Solving One-Electron Systems in a Novel Gaussian-Sinc Mixed Basis Set

Abstract

A novel Gaussian-Sinc mixed basis set for the calculation of the one-electron electronic structure within a uniform magnetic field in three dimensions is presented. The one-electron system is used to demonstrate the utility of this new methodology and is a first step in laying the foundation for further development of many-electron atomic and molecular methodology. It is shown in this manuscript how to effectively calculate all basis set integrals, which includes the mixed Gaussian-Sinc integrals, with a fast and accurate method. The Sinc basis is invariant to the choice of the position of the Coulomb potential, as opposed to traditional grid based methods. This invariance guarantees that the choice of the grids origin has no effect on the electronic structure calculation. This is because the Coulomb potential is treated properly in this methodology, as opposed to DVR methodologies. The off-diagonal terms are sparse but very important around the Coulomb singularity. In general, five to six significant digits of accuracy on all converged results without the linear dependency problems of the Gaussian methodologies are achievable. This methodology is applied to calculate the ground state energy of H atom, $H_2^{+}$ ion and $H_3^{2+}$ ion in magnetic fields up to a magnetic field strength of 2.35x$10^13$ G (10,000 au). From these calculations it is shown that $H_3^{2+}$ ion is unstable without relativistic considerations.