Hyperspherical symmetry of hydrogenic orbitals and recoupling coefficients among alternative bases

Fock's representation of momentum space hydrogenic orbitals in terms of harmonics on the hypersphere S(3) Of a four-dimensional space is extended to classify alternative bases. These orbitals are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. Because of the correspondence between the S(3) manifold and the SU(2) group, new sum rules are established which are of relevance for the connection, not only among hydrogen atom orbitals in different bases, but also among the usual vector coupling coefficients and rotation matrix elements.