Exact Quantum Dynamics (Wave Packets) in Reduced Dimensionality

Abstract

The present chapter outlines basic and advanced concepts of grid-based quantum dynamics for molecular systems. Simulations within this framework are used to investigate the time-evolution of a molecular quantum system during a physical or chemical process of interest. The goal is to give the reader a concise introduction to wave packet simulations and strategies to reduce complex systems to few coordinates. Strengths and limitations are discussed using applied examples. A comprehensive flowchart on how to set up a wave packet simulation is given. Being principally exact, a solution of the time-dependent Schrödinger equation for the nuclear dynamics is only feasible in few dimensions due to exponential computational cost. This means the most crucial step is to find a representation of the molecular process using only few important coordinates. The concept of reactive coordinates is introduced, being determined either by chemical intuition or the adaptation of machine learning techniques. Using a certain reduced-dimensional representation, all terms within the molecular Hamiltonian are discussed along with the means to obtain them. A special focus lies on the kinetic energy operator, where the G-Matrix formalism is introduced as a very general scheme to transform it from cartesian coordinates to an arbitrary set of linear or non-linear reactive coordinates. Another focus lies on the evaluation of non-adiabatic coupling matrix elements and their implementation within the reduced-dimensional quantum dynamical framework. This allows for the simulation of wave packets passing through conical intersections, a feature determining the outcome of virtually all fast photochemical processes. Applied examples using the introduced concepts are illustrated.