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.

Recent publications

• Radial basis functions for the time-dependent Schrödinger equation. Katharina Kormann and Elisabeth Larsson. In Numerical Analysis and Applied Mathematics: ICNAAM 2011, volume 1389 of AIP Conference Proceedings, pp 1323-1326, American Institute of Physics, Melville, NY, 2011. (DOI).
• Nonmonotonic recursive polynomial expansions for linear scaling calculation of the density matrix. Emanuel H. Rubensson. In Journal of Chemical Theory and Computation, volume 7, pp 1233-1236, 2011. (DOI).
• Methods for Hartree–Fock and density functional theory electronic structure calculations with linearly scaling processor time and memory usage. Emanuel H. Rubensson, Elias Rudberg, and Pawel Salek. In Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications, pp 263-300, Springer, Dordrecht, The Netherlands, 2011. (DOI).
• Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian. Katharina Kormann, Sverker Holmgren, and Hans O. Karlsson. In Journal of Computational Science, volume 2, pp 178-187, 2011. (DOI).
• Assessment of density matrix methods for linear scaling electronic structure calculations. Elias Rudberg and Emanuel H. Rubensson. In Journal of Physics: Condensed Matter, volume 23, pp 075502:1-10, 2011. (DOI).

Full list of publications and conference contributions

Group information

• Participants
  • Senior researchers: Sverker Holmgren, Hans Karlsson, and Gunilla Kreiss.
  • Post Docs: Emanuel Rubensson and Elias Rudberg.
  • PhD Students: Kenneth Duru, Magnus Gustafsson, Emil Kieri, Katharina Kormann, and Anna Nissen.
• Collaboration
  • Vasile Gradinaru, Seminar for applied mathematics, ETH, Zürich.
  • Pawel Salek, PS Consulting, Krakow.
  • Anders Szepessy, Numerical Analysis, KTH, Stockholm.
• Events
  • NQD colloquium: Third Friday each month at 1:15 pm. Contact: Elias Rudberg.
  • Workshop on quantum dynamics and semiclassical methods: June 15, 2011. Program. Contact: Katharina Kormann.
• Contact: Sverker Holmgren

Internal

NQD group internal web page, only accessible to group members: http://www.it.uu.se/research/project/qd/internal/