Unified treatment of complete orthonormal sets for exponential type vector orbitals of particles with spin 1 in coordinate, momentum and four-dimensional spaces

Abstract

The new formulas are obtained for complete orthonormal sets of exponential type vector orbitals of a particle with spin 1 in coordinate, momentum and four-dimensional spaces using the properties of spherical vectors and complete orthonormal scalar basis sets of $\psi^{\alpha}$ -exponential type orbitals ($\psi ^{\alpha }$ -ETO), $\phi^{\alpha}$ -momentum space orbitals ($\phi ^{\alpha}$ -MSO) and $z^{\alpha}$ -hyperspherical harmonics ($z^{\alpha}$ -HSH) introduced by the author for particles with spin s = 0, where $\alpha=1,0,-1,-2,\ldots$ These vector orbitals are complete without the inclusion of the continuum and, therefore, their group of transformation is the four-dimensional rotation group of O(4). For overlap integrals over vector Slater orbitals with the same screening constant the analytical relations in coordinate space are also derived. It should be noted that the new idea presented in this study is the combination of spherical vectors with complete orthonormal scalar sets for radial parts of $\psi^{\alpha}-,\phi ^{\alpha}-,z^{\alpha}$ -orbitals.