Numerical methods for Quantum Dynamics

The dynamics of chemical reactions on a molecular level is a very active field in chemistry. Using ultrafast lasers it is possible to follow a reaction from reactants to products. Ahmed Zewail was awarded the Nobel Prize in Chemistry 1999 for his studies within this field. From a theoretical point of view, the dynamics of a chemical reaction is governed by the time dependent Schrödinger equation (TDSE). The goal of this project is to develop and analyze new accurate numerical techniques for solving the TDSE, using different levels of approximations for describing molecular systems with several coupled electronic states and many degrees of freedom.

Simulation of chemical reactions

For an accurate description of the dynamics of chemical reaction quantum mechanics must be used. This means solving the time dependent Schrödinger equation and following the nuclear dynamics from reactants to products, possibly including the interaction with external time-dependent fields. For systems with many degrees of freedom semiclassical methods must be used, leading to a less exact description of the dynamics.

We are particularly interested in the following aspects.

Electronic structure calculations Computation of the electronic structure allows for studies of molecular properties and computing potential energy surfaces.
Dissociative systems For chemical reactions that can lead to dissociation scattering boundary conditions must be enforced, which pose difficulties at the numerical boundaries.
Quantum optimal control Time dependent fields can be tuned to initiate and control the outcome of chemical reactions. The problem can be reformulated in Fourier space and a quasi-Newton method can be used as a powerful tool to solve the optimization problem.
Semiclassical methods Semiclassical methodologies can be used to reduce the computational complexity of a large molecular system, making the study of more complex molecules possible.

High-performance tools

Solving the Schrödinger equation for large molecular systems leads to high-dimensional problems that become computationally demanding. We develop and implement methods optimized for modern computer architectures.

Time propagators Exponential integrators are a suitable tool for numerical simulations of the Schrödinger equation. Error estimation, adaptive step size control, and parallel scalability are analyzed and improved.
Spatial discretization schemes Since solutions to the TDSE are usually very smooth, high-order methods are convenient for this type of problems. We analyze and devise methods based on summation-by-parts difference methods, radial basis functions, and spectral element methods.
The Chunks & Tasks framework Computer programs are written in terms of subtasks that operate on chunks of data. This allows for efficient use of computers with many processors.

Software

Ergo, an open source (GPL) program for large-scale electronic structure calculations. The source code is available for download at www.ergoscf.org.
HAParaNDA, an implementation framework for high-dimensional time-dependent PDEs. To be made publicly available in a near future.