Quantum Dynamical Behaviour in Complex Systems - A Semiclassical Approach

Abstract

One of the biggest challenges in Chemical Dynamics is describing the behavior of complex systems accurately. Classical MD simulations have evolved to a point where calculations involving thousands of atoms are routinely carried out. Capturing coherence, tunneling and other such quantum effects for these systems, however, has proven considerably harder. Semiclassical methods such as the Initial Value Representation (SC-IVR) provide a practical way to include quantum effects while still utilizing only classical trajectory information. For smaller systems, this method has been proven to be most effective, encouraging the hope that it can be extended to deal with a large number of degrees of freedom. Several variations upon the original idea of the SCIVR have been developed to help make these larger calculations more tractable; these range from the simplest, classical limit form, the Linearized IVR (LSC-IVR) to the quantum limit form, the Exact Forward-Backward version (EFB-IVR). In this thesis a method to tune between these limits is described which allows us to choose exactly which degrees of freedom we wish to treat in a more quantum mechanical fashion and to what extent. This formulation is called the Tuning IVR (TIVR). We further describe methodology being developed to evaluate the prefactor term that appears in the IVR formalism. The regular prefactor is composed of the Monodromy matrices (jacobians of the transformation from initial to finial coordinates and momenta) which are time evolved using the Hessian. Standard MD simulations require the potential surfaces and their gradients, but very rarely is there any information on the second derivative. We would like to be able to carry out the SC-IVR calculation without this information too. With this in mind a finite difference scheme to obtain the Hessian on-the-fly is proposed. We also apply the IVR formalism to a few problems of current interest. A method to obtain energy eigenvalues accurately for complex systems is described. We proposed the use of a semiclassical correction term to a preliminary quantum calculation using, for instance, a variational approach. This allows us to increase the accuracy significantly. Modeling Nonadiabatic dynamics has always been a challenge to classical simulations because the multi-state nature of the dynamics cannot be described accurately by the time evolution on a single average surface, as is the classical approach. We show that using the Meyer-Miller-Stock-Thoss (MMST) representation of the exact vibronic Hamiltonian in combination with the IVR allows us to accurately describe dynamics where the non Born-Oppenheimer regime. One final problem that we address is that of extending this method to the long time regime. We propose the use of a time independent sampling function in the Monte Carlo integration over the phase space of initial trajectory conditions. This allows us to better choose the regions of importance at the various points in time; by using more trajectories in the important regions, we show that the integration can be converged much easier. An algorithm based loosely on the methods of Diffusion Monte Carlo is developed that allows us to carry out this time dependent sampling in a most efficient manner.