**Sonja Eskelinen**

Höynälänmaa examined the use of functions called wavelets for calculation of the properties of atoms and molecules in his study. He combined wavelets with the existing calculation methods of quantum physics and developed algorithms needed for that.

Höynälänmaa did also some mathematical research, in which he developed the generalization of wavelets into multiple dimensions needed in his study. He did the computational research by writing and running computer programs using the C++ programming language and Octave application.

Wavelets are functions that are obtained by translating and shrinking so called mother scaling function and mother wavelet in the real axis. Höynälänmaa examines the application of both one- and three-dimensional wavelets into the calculation of atoms and two-atom molecules.

“I did most of the calculations using the so called self-consistent field method, which means iterating the equation describing a system so that the electric potential is computed from the solution of an iteration step and that potential is used as input for the next iteration step. For example, the bond lengths of the hydrogen molecule and lithium hydride molecule were computed in the study”, he says.

Höynälänmaa also combined wavelets with the path integral formulation, which is a method to compute the time evolution of a quantum physical system. He applied this method for the computation of harmonic oscillator and hydrogen atom. Harmonic oscillator is a system where a particle is affected by a force proportional to the deviation of the particle from the equilibrium position and the force tries to resist the movement of the particle. An example of a classical harmonic oscillator is a particle oscillating in the head of a spring.

“The most important challenge in my study turned out to be the singularity of nuclear potential at the nuclei, which means that the potential tends to infinity at the nuclei. The slowness of the three-dimensional calculations, which is caused by the large number of necessary basis functions, was also a challenge."

Höynälänmaa believes that the developed calculation methods may have some applications to solving differential equations in engineering.

Höynälänmaa comes originally from Kemi.

## Public defence on Friday 5 July

The dissertation *Wavelet Approach to Electronic Structure Calculations *written by M.Sc. Tommi Höynälänmaa and belonging to the field of computational physics will be presented for public discussion in the Faculty of Engineering and Natural Sciences of Tampere University on Friday 5.7.2024 at 12 o'clock in the old hall of Niuvanniemi hospital (Niuvankuja 65, Kuopio).

The opponent is Dr. **Mikael Kuisma** from the Technical University of Denmark. The custos is Professor **Tapio Rantala** from the Faculty of Engineering and Natural Sciences, Tampere University.