (c) Sodalite-like (AlOMe)12,c. One of the bonds belonging to two hexagonal faces (h−h) is colored green. (d) Nanotubular (AlOMe)12,t oligomer. One of the four (s−s) bonds is colored green. Aluminum/oxygen/carbon/hydrogen atoms are gray/red/black/white.

different solvents or via the addition of acid.23 Some experiments suggest that the size of an average MAO oligomer, (AlOMe)n, is temperature dependent ranging from n = 9−30,24 while others have proposed that up to 50−60 Al atoms comprise a typical species in MAO.25 These hypothetical (AlOMe)n are reactive and exhibit latent Lewis acidity (LLA) because of the ring strain present in some of the Al−O bonds.26,27 As a result, the residual TMA that is present in all MAO solutions interacts with the MAO oligomers such that the average C:Al:O ratio in MAO is estimated as being 1.5:1:0.75.28,29 Indeed, the reaction of the hexameric cage [(tBu)Al(μ3-O)]6 with TMA and with Cp2ZrMe2 results in breaking a strained bond possessing LLA.26,30 Recent studies have suggested that other structures might be present in MAO as well. For example, electrospray ionization mass spectrometry experiments on isobutylaluminoxanes by McIndoe and co-workers provided evidence for cyclic and cagelike structures.8 Further experiments by these same authors suggested that MAO exists as an ion-pair in sufficiently polar media, with the general formula [Me 2 Al][(MeAlOx(AlMe3)yMe].31 On the theoretical front, computational experiments are also helping to demystify MAO.33 First-principles calculations on a number of (AlOMe)n structural alternatives including linear,34 cyclic,34−36 cage,35−39 linked-cage,40 and even nanotubular structures32,41 have been carried out. MAO formation via the hydrolysis of TMA has been studied32,42−44 The interaction of (AlOMe)n with TMA,36,37,39,45−47 as well as MAO with the metallocene and potential active and dormant species in olefin polymerization were examined.32,48−54 A few studies have looked at the mechanism of olefin polymerization.7,55−57 One of us has performed density functional theory (DFT) calculations on a variety of (AlOMe)n oligomers, and cage-like structures resembling those synthesized by Barron and coworkers were found to be thermodynamically preferred.35 The most stable cages consisted of square and hexagonal faces. Because edges belonging to two square faces possess a high degree of ring strain, polyhedra with the fewest number of such B


Macromolecules

Article

Table 1. Thermodynamic Quantities (Change in Energy, ΔE; Enthalpy, ΔH; Gibbs Free Energy, ΔG) for the Dimerization of TMA, and Geometric Properties of the TMA Dimer Calculated with Different Computational Methodsa method

ΔE

ΔHb

ΔGb

C1−Alc

C2−Alc

Al−C2−Alc

B3LYP/6-311G(d,p)d rev-PBE/TZP rev-PBE+D3/TZP M06/TZP M06-L/6-311G(d,p)e M06-HF/6-311G(d,p)e M06-2X/6-311G(d,p)e MP2/TZVP CCSD(T)/def2-QZVPPf experimentalg

−8.4 −10.1 −19.2 −20.2 − − − −21.3 −22.3 −

−6.6 −6.9 −14.8 −14.7 −17.6 −24.9 −19.5 −19.5 −20.7 −20.4

12.0 6.9 −4.1 −5.5 −5.3 −6.4 −4.5 −3.5 −4.5 −7.5

− 1.979 1.975(3) 1.951 1.953 1.966 1.958 1.972 1.972(69) −

− 2.171 2.161 2.129 2.138 2.148 2.140 2.148 2.148 −

− 74.30 73.90 74.70 74.90 75.05 75.21 75.00 75.00 −

a

Energies presented in kcal/mol, bond distances in Ångströms, and angles in degrees. bP = 1 atm, T = 298.15 K. cSee Figure 2 for the atom labeling. Reference 32. eReference 60. fSingle-point calculation with CCSD(T) from the structure optimized with MP2/TZVP, thermodynamic corrections from the MP2/TZVP calculations were employed. gReference 62.

d

semiempirical dispersion correction63 (rev-PBE+D3) yield thermochemical quantities that differ substantially from those computed with rev-PBE alone, and approach those obtained with the M06 functional. Although M06 gives slightly better results than revPBE+D3, and has been successfully employed to study (AlOMe) n oligomers by Boudene et al.,60 it is computationally more demanding. In particular, the metaGGA functional requires the use of an ultrafine integration grid in order to avoid errors such as spurious imaginary frequencies.64,65 We believe that the loss of about 1 kcal/mol in accuracy for ΔG is a reasonable price to pay for the use of a more efficient method such as revPBE+D3. Since the reactions investigated in this manuscript involve the addition of TMA to a MAO cage via breaking a Lewis acidic bond and the formation of species similar to those illustrated in Figure 2d,e, further tests of the computational method must be

the bonding in the TMA dimer. Both cages and capped nanotubes are shown to be important components of the MAO mixture at ambient conditions, but the novel Td-(AlOMe)16,c structure is the most abundant. Increasing temperature shifts the equilibrium toward the formation of the cage-like species with the formula (AlOMe)n,c.

2. RESULTS AND DISCUSSION 2.1. Benchmarking a Fast Computational Method. Before beginning our theoretical investigation of the various species that may be important components of a MAO solution, an appropriate theoretical method must be chosen for the task at hand. The method should be fast enough to optimize the geometries and calculate the vibrational frequencies of the plethora of potential species with reasonable computational cost, while at the same time being able to reproduce thermochemical quantities for important reactions with sufficient accuracy. Typically DFT is the method of choice in such a situation. But, as Table 1 shows, standard gradientcorrected hybrid and nonhybrid functionals such as rev-PBE and the very popular B3LYP are unable to accurately reproduce parameters associated with the dimerization of TMA. In fact, the free energy calculated with both of these methods at ambient conditions suggests that the monomer ought to be the dominant species in solution, contradicting experimental observations. Using the current computational chemistry “gold standard”, CCSD(T) with a large basis set, yields values for the dimerization of TMA in very good agreement with experiment, see Table 1. The change in enthalpy is calculated to within chemical accuracy, whereas the computed and experimental changes in the Gibbs free energy differ by 3 kcal/mol. The thermochemical properties and geometrical parameters calculated with MP2, and various Gaussian type basis sets32,47 agree well with CCSD(T). Combining MP2 with larger basis sets, such as def-TZVP, used by Linnolahti and co-workers in their studies of (AlOMe)n·(AlMe3)m species, further improves agreement between theory and experiment.32 An even computationally cheaper alternative was proposed by Boudene et al., who showed that various functionals belonging to the M06 suite could obtain reasonable values of ΔH and ΔG for the dimerization of TMA.60 Is it possible to decrease the computational cost further, while maintaining similar accuracy? As shown in Table 1, our tests using the rev-PBE functional supplemented by Grimme’s

Figure 2. Optimized geometries of (a) the TMA dimer, (AlMe3)2, (b) (AlOH)4,c, (c) (AlOH)6,c, (d) (AlOH)4,c·(AlH3), and (e) (AlOH)6,c· (AlH3).

performed. Because there is no experimental data available for these types of reactions, MP2 and M06 were used as references. Table 2 shows that the thermochemical quantities calculated with MP2 and M06 for these reactions differed by less than 3.5 kcal/mol. Results computed with the semiempirical dispersion correction differed somewhat more from MP2, in all cases underestimating ΔE, ΔH, and ΔG. Nevertheless, the deviation between the values of the Gibbs free energies computed with rev-PBE+D3 and MP2 is acceptably small. Changing the hydrogens to methyls in the cubic cage does not affect the error in ΔG. This would be important if one needs to compare results for larger structures where the MP2 method becomes too expensive. Therefore, we conclude that C


Macromolecules

Article

Table 2. Thermodynamic Quantities (kcal/mol) for the Reaction (AlOR)n,c + (1/2) (AlR3)2 → (AlOR)n,c·(AlR3), with R = H, CH3, and Selected Geometric Parameters (in Å) for (AlOR)n,c as Calculated with Different Computational Methods structure

method

ΔE

ΔHa

ΔGa

Al−O (s−s)b

Al−O (s−h)b

Al−Cb

Al−Hb

(AlOH)4,c

MP2/TZVP M06/6-311G(d,p) rev-PBE+D3/TZP MP2/TZVP M06/6-311G(d,p) rev-PBE+D3/TZP MP2/TZVP M06/6-311G(d,p) rev-PBE+D3/TZP

−26.2 −27.4 −21.5 −22.5 −24.7 −18.9 −23.2 −26.3 −20.6

−25.4 −26.4 −20.3 −22.0 −23.8 −18.5 −22.3 −25.5 −19.8

−20.7 −22.1 −15.8 −18.1 −18.2 −13.2 −15.7 −18.8 −14.4

1.801 1.849 1.873 1.841 1.857 1.876 1.921 1.904 1.844

− − − − − − 1.822 1.807 1.840

− − − 1.972 1.924 1.953 − − −

1.578 1.555 1.573 − − − 1.574 1.557 1.583

(AlOMe)4,c

(AlOH)6,c

a

P = 1 atm, T = 298.15 K. bSee Figure 1 and Figure 2 for an illustration of the bond types, and the (AlOR)n,c structures.

where ADav is the average distortion, Ai is an angle between the aluminum atom and two of the four atoms to which it is coordinated, A0 = 109.5°, and m = 6. A similar relationship was employed to calculate the average distortion for the oxygen environments, ODav, except the deviation was calculated with respect to an ideal planar geometry with A0 = 120°, and m = 3. A number of other descriptors were examined, but the best correlation was found for ADav. The most stable environments had the smallest ADav, highlighting the importance of the tetrahedral environment around the aluminum atoms. This descriptor provides a more powerful and general method to estimate electronic energies as opposed to simply considering the number of (h−h−h), (s−h−h), (s−s−h), and (s−s−s) environments. It should be noted that for square and hexagonal faces alone, the order of stability of the different environments was found to be the same in the two studies (differences arose when octagonal faces were considered as well).35,60 In this section, we re-examine the relative energies and free energies of various (AlOMe)n oligomers. Moreover, we present a newly found oligomer, Td-(AlOMe)16,c, that turns out to be more stable at room temperature than any other structure studied so far. The two main classes of (AlOMe)n oligomers likely to be important components of MAO are cages, (AlOMe)n,c, similar to those synthesized by Barron et al.,21,22 and (2,2) nanotubes, (AlOMe)n,t, such as those investigated extensively by Linnolahti and co-workers.32,41 Because the nanotubular systems must contain an even number of oligomeric units, in this manuscript we restrict our computational investigations to species with even n. Various (AlOMe)n,c with odd n have been studied in refs 35, 57 and 60. Because of the large computational expense required for frequency calculations, systems containing up to 20 oligomeric units are considered herein. We note that the aforementioned geometry-based descriptors can be used to estimate energies and free energies of larger (AlOMe)n structures.35,60 Figure 3 illustrates the cage-like oligomers for a given n that were found to be the most stable in this study. The tert-butyl analogues of (AlOMe)6,c, (AlOMe)8,c and (AlOMe)12,c were synthesized by Barron.21,22 Galloxane and alumoxane hydroxides with the formulas [Ga 1 2 t Bu 1 2 O 1 0 (OH) 4 ] and [Al6tBu6O4(OH)4] have also been synthesized.61 The bodies of these species resembled the (AlOMe)n cages, but there were some differences as well. For example, the galloxane was composed solely of hexagonal faces and a few of the oxygen atoms comprising the cage were two coordinate. The alumoxane, on the other hand, had five coordinate aluminum atoms and all of the faces were square. There is only one way to

the rev-PBE+D3 combination provides a good balance of accuracy and computational cost for this study. In the minimum energy configurations of (AlOH)n,c·(AlH3) (n = 4, 6) the −O−AlH2 groups are oriented perpendicular to −AlH2, as shown in Figure 2d,e. When the hydrogens were replaced by methyl groups, this perpendicular orientation remained the most stable one for (AlOMe)4,c·(AlMe3), but as we will see below, a geometry where the −O−AlMe2 groups are parallel to −AlMe2 is preferred for (AlOMe)6,c·(AlMe3). Sections 2.4, 2.5 and the Supporting Information contain a detailed exploration of the interaction of TMA with larger (AlOMe)n oligomers. 2.2. Cage and Nanotubular Structures with the Formula (AlOMe)n. A plethora of (AlOMe)n oligomers have been investigated in refs 35 and 60. One of the goals of these studies was to find a relationship between the electronic energy of these species and some set of geometric parameters or descriptors. Oligomers containing octagonal faces were found to be destabilized as compared to those comprised of square and hexagonal faces alone.35 In polyhedra made up of four, six, and eight membered faces, it can be shown that the number of square faces must be equal to the number of octagonal faces plus six. So, introducing octagonal faces into the polyhedron necessitates an increase in the number of destabilizing, strained square faces. For this reason, we restrict our study toward species without octagonal faces. In ref 35, it was also illustrated that the electronic energy of the (AlOMe)n oligomers could be estimated by considering the bonding environments of the atoms in the polyhedra. These atoms could be classified according to the types of faces to which they belonged: for example, (s−s−s) refers to a vertex joining three square faces, (s−s−h) two square faces and one hexagonal face, and so on. The order of stability of the different environments, in decreasing order, was found to be (h−h−h) > (s−h−h) > (s−s−h) > (s−s−s). One of the problems with using this descriptor is that two oligomers may have the same types of bonding environments, but different electronic energies. Boudene and co-workers carried out a more sophisticated analysis: they calculated the correlation of the electronic energy per AlOMe unit (normalized electronic energies) with a number of geometric parameters.60 One of the parameters they considered was the deviation between the angles of the aluminum atoms and the ideal tetrahedral geometry using the equation: m

ADav =

∑ |A i − A 0 | i=1

(1) D


Macromolecules

Article

Comparison of the energies and Gibbs free energies per AlOMe unit shows that for n > 10 the cage-like structures are more stable than the nanotubes. This has been noted previously in calculations using the BP86 functional, where the decreased stability of the nanotubular oligomers was explained by the fact that each tube contained eight atoms in (s−s−h) environments.35 These atoms are found on either end of the tube, and the bonds between them are the most strained and therefore are the most reactive with TMA yielding species such as (AlOMe)n,t·(AlMe3)m.45 Out of all of the oligomers studied in ref 35 the Th symmetry (AlOMe)12,c, whose body resembles a truncated octahedron, was found to have the lowest normalized free energy in a wide temperature range from 198 to 598 K. At 298 K Boudene and co-workers found another structure to be preferred, the (AlOMe)18,c cage, but by 498 K (AlOMe)12,c had regained its reign as the most stable oligomer.60 This shift of the equilibrium toward smaller n can be explained by the increasing importance of the entropy, which favors smaller structures, toward the total free energy at higher temperatures. In agreement with Boudene’s results, we find that (AlOMe)18,c has a lower normalized free energy than (AlOMe)12,c at ambient conditions. But, we also find a hitherto unconsidered cage-like oligomer to be slightly more stable than (AlOMe)18,c at 298 K. This Td symmetry (AlOMe)16,c structure, illustrated in Figure 5a, was not scrutinized in previous studies,35,60 which employed the C3v isomer shown in Figure 5b instead. Td-(AlOMe)16,c is related to the chamfered cube, otherwise known as the truncated rhombic dodecahedron. The main difference is that whereas the chamfered cube has Oh symmetry, this symmetry is broken in Td-(AlOMe)16,c because half of the vertices are occupied by oxygen and the other half by aluminum atoms. Some zeolite crystals assume the chamferededge cube geometry.66

Figure 3. Optimized geometries of the cage-like (AlOMe)n,c oligomers with n = 6, 8, 10, 12, 14, 16, 18, 20. The red vertices correspond to oxygen atoms, and the gray ones aluminum atoms. Methyl groups have been removed for clarity. The energies and Gibbs free energies at 298 K and 1 atm are given with respect to Td-(AlOMe)16,c, in units of kcal/ (mol n).

Figure 5. Optimized geometries of the (A) novel Td-symmetry (AlOMe)16,c, and the (B) C3v-symmetry (AlOMe)16,c considered in previous studies.33,60 The energies and Gibbs free energies at 298 K and 1 atm are given with respect to the more stable of the two species, Td-(AlOMe)16,c, in units of kcal/(mol n).

There are other slight differences between the relative stabilities of the various oligomers at 298 K calculated here, as compared to the normalized free energy orderings obtained in prior investigations. For example, we find (AlOMe)12,c to be less stable than (AlOMe)14,c by 0.3 kcal/(mol n), whereas the opposite was found in refs 35 and 60. There could be a number of reasons for this discrepancy, including the fact that different functionals and basis sets were used in these three studies. Another difference is that, in ref 35, vibrational frequencies, which were subsequently used to determine the free energies, were calculated via a force field that had been specifically parametrized for these particular systems. And, in ref 60, the energies and free energies of species with n > 12 were estimated using values associated with geometric descriptors. To verify the stability ordering calculated here, we optimized the

Figure 4. Same as Figure 3, but for the nanotubular (AlOMe)n,t cages with n = 10, 12, 14, 16, 18, and 20.

construct polyhedral structures with the formulas (AlOMe)6 and (AlOMe)8 that are comprised of square and hexagonal faces. Adding two AlOMe units to a single square face of (AlOMe)6 yields (AlOMe)8, and subsequent growth in onedimension results in the nanotubes35,41 shown in Figure 4. Because of this the six-mer and eight-mer can be thought of as the smallest nanotubes. But, because these oligomers are not particularly long in a single dimension, they can also be thought of as cages. Within this work we refer to these species as cages, keeping in mind that this designation is arbitrary. E


Macromolecules

Article

Figure 6. (a) Predicted abundances of the cage and nanotubular (AlOMe)n oligomers from Figure 3 and Figure 4 at 298 K. (b) Same as part a, but including both the Td-(AlOMe)16,c and C3v-(AlOMe)16,c cages in Figure 5. (c) Same as part b, but at 398 K. (d) Same as part b, but at 498 K.

geometries of the n = 12, 14, and 16 structures with MP2/ccpVDZ. Frequency calculations at this level of theory would have been prohibitively expensive for these systems. Nonetheless, the 0 K relative energies computed with MP2 were in full agreement with those obtained using rev-PBE+D3: (AlOMe)14,c was more stable than (AlOMe)12,c by 0.3 kcal/(mol n) and (AlOMe)16,c was more stable than (AlOMe)14,c by 1.4 kcal/ (mol n). The percent abundance of various (AlOMe)n oligomers in a hypothetical mixture at equilibrium can be estimated using the free energies per monomer and assuming a Boltzmann distribution, as described in Section 4. This procedure was carried out for the cage and nanotubular structures shown in Figure 3 and Figure 4, and initially did not include the C3v(AlOMe)16,c illustrated in Figure 5b. Other isomers, species with odd n, and those with n > 20 were not considered. The data at 298 K, see Figure 6a, illustrate that the nanotubular systems are not important components of such a hypothetical mixture, comprising less than 3.4% of the total. The cage-like structures with n = 16, 18, and 20 make up nearly 87.8%, on the other hand. In constructing Figure 6a, we have considered only the single most stable cage-like oligomer for a given number of AlOMe units. Of course there may be a number of isomers, potentially close in free energy, for the larger cages. A case in point are the two (AlOMe)16,c isomers illustrated in Figure 5. How does the distribution change if both of these structures are included in calculating the percent composition? Figure 6b shows that, at 298 K, C3v-(AlOMe)16,c becomes the fourth most stable structure, and the abundance of the other oligomers decreases somewhat. For example, Td-(AlOMe)16,c decreases from 33.5% to 27.6%. The two cage-like 16-mers comprise 45.4% of this hypothetical mixture, whereas (AlOMe)n,c with n = 16, 18, and 20 constitute 89.9%.

At 298 K the average molecular formula (including all of the cage and nanotubular structures explicitly considered) is calculated as being (AlOMe)17.08. Because entropy favors smaller structures, the average size of the oligomers decreases with increasing temperature to (AlOMe)17.02 and (AlOMe)16.85 at 398 and 498 K, respectively. These numbers are in quite good agreement with the average formula of (AlOMe)17.23 at 298 K, and (AlOMe)16.89 at 398 K calculated previously.35 Our calculations find that cage-like oligomers with n = 16, 18, and 20 are more stable than the structure with the lowest free energy identified in ref 35, Th-(AlOMe)12. But, the (estimated) free energies of cages with n up to 30 were included in the Boltzmann average in ref 35. These two factors counterbalance each other, and this is the reason for the very similar average molecular formulas in the two studies, even though the structures identified as being the most stable differ. At lower temperatures, the equilibrium will shift to favor (AlOMe)18,c, since this is the oligomer with the lowest normalized electronic energy at 0 K. It is likely that we did not consider every important isomer in constructing the percent abundances illustrated in Figure 6. And because the free energies fall in the exponential in the Boltzmann equation changing them slightly (e.g., because a different level of theory is used) will also have a non-negligible effect on the computed distribution. Moreover, all the percentages of the (AlOMe)n oligomers (as well as for their TMA-bound variants, discussed later in the text) are an approximation for the actual composition, because MAO may change over time. For instance, in the case of CoCl2-MAO, used for Ziegler−Natta olefin polymerization, aging of this catalytic system before addition of 1,3-butadiene showed accelerated deterioration of the catalyst.67 Conversely, in a recent study by Hu et al.,68 the authors reported that neodymium isopropoxide catalyst activated by MAO led to highly activated production of 1,3-butadiene polymer and 1,3F


Macromolecules

Article

oxygen atom on the square and hexagonal faces. For a perfect tetrahedron the twist angle is 60.0°. In Table 3 the deviation from the perfect tetrahedral environment, Δ(Td) = |TA − 60°|, is also provided. This measure is in particular useful for oxygen, for which only three angles are available instead of six to determine the variance from the ideal tetrahedral geometry using an equation similar to eq 1. An oxygen atom in a tetrahedral environment will experience enhanced ring strain, so it would be the most likely to undergo reaction with TMA. The NBO analysis shows that each oxygen atom has one lone pair that is oriented roughly orthogonal to the the body of the (AlOMe)n oligomers. Let us first consider the data for (AlOMe)12,t, since it has atoms in three distinct sites. The three geometrical descriptors, ADav, ODav, and Δ(Td), illustrate that an aluminum and oxygen atom in an h−h−h site are the closest to achieving the perfect tetrahedral and trigonal planar environments, respectively. The average hybridizations are in full agreement with these descriptors, such that the aluminum atom is calculated as being sp3.0 hybridized, and oxygen attains sp2.1 hybridization. The descriptors show that the geometries around atoms in s− h−h, and even more so around s−s−h sites, differ significantly from the ideal. Interestingly, even though ADav and Δ(Td) clearly indicate that the aluminum atom in an h−h−h site is closest to, and the one in an s−s−h site is furthest from, the perfect tetrahedral environment, the average hybridizations as calculated with the NBO program are virtually the same. Thus, the average hybridizations of the aluminum atoms appear to be quite insensitive to their surrounding geometry. The average hybridizations of the oxygen atoms, on the other hand, have a much broader range from sp2.7 in the strained s−s−h site, to sp2.1 in the stable h−h−h vertex. Even though (AlOMe)12,c does not have atoms in the highly preferable h−h−h environments, it does not possess any of the unstable s−s−h vertices. Overall, it has a lower ADtot av than (AlOMe)12,t, but ODtot is higher. The average hybridizations for av all of the aluminum atoms in both the cage and the nanotube are very close to the ideal sp3, but the average hybridization averaged over all oxygen atoms for (AlOMe)12,c is sp2.2, whereas for (AlOMe)12,t it is sp2.4. Since the magnitude of the normalized electronic energy of the nanotube is lower than that of the cage, this suggests that the hybridization of the oxygen atoms is likely a very important factor in determining which structures are the most stable, and which bonds are the most reactive. Next, we examined two very similar structures: the cage-like (AlOMe)16,c isomers illustrated in Figure 5. These two species are particularly stable and are predicted to be quite abundant in a hypothetical (AlOMe)n mixture, as illustrated in Figure 6. Both oligomers contain eight atoms in an h−h−h site and 24 in

butadiene/isoprene copolymer when aged. Despite the limitations mentioned above, a number of conclusions can be drawn regarding the species present in a hypothetical mixture of (AlOMe)n oligomers at equilibrium: (i) the abundance of nanotubular structures is low, (ii) at 298 K the distribution is likely to have a maximum around n = 16 and somewhat larger cages will also be important components of the mixture, (iii) at 298 K cages with n < 14 will not be abundant, (iv) at 398 K and higher the percentage of nanotubular structures approaches zero, (v) increasing the temperature results in a slight decrease of the size of the average oligomer, and (vi) the relative abundance of Td-(AlOMe)16,c increases with respect to larger oligomers at higher temperatures (in the temperature range studied). 2.3. Exploring Aluminum and Oxygen Environments. Can we understand the enhanced stability of the cage-like structures over the nanotubes in terms of the descriptors outlined at the beginning of section 2.2? In order to do so geometrical parameters (ADav and ODav), and the average hybridizations of the aluminum and oxygen atoms in the (AlOMe)12,t and (AlOMe)12,c oligomers were compared. These two isomers were chosen because of the very different bonding environments within them, which are representative of those found in nanotubes and cages, respectively.

Figure 7. revPBE+D3/TZP optimized geometries of the (a) nanotubular and (b) cage-like (AlOMe)12 isomers. The symmetryinequivalent oxygen and aluminum atoms in the body of the oligomer are labeled.

Table 3 provides ADav, ODav and the average hybridizations, as calculated with the natural bond orbital (NBO) program, of the oxygen and aluminum atoms labeled in Figure 7. The tetrahedral twist angle (TA) is another measure useful in determining the deviation from perfect tetrahedral geometry.69 In order to calculate the TA, the Mercury 2.4 program70 was employed to define the angles between planes containing Al and O atoms in various environments. For example, the TA for Al2 in (AlOMe)n,t in the vertex of an s−h−h site would be measured by considering the angle between the plane containing Al2 and two oxygen atoms in the square face, with the plane that passes through Al2 and two oxygens in the hexagonal face. Similarly, one can obtain the twist angles for O2, but here the planes would include two aluminums and one

Table 3. Geometrical Descriptors Characterizing the Nanotubular, (AlOMe)12,t, and Cage-Like, (AlOMe)12,c, Oligomers Illustrated in Figure 7 structure (AlOMe)12,t

(AlOMe)12,c

atoma,b Al1 Al2 Al3 Al1

(4) (4) (4) (12)

site s−s−h s−h−h h−h−h s−h−h

ADavc (deg) 78.2 57.0 17.0 43.2

Δ(Td)d (deg)

hybe 3.1

26.9 19.5 9.7 9.6

sp sp3.0 sp3.0 sp3.1

atom O1 O2 O3 O1

(4) (4) (4) (12)

site

ODavc (deg)

Δ(Td)d (deg)

hybe

s−s−h s−h−h h−h−h s−s−h

62.1 33.5 10.3 43.5

4.0 11.9 36.4 11.1

sp2.7 sp2.3 sp2.1 sp2.2

a

tot,c See Figure 7 for the atom labeling. bThe number in brackets indicates the total number of atoms in the given environment. cADtot,t av = 43.2°, ODav d e tot,t = 43.5° and ADtot,t av = 50.7°, ODav = 35.3°. Deviation of the tetrahedral twist angle from 60° (the ideal for a perfect tetrahedron). Average hybridization as determined with the NBO analysis.

G


Macromolecules

Article

Table 4. Geometrical Descriptors Characterizing the Two Cage-Like Oligomers Illustrated in Figure 5 structure Td-(AlOMe)16 C3v-(AlOMe)16

atoma Al1 Al2 Al1 Al2 Al3

(12) (4) (12) (3) (1)

site s−h−h h−h−h s−h−h h−h−h h−h−h

ADavb (deg)

Δ(Td)d (deg)

41.9 20.7 41.9 16.2 28.2

hybc 3.0

9.6 4.6 10.6 6.5 10.0

sp sp3.1 sp3.0 sp3.1 sp3.1

atoma O1 O2 O1 O2 O3

(12) (4) (12) (3) (1)

site

ODavb (deg)

Δ(Td)d (deg)

hybc

s−h−h h−h−h s−h−h h−h−h h−h−h

49.0 0.0 46.3 14.7 0.6

13.1 50.8 15.1 51.0 50.5

sp2.2 sp2.0 sp2.2 sp2.0 sp2.0 tot,C3v

a

d d The number in brackets indicates the total number of atoms in the given environment. bADtot,T = 36.6°, ODtot,T = 37.5° and ADav av av c 3v = 31.4°. Average hybridization as determined with the NBO analysis. ODtot,C av

an s−h−h environment, see Table 4. Moreover, their ADtot av / ODtot av are nearly identical measuring 36.3°/31.4° for C3v(AlOMe)16,c and 36.6°/37.5° for Td-(AlOMe)16,c. The average hybridizations of the oxygen and aluminum atoms averaged over all atoms are the same for the two structures. In this case neither the geometric descriptors of Boudene, nor the average hybridizations are able to rationalize the preference of one isomer over the other. Nonetheless, the Td oligomer is slightly more stable than the C3v one: MP2/cc-pVDZ results show that the electronic energy of the former is 0.3 kcal/(mol n) lower than that of the latter, in full agreement with the results obtained from revPBE+D3/TZP. 2.4. How Does (AlMe3)2 Interact with (AlOMe)n? Even though the (AlOMe)n cages and nanotubes discussed above are beautiful and fascinating, a mixture comprised solely of these systems has not yet been made. Real MAO solutions always contain residual TMA, and a dynamic equilibrium ensues between (AlOMe)n, (AlMe3)2, and species with the general formula (AlOMe)n·(AlMe3)m. Some authors have proposed that entities with different compositions may be important components of MAO as well (see for example refs 8 and 31), but these will not be considered here. Let us get a better understanding of the interaction of (AlOMe)n with (AlMe3)2, and compare our results to previous studies that employed the BP86 functional,45 and second order Møller−Plesset perturbation theory.32 Despite the fact that (AlOMe)6,c is not likely to be a major component of MAO, its strained LLA bonds render it a useful model system for studying the interaction of TMA with (AlOMe)n. In ref 45, a number of possible products of the reaction (AlOMe)n , c +

m (AlMe3)2 → (AlOMe)n , c ·(AlMe3)m 2

= 36.3°,

Figure 8. Optimized geometries for (AlOMe)6,c·(AlMe3) where the OAlMe2 is oriented (a) parallel and (b) perpendicular to methyl groups in Al(Me)2. Frequency calculations showed that both configurations are minima. The change in the Gibbs free energy at 298 K and 1 atm for the formation of these species according to reaction 2 is provided. Most of the methyl groups on (AlOMe)6,c have been omitted for clarity, and only the methyls originating from TMA and one methyl on the cage are shown as black lines. Average hybridizations of select oxygen and aluminum atoms are shown.

little TMA coordinates to (AlOMe)n, whereas MP2 calculations came to the opposite conclusion.32 The average hybridizations for the oxygen and aluminum atoms in (AlOMe)6,c were calculated as being sp2.7 and sp3.1, respectively. These values are exactly the same as the ones obtained for the oxygens and aluminums in (AlOMe)6,c· (AlMe3) that are at least one atom removed from the newly formed hexagonal face, see Figure 8. The addition of TMA to the cage primarily affects the average hybridization of the oxygen atoms in the hexagonal face, which become sp2.1 (OAlMe2) and sp2.4/2.5 (O−Al3) hybridized. By reacting with TMA, the hybridization of these oxygen atoms can approach the ideal value of sp2 expected for a trigonal planar geometry. The hybridization of the aluminum atom in O2−Al-Me2 is sp3.2, in line with a tetrahedral coordination, and it increases only slightly from the value calculated for (AlOMe)6,c. Because the hybridizations of the “parallel” and “perpendicular” arrangements in Figure 8 are nearly identical, the difference in the free energy of the two species must be attributed in part to other effects such as sterics. Let us consider the addition of the second AlMe3 to (AlOMe)6,c·(AlMe3) (shown in Figure 8a) across the hexagonal face that results from the first addition of AlMe3 (i.e., to O-sp2.4 and Al-sp3.0). This yields a configuration where the aluminum in the O−AlMe2 group lies close to the carbon in AlMe2. Figure 9b shows that this results in the formation of an Al−C bond measuring 2.268 Å, and the initially three-coordinate aluminum increases its coordination by one. The average hybridization of this aluminum increases from sp2.0 to sp2.2. Because the carbon atom is now five-coordinate, the bond to the parent aluminum weakens and lengthens to 2.107 Å. Both of these Al−C bonds are somewhat longer than the other Al−C bonds within

(2)

with n = 6, m = 1 were considered. In the most stable configurations a strained bond belonging to two square faces was cleaved, yielding a hexagonal face, and O−AlMe2 and (O2Me)Al−Me bonds were formed. We have reoptimized the two most stable structures obtained in ref 45, and their geometries are shown in Figure 8. The ambient-temperature Gibbs free energy of the species where AlMe2 is oriented parallel to the recently added methyl group (Figure 8a) is calculated as being 10.9 kcal/mol lower than the perpendicular configuration (Figure 8b). The stability ranking obtained here agrees with ref 45, but there is one crucial difference. The change in the Gibbs free energy at 298 K for reaction 2 in the parallel configuration was found to be less than −5 kcal/mol with BP86, whereas rev-PBE+D3 yields −15.9 kcal/mol. The GGA underestimates the ΔG for this reaction, as it does the enthalpy and free energy associated with the dimerization of TMA. This is likely the reason why ref 45 concluded that very H


Macromolecules

Article

Figure 9. Optimized geometries and select C−Al bond lengths in (a) (AlMe3)2, (b) (AlOMe)6,c·(AlMe3)2, (c) (AlOMe)12,t·(AlMe3)4 in the “bending in” and the (d) “bending out” configuration. The ΔG values are given for the formation of (AlOMe)12,t·(AlMe3)4 from (AlOMe)12,t and 2(AlMe3)2 at 298 K and 1 atm. The yellow boxes highlight the Al−μ-Me−Al bonds.

Figure 10. Optimized geometries of (AlOMe)n,c·(AlMe3)m. The parent cage structures are illustrated in Figure 3.

Table 5. Changes in the Energy (ΔE), Enthalpy (ΔH), Temperature Multiplied by the Entropy (TΔS), and Gibbs Free Energy (ΔG) at 298.15 K and P = 1 atm for Reaction 2a

(AlOMe)6,c·(AlMe3)2 and the terminal Al−C bonds in (AlMe3)2, which all measure between 1.96 and 1.98 Å. In fact, they more closely resemble the 2.161 Å (or 2.2−2.17 Å71) distance between the carbon atoms in the bridging methyl groups and aluminum in the TMA dimer (see Figure 9a). From now on this newly formed motif will be referred to as a Al-μMe-Al “bending in” interaction. For more details see the discussion in the Supporting Information. When two AlMe3 groups are added to the same side of one of the nanotubular structures shown in Figure 4, one can obtain either a "bending in" or a "bending out" motif. The difference between the two is that in the former an Al−μ-Me−Al bond is formed with a methyl group that originated from the addition of TMA, whereas in the latter the Al-μ-Me-Al bond comprises a carbon atom on a methyl group originating from the (AlOMe)n,t “backbone”. These two configurations are shown in Figure 9c,d. We have optimized the geometries of a few species where three or four TMA monomers have been added to an (AlOMe)n,t nanotube in these two configurations, and in all cases the “bending in” geometry was favored over “bending out” by 10 kcal/mol or more depending on the structure. On the basis of the results of our computations (see the Supporting Information for further details) the following rules have been employed to create the structures discussed in this manuscript: (i) AlMe3 was preferentially added to strained bonds belonging to two square faces over those comprising one square and one hexagonal face, (ii) the O−AlMe2 groups were aligned parallel to the AlMe2 groups as in Figure 8A, (iii) the second AlMe3 was added so as to minimize steric interactions (see the Supporting Information for further details), and (iv) bridging methyl groups were formed via the “bending out” and “bending in” configurations in Figure 9c,d, with the latter being favored if geometrically possible. 2.5. Stabilities of Nanotubular and Cage-Like (AlOMe)n·(AlMe3)m Species. Now that we have determined the most favorable ways that TMA can coordinate with (AlOMe)n, let us take a closer look at the (AlOMe)n·(AlMe3)m species. The optimized geometries resulting from the reaction of the smallest members of the cage family in Figure 3 are shown in Figure 10, and the ΔGs for reaction 2 at ambient conditions are

n

m

ΔE

ΔH

TΔS

ΔG

6

1 2 3 4 1 2 3 4 1 2 3 1 1 2

−19.4 −36.6 −52.8 −56.7 −15.5 −34.9 −40.7 −65.9 −15.8 −34.0 −43.2 −11.4 −15.4 −28.6

−19.0 −35.4 −50.1 −53.8 −14.8 −33.6 −45.4 −63.3 −15.0 −32.2 −40.0 −10.1 −14.7 −25.8

−3.1 −10.5 −22.6 −31.4 −6.1 −11.7 −21.2 −31.0 −6.1 −14.6 −29.3 −9.7 −9.9 −20.4

−15.9 −25.0 −27.5 −20.1 −8.8 −22.0 −24.3 −32.3 −8.8 −17.6 −10.7 −0.4 −4.8 −5.4

8

10

14 20 a

Thermochemical data given in units of kcal/mol. The parent (AlOMe)n,c structures are illustrated in Figure 3, and (AlOMe)n,c· (AlMe3)m are shown in Figure 10.

provided in Table 5. Just like (AlOMe)6,c, (AlOMe)8,c is able to coordinate up to four AlMe3 groups, two on either end of the structure. But, unlike its smaller brethren the reaction becomes less exgergonic after the second successive addition. The distribution of LLA sites around (AlOMe)10,c differs from the n = 6, 8 cages, and this structure only has three strained bonds belonging to two square faces. Adding the first and second AlMe3 group to this cage is favored, but the third addition is not because of the increased sterics. We were not able to find any (AlOMe)12,c·(AlMe3) structure whose free energy was lower than that of the isolated (AlOMe)12,c and (1/2)(AlMe3)2, likely because the parent cage does not have any LLA sites and because all of the possible structures have unfavorable steric interactions between methyl groups. The only other cages that possess bonds belonging to two square faces are (AlOMe)14,c (1 bond) and (AlOMe)20 (2 bonds). As Table 5 illustrates, the addition of TMA to these bonds is favorable. I


Macromolecules

Article

successive addition of AlMe3. However, only a few of the reactions in Table 6 are found to be exergonic. Specifically, this is the case for the formation of (AlOMe)10,t·(AlMe3)m with m = 1−4, (AlOMe)12,t·(AlMe3)m with m = 3,4, and (AlOMe)14,t· (AlMe3)m with m = 3,4. The formation of (AlOMe)n,t·(AlMe3)m systems with n > 14 is not found to be exergonic because of the high stability of the parent cages (see for example the computed percent abundances in Figure 6). In fact, the Gibbs free energies of (AlOMe)14,c·(AlMe3), (AlOMe)20,c·(AlMe3) and (AlOMe)20,c·(AlMe3)2 are 11.0, 30.9, and 25.5 kcal/mol lower than those of the corresponding TMA-capped nanotubes. Using MP2 calculations, Linnolahti and co-workers calculated ΔE and ΔG for the formation of (AlOMe)12,t·(AlMe3)4 from free (AlMe3)2 and (AlOMe)12,c as being −45.3 and −7.7 kcal/mol.32 These values are in good agreement with the results obtained here (−48.8 and −12.3 kcal/mol), confirming that rev-PBE+D3/TZP affords a good balance between computational expense and accuracy for these types of systems. On the other hand, because the BP86 functional is unable to treat dispersion, the addition of AlMe3 to (AlOMe)n was in general found to be unfavorable in ref 45. Linnolahti et al. speculated that due to its high stability (AlOMe)12,t·(AlMe3)4 may be a very important component of MAO.32 Its molecular weight, and C:Al:O ratio of 1.5:1.0:0.75 are in excellent agreement with experimental estimates regarding the average MAO oligomer. Moreover, the O−AlMe2 groups at the end of the tube can coordinate Cp2ZrMe2, giving rise to systems that are likely to be active in polymerization.7,49 2.6. The MAO Mixture. Because Linnolahti and coworkers32 did not carry out calculations on some of the species found to be particularly stable here, we decided to re-explore the equilibrium between cage-like structures, nanotubes where TMA is coordinated at the ends, and those cages that have Lewis acid sites that have reacted with TMA. In order to compare the relative stabilities of these two types of species, the energy of m/2 (AlMe3)2 was subtracted from (AlOMe)n,t· (AlMe3)m, and the result was divided by n. This procedure yields a normalized energy per monomer unit. In the Supporting Information we plot these normalized energies, and the normalized 298 K free energies for (AlOMe)n,t· (AlMe3)4 (n = 6−20) relative to (AlOMe)12,c. In general, these results are in agreement with those found previously:32 the fully saturated nanotubes are more stable than (AlOMe)12,c, but the cage becomes preferred upon increasing temperature. There are slight differences, however. For example, we find (AlOMe)10,t· (AlMe3)4, which has not yet been studied computationally, to be slightly more stable than (AlOMe)12,t·(AlMe3)4. But, (AlOMe)12,c is not the predominant species with the formula (AlOMe)n at 298 K. Instead, we found the novel Td(AlOMe)16,c oligomer to be the most abundant system at this temperature, see Figure 6. A plot of the normalized (free) energies of the TMA-capped nanotubes, with respect to Td(AlOMe)16,c at temperatures ranging from 298 to 498 K, is provided in Figure 11. Because of the enhanced stability of Td(AlOMe)16,c over (AlOMe)12,c the picture totally changes, such that none of the ΔGs are computed to be negative. So, even though the reaction of the n = 10, 12, 14 cages with TMA is in some cases exergonic, overall the formation of the most stable cage, Td-(AlOMe)16,c, and free TMA is preferred. The normalized free energies of the various (AlOMe)n· (AlMe3)m considered herein were employed to estimate their percent abundance assuming a Boltzmann distribution. The results, given in Figure 12, confirm the enhanced stability of Td-

Above we have shown that the nanotubular species illustrated in Figure 4 are less stable than the cage-like alternatives in Figure 3. However, each tube has strained LLA sites on both ends that allow for the addition of up to four AlMe3 groups to the structure, in a fashion similar to the (AlOMe)6,c·(AlMe3)m and (AlOMe)8,c·(AlMe3)m structures in Figure 10. Since the coordination of TMA to the ends of these tubes is likely to be exergonic, it may be that species with the formula (AlOMe)n,t· (AlMe3)m are more stable than (AlOMe)n,c and free (AlMe3)2 (or (AlOMe)n,c·(AlMe3)m). In order to explore this possibility, we have optimized the geometries of (AlOMe)n,t·(AlMe3)m with n = 10, 12, 14, 16, 18, 20 and m = 1−4. The optimized geometries are not shown (coordinates may be found in the Supporting Information), because they resemble the (AlOMe)6,c·(AlMe3)m and (AlOMe)8,c·(AlMe3)m structures in Figure 10, but the “tube” portion in the middle is longer. The thermochemical parameters associated with formation of these species via the reaction: (AlOMe)n , c +

m (AlMe3)2 → (AlOMe)n , t ·(AlMe3)m 2

(3)

are given in Table 6. Note that they gauge which species are thermodynamically favored: cage-like structures and free TMA dimer, or nanotubes that have been capped with TMA at their ends. At room temperature and pressure, the changes in the Gibbs free energies are computed to be less positive for each Table 6. Same as Table 5 Except for Reaction 3a n

m

ΔE

ΔH

TΔS

ΔG

10

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

3.7 −14.7 −33.8 −49.9 −65.6 18.3 0.7 −18.0 −33.9 −48.8 16.4 −1.9 −20.4 −34.7 −47.0 29.6 10.8 −7.4 −21.0 −33.0 33.1 13.6 −5.2 −18.7 −33.5 28.7 7.8 −6.3 −22.8 −36.7

3.9 −13.7 −32.7 −47.2 −62.6 19.1 1.6 −16.3 −31.3 −46.5 15.7 −1.2 −18.2 −31.4 −42.6 30.8 12.6 −4.7 −17.4 −28.7 33.7 15.1 −3.6 −16.4 −29.1 29.8 13.0 −0.2 −18.5 −32.1

−1.4 −9.2 −16.6 −21.8 −30.0 −1.7 −10.5 −18.2 −25.0 −34.2 −6.1 −11.8 −20.2 −26.7 −32.9 −6.7 −14.0 −22.3 −26.9 −38.1 −3.8 −11.0 −20.5 −30.1 −35.2 −5.1 −11.8 −19.0 −30.9 −41.5

5.3 −4.5 −16.1 −25.4 −32.6 20.8 12.1 1.9 −6.3 −12.3 21.8 10.6 2.0 −4.8 −9.7 37.6 26.6 17.6 9.4 9.4 37.5 26.0 16.9 13.7 6.1 35.0 24.8 18.9 12.4 9.4

12

14

16

18

20

a

The parent (AlOMe)n,t structures are illustrated in Figure 4. J


Macromolecules

Article

Al17.42O15.93Me20.38, Al17.31O17.03Me17.88, and Al16.94O16.86Me17.12 at 358, 398, and 498 K. These correspond to molecular weights of 1031, 1009, and 984 g/mol and C:Al:O ratios of 1.17:1:0.91, 1.03:1:0.98, and 1.01:1:1.00, at these temperatures. Notably, higher temperatures shift the equilibrium toward the formation of cage-like structures, such that the number of species with which TMA has reacted comprising the mixture decreases (47.3%, 7.1%, and 2.2% at 358, 398, and 498 K). The abundance of bare nanotubular species is virtually negligible at the highest temperature studied. The only other computational study that used a statistical approach to obtain an average formula for species in the MAO mixture employed free energies calculated with the BP86 functional. Since this method underestimates the interaction of TMA with (AlOMe)n, it is not surprising that the calculated C:Al:O ratio was close to unity at 298 K.45 Linnolahti and coworkers carried out MP2 calculations and obtained accurate free energies, but they did not use these to calculate an average formula for MAO.32 Their work was seminal in proposing that nanotubes capped with TMA are an important component of MAO, but they focused primarily on the (AlOMe)12,t·(AlMe3)4 structure. We have, for the first time, combined a quantum mechanical approach that is able to treat dispersion interactions with sufficient accuracy, with a statistical mechanical approach, in order to estimate the percent abundance of a plethora of species likely to be present in the MAO mixture as a function of temperature. On the basis of the ΔG computed for the formation of (AlOMe)12,t·(AlMe3)4 from (AlOMe)12,c and (AlMe3)2, Linnolahti et al. postulated that at higher temperatures the cages and free TMA dimers would become preferred over the capped nanotubes.32 Indeed, our results confirm this prediction. For example, at 398 K we find that only 7.1% of the species in solution are tubes or cages that have reacted with TMA. Experiments have demonstrated that heating MAO to ∼370 K

Figure 11. Changes in the electronic energy (0 K) and the Gibbs free energy for the reaction: (AlOMe)16,c+2(AlMe3)2 → (AlOMe)n,t· (AlMe3)4 given in kcal/(mol n), at 298, 398 and 498 K.

(AlOMe)16,c, which comprises 8.5% of the mixture at 298 K. (AlOMe) 18,c and (AlOMe)20,c are also very important components, such that the abundance of these species alone is 16.7%. In total the cage-like structures constitute 29.9% of the mixture, and the bare nanotubes only 0.8%. These systems are unable to react with the catalyst to form the active species. The cages and nanotubes which have reacted with TMA are prerequisites to formation of the active species, and they sum up to ∼69% of the total. The most prevalent of these is (AlOMe)20,c·(AlMe3) 2 (8.6%) followed by (AlOMe)20,c· (AlMe3), (AlOMe)10,t·(AlMe3)4, (AlOMe)12,t·(AlMe3)4, and others. The average formula at this temperature is Al18.07O16.12Me22.00, with the molecular weight of 1077 g/mol and C:Al:O ratio of 1.22:1:0.89. This C:Al:O ratio matches relatively well with the one obtained experimentally: 1.5:1:0.75.28,29 Our average molecular weight is also in good agreement with experimental estimates of ∼1000 g/mol.29 Increasing the temperature leads to a decrease in the average molecular weight, as expected. The average formula becomes

Figure 12. Predicted abundances of the cage-like structures, (AlOMe)n,c in Figure 3 and Figure 5, those cages which have reacted with TMA in Figure 10 whose contributions are non-negligible, and the nanotubular systems from Figure 4 to which AlMe3 is coordinated at (a) 298 K, (b) 358 K, (c) 398 K, and (d) 498 K. K


Macromolecules

Article

• Increasing temperature shifts the equilibrium toward the formation of cages and free TMA dimer such that the average formula is Al17.31O17.03Me17.88 by 398 K. The release of TMA into the MAO solution explains the drop in polymerization observed above 370 K. MAO, one of the industrially most important cocatalysts in olefin polymerization, harbors many mysteries. First-principles calculations are slowly unveiling the types of species that may be present in a MAO solution as a function of temperature. Such information is important for studying the reaction mechanism in olefin polymerization, and for designing cocatalysts with higher activities.

resulted in a sudden decrease in the polymerization rate.38 The active species in polymerization is likely formed from the interaction of (AlOMe)n·(AlMe3)m with the metallocene via a Zr−μ-Me−Al bond.49 Our data suggests that very few active sites will be present at 398 K, as compared to lower temperatures, thereby providing an explanation for the sudden drop in the rate of polymerization. Moreover, our data shows that higher temperatures will favor the liberation of TMA into the MAO solution. It has previously been proposed that TMA can form methyl-bridged dinuclear species with the metallocene that are catalytically inactive,72 so this is likely another reason for the decrease in the activity at higher temperatures.

3. CONCLUSIONS In an attempt to assist experimental characterization techniques, a number of excellent theoretical investigations have previously been carried out on structures likely to be important components of the MAO mixture. We have, for the first time, used a combined a quantum mechanical method that is able to account for dispersion with a statistical mechanical approach to study the relative abundance of various (AlOMe)n· (AlMe3)m species that may be present in MAO. In addition to confirming what is already known about MAO, we have found the following new points: • revPBE+D3/TZP is a cost-efficient computational method that is able to calculate ΔEs and ΔGs for reactions likely to be important in the MAO mixture to within a few kcal/mol/n of the results obtained with second order Møller−Plesset perturbation theory and the M06 functional. • A novel (AlOMe)16,c cage with Td symmetry is the most stable oligomer with the formula (AlOMe)n in a temperature range of at least 298−498 K. • An NBO analysis was employed to confirm that the most stable (AlOMe)n oligomers contain aluminum and oxygen atoms in environments that are close to being sp3 and sp2 hybridized, respectively. • When AlMe3 coordinates to (AlOMe)n by breaking a strained bond belonging to two square faces, the oxygen atoms in the newly formed hexagonal face attain a hybridization that approaches the ideal for a trigonal planar geometry, and the aluminum atoms in O−AlMe2 are sp2 hybridized. • The aluminum atoms in O−AlMe2 groups, which form via the reaction with TMA, can increase their coordination by bonding with carbons in methyl groups bridging two aluminum atoms. In such a way the carbons become five-coordinate, and the average hybridization of the aluminums increases to sp2.2. This type of bonding is reminiscent of the Al−μ-Me−Al bonds present in the TMA dimer. • at 298 K the six components of the MAO mixture with the lowest free energies are predicted to be (in order of decreasing stability): (AlOMe)20,c·(AlMe3)2, Td-(AlOMe) 16,c , (AlOMe) 18,c , (AlOMe) 20,c ·(AlMe 3 ), (AlOMe)10,t·(AlMe3)4, and (AlOMe)20,c. TMA-containing species, prerequisites to the active species, comprise 69% of the mixture. The average formula of Al18.07O16.12Me22.00, molecular weight of 1077 g/mol, and C:Al:O ratio of 1.22:1:0.89 match well with experimental results.

4. COMPUTATIONAL DETAILS The majority of the computations (geometry optimizations, frequency calculations) were performed using the rev-PBE73−76 nonhybrid generalized gradient density functional coupled with Grimme’s D3 dispersion correction63 (rev-PBE+D3) as implemented in the Amsterdam Density Functional (ADF) package.77,78 The basis functions consisted of a triple-ζ Slater-type basis set with polarization functions from the ADF basis−set library. The core-shells up to 1s for C, O and 2p for Al were kept frozen. We have tested the accuracy of the rev-PBE+D3/TZP results by comparing them to those obtained with a number of different methods (see Table 1 and Table 2). Computations performed with the M06 meta-hybrid GGA functional from the Minnesota family79 and a TZP basis set were also carried out using ADF. The ΔG for the dimerization of TMA calculated with M06/TZP was within 0.3 kcal/mol of the results previously obtained with M06/6-311G**,60 showing that these two basis sets give comparable results. The remaining calculations were performed using the Gaussian 09 software package.80 This includes geometry optimizations and frequency calculations carried out with second order Møller−Plesset perturbation theory (MP2),81 and the M06 functional.79 For these calculations we used Gaussian-type valence TZP (TZVP)82,83 or 6311G** (triple-ζ with polarization functions) basis sets. In situations where MP2/TZVP calculations were prohibitively expensive (i.e., geometry optimizations of large cage structures), the cc-pVDZ basis was used instead. The CCSD(T) calculations84 were performed using the def2-QZVPP85,86 basis set. This level of theory was employed on the geometry optimized with MP2/TZVP to obtain a single point energy for the dimerization of TMA. This procedure, i.e., performing single point energy calculations with a higher level of theory on a geometry optimized within a less demanding method, is often used to recover the correlation energy more completely. It is worth mentioning that it is not possible to calculate analytical frequencies using the CCSD(T) code, hence the thermal corrections for the dimerization of TMA were extracted from MP2/TZVP. The natural bond orbital (NBO)87−89 analysis was performed with PBE+D3/ TZVP via NBO 3.190 as implemented in Gaussian 09 (on the ADF optimized geometries). The normalized electronic energies, E(n,m)/n, for species with the general formula (AlOMe)n · (AlMe3)m (n = 6−20, m = 0−4) were calculated via:

(E((AlOMe) ·(AlMe ) ) − E( E(n , m)/n = n

3 m

m (AlMe3)2 2

))

n

(4) where E((AlOMe)n,c) and E((AlOMe)n,t) refer to the energies of the cage and nanotube structures illustrated in Figure 3 and Figure 4, respectively. E((AlOMen)·(AlMe3)m) is the energy of a tube or cage to which TMA has coordinated, and E((AlMe3)2) is the energy of a TMA dimer. The normalized Gibbs free energies (G(n,m,T)/n) = (H(n,m,T) − (TS)(n,m,T)/n), where H(n,m,T)/n is the normalized enthalpy, T is the temperature, and S(n,m,T)/n is the normalized entropy) were calculated in an analogous fashion. L


Macromolecules

Article

To determine the percent abundance of a given species at 1 atm of pressure and at a temperature T, we followed the procedure outlined in ref 35. The difference between the normalized free energy of this oligomer and a single AlOMe monomer was calculated via

ΔG(n , m , T )/n = G(n , m , T )/n − G(AlOMe, T )

(17) Shoken, D.; Sharma, M.; Botoshansky, M.; Tamm, M.; Eisen, M. S. J. Am. Chem. Soc. 2013, 135, 12592−12595. (18) Atwood, J. L.; Hrncir, D. C.; Priester, R. D.; Rogers, R. D. Organometallics 1983, 2, 985−989. (19) Pasynkiewicz, S. Polyhedron 1990, 9, 429−453. (20) Stellbrink, J.; Niu, A.; Allgaier, J.; Richter, D.; Koenig, B.; Hartmann, R.; Coates, G.; Fetters, L. Macromolecules 2007, 40, 4972− 4981. (21) Mason, M. R.; Smith, J. M.; Bott, S. G.; Barron, A. R. J. Am. Chem. Soc. 1993, 115, 4971−4984. (22) Harlan, C. J.; Mason, M. R.; Barron, A. R. Organometallics 1994, 13, 2957−2969. (23) Barbarich, T. J.; Bott, S. G.; Barron, A. R. J. Chem. Soc., Dalton Trans. 2000, 1679−1680. (24) Babushkin, D. E.; Semikolenova, N. V.; Panchenko, V. N.; Sobolev, A. P.; Zakharov, V. A.; Talsi, E. P. Macromol. Chem. Phys. 1997, 198, 3845−3854. (25) Ghiotto, F.; Pateraki, C.; Tanskanen, J.; Severn, J. R.; Luehmann, N.; Kusmin, A.; Stellbrink, J.; Linnolahti, M.; Bochmann, M. Organometallics 2013, 32, 3354−3362. (26) Harlan, C. J.; Bott, S. G.; Barron, A. R. J. Am. Chem. Soc. 1995, 117, 6465−6474. (27) Koide, Y.; Bott, S. G.; Barron, A. R. Organometallics 1996, 15, 5514−5518. (28) Imhoff, D. W.; Simeral, L. S.; Sangokoya, S. A.; Peel, J. H. Organometallics 1998, 17, 1941−1945. (29) Sinn, H. Macromol. Symp. 1995, 97, 27−52. (30) Watanabi, M.; McMahon, C. N.; Harlan, C. J.; Barron, A. R. Organometallics 2001, 20, 460−467. (31) Trefz, T. K.; Henderson, M. A.; Wang, M. Y.; Collins, S.; McIndoe, J. S. Organometallics 2013, 32, 3149−3152. (32) Linnolahti, M.; Severn, J. R.; Pakkanen, T. A. Angew. Chem., Int. Ed. 2008, 47, 9279−9283. (33) Zurek, E.; Ziegler, T. Prog. Polym. Sci. 2004, 29, 107−148. (34) Luhtanen, T. N. P.; Linnolahti, M.; Pakkanen, T. A. J. Organomet. Chem. 2002, 648, 49−54. (35) Zurek, E.; Woo, T. K.; Firman, T. K.; Ziegler, T. Inorg. Chem. 2001, 40, 361−370. (36) Zakharov, I. I.; Zakharov, V. A.; Potapov, A. G.; Zhidomirov, G. M. Macromol. Theory Simul. 1999, 8, 272−278. (37) Zakharov, I. I.; Zakharov, V. A. Macromol. Theory Simul. 2001, 10, 108−116. (38) Panchenko, V. N.; Zakharov, V. A.; Danilova, I. G.; Paukshtis, E. A.; Zakharov, I. I.; Goncharov, V. G.; Suknev, A. P. J. Mol. Catal. A: Chem. 2001, 174, 107−117. (39) Ystenes, M.; Eilertsen, J. L.; Liu, J. K.; Ott, M.; Rytter, E.; Stovneng, J. A. J. Polym. Sci., Polym. Chem. 2000, 38, 3106−3127. (40) Bryant, P. L.; Harwell, C. R.; Mrse, A. A.; Emery, E. F.; Gan, Z. H.; Caldwell, T.; Reyes, A. P.; Kuhns, P.; Hoyt, D. W.; Simeral, L. S.; Hall, R. W.; Butler, L. G. J. Am. Chem. Soc. 2001, 123, 12009−12017. (41) Linnolahti, M.; Severn, J. R.; Pakkanen, T. A. Angew. Chem., Int. Ed. 2006, 45, 3331−3334. (42) Negureanu, L.; Hall, R. W.; Butler, L. G.; Simeral, L. A. J. Am. Chem. Soc. 2006, 128, 16816−16826. (43) Glaser, R.; Sun, X. J. Am. Chem. Soc. 2011, 133, 13323−13336. (44) Linnolahti, M.; Laine, A.; Pakkanen, T. A. Chem.Eur. J. 2013, 19, 7133−7142. (45) Zurek, E.; Ziegler, T. Inorg. Chem. 2001, 40, 3279−3292. (46) Rytter, E.; Stovneng, J. A.; Eilertsen, J. L.; Ystenes, M. Organometallics 2001, 20, 4466−4468. (47) Tossell, J. A. Organometallics 2002, 21, 4523−4527. (48) Vanka, K.; Chan, M. S. W.; Pye, C. C.; Ziegler, T. Organometallics 2000, 19, 1841−1849. (49) Zurek, E.; Ziegler, T. Organometallics 2002, 21, 83−92. (50) Xu, Z.; Vanka, K.; Firman, T. K.; Michalak, A.; Zurek, E.; Zhu, C.; Ziegler, T. Organometallics 2002, 21, 2444−2453. (51) Zakharov, I. I.; Zakharov, V. A. Macromol. Theory Simul. 2002, 11, 352−358.

(5)

A Boltzmann distribution was assumed. The percent abundance of the species whose ΔG(n,m,T) is in the numerator of eq 6 was calculated using the formula exp %((AlOMe)n ·(AlMe3)m , T ) =

(

−ΔG(n , m , T ) nRT

(

∑i exp

)

−ΔGi(n , m , T ) nRT

)

× 100%

(6) where the summation in the denominator is over all of the oligomers assumed to be in the mixture (i), and R is the ideal gas constant.

■

ASSOCIATED CONTENT

■

AUTHOR INFORMATION

S Supporting Information *

Optimized coordinates, a figure analogous to Figure 11 but for (AlOMe)12,c, and a discussion of the addition of TMA to (AlOMe)n with “bending out” versus “bending in” configurations. This material is available free of charge via the Internet at http://pubs.acs.org.

Corresponding Author

*(E.Z.) E-mail: ezurek@buffalo.edu. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund, administered by the American Chemical Society, for support of this research (Grant 51672-DNI6). We acknowledge support from the Center of Computational Research at SUNY Buffalo. E.Z. thanks the Alfred P. Sloan Foundation for a Research Fellowship (2013-2015).

■

REFERENCES

(1) Sinn, H.; Kaminsky, W.; Vollmer, H. J.; Woldt, R. Angew. Chem., Int. Ed. 1980, 19, 390−392. (2) Kaminsky, W.; Sinn, H. Adv. Organomet. Chem. 1980, 18, 99− 149. (3) Kaminsky, W. Macromolecules 2012, 45, 3289−3297. (4) Cossée, P. J. Catal. 1964, 3, 80−88. (5) Arlman, E. J. J. Catal. 1964, 3, 89−99. (6) Arlman, E. J.; Cossée, P. J. Catal. 1964, 3, 99−104. (7) Zurek, E.; Ziegler, T. Faraday Discuss. 2003, 124, 93−109. (8) Henderson, M. A.; Trefz, T. K.; Collins, S.; Wang, M. Y.; McIndoe, J. S. Organometallics 2013, 32, 2079−2083. (9) Angermund, K.; Fink, G.; Jensen, V. R.; Kleinschmidt, R. Chem. Rev. 2000, 100, 1457−1470. (10) Coates, G. W. Chem. Rev. 2000, 100, 1223−1252. (11) Coates, G. W. J. Chem. Soc., Dalton Trans. 2002, 4, 467−475. (12) Chen, E. Y. X.; Marks, T. J. Chem. Rev. 2000, 100, 1391−1434. (13) Rappe, A. T.; Skiff, W. M.; Casewit, C. J. Chem. Rev. 2000, 100, 1435−1456. (14) Annunziata, L.; Pragliola, S.; Pappalardo, D.; Tedesco, C.; Pellecchia, C. Macromolecules 2011, 44, 1934−1941. (15) Baier, M. C.; Zuideveld, M. A.; Mecking, S. Angew. Chem., Int. Ed. 2014, 53 (37), 9722−9744. (16) Schwerdtfeger, E. D.; Price, C. J.; Chai, J.; Miller, S. A. Macromolecules 2010, 43, 4838−4842. M


Macromolecules

Article

(52) Ferreira, M. L.; Belelli, P. G.; Damiani, D. E. Macromol. Chem. Phys. 2001, 202, 495−511. (53) Fusco, R.; Longo, L.; Masi, F.; Garbassi, F. Macromolecules 1997, 30, 7673−7685. (54) Belelli, P. G.; Branda, M. M.; Castellani, N. J. J. Mol. Catal. A: Chem. 2003, 192, 9−24. (55) Ferreira, M. L.; Branda, M. M.; Juan, A.; Damiani, D. E. J. Mol. Catal. A: Chem. 1997, 122, 51−60. (56) Ferreira, M. L. Macromol. Theor. Simul. 2002, 11, 250−266. (57) van Rensburg, W. J.; van den Berg, J. A.; Steynberg, A. R. Organometallics 2007, 26, 1000−1013. (58) Carrasco, J.; Illas, F.; Bromley, S. T. Phys. Rev. Lett. 2007, 99, 235502. (59) Wang, H.; Tse, J. S.; Tanaka, K.; Iitaka, T.; Ma, Y. Proc. Natl. Acad. Sci. U.S.A. 2012, 109, 6463−6466. (60) Boudene, Z.; De Bruin, T.; Toulhoat, H.; Raybaud, P. Organometallics 2012, 31, 8312−8322. (61) Landry, C. C.; Harlan, J.; Bott, S. G.; Barron, A. R. Angew. Chem., Int. Ed. Engl. 1995, 34, 1201−1202. (62) Smith, M. B. J. Organomet. Chem. 1972, 46, 31−49. (63) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. J. Chem. Phys. 2010, 132, 154104. (64) Wheeler, S. E.; Houk, K. J. Chem. Theory Comput. 2010, 6, 395− 404. (65) Simón, L.; Goodman, J. M. Org. Biomol. Chem. 2011, 9, 689− 700. (66) Basaldella, E. I.; Tara, J. C. Mater. Lett. 1998, 34, 119−123. (67) Nath, D. C. D.; Fellows, C. M.; Shiono, T. Macromol. Res. 2006, 14, 338−342. (68) Hu, Y.; Dong, W.; Masuda, T. Macromol. Chem. Phys. 2013, 214, 2172−2180. (69) Bovino, M. T.; Liwosz, T. W.; Kendel, N. E.; Miller, Y.; Tyminska, N.; Zurek, E.; Chemler, S. R. Angew. Chemie. Int. Ed. 2014, 53, 6383−6387. (70) Macrae, C. F.; Bruno, I. J.; Chisholm, J. A.; Edgington, P. R.; McCabe, P.; Pidcock, E.; Rodriguez-Monge, L.; Taylor, R.; van de Streek, J.; Wood, P. A. J. Appl. Crystallogr. 2008, 41, 466−470. (71) Berthomieu, D.; Bacquet, Y.; Pedocchi, L.; Goursot, A. J. Phys. Chem. A 1998, 102, 7821−7827. (72) Busico, V.; Cipullo, R.; Cutillo, F.; Friederichs, N.; Ronca, S.; Wang, B. J. Am. Chem. Soc. 2003, 125, 12402−12403. (73) Perdew, J. P.; Burke, K.; Wang, Y. Phys. Rev. B 1996, 54, 16533− 16539. (74) Zhang, Y.; Yang, W. Phys. Rev. Lett. 1998, 80, 890−890. (75) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1998, 80, 891−891. (76) Hammer, B.; Hansen, L. B.; Nørskov, J. K. Phys. Rev. B 1999, 59, 7413−7421. (77) Baerends, E.; Autschbach, J.; Bérces, A.; Bo, C.; Boerrigter, P.; Cavallo, L.; Chong, D.; Deng, L.; Dickson, R.; Ellis, D. ADF2010.01; Vrije Universiteit: Amsterdam, 2010. (78) te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C.; van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T. J. Comput. Chem. 2001, 22, 931−967. (79) Zhao, Y.; Truhlar, D. G. Theor. Chem. Acc. 2008, 120, 215−241. (80) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, Ö .;

Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Revision D.01. Gaussian Inc.: Wallingford CT, 2009. (81) Head-Gordon, M.; Pople, J. A.; Frisch, M. J. Chem. Phys. Lett. 1988, 153, 503−506. (82) Schaefer, A.; Horn, H.; Ahlrichs, R. J. Chem. Phys. 1992, 97, 2571−2577. (83) Schaefer, A.; Horn, H.; Ahlrichs, R. J. Chem. Phys. 1994, 100, 5829−5835. (84) Pople, J. A.; Head-Gordon, M.; Raghavachari, K. J. Chem. Phys. 1987, 87, 5968−5675. (85) Feller, D. J. Comput. Chem. 1996, 17, 1571−1586. (86) Schuchardt, K. L.; Didier, B. T.; Elsethagen, T.; Sun, L.; Gurumoorthi, V.; Chase, J.; Li, J.; Windus, T. L. J. Chem. Inf. Model. 2007, 47, 1045−1052. (87) Reed, A. E.; Weinstock, R. B.; Weinhold, F. J. Chem. Phys. 1985, 83, 735−746. (88) Reed, A. E.; Weinhold, F. J. Chem. Phys. 1985, 83, 1736−1740. (89) Foster, J. P.; Weinhold, F. J. Am. Chem. Soc. 1980, 102, 7211− 7218. (90) Glendening, E. D.; Reed, A. E.; Carpenter, J. E.; Weinhold, F. NBO Version 3.1; Theoretical Chemistry Institute, University of Wisconsin: Madison, WI, 1998.

N


The Dynamic Equilibrium Between (AlOMe)n Cages and - American

Recommend Documents