Periodic Gaussian Expansions of Hydrogenic Orbitals: A Spatial and Frequency Domain Analysis

Abstract: In this paper, we investigate the properties and applications of the Periodic Gaussian basis function for Hydrogenatom wavefunctions. We derive analytical expressions for the expansions of 1s, 2s, 2p, 3s, 3p and 3d-type orbitals in one, two and three dimensions, and compare their accuracy and efficiency with the conventional Slater-type orbitals. We show that the Periodic Gaussian function can represent periodic Hydrogenic orbitals in Cartesian 3D space with fewer basis functions, and that its Fourier transform reveals interesting insights into the physical characteristics of Hydrogen-atom wavefunctions, such as the quantization of energy levels. Our results demonstrate the potential of the Periodic Gaussian basis function for computational chemistry applications involving periodic systems.