# Numerical Quantum Dynamics (NQD)

The dynamics of chemical reactions on a molecular level is a very active field in chemistry, and both experimental and theoretical methods are being constantly developed to follow a reaction from reactants to products, see e.g. the Nobel prices in Chemistry in 1998, 1999 and 2013.

At a fundamental level, the dynamics of a chemical reaction is governed by the time dependent Schrödinger equation (TDSE). The goal of the research in the NQD group is to develop and analyze new accurate numerical techniques for simulating chemical reactions, using different levels of approximations of the Schrödinger equation for describing molecular systems.

## Simulation of chemical reactions

For an accurate description of the dynamics of chemical reactions 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. To be able to compute the nuclear dynamics, accurate descriptions of the electron structures for the system is needed. This can be achieved by parametric potential surface models or by using an ab-initio model, where the time-independent Schrödinger equation is solved to determine the electron structure.

## Subprojects

### Active

- Electronic structure calculations Computation of the electronic structure allows for studies of molecular properties and computing potential energy surfaces.
- 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.

### Finished or dormant

- Semiclassical methods If the atoms are heavy, solutions to the Schrödinger equation become highly oscillatory. Solution with standard methods is then expensive or intractable. Asymptotic methods which are valid in the high frequency regime can reduce the computational complexity. Such methods are called semiclassical.
- 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.
- Dissociative systems For chemical reactions that can lead to dissociation scattering boundary conditions must be enforced, which pose difficulties at the numerical boundaries. We have formulated perfectly matched layers for the Schrödinger equations which work well as boundary closures.
- Quantum optimal control Laser pulses can be used to control chemical reactions. The outcome is affected by the shape of the pulse. The search for the ideal pulse can be formulated as an optimization problem.

## Parallel implementation and software

- The Chunks and Tasks programming model Computer programs are written in terms of subtasks that operate on chunks of data. This allows for efficient use of computers with many processors. Source code for Chunks and Tasks library implementations available at chunks-and-tasks.org.
- Ergo, an open source (GPL) program for large-scale electronic structure calculations. The source code is available for download at www.ergoscf.org. Work is ongoing to parallelize Ergo using the Chunks & Tasks programming model.
- HAParaNDA, an implementation framework for high-dimensional time-dependent partial differential equations, targeting large-scale parallel computers. A solver for the TDSE is implemented as a pilot application problem.

## Group information

Full list of publications and conference contributions

### Participants

- Sverker Holmgren (Professor)
- Katharina Kormann (Associate Professor)
- Gunilla Kreiss (Professor)
- Hans Karlsson (Professor)
- Emanuel Rubensson (Associate Professor, Docent)
- Elias Rudberg (Researcher)

### Alumni

- Anastasia Kruchinina (PhD thesis 2019, Principal advisor: E. Rubensson)
- Emil Kieri (PhD thesis 2016, Prinicipal advisor: S. Holmgren)
- Magnus Grandin (PhD thesis 2014, Principal advisor: S. Holmgren)
- Katharina Kormann (PhD thesis 2012, Principal advisor: S. Holmgren)
- Anna Nissen (PhD thesis 2011, Principal advisor: G. Kreiss).

### Collaboration

- Vasile Gradinaru, Seminar for applied mathematics, ETH, Zürich.
- Olof Runborg, KTH, Stockholm.
- Anders Niklasson, Los Alamos National Laboratory, NM, USA.

### Contact

## Internal

The NQD group internal web page is accessible to group members only.