Unified treatment of complete orthonormal sets of nonrelativistic, quasirelativistic and relativistic sets of spinor wave functions, and Slater spinor orbitals in coordinate, momentum and four-dimensional spaces

Abstract

Using the properties of tensor spherical harmonics introduced by the author in previous paper (Guseinov, Phys Lett A 372:44, 2007) and complete orthonormal scalar basis sets of nonrelativistic $\psi^{\alpha}$ -exponential type orbitals ($\psi^{\alpha}$ -ETO), $\phi^{\alpha }$ - momentum space orbitals ($\phi^{\alpha}$ -MSO) and z ^α-hyperspherical harmonics (z ^α-HSH) for particles with spin s = 0 the new analytical relations for the quasirelativistic and relativistic spinor wave functions and Slater spinor orbitals in coordinate, momentum and four-dimensional spaces are derived, where α = 1, 0, −1, −2,.... The 2-component quasirelativistic and 4-component relativistic spinor wave functions obtained are complete without the inclusion of the continuum. The relativistic spinor wave function sets and Slater spinor orbitals are expressed through the corresponding quasirelativistic spinor wave functions and Slater spinor orbitals, respectively. The analytical formulas for overlap integrals over quasirelativistic and relativistic Slater spinor orbitals with the same screening constants in coordinate space are also derived.