# Dynamics and time development

Although the accurate calculation of energies and properties of molecules and solids is a prerequisite for reliable modeling, the time evolution of a system must be taken into account to model chemical reactions and rearrangement processes.

Within the Born Oppenheimer approximation, the actual dynamical course of a molecular system in real time can be described by a trajectory in phase space, Γ(p,r,t), in which p and r is the 3N dimensional vector representing the momenta and co-ordinates of the nuclei, respectively, and t is the time. To simplify matters, we may regard the time evolution of the system, for example during a chemical reaction, as a movie. There are several recipies on how to obtain a trajectory. In molecular mechanics, the potential energy surface V(r) is mapped using a simple empirical potential energy function, and the time development is obtained by numerical integration. More sophisticated potential energy surfaces can be obtained by fitting the results of extensive ab initio calculations to a more general functional form. This application is sometimes described as direct dynamics, and requires that the potential energy surface is known prior to the trajectory calculation. Yet another approach is to obtain the potential energy surface on-the-fly. This means that the potential energy is calculated (usually the gradient and the Hessian are also calculated) at the current point, V(ri). By using this information an infinitisimal step (Δr and Δt) is taken to the next point ri+1 by integration of the classical equations of motion, whereupon the potential energy is calculated again, V(ri+1). This procedure is followed step-by-step, until a complete trajectory has been obtained. The CTCC members have extensive experience in modeling chemical reactions using this approach, pioneered in a paper published in 1990.*

One advantage of this approach is that it requires no a priori assumptions on the exact form of the surface, and any ab initio wave function can be applied depending on the accuracy one wants. Compared to potential energy surface mapping, computing a sufficiently large number of reaction trajectories is, however, a far more ambitious task. Typically, one trajectory requires thousands of ab initio calculations, and very often hundreds of trajectories are required to get statistically significant simulations. This puts computational methods to their limits, and it will be extremely important to develop more efficient procedures.

To extend direct dynamics to more complicated reactions, involving long reaction times, multiscaling methods must be developed also in the time domain, recognizing that multi-step reactions may consist of a combination of slow and fast processes. To model the dynamics of solvation processes, temperature and dynamical effects must be considered~Wfor instance, through integration of ab initio methods with molecular dynamics. Since small time steps must be used in these very long simulations, they constitute an immense computational challenge. The success of such studies will rely on the fast evaluation of single-point energetics and properties, as an important first step toward the study of complex materials at working conditions. The CTCC will explore this important field of computational modeling further, with an aim to make significant contributions to chemical problems that require such models.

Within the CTCC a focus will be on simple gas phase reactions of ions and radicals. Particular emphasis will be put on understanding chemical reaction dynamics as an interplay between the initial conditions (po,ro) and the topography of the potential energy surface. Among other topics, the validity of statistical theories on reaction kinetics will be probed.

*T. Helgaker, E. Uggerud and H. J. Aa. Jensen, Chem. Phys. Lett. 173, 145 (1990).