Solving the Time-Dependent Schrödinger Equation Using the Multi-Configurational Time-Dependent Hartree Approach

Abstract: Solving the Time-Dependent Schrödinger Equation (TDSE) is required to simulate non-equilibrium molecular dynamics. While approximations, such as swarms of classical trajectories, can be used to alleviate the inherent exponential growth of resources required with system size, in many systems the TDSE must be solved so that quantum effects are correctly described. This is the case for chemical systems involving tunneling (proton transfer) or non-adiabatic couplings (curve crossing). In this lecture I will introduce and explain the properties of the Multi-Configuration Time-Dependent Hartree (MCTDH) [1]. This is an algorithm that uses tensor contraction to optimally compress the wavefunction and allow the propagation of many degrees of freedom. Variants such as Multi-Layer MCTDH [2] and G-MCTDH [3] make the algorithm more powerful and more flexible, leading to the Direct Dynamics variational Multi-Configurational Gaussian (DD-vMCG) [4] that enables complete on-the-fly quantum dynamics simulations. As well as showing how the different methods link together, their properties will be discussed and examples given in how they can be used to study chemical reaction dynamics. References M. H. Beck, A. Jäckle, G. A. Worth and H.-D. Meyer, Phys. Rep. 324 (2000) 1. H. Wang and M. Thoss, J. Chem. Phys. 119 (2003) 1289. I. Burghardt, H.-D. Meyer and L. S. Cederbaum, J. Chem. Phys. 111 (1999) 2927. G. A. Worth, M. A. Robb and I. Burghardt, Faraday Discuss. 127 (2004) 307.