Chapter 1 Introduction 1 General Introduction

The evolution of the average constraint force, , during the two undecane simulations is plotted in Figure 5.17. Since a high energy starting structure of the backbone was selected, the force on the constraint is initially high. As the backbone structure equilibrates, the force on the constraint should relax to a steady state value. The average constraint force decays much more rapidly in the multiple time-step simulation than in the standard simulation. This is because the heavier MM subsystem in the standard simulation is slower to settle into an equilibrium conformation. Figure 5.16 shows that the constraint force in the multiple time-step simulation achieves a steady state value within approximately 4000 fs. The constraint force in the standard simulation, which should converge to the same value, is slow to decay and convergence is not observed in the first 8000 fs of the simulation. The simulation was stopped after 50000 time steps, since it was evident that there is an accelerated relaxation of the average force with the multiple time step method. Furthermore, since temperature equilibration in the standard simulation does not begin to arise until the end of the 8 ps simulation, the convergence of the constraint force will not occur for some time later.

When using the slow growth technique to map out free energy surfaces, it is actually the ensemble averaged force on the constraint that is the ideal force used in the integration. Thus, the evolution of the averaged force plotted in Figure 5.16 emphasis that the multiple time-step QM/MM method can provide for better sampling and smaller errors. Uncertainty estimates in a slow growth simulation are determined by the amplitude of the fluctuations in the constraint force.¹⁷⁴ The larger the fluctuations the larger the uncertainties are in the calculated relative free energies. Examination of the unaveraged forces on the constraint (not plotted) show the amplitude of the fluctuations is approximately 50% smaller in the multiple time-step simulation compared to the standard simulation. This corresponds to roughly 20% smaller error bars in the multiple time-step simulation.

The undecane simulation was designed such that the structure of the MM system strongly influenced the constraint force. Therefore, a significant acceleration in the convergence of the constraint force was observed with the multiple time-step method. However, most systems are more balanced in that the QM system has a stronger influence on the constraint force. Thus, the benefit of the multiple time-step method will be less pronounced for most simulations. On the other hand, even if the improvement is small, the additional computational effort is minimal(assuming that the computational expense of the MM system is negligible compared to the QM system). In the Car-Parrinello QM/MM simulation of the Brookhart catalyst described in the previous section, the calculation of the MM forces amounted to less than 0.01% of the total computational effort.^◊ Therefore, if the multiple time-step technique were to be applied to the system with a high oversampling ratio of 100:1, the 100 fold increase in computing the MM forces would still be negligible. Therefore, in these cases where the computational expense of the MM subsystem is negligible and were time-dependent properties are not being investigated, there is no reason not to apply the multiple time-step procedure.

A new implementation to carry out Car-Parrinello ab initio molecular dynamics simulations of extended systems using a combined quantum mechanics and molecular mechanics potential is presented. Our implementation allows the QM/MM boundary to cross covalent bonds such that the potential surface of a single molecular system is described by a hybrid potential. Since the potential surface of the molecular mechanics region is usually much less computationally demanding to calculate than that in the QM region, we have implemented a multiple time step technique to over-sample the MM region relative to the QM region. The goal here is to provide better ensemble averaging in the MM region which is usually larger in size and therefore usually has a higher degree of configurational variability. We have demonstrated the multiple time step integrator will generate the same trajectory as a standard molecular dynamics integrator. Moreover, with a gradual rescaling of masses the energy conservation of a multiple time step simulation can be brought to the same level as a standard simulation. Finally we have demonstrated that the multiple time step QM/MM method can accelerate the equilibration and configurational sampling of a molecular dynamics simulation.

In this chapter we have also demonstrated the applicability of the combined QM/MM AIMD approach for studying transition metal based catalytic systems. In particular, we have examined the chain termination process in Brookhart's Ni(II) diimine olefin polymerization catalyst. In this simulation the QM/MM boundary crosses several covalent bonds, such that the Ni diimine core is treated by the DFT potential while the large bulky substituents are treated with the AMBER molecular mechanics force field. For the chain termination, the calculated free energy barrier of ∆F^‡ = 14.8 kcal/mol at 25°C is in good agreement with the experimental result of ∆G^‡ = 15.5-16.5 kcal/mol at 0°C.¹²²
Chapter 6

Monomer Capture in Brookhart's Ni(II) Diimine Olefin Polymerization Catalyst: A Static and Dynamic QM/MM Study.
6.1 Summary

Mechanistic aspects of transition metal catalyzed olefin polymerization have been extensively studied, both experimentally and theoretically.¹²³ The focus of theoretical studies is most often the monomer insertion and chain termination processes. The capture or ejection of the monomer are either neglected or not thoroughly examined. However, the capture and expulsion are also integral parts of the catalytic cycle. Obviously, for chain growth to occur, the monomer must be captured and inserted into the growing polymer chain. For chain termination which is believed to occur via either unimolecular or bimolecular -elimination in single-site olefin polymerization catalyst systems, the ultimate result is the expulsion of the olefinic terminated polymer chain that was grown at that center. In the case of Brookhart's relatively new Ni diimine olefin polymerization catalyst which can produce polyethylene with a controlled level of short chain branching (Scheme 6.1), the capture process may play a key role controlling the extent of the chain branching. To date, no detailed theoretical studies of the capture/ejection processes in olefin polymerization catalysts have appeared. This is in part due to the scarcity of experimental data and due in part to the theoretical difficulty of treating the process. In this chapter we examine the monomer capture/ejection process with Brookhart's Ni-diimine catalyst with a combination of the ADF QM/MM and PAW QM/MM methods.

One of the most unique aspects of Brookhart's Ni diimine catalyst system, is that a controlled level of short chain branching is possible with the homopolymerization of ethylene (Scheme 6.1). Normally, branching is introduced in polyethylene via the introduction of short chain -olefin comonomers, such as 1-hexene. Thus, this property is of commercial interest because it offers potential economic advantages. The degree of chain branching is observed to decrease with increasing monomer concentration whereas both the activity and molecular weights are found to be more or less independent of the monomer concentration.¹ Consistent with these observations, Johnson et al.¹ proposed the mechanism depicted in Figure 6.1. Both the insertion and termination proceed from the -complex which has been identified as the catalyst resting state. The branching, however, is believed to proceed from the 'naked' metal alkyl complex via an isomerization mechanism as previously examined in Chapter 3. The monomer concentration effect on the chain branching can then be explained in terms of the equilibrium or kinetics involving the metal alkyl complex and the -complex. The higher monomer concentrations, the faster the -complex is formed and the less time there is to allow the isomerization process to take place from the metal alkyl complex. Therefore, the observed amount of branching diminishes with increased monomer concentration.