Deactivation of Ti and Zr half-metallocene complexes activated with B(C6F5)3: a case study in constructing DFT-based QSARs to predict unimolecular rate constants

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

Received 10th January 2015 , Accepted 19th May 2015

First published on 19th May 2015


Abstract

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.


1. Introduction

Studying chemical reactions is one of the primary uses of ab initio quantum chemistry methods like density functional theory (DFT). A common goal is to predict the relative or absolute rates of several alternate reactions. Although the reaction with a lower computed barrier height is usually the preferred pathway, rate constant computation is required for a proper prediction of the preferred reaction pathway. In conventional transition state theory (CTST), the rate constant takes the form1,2
 
image file: c5ra00546a-t1.tif(1)
where Q and Qr are the partition functions of the activated complex and reactant, T is absolute temperature, ΔG is the free energy of the transition state (TS) minus reactants, h is the Planck constant and kB is the Boltzmann constant. The computational prediction of rate constants with CTST is difficult, because modest ΔG errors produce large rate constant errors. For reactions in solution, the difficulty of modeling solute–solvent interactions makes computational prediction of rate constants especially difficult.3–6 For complexes with low frequency modes, the entropy portion of ΔG is difficult to model both in the gas phase and in solution.

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


image file: c5ra00546a-f1.tif
Fig. 1 Catalyst activation and deactivation reactions.

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

2. Results and discussion

Twenty-seven catalysts were studied with DFT calculations in Gaussian 03 (ref. 33) using the OLYP34,35 exchange–correlation functional and LANL2DZ basis sets. Solvent dielectric screening was modeled with the polarizable continuum medium (PCM) model.36 Our reasons for choosing this level of theory are now summarized. First, the OLYP exchange–correlation functional combined with the PCM model has been shown to yield relative stabilities of different ion pair forms that are in close agreement with available experimental data for these catalysts.9 Second, a previous study of 18 of these systems using the OLYP exchange–correlation functional showed the LANL2DZ double zeta basis set gives ion pair separation energies in vacuum and toluene solvent that are within 1 (average) ± 1 (standard deviation) kcal mol−1 of those computed using the 6-311++G** triple zeta basis set with polarization and diffuse functions.8 Due to its smaller size, the LANL2DZ basis set enables faster transition state optimizations than the 6-311++G** basis set. Geometries were optimized in vacuum to better than 0.005 Å on atom positions, and 0.0025 au on forces. For ground states, different initial conditions were explored and the lowest energy conformation was selected. Transition states were optimized by the following procedure. First, a series of geometries intermediate between reactant and product were optimized with constraints placed on bond distances involved in the reaction. After optimization, the structure with the highest energy was used as an initial guess for a subsequent quadratic synchronous transit (QST3) calculation. After QST3 optimization, frequency analysis was performed. If more than one imaginary frequency was present (subject to a computational tolerance of 30 cm−1), the geometry was manually modified and subsequent geometry optimization was performed using the QST3 or TS algorithms until only one imaginary frequency was present and this mode was along the reaction coordinate.

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.

Table 1 DFT-computed energies (kcal mol−1) for CpTi(OC6H2-2,6-Me2-4-Br)Me2/B(C6F5)3 (catalyst 1)
  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.

Table 2 DFT-computed transition state frequencies and energy barriers (kcal mol−1) with and without dispersion corrections
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



image file: c5ra00546a-f2.tif
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

 
image file: c5ra00546a-t2.tif(2)

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.

Table 3 DFT-computed free solid angles, tunneling crossover temperatures, and kd values. Boldface entries denote reaction pathway (i.e., H or C6F5 transfer) with largest kd for QSAR with implicit dispersion
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.


image file: c5ra00546a-f3.tif
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[thin space (1/6-em)]exp(−Ea/(kBT)) (3)
where A and Ea are approximately (but not necessarily strictly) temperature independent.50 A QSAR for catalyst deactivation rate constants is now constructed by correlating the pre-exponential factor (A) and the effective activation energy (Ea) to DFT-computed chemical descriptors. Because the pre-exponential factor quantifies the reaction's intrinsic frequency factor, we correlated it to the imaginary frequency mode along the reaction coordinate using the model equation
 
A = c1ν (4)
where c1 is a fitted constant for each reaction. Because the activation energy quantifies the effective barrier height, we correlated it to the DFT-computed barrier height using the model equation
 
Ea = c2 × barrier height (5)
where c2 is a fitted constant for each reaction.

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).

Table 4 Temperature dependence of the H transfer rate constant for catalyst 19
T (°C) Experimentala kd (h−1)
a Experimental data from ref. 14.
−6 0.12
10 0.51
25 2.7
35 5.6


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

3. Conclusions

In summary, density functional theory was used to study [Cp′m(OAr)Me]+ [MeB(C6F5)3] deactivation by two different pathways. The barrier height for C6F5 transfer was found to linearly increase as steric congestion at the metal increased, and tunneling was found to be unimportant for both pathways. Quantitative structure activity relationships (QSARs) were constructed to estimate each deactivation rate constant from the DFT-computed energy barrier and vibrational frequency along the reaction coordinate. This method of using ab initio calculations coupled with experimental reference data to construct QSARs for estimating rate constants should find widespread applications to related processes. For each catalyst whose deactivation mechanism had been experimentally determined, the rate constants estimated with the implicit dispersion QSAR correctly predicted which decay pathway is dominant and the relative stabilities of different catalysts. Computations identified catalyst 22 with low deactivation rates that is known to have high olefin polymerization rates, and this catalyst would be a good candidate for follow-up studies investigating counterion, solvent, or monomer variations.

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.

Acknowledgements

Computational resources were provided by the Teragrid Network, Purdue University, and the Extreme Science and Engineering Discovery Environment (XSEDE). Thanks to Dr James Caruthers and Dr Kendall Thomson, my former Ph.D. advisors, for introducing me to the subject of building QSARs for single-site olefin polymerization catalysts, which provided some of the motivation for this work.

References

  1. R. Hernandez and W. H. Miller, Chem. Phys. Lett., 1993, 214, 129–136 CrossRef CAS.
  2. K. J. Laidler and M. C. King, J. Phys. Chem., 1983, 87, 2657–2664 CrossRef CAS.
  3. J. N. Harvey, Faraday Discuss., 2010, 145, 487–505 RSC.
  4. R. Hernandez, T. Uzer and T. Bartsch, Chem. Phys., 2010, 370, 270–276 CrossRef CAS PubMed.
  5. D. G. Truhlar, B. C. Garrett and S. J. Klippenstein, J. Phys. Chem., 1996, 100, 12771–12800 CrossRef CAS.
  6. H. Azzouz and D. Borgis, J. Chem. Phys., 1993, 98, 7361–7375 CrossRef CAS PubMed.
  7. T. A. Manz, K. Phomphrai, G. Medvedev, B. B. Krishnamurthy, S. Sharma, J. Haq, K. A. Novstrup, K. T. Thomson, W. N. Delgass, J. M. Caruthers and M. M. Abu-Omar, J. Am. Chem. Soc., 2007, 129, 3776–3777 CrossRef CAS PubMed.
  8. T. A. Manz, S. Sharma, K. Phomphrai, K. A. Novstrup, A. E. Fenwick, P. E. Fanwick, G. A. Medvedev, M. M. Abu-Omar, W. N. Delgass, K. T. Thomson and J. M. Caruthers, Organometallics, 2008, 27, 5504–5520 CrossRef CAS.
  9. T. A. Manz, J. M. Caruthers, S. Sharma, K. Phomphrai, K. T. Thomson, W. N. Delgass and M. M. Abu-Omar, Organometallics, 2012, 31, 602–618 CrossRef CAS.
  10. K. Nomura and A. Fudo, Inorg. Chim. Acta, 2003, 345, 37–43 CrossRef CAS.
  11. K. Nomura, Dalton Trans., 2009, 8811–8823 RSC.
  12. K. Nomura and J. Liu, Dalton Trans., 2011, 7666–7682 RSC.
  13. A. E. Fenwick, K. Phomphrai, M. G. Thorn, J. S. Vilardo, C. A. Trefun, B. Hanna, P. E. Fanwick and I. P. Rothwell, Organometallics, 2004, 23, 2146–2156 CrossRef CAS.
  14. K. Phomphrai, A. E. Fenwick, S. Sharma, P. E. Fanwick, J. M. Caruthers, W. N. Delgass, M. M. Abu-Omar and I. P. Rothwell, Organometallics, 2006, 25, 214–220 CrossRef CAS.
  15. A. Al-Humydi, J. C. Garrison, W. J. Youngs and S. Collins, Organometallics, 2005, 24, 193–196 CrossRef CAS.
  16. D. W. Stephan, Macromol. Symp., 2001, 173, 105–115 CrossRef CAS.
  17. D. W. Stephan, Organometallics, 2005, 24, 2548–2560 CrossRef CAS.
  18. M. Bochmann and S. J. Lancaster, Angew. Chem., Int. Ed., 1994, 33, 1634–1637 CrossRef PubMed.
  19. E. Y. X. Chen and T. J. Marks, Chem. Rev., 2000, 100, 1391–1434 CrossRef CAS PubMed.
  20. V. Tabernero, C. Maestre, G. Jimenez, T. Cuenca and C. R. de Arellano, Organometallics, 2006, 25, 1723–1727 CrossRef CAS.
  21. T. Wondimagegn, Z. T. Xu, K. Vanka and T. Ziegler, Organometallics, 2004, 23, 3847–3852 CrossRef CAS.
  22. T. Wondimagegn, K. Vanka, Z. T. Xu and T. Ziegler, Organometallics, 2004, 23, 5737–5743 CrossRef CAS.
  23. T. Wondimagegn, K. Vanka, Z. T. Xu and T. Ziegler, Organometallics, 2004, 23, 2651–2657 CrossRef CAS.
  24. T. Wondimagegn, Z. T. Xu, K. Vanka and T. Ziegler, Organometallics, 2005, 24, 2076–2085 CrossRef CAS.
  25. N. Yue, E. Hollink, F. Guerin and D. W. Stephan, Organometallics, 2001, 20, 4424–4433 CrossRef CAS.
  26. S. Zhang, W. E. Piers, X. L. Gao and M. Parvez, J. Am. Chem. Soc., 2000, 122, 5499–5509 CrossRef CAS.
  27. T. A. Manz, A. E. Fenwick, K. Phomphrai, I. P. Rothwell and K. T. Thomson, Dalton Trans., 2005, 668–674 RSC.
  28. P. M. Gurubasavaraj, H. W. Roesky, B. Nekoueishahraki, A. Pal and R. Herbst-Irmer, Inorg. Chem., 2008, 47, 5324–5331 CrossRef CAS PubMed.
  29. J. M. Caruthers, J. A. Lauterbach, K. T. Thomson, V. Venkatasubramanian, C. M. Snively, A. Bhan, S. Katare and G. Oskarsdottir, J. Catal., 2003, 216, 98–109 CrossRef CAS.
  30. T. A. Manz, QSARs for Olefin Polymerization Catalyzed by Ti and Zr Complexes, Scholars' Press, Saarbrucken, Germany, 2013, ISBN 978-3-639-51617-3 Search PubMed.
  31. V. L. Cruz, J. Ramos, S. Martinez, A. Munoz-Escalona and J. Martinez-Salazar, Organometallics, 2005, 24, 5095–5102 CrossRef CAS.
  32. V. L. Cruz, S. Martinez, J. Ramos and J. Martinez-Salazar, Organometallics, 2014, 33, 2944–2959 CrossRef CAS.
  33. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. Montgomery, T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez and J. A. Pople, GAUSSIAN 03, 2003 Search PubMed.
  34. N. C. Handy and A. J. Cohen, Mol. Phys., 2001, 99, 403–412 CrossRef CAS PubMed.
  35. W. M. Hoe, A. J. Cohen and N. C. Handy, Chem. Phys. Lett., 2001, 341, 319–328 CrossRef CAS.
  36. M. Cossi, V. Barone, R. Cammi and J. Tomasi, Chem. Phys. Lett., 1996, 255, 327–335 CrossRef CAS.
  37. S. Grimme, S. Ehrlich and L. Goerigk, J. Comput. Chem., 2011, 32, 1456–1465 CrossRef CAS PubMed.
  38. S. Grimme, DFT-D3 Program Version 3.1 Revision 0, June 20, 2014, http://www.thch.uni-bonn.de/tc/index.php?section=downloads%26subsection=DFT-D3%26lang=english, accessed March 2015 Search PubMed.
  39. S. Grimme, J. Antony, S. Ehrlich and H. Krieg, J. Chem. Phys., 2010, 132, 154104 CrossRef PubMed.
  40. A. D. Becke and E. R. Johnson, J. Chem. Phys., 2005, 122, 154104 CrossRef PubMed.
  41. A. D. Becke and E. R. Johnson, J. Chem. Phys., 2005, 123, 154101 CrossRef PubMed.
  42. E. R. Johnson and A. D. Becke, J. Chem. Phys., 2006, 124, 174104 CrossRef PubMed.
  43. D. A. McQuarrie and J. D. Simon, Molecular Thermodynamics, University Science Books, Sausalito, CA, 1999 Search PubMed.
  44. J. W. Ochterski, Thermochemistry in Gaussian, April 19, 2000, http://www.gaussian.com/g_whitepap/thermo.htm, accessed April 2015 Search PubMed.
  45. J. T. Fermann and S. Auerbach, J. Chem. Phys., 2000, 112, 6787–6794 CrossRef CAS PubMed.
  46. D. G. Truhlar, B. C. Garrett, P. G. Hipes and A. Kuppermann, J. Chem. Phys., 1984, 81, 3542–3545 CrossRef CAS PubMed.
  47. J. Rehbein and B. K. Carpenter, Phys. Chem. Chem. Phys., 2011, 13, 20906–20922 RSC.
  48. Y. Oyola and D. A. Singleton, J. Am. Chem. Soc., 2009, 131, 3130–3131 CrossRef CAS PubMed.
  49. J. R. E. T. Pineda and S. D. Schwartz, Philos. Trans. R. Soc., B, 2006, 361, 1433–1438 CrossRef CAS PubMed.
  50. S. Arrhenius, J. Phys. Chem., 1889, 4, 226 Search PubMed.

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

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