Wavelet Approach to Electronic Structure Calculations

Standard approaches to calculate properties of atoms, molecules and models of nano-structures in quantum chemistry are Hartree–Fock (HF) and density functional theory (DFT). These methods give numerical solutions of the Schrödinger equation for the wavefunctions and energies of electrons, i.e., the electronic structure. Majority of numerical algorithms are based on finding the wavefunctions or orbitals as series expansion of basis functions, the basis set. In this study we consider different wavelets as basis functions.

Wavelets are functions constructed as translations and dilatations of the so called mother scaling function and mother wavelet. Orthonormal wavelets are orthonormal to each other and interpolating wavelets satisfy biorthogonality relations involving dual wavelets.

In order to calculate the electronic structure of atoms the radial Schrödinger and Hartree–Fock equations are solved using a Galerkin method with a Deslauriers–Dubuc wavelet basis. Several light atoms are considered as examples.

We introduce new approach called the Exact Pseudopotential (EPP), which is one method to remove the Coulomb singularity of atomic nuclei. This is tested with hydrogen and helium atom orbitals.

A three-dimensional Deslauriers–Dubuc wavelet basis is used to compute the energetics of the hydrogen and helium atoms, hydrogen molecule ion, hydrogen molecule, and lithium hydride molecule. The internuclear equilibrium distance is determined for the molecules, too.

Wavelets are applied to the path integral formulation and this method is used to calculate one- and three-dimensional harmonic oscillator and hydrogen atom in one and three dimensions.

As a conclusion, we found the Deslauriers–Dubuc wavelets a practical basis set for solving wavefunctions of electronic systems. Furthermore, Daubechies wavelets were found appropriate with the numerical path integral approach.