Thomas A. Manz*abc
aDepartment of Chemical & Materials Engineering, New Mexico State University, Las Cruces, NM 88003-8001, USA. E-mail: tmanz@nmsu.edu
bSchool of Chemical Engineering, Purdue University, West Lafayette, IN 47907-2100, USA
cSchool of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA
First published on 19th May 2015
Two deactivation pathways of Ti and Zr half-metallocene complexes activated with B(C6F5)3 in toluene solvent were studied using Density Functional Theory (DFT) with dispersion corrections: (a) H transfer from the counterion to Me initiating group to release methane and (b) C6F5 transfer from the counterion to the metal. Transition state geometries and energies were computed for twenty-seven complexes, and the barrier height for the C6F5 transfer pathway was linearly correlated to the amount of steric congestion near the metal. Unimolecular rate constants for catalyst deactivation were predicted for all 27 catalysts by constructing a DFT-based quantitative structure activity relationship (QSAR). This QSAR was constructed by using the DFT-computed energy barrier (ΔV0) and vibrational frequency along the reaction coordinate (ν‡) as chemical descriptors and fitting QSAR parameters to experimental data for reference systems. The computed rate constants were in excellent agreement with the available experimental data. Specifically, the dominant deactivation pathway for each catalyst and the relative deactivation rates of different catalysts were correctly predicted. Of note, the IndTi(OC6H-2,3,5,6-Ph4)Me2/B(C6F5)3 system is predicted to have a good combination of slow deactivation and high olefin polymerization rates.
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In this article, deactivation of [Cp′m(OAr)Me]+ [MeB(C6F5)3]− (m = Ti, Zr; Cp′ = Cp (C5H5), Cp* (C5Me5), Ind (C9H7)) complexes is studied as an example. These are active catalysts for polymerizing olefins like 1-hexene in aprotic solvents such as toluene, bromobenzene, and 1,2-dichlorobenzene.7–12 As shown in Fig. 1, these catalysts are formed by activating a Cp′m(OAr)Me2 precatalyst with B(C6F5)3. Experiments showed the two main deactivation pathways for CpTi(OAr)Me2/B(C6F5)3 complexes are (a) H transfer from the Me group of the [MeB(C6F5)3]− counterion to the Ti-bound Me group of the cation to release methane gas, and (b) C6F5 transfer from the counterion to Ti.13,14 The H transfer product Cp′m(OAr)CH2B(C6F5)3 can subsequently rearrange to form Cp′m(OAr) (C6F5) (CH2B(C6F5)2).13,14 In addition to irreversible deactivation, Fig. 1 also shows two reversible side reactions: (c) the double activation of catalyst that occurs when two Me groups of the precatalyst are abstracted by two B(C6F5)3 activator molecules15–17 and (d) the reaction of activated catalyst with precatalyst to form a complex with two metal centers.18 Each of the reactions shown in Fig. 1 occurs in a single chemical step, where a single chemical step is taken to mean a process involving the simultaneous rearrangement of chemical bonds. Deactivation for these complexes is similar to deactivation processes in related half-metallocene complexes.19–27 To be viable for commercial-scale use, an olefin polymerization catalyst should have both high activity and high stability.28
An important question is how to correlate structural features of single-site olefin polymerization catalysts to their performance. This can be done by using a combination of experiments and computations to build quantitative structure activity relationships (QSARs) that correlate catalyst performance to chemical descriptors of material properties.7–9,19,29–32 When modeling catalytic processes, QSARs should preferably be constructed to predict rate constants (which should be independent of reagent concentrations) as opposed to catalyst activity or selectivity (which are highly dependent on reagent concentrations).7–9,29,30
Dispersion interactions are the attractive forces between atoms in materials caused by fluctuating dipoles and higher-order fluctuating multipole moments. Local DFT functionals such as OLYP do not include these dispersion interactions. Grimme et al.'s DFT+D3(BJ) method37,38 was used to compute dispersion corrections. This method combines the DFT-D3 method39 with the Becke–Johnson damping functional.40–42 This DFT-D3(BJ) method includes both sixth- and eighth-order two-body terms (i.e., terms proportional to RAB−6 and RAB−8) and coordination number dependence, but not the ninth-order three-body terms39 (the D3 method may be modified to include ninth-order three-body interactions, but these are not recommended to be included by default37,39). This method requires only the atomic coordinates as input to compute a semi-empirical dispersion correction to the DFT energy. The optimized OLYP/LANL2DZ geometries were used as input and the DFT+D3(BJ) dispersion energies for each geometry are listed in the ESI.†
Catalyst numbers follow the same order as used previously7–9 and are displayed in boldface type when referred to in the text. For catalysts with substitution on only one side of the aryloxide ligand, there are two possible conformations. The proximal (p) conformation occurs when the substituent is located on the side closest to the metal-bound Me group, and the distal (d) conformation occurs when the substituent is located on the opposite side. For catalyst 15, the computed rotational barrier from proximal to distal conformations is 4 (without dispersion or zero-point corrections), 3.8 (with dispersion corrections), 3.7 (with zero-point corrections), and 3.6 (with dispersion and zero-point corrections) kcal mol−1.
Table 1 shows DFT-computed activation barriers and Erxn values for several reactions involving the CpTi(OC6H2-2,6-Me2-4-Br)Me2/B(C6F5)3 system, where the activation barrier is the electronic energy from reactant to TS. In contrast to the large solvent effect for monomer coordination,8 solvent dielectric screening had little effect on the computed activation barrier and Erxn values for catalyst activation and deactivation. Unlike monomer coordination, the activation and deactivation pathways in Table 1 do not require ion pair separation. The computed barrier for catalyst activation (∼10 kcal mol−1) is low compared to the barriers for catalyst deactivation (∼20–30 kcal mol−1). Experimentally, the pre-catalysts are observed to activate immediately upon B(C6F5)3 addition.14 In experiments, B(C6F5)3 is added in only slight excess (e.g., B(C6F5)3 to precatalyst ratio = ∼1.0–1.1), thereby restricting double activation to a negligible amount. NMR spectroscopy shows singly activated catalyst is the dominant species.14 As shown in Table 1, zero-point vibration corrections had only a small effect on the energetics, while dispersion and free energy corrections had significant effects. According to the DFT+dispersion free energies, formation of the H transfer product is energetically favorable (ΔGrxn = −7.2 kcal mol−1) and formation of the C6F5 transfer product is almost energetically neutral (ΔGrxn = 0.1 kcal mol−1). Catalyst dimerization is predicted to be unfavorable (ΔGrxn = 6.8 kcal mol−1). In summary, experiments and DFT calculations show the two most important deactivation processes for these catalysts are (a) H transfer and (b) C6F5 transfer.
No zero-point no dispersion | With zero-point no dispersion | With zero-point with dispersion | Free energy no dispersion | Free energy with dispersion | ||||
---|---|---|---|---|---|---|---|---|
Vacuum | Toluene | Vacuum | Toluene | Vacuum | Toluene | Vacuum | Vacuum | |
Catalyst activation barrier | 8.8 | 10.4 | 7.4 | 9.0 | 11.1 | 12.8 | 6.0 | 9.8 |
Catalyst activation Erxn | 2.5 | 1.4 | 5.1 | 4.0 | −21.6 | −22.7 | 21.4 | −5.4 |
Dimerization Erxn | 2.3 | −0.8 | 2.9 | −0.2 | −4.0 | −7.4 | 13.7 | 6.8 |
Double activation Erxn | 22.4 | 27.1 | 23.9 | 28.5 | −10.6 | −5.9 | 37.7 | 3.2 |
H transfer Erxn | −3.5 | −3.2 | −5.7 | −5.3 | 1.9 | 2.2 | −14.7 | −7.2 |
H transfer barrier | 29.1 | 28.4 | 26.4 | 25.7 | 21.3 | 20.6 | 26.6 | 21.5 |
C6F5 transfer Erxn | −8.6 | −8.5 | −10.1 | −9.9 | 16.0 | 16.1 | −25.9 | 0.1 |
C6F5 transfer barrier | 33.3 | 34.7 | 32.8 | 34.2 | 21.5 | 22.9 | 33.7 | 22.4 |
Computed barrier heights in vacuum and ν‡ (in wavenumber) are shown in Table 2. Here, ΔV0 is the energy from reactant to transition state (TS) without zero-point corrections, ΔE0 is the zero-point corrected energy from reactant to TS, and ν‡ is the magnitude of the imaginary frequency along the reaction coordinate. In Table 2, the entries marked ‘+disp.’ include dispersion corrections, while the others do not. Energy barriers in vacuum and in toluene solvent are marked ‘vac.’ and ‘tol.’, respectively. For each catalyst, the difference between ΔV0 and ΔE0 was small. ΔV0 varied from 33.3 to 45.6 kcal mol−1 for C6F6 transfer but only from 27.2 to 32.6 kcal mol−1 for H transfer. For each reaction, ν‡ did not vary much from catalyst to catalyst with values of ∼230 (C6F5 transfer) and ∼1000 (H transfer) cm−1. C6F5 transfer for catalyst 16 was the one exception with a ν‡ of only 98 cm−1. Since they differ by only a small amount from vacuum values, ΔV0 values in toluene solvent are given in the ESI.† Selected TS geometries are shown in Fig. 2. In the C6F5 transfer TS, the Ti–C* distance is 2.57 (catalyst 1) and 4.20 (catalyst 16) Å and the C*–B distance is 2.16 (catalyst 1) and 2.93 (catalyst 16) Å. The much larger C6F5 transfer distance for 16 probably led to its much lower ν‡.
Cat no.a | C6F5 transfer | H transfer | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
ν‡, cm−1 | ΔV0 | ΔV0+disp. | ΔE0 | ΔE0+disp. | ΔG‡ | ΔG‡+disp. | ν‡, cm−1 | ΔV0 | ΔV0+disp. | ΔE0 | ΔE0+disp. | ΔG‡ | ΔG‡+disp. | |
vac. | vac. | vac. | vac. | tol. | tol. | vac. | vac. | vac. | vac. | tol. | tol. | |||
a Distal (d) or proximal (p) conformation.b Numerous optimizations did not show a H transfer pathway exists for the distal conformation of 21. | ||||||||||||||
1 | 199 | 33.3 | 22.1 | 32.8 | 21.6 | 35.0 | 23.7 | 1067 | 29.1 | 24.0 | 26.4 | 21.3 | 25.9 | 20.8 |
2 | 202 | 33.8 | 21.7 | 33.4 | 21.3 | 39.1 | 27.0 | 1067 | 29.5 | 25.1 | 26.8 | 22.4 | 29.6 | 25.2 |
3 | 205 | 34.5 | 23.4 | 33.8 | 22.7 | 37.9 | 26.7 | 1051 | 28.7 | 24.8 | 25.4 | 21.4 | 24.8 | 20.8 |
4 | 225 | 38.1 | 27.9 | 37.1 | 26.8 | 39.7 | 29.4 | 984 | 32.0 | 26.2 | 29.1 | 23.2 | 29.3 | 23.4 |
5 | 228 | 38.0 | 27.9 | 37.0 | 26.8 | 40.0 | 29.8 | 1062 | 30.7 | 25.4 | 27.7 | 22.4 | 36.3 | 31.0 |
6 | 228 | 38.2 | 28.1 | 37.0 | 26.9 | 39.6 | 29.5 | 900 | 32.0 | 26.3 | 29.3 | 23.6 | 30.1 | 24.4 |
7 | 220 | 38.0 | 27.6 | 37.0 | 26.6 | 38.1 | 27.7 | 1068 | 31.0 | 25.4 | 28.0 | 22.5 | 26.6 | 21.1 |
8 | 224 | 38.1 | 27.5 | 36.9 | 26.3 | 39.7 | 29.1 | 970 | 31.8 | 26.2 | 28.8 | 23.2 | 28.6 | 23.0 |
9 | 223 | 37.8 | 27.3 | 36.8 | 26.4 | 40.4 | 29.9 | 975 | 31.9 | 26.6 | 29.0 | 23.7 | 30.4 | 25.1 |
10 | 240 | 41.7 | 35.7 | 41.2 | 35.2 | 44.9 | 38.9 | 1008 | 29.7 | 24.4 | 27.2 | 21.9 | 29.5 | 24.2 |
11 | 240 | 42.4 | 37.2 | 41.4 | 36.1 | 42.9 | 37.6 | 964 | 31.9 | 26.9 | 28.4 | 23.4 | 28.6 | 23.6 |
12 | 237 | 44.7 | 42.9 | 43.6 | 41.9 | 45.8 | 44.1 | 1004 | 29.4 | 23.8 | 26.8 | 21.2 | 28.8 | 23.2 |
13d | 240 | 41.4 | 34.2 | 40.4 | 33.3 | 42.9 | 35.7 | 1021 | 29.9 | 25.3 | 27.3 | 22.8 | 29.0 | 24.5 |
13p | 238 | 41.7 | 32.9 | 40.6 | 31.8 | 45.3 | 36.5 | 1033 | 31.3 | 25.7 | 28.5 | 22.9 | 30.2 | 24.6 |
14 | 241 | 41.7 | 35.8 | 40.8 | 35.0 | 44.0 | 38.2 | 1025 | 29.6 | 24.4 | 27.0 | 21.8 | 29.1 | 23.9 |
15d | 235 | 40.7 | 31.8 | 39.2 | 30.3 | 43.0 | 34.1 | 962 | 30.2 | 24.3 | 27.2 | 21.3 | 27.7 | 21.8 |
15p | 230 | 39.8 | 29.8 | 38.8 | 28.8 | 42.6 | 32.6 | 876 | 31.4 | 25.9 | 28.9 | 23.4 | 30.9 | 25.4 |
16 | 98 | 42.3 | 64.1 | 39.1 | 60.9 | 35.8 | 57.5 | 995 | 27.4 | 19.9 | 24.7 | 17.2 | 26.5 | 19.0 |
17d | 226 | 38.1 | 27.3 | 37.0 | 26.2 | 39.5 | 28.6 | 1048 | 30.9 | 25.8 | 28.0 | 23.0 | 28.5 | 23.4 |
17p | 230 | 38.3 | 28.5 | 37.3 | 27.4 | 38.9 | 29.1 | 1066 | 30.9 | 25.7 | 28.1 | 22.9 | 27.7 | 22.5 |
18 | 233 | 38.2 | 28.4 | 37.2 | 27.4 | 41.4 | 31.6 | 912 | 31.8 | 26.3 | 28.7 | 23.2 | 29.4 | 23.9 |
19 | 214 | 38.2 | 28.3 | 37.2 | 27.3 | 40.7 | 30.8 | 1063 | 27.2 | 22.9 | 24.3 | 20.0 | 24.2 | 20.0 |
20d | 233 | 41.2 | 31.8 | 40.1 | 30.6 | 42.5 | 33.0 | 1033 | 30.6 | 25.6 | 27.7 | 22.7 | 28.6 | 23.6 |
20p | 230 | 39.8 | 29.9 | 40.9 | 30.9 | 42.8 | 32.9 | 1082 | 29.9 | 24.7 | 28.5 | 23.3 | 28.5 | 23.3 |
21d | 240 | 41.9 | 33.4 | 41.1 | 32.7 | 44.1 | 35.7 | b | b | b | b | b | b | b |
21p | 236 | 41.0 | 31.6 | 40.1 | 30.6 | 43.2 | 33.7 | 1056 | 29.8 | 24.2 | 27.2 | 21.6 | 29.4 | 23.7 |
22 | 229 | 39.7 | 31.1 | 38.4 | 29.8 | 42.6 | 34.0 | 933 | 32.6 | 23.5 | 29.9 | 20.8 | 34.4 | 25.2 |
23 | 217 | 36.0 | 25.8 | 35.1 | 24.9 | 37.1 | 26.9 | 1019 | 29.2 | 23.2 | 26.5 | 20.5 | 26.5 | 20.5 |
24 | 234 | 44.3 | 36.0 | 43.1 | 34.9 | 47.0 | 38.7 | 1063 | 30.6 | 23.8 | 27.8 | 21.0 | 28.1 | 21.2 |
30 | 241 | 41.9 | 34.8 | 40.6 | 33.5 | 42.8 | 35.6 | 1075 | 29.8 | 25.1 | 26.9 | 22.2 | 27.7 | 23.0 |
31 | 232 | 45.6 | 46.0 | 44.4 | 44.7 | 46.8 | 47.2 | 1010 | 29.3 | 24.0 | 26.5 | 21.2 | 28.2 | 23.0 |
33 | 202 | 33.5 | 22.2 | 33.3 | 22.0 | 38.4 | 27.1 | 1095 | 29.2 | 24.2 | 26.8 | 21.8 | 30.1 | 25.0 |
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Fig. 2 Selected transition state geometries (small pink = H, large pink = B, gray = C, red = O, cyan = F, yellow = Ti, large maroon = Br). |
ΔG‡ is the computed difference between the TS and reactant Gibbs free energies in solution. To compute these Gibbs free energies, vibrational analysis was performed on each vacuum optimized geometry and the corresponding gas phase free energy was computed in Gaussian 03 using statistical thermodynamics based on the harmonic approximation.43,44 Then, the free energy in solution was estimated by adding the difference between the DFT self-consistent electronic energy in toluene (PCM model) and vacuum to each gas phase free energy. All free energies reported in this paper use a standard state of 1 atmosphere pressure and a temperature of 298.15 K, which corresponds to a standard state concentration of 1 mol/22.4 L = 0.045 M for both gas and solution phase free energies. (The solution standard state is not set to 1 M, because the catalyst concentrations are typically much less than 1 M in experiments.) In previous DFT studies of these catalysts, free energies and enthalpies utilized this same standard state of 1 atmosphere pressure at 298.15 K, which corresponds to a standard state concentration of 1 mol/22.4 L = 0.045 M for both gas and solution phases.7–9,30
To determine the relative importance of tunneling during H transfer, the tunneling crossover temperature, Tx, was computed:45
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Tunneling is important below Tx, but not above Tx. Tx was computed without (i.e., using ‘ΔE0’ values from Table 2) and with (i.e., using ‘ΔE0+disp.’ values from Table 2) dispersion corrections. As shown in Table 3, including dispersion corrections had negligible impact on Tx. As shown in Table 3, tunneling is not important for any of these systems above −18 °C. Due to the larger mass of C compared to H, tunneling is also insignificant for C6F5 transfer.
Cat no.a | m | Cp′ | OAr substituents | Ωf | C6F5 transfer to m (kd, h−1 at 25 °C) | H transfer to initiating group | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Unscaled DFT | QSAR | Tx, K | kd, h−1 at 25 °C | |||||||||||
No disp. | With disp. | Implicit disp. | Explicit disp. | No disp. | With disp. | Unscaled DFT | QSAR | |||||||
No disp. | With disp. | Implicit disp. | Explicit disp. | |||||||||||
a Distal (d) or proximal (p) conformation.b Numerous optimizations did not show a H transfer pathway exists for the distal conformation of 21. | ||||||||||||||
1 | Ti | Cp | 2,6-Me2-4-Br | 2.39 | 5 × 10−10 | 9 × 10−02 | 4 × 10+01 | 2 × 10+01 | 247 | 248 | 2 × 10−03 | 1 × 10+01 | 4 × 10−01 | 7 × 10−01 |
2 | Ti | Cp | 2,6-Et2 | 2.33 | 4 × 10−13 | 3 × 10−04 | 2 × 10+01 | 3 × 10+01 | 247 | 248 | 4 × 10−06 | 8 × 10−03 | 3 × 10−01 | 2 × 10−01 |
3 | Ti | Cp | 2,6-iPr2 | 2.01 | 4 × 10−12 | 6 × 10−04 | 1 × 10+01 | 3 × 10+00 | 244 | 244 | 1 × 10−02 | 1 × 10+01 | 5 × 10−01 | 3 × 10−01 |
4 | Ti | Cp* | None | 2.16 | 2 × 10−13 | 6 × 10−06 | 5 × 10−01 | 5 × 10−03 | 228 | 228 | 8 × 10−06 | 1 × 10−01 | 2 × 10−02 | 5 × 10−02 |
5 | Ti | Cp* | 4-F | 2.13 | 1 × 10−13 | 3 × 10−06 | 5 × 10−01 | 5 × 10−03 | 246 | 247 | 5 × 10−11 | 4 × 10−07 | 8 × 10−02 | 1 × 10−01 |
6 | Ti | Cp* | 4-Cl | 2.16 | 2 × 10−13 | 5 × 10−06 | 5 × 10−01 | 4 × 10−03 | 208 | 209 | 2 × 10−06 | 3 × 10−02 | 2 × 10−02 | 4 × 10−02 |
7 | Ti | Cp* | 4-Br | 2.15 | 3 × 10−12 | 1 × 10−04 | 5 × 10−01 | 7 × 10−03 | 248 | 248 | 7 × 10−04 | 8 × 10+00 | 6 × 10−02 | 1 × 10−01 |
8 | Ti | Cp* | 4-Ph | 2.15 | 2 × 10−13 | 9 × 10−06 | 5 × 10−01 | 7 × 10−03 | 225 | 225 | 2 × 10−05 | 3 × 10−01 | 2 × 10−02 | 5 × 10−02 |
9 | Ti | Cp* | 4-tBu | 2.16 | 5 × 10−14 | 2 × 10−06 | 6 × 10−01 | 1 × 10−02 | 226 | 226 | 1 × 10−06 | 9 × 10−03 | 2 × 10−02 | 3 × 10−02 |
10 | Ti | Cp* | 2,6-Me2 | 1.57 | 3 × 10−17 | 7 × 10−13 | 2 × 10−02 | 7 × 10−08 | 234 | 234 | 5 × 10−06 | 4 × 10−02 | 2 × 10−01 | 4 × 10−01 |
11 | Ti | Cp* | 2,6-Et2 | 1.51 | 8 × 10−16 | 5 × 10−12 | 9 × 10−03 | 8 × 10−09 | 223 | 224 | 2 × 10−05 | 1 × 10−01 | 2 × 10−02 | 2 × 10−02 |
12 | Ti | Cp* | 2,6-iPr2 | 1.25 | 6 × 10−18 | 1 × 10−16 | 1 × 10−03 | 2 × 10−12 | 233 | 233 | 2 × 10−05 | 2 × 10−01 | 3 × 10−01 | 8 × 10−01 |
13d | Ti | Cp* | 2-Cyclohexyl | 1.64 | 8 × 10−16 | 1 × 10−10 | 2 × 10−02 | 5 × 10−07 | 237 | 237 | 1 × 10−05 | 3 × 10−02 | 2 × 10−01 | 1 × 10−01 |
13p | Ti | Cp* | 2-Cyclohexyl | 1.58 | 1 × 10−17 | 4 × 10−11 | 2 × 10−02 | 4 × 10−06 | 239 | 240 | 2 × 10−06 | 2 × 10−02 | 4 × 10−02 | 9 × 10−02 |
14 | Ti | Cp* | 2,6-Me2-4-Br | 1.56 | 1 × 10−16 | 2 × 10−12 | 2 × 10−02 | 5 × 10−08 | 238 | 238 | 1 × 10−05 | 6 × 10−02 | 2 × 10−01 | 4 × 10−01 |
15d | Ti | Cp* | 2-CH2Ph | 1.85 | 6 × 10−16 | 2 × 10−09 | 4 × 10−02 | 2 × 10−05 | 223 | 223 | 1 × 10−04 | 2 × 10+00 | 1 × 10−01 | 4 × 10−01 |
15p | Ti | Cp* | 2-CH2Ph | 1.84 | 1 × 10−15 | 3 × 10−08 | 1 × 10−01 | 3 × 10−04 | 203 | 203 | 5 × 10−07 | 6 × 10−03 | 3 × 10−02 | 6 × 10−02 |
16 | Ti | Cp* | 2,3,5,6-Ph4 | 1.05 | 1 × 10−10 | 1 × 10−26 | 4 × 10−03 | 6 × 10−26 | 231 | 232 | 8 × 10−04 | 3 × 10+02 | 2 × 10+00 | 7 × 10+01 |
17d | Ti | Cp* | 3-OMe | 2.17 | 3 × 10−13 | 2 × 10−05 | 5 × 10−01 | 1 × 10−02 | 243 | 243 | 3 × 10−05 | 1 × 10−01 | 6 × 10−02 | 8 × 10−02 |
17p | Ti | Cp* | 3-OMe | 2.17 | 6 × 10−13 | 1 × 10−05 | 4 × 10−01 | 2 × 10−03 | 247 | 248 | 1 × 10−04 | 7 × 10−01 | 6 × 10−02 | 1 × 10−01 |
18 | Ti | Cp* | 4-OMe | 2.17 | 1 × 10−14 | 2 × 10−07 | 5 × 10−01 | 2 × 10−03 | 211 | 211 | 6 × 10−06 | 6 × 10−02 | 2 × 10−02 | 4 × 10−02 |
19 | Ti | Cp | 2,3,5,6-Ph4 | 1.88 | 3 × 10−14 | 5 × 10−07 | 4 × 10−01 | 2 × 10−03 | 247 | 248 | 4 × 10−02 | 5 × 10+01 | 2 × 10+00 | 2 × 10+00 |
20d | Ti | Cp* | 2-Br | 1.79 | 2 × 10−15 | 1 × 10−08 | 3 × 10−02 | 2 × 10−05 | 239 | 240 | 2 × 10−05 | 1 × 10−01 | 9 × 10−02 | 1 × 10−01 |
20p | Ti | Cp* | 2-Br | 1.80 | 8 × 10−16 | 2 × 10−08 | 1 × 10−01 | 3 × 10−04 | 251 | 251 | 3 × 10−05 | 2 × 10−01 | 2 × 10−01 | 3 × 10−01 |
21d | Ti | Cp* | 2-Ph | 1.57 | 1 × 10−16 | 1 × 10−10 | 2 × 10−02 | 2 × 10−06 | b | b | b | b | b | b |
21p | Ti | Cp* | 2-Ph | 1.57 | 5 × 10−16 | 4 × 10−09 | 3 × 10−02 | 2 × 10−05 | 245 | 246 | 6 × 10−06 | 9 × 10−02 | 2 × 10−01 | 5 × 10−01 |
22 | Ti | Ind | 2,3,5,6-Ph4 | 1.73 | 1 × 10−15 | 3 × 10−09 | 1 × 10−01 | 5 × 10−05 | 216 | 217 | 1 × 10−09 | 7 × 10−03 | 1 × 10−02 | 1 × 10+00 |
23 | Ti | Ind | 2,6-iPr2 | 1.88 | 1 × 10−11 | 4 × 10−04 | 3 × 10+00 | 8 × 10−02 | 236 | 237 | 8 × 10−04 | 2 × 10+01 | 3 × 10−01 | 2 × 10+00 |
24 | Zr | Cp* | 2,3,5,6-Ph4 | 1.51 | 8 × 10−19 | 9 × 10−13 | 2 × 10−03 | 4 × 10−08 | 246 | 247 | 6 × 10−05 | 6 × 10+00 | 9 × 10−02 | 9 × 10−01 |
30 | Ti | Cp* | 2,6-(OMe)2 | 1.74 | 1 × 10−15 | 2 × 10−10 | 1 × 10−02 | 2 × 10−07 | 249 | 250 | 1 × 10−04 | 3 × 10−01 | 2 × 10−01 | 2 × 10−01 |
31 | Ti | Cp* | 2,6-iPr2-4-Br | 1.24 | 1 × 10−18 | 5 × 10−19 | 5 × 10−04 | 3 × 10−14 | 234 | 235 | 4 × 10−05 | 3 × 10−01 | 3 × 10−01 | 6 × 10−01 |
33 | Ti | Cp | 2,6-Me2 | 2.44 | 1 × 10−12 | 3 × 10−04 | 3 × 10+01 | 1 × 10+01 | 254 | 255 | 2 × 10−06 | 1 × 10−02 | 3 × 10−01 | 6 × 10−01 |
Steric congestion has a strong effect on the rate of C6F5 transfer.13,14,21 To quantify this, the free solid angle, Ωf, of [Cp'm(OAr)Me]+ [MeB(C6F5)3]− was computed.9 Ωf equals 4π times the illuminated fraction of a spherical screen around the metal complex if a point source of light is placed at the metal's center and all portions of the complex's van der Waals surface outside the metal's van der Waals radius are made opaque.9 As shown in Fig. 3, there is a linear relationship between Ωf and ΔV0 for C6F5 transfer. With the exception of catalyst 16 mentioned above, the linear fit had a squared correlation coefficient R2 = 0.81 without dispersion (i.e., using Ea = ‘ΔV0’ values from Table 2) and 0.83 when dispersion corrections were included (i.e., using Ea = ‘ΔV0+disp’ values from Table 2). Ligand cone angles are another way to quantify steric congestion in organometallic complexes. The cyclopentadienyl ligand cone angle (ΘCp′) and aryloxide ligand cone angle (ΘOAr) computed by the method of Manz et al.8 are listed in Table S2 of the ESI.† The linear correlation between ΔV0 and the ligand cone angle sum (ΘCp′ + ΘOAr) was weak and had a correlation coefficient R2 = 0.42. This shows the free solid angle was more strongly correlated than the ligand cone angle sum to the C6F5 transfer barrier heights.
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Fig. 3 QSAR for the C6F5 transfer reaction barrier. The two lines are fit to the vacuum data (without and with dispersion corrections) excluding catalyst 16. |
Catalyst deactivation rate constants (kd) were first estimated without any adjustable parameters by inserting ΔG‡ into eqn (1) using CTST. The resulting kd values, listed in Table 3 under the columns labeled ‘unscaled DFT’, predict H transfer to initiating group is the dominant decay pathway for every catalyst when no dispersion corrections are included and for all catalysts except catalyst 5 when dispersion corrections are included. This prediction is wrong, because experiments show C6F5 transfer to metal is the dominant decay pathway for several catalysts.13,14 Careful examination shows the ‘unscaled DFT’ kd values are erratic and span more than 28 orders of magnitude! There are several potential sources for this discrepancy: (a) errors introduced by the solvation model (e.g., PCM), (b) errors introduced by the density functional approximation (e.g., OLYP) to the exchange–correlation energy, (c) basis set limitations, (d) errors introduced by the thermochemistry model (e.g., harmonic approximation), and (e) errors introduced by the CTST approximation itself. Of note, CTST rate constant predictions would not be exact (even for an elementary reaction) if the exact ΔG‡ were known, because of effects like transition state recrossing.46–49
Repeating the calculations with alternate solvation models, exchange–correlation theories, basis sets, statistical thermochemistry models, and transition state theories would be a tedious and time-consuming process. In this article, our goal is to develop a practical model that yields accurate results with minimal effort and without redoing the DFT calculations at various levels of theory. In general, the temperature dependence of the rate constant can be approximated by the Arrhenius equation
k = A![]() | (3) |
A = c1ν‡ | (4) |
Ea = c2 × barrier height | (5) |
The barrier height appearing on the right-hand side of eqn (5) could alternatively include (or not) zero-point energies, solvation effects, and/or dispersion effects. Two opposing philosophies are possible. The first philosophy, called the Principle of Parsimony, posits that when two competing theories explain available experiments the theory making the fewer assumptions or having a simpler mathematical form is preferable. The second philosophy posits that when two competing theories explain available experiments the theory explicitly including more interactions is preferable. At first it may appear that using a higher level of theory that explicitly includes more interactions will always produce more accurate correlations (e.g., QSARs) than a lower level of theory that explicitly includes fewer interactions, but this is not necessarily true. Including more explicit interactions comes with a trade-off. Specifically, the uncertainty in including those explicit interactions will add to the model's overall uncertainty. Thus, in cases where the uncertainty in explicitly including additional interactions (aka ‘the cost’) exceeds the gain in precision (aka ‘the benefit’), the model will be worsened by going to a higher level of theory. On the other hand, using a lower level of theory will worsen the results if the gain in precision from explicitly including additional interactions exceeds the uncertainty costs of including those new interactions. Therefore, one cannot make universal statements of the type “higher levels of theory are better” or “lower levels of theory are better”, but rather one must determine appropriate levels of theory on a case-by-case basis.
This is more subtle than it first appears, because QSARs can implicitly include many interactions that are not explicitly included. A key question when developing a QSAR is whether implicit or explicit inclusion of each interaction type is optimal. Examining Table 2, including dispersion interactions reduces the H transfer barrier heights for all catalysts and the C6F5 transfer barrier heights for all catalysts except catalysts 16 and 31. This can be accounted for explicitly by using ‘ΔV0+disp’ or ‘ΔE0 + disp’ as the barrier height on the right-hand side of eqn (5). Dispersion effects can be accounted for implicitly by using ‘ΔV0’ or ‘ΔE0’ without dispersion corrections as the barrier height on the right-hand side of eqn (5) and reducing the value of parameter c2. With the exception of a slight increase for catalyst 20p C6F5 transfer, including zero-point vibrations leads to small decrease in the C6F5 and H transfer barrier heights. Therefore, the effects of zero-point vibrations on Ea could be modeled either (a) explicitly by using ‘ΔE0’ or ‘ΔE0+disp’ as the barrier height on the right-hand side of eqn (5) or (b) implicitly by using ‘ΔV0’ or ‘ΔV0+disp’ as the barrier height on the right-hand side of eqn (5) with an associated decrease in c2 compared to case (a). Including dispersion and zero-point effects implicitly in this manner is equivalent to assuming that the net effect of these interactions is proportional to the barrier height. For C6F5 transfer, solvation increases the barrier height by 0.6 (average) ± 0.6 (standard deviation) kcal mol−1. For H transfer, solvation increases the barrier height by −0.4 (average) ± 1.1 (standard deviation) kcal mol−1. Therefore, solvation does not have a significant effect on the barrier heights.
Because the zero-point and solvation effects are small in magnitude, the QSARs described below did not explicitly include them and the barrier height in vacuum without zero-point corrections was utilized. To estimate the values of c1 and c2 for each reaction, eqn (3)–(5) were compared to experimental data for selected reference systems. Table 4 summarizes H transfer deactivation rates measured by Phomphrai et al. for catalyst 19.14 Directly fitting this experimental data gives the Arrhenius parameters A = 7.2 × 1011 h−1 and Ea = 15.7 kcal mol−1. Inserting these into eqn (4) and (5) with the DFT-computed ν‡ and ΔV0 for this catalyst gives c1 = 6.26 × 10−6 (dimensionless) and c2 = 0.576 (dimensionless) for the H transfer catalyst deactivation reaction. These same Arrhenius parameters were fit to eqn (4) and (5) using ‘ΔV0+disp’ as the barrier height to explicitly include dispersion corrections and generate a second model with c1 = 6.26 × 10−6 (dimensionless) and c2 = 0.683 (dimensionless). Rate constant data for a single catalyst at various temperatures is not readily available for the C6F5 transfer reaction. However, Phomphrai et al. reported approximate deactivation rates for catalysts 33 (−20 °C), 2 (−10 °C), 3 (10 °C),14 which we used to estimate c1 = 4.63 × 10−2 (dimensionless) and c2 = 0.55 (dimensionless) for the C6F5 transfer reaction when using ΔV0 as the barrier height in eqn (5) and c1 = 4.63 × 10−2 (dimensionless) and c2 = 0.85 (dimensionless) when using ‘ΔV0+disp’ as the barrier height in eqn (5).
The columns labeled ‘QSAR’ in Table 3 list the kd values computed using these parameters for every catalyst. The columns marked ‘implicit disp.’ used ΔV0 as the barrier height in eqn (5) and the columns marked ‘explicit disp.’ used ‘ΔV0+disp’ as the barrier height in eqn (5). As discussed above, the implicit dispersion models had smaller c2 values (than the explicit dispersion models) to implicitly correct the barrier heights for dispersion. The explicit dispersion QSAR is ill-behaved for the C6F5 transfer reaction, because it (a) predicts the kd values span more than 26 orders of magnitude which is too large a range and (b) incorrectly predicts the more sterically hindered catalyst 2 has a higher C6F5 transfer rate than the less sterically hindered catalyst 33. These discrepancies can be attributed to uncertainties associated with constructing DFT+dispersion exchange–correlation functionals, which is intrinsically challenging owing to the need to correctly damp the add-on dispersion correction at close distances to avoid double-counting electron correlations.37,39 Alternatively, dispersion interactions from solvation may have partially compensated for changes in dispersion interactions due to geometric changes in the organometallic complex.
For each catalyst, the pathway with largest kd for the implicit dispersion QSAR is shown in boldface type. Steric congestion was the main determinant of the dominant pathway. Using the implicit dispersion QSAR, all twelve catalysts with Ωf > 2.0 are predicted to decay by C6F5 transfer, and all nine catalysts with Ωf < 1.7 are predicted to decay by H transfer. Comparisons to experimental data can be made as follows. Phomphrai et al. report that the deactivation rates for 33, 2, 3, and 19 are similar at −20, −10, 10, and 30 °C, respectively, where 33, 2, and 3 decay by C6F5 transfer and 19 decays by H transfer.14 The implicit dispersion QSAR predicts the correct decay pathway for each of these catalysts, and the computed kd values at 25 °C follow the same trend in deactivation rates: 33 (31 h−1) > 2 (24 h−1) > 3 (12 h−1) > 19 (2 h−1). Experiments show 12 decomposes by H transfer,10 the same pathway our calculations predict. The implicit dispersion QSAR predicts catalysts 11, 15, 22, and 24 to be the most stable with a predicted kd ≤ 0.1 h−1 at 25 °C. Using the implicit dispersion QSAR as the lower bound on kd and the explicit dispersion QSAR as the upper bound on kd leads to the predictions kd = 0.02 (cat. 11), 0.1–0.4 (cat. 15), 0.1–1.0 (cat. 22), and 0.09–0.9 (cat. 24) h−1. Catalysts 20 and 22 were previously found to have the highest olefin polymerization rates of all catalysts in Table 3 and exhibit opportunistic ligand coordination that lowers the ion pair separation energy to facilitate monomer coordination.8 Catalyst 22 is also highly desirable because it can be purified by crystallization during synthesis and gives facile chain initiation during olefin polymerization.8,9
A detailed analysis of dispersion, zero-point, and solvation energies showed including these in the QSAR implicitly was more expedient than including them in the QSAR explicitly for these particular reactions. This illustrates the general principle that models explicitly including more interactions (aka ‘higher levels of theory’) are better if and only if the additional uncertainty associated with explicitly including these new interactions is outweighed by an increase in precision when the new interactions are explicitly included.
Footnote |
† Electronic supplementary information (ESI) available: Optimized geometries, energies in vacuum and toluene, ligand cone angles, and frequencies for reactants, transition states, and products. See DOI: 10.1039/c5ra00546a |
This journal is © The Royal Society of Chemistry 2015 |