Chapter 1 Introduction 1 General Introduction

The recent development of the Projector Augmented Wave (PAW)⁴¹ Car-Parrinello¹³⁰ method has allowed for practical ab initio molecular dynamics simulations of transition metal complexes to be performed.^131,132 We have recently applied the method to study olefin polymerization catalysis.^121,133-135 In this chapter, a summary how we^121,133,135 have found the ab initio molecular dynamics method to be a practical computational tool for studying homogenous catalysis will be provided. Since the ab initio molecular dynamics is not widely utilized by quantum chemists, a succinct review of the method^◊ will first be presented with a particular focus on the aspects and methodologies that we have applied to study transition metal based catalysis.

Conventional electronic structure calculations can be classified as static simulations. In these calculations the nuclear positions are optimized to locate local minima and transition states on the potential surface at the zero temperature limit. This involves, for each nuclear geometry, converging the electronic structure in order to determine the energy and forces on the nuclei. Using special algorithms, this information is then used to move the nuclei to a more optimal geometry. In classical molecular dynamics the nuclei are allowed to move on the potential surface according to Newton's classical laws of motion (Eqn. 4-1) as to simulate nuclear motion at finite (non-zero) temperatures.

The nuclear motion generated in a molecular dynamics simulation can be utilized for a variety of purposes. Stationary points can be optimized by simply applying friction to the nuclear motion, thereby causing the system to settle into a local minimum. The motion can also be used to sample configuration space as to construct a partition function from which properties can be derived rigorously from statistical mechanics. There are also global minimization schemes which utilize molecular dynamics, such as simulated annealing.

With each new time step, a new geometry is generated and therefore the forces on the nuclei have to be recalculated. In order to simulate molecular vibrations, the time step ∆t must be at least a third smaller than the period of the fastest vibration. Thus in order to simulate C-H bond stretching which has a period of 0.01 psec, the ∆t must be less than 0.003 psec. Even a relatively short 10 ps simulation requires over 3300 time steps and therefore 3300 gradient calculations. (To put this in perspective, even the fastest enzyme catalyzed reactions have turnover periods of greater than 1000 picoseconds.^◊ ) For this reason, most MD simulations are performed with a molecular mechanics force field. Their computational simplicity allows for extremely large simulations to be performed. An impressive example of this is the molecular dynamics simulations protein folding processes, which often involve 10⁴ atoms and require simulation times of over 10^-3 seconds. However, the empirical nature of molecular mechanics force fields has its limitations. For example, transition metal complexes are problematic because there is currently no force field general enough to handle all of the complicated bonding schemes available to these complexes. Moreover, chemical reactions cannot be simulated accurately (if at all) with a molecular mechanics force field. To overcome these problems ab initio or QM potentials can be used in conventional classical molecular dynamics. With the ab initio molecular dynamics (AIMD) method the forces are not determined from a molecular mechanics force field, but rather they are derived from a full electronic structure calculation at each time step. Although this is computationally demanding, it allows for bond breaking and forming processes to be simulated and it allows for molecular dynamics to be performed on systems where a molecular mechanics force field is not readily applicable.