Development and Implementation of a Quantum-Stochastic Method for Electron-Nuclear Coupled Dynamics

The simulation of processes characterized by a strong coupling between electrons and nuclei represents one of the most difficult tasks in theoretical and computational chemistry. The difficulty lies in the breakdown of the Born-Oppenheimer approximation, also known as adiabatic approximation, which allows for the separation between the electronic and nuclear motions. For this reason, the simulations of the so-called non-adiabatic phenomena must take into account the interaction between the two classes of particles, using an appropriate level of theory. The most used methods for the simulation of non-adiabatic processes belong the non-adiabatic mixed quantum-classical category. The strength of these methods lies in their multiscale nature, which allows for the different treatment of the nuclear and electronic coordinates. In particular, the former are consider classical, while the latter are treated at quantum mechanical level. In this way, each set of degrees of freedom is treated in the most computational efficient way, overcoming the computational limits that, at present, do not allow the computation of the nuclear-electronic Schrödinger equation. However, these methods are still very resource demanding, so they can be effectively applied only to systems of up to hundreds of atoms. The aim of the Thesis is the derivation and implementation of a new non-adiabatic mixed quantum-classical approach for simulating the evolution of electronic populations coupled with the nuclear dynamics of a molecule in a dissipative environment. The starting point is the quantum-classical Liouville equation formulated by R. Kapral and G. Ciccotti [R. Kapral, G. Ciccotti, J. Chem. Phys. 110, 8919 (1999)], that is recast here in internal natural coordinates, allowing for the description of structure and dynamics in a way that chemists are accustomed to when describing molecular geometry and its changes. Then, the projection of the bath coordinates is performed, providing an important reduction of both complexity and computational cost. This leads to the formulation of the quantum-stochastic Liouville equation, that can be used directly in the statistical thermodynamics description of chemical systems. The derivation is followed by the parameterization of the method, with particular focus on the transition rates, i.e. the terms that comprise the electronic-nuclear coupling. Finally, a software package based on this approach is developed and tested. The software is currently able to simulate a two-states system regulated by one angular degree of freedom, but more options will be soon available. It is designed to be used even without deep knowledge of the quantum-stochastic Liouville equation theory, also giving the user access to post-processing tools.