Hydrogenic orbitals in momentum space and hyperspherical harmonics: Elliptic Sturmian basis sets

Momentum space hydrogenic orbitals can be regarded as orthonormal and complete Sturmian basis sets and explicitly given in terms of (hyper)-spherical harmonics on the 4-D hypersphere S-3. Among the alternative coordinate systems, that allow separation of variables, the usual ones involving parameterizations of the sphere S-3 by circular functions correspond to canonical subgroup reduction chains; we also investigate harmonic "elliptic" sets (as, e.g., obtained by parameterizations in terms of Jacobi elliptic functions), In this article we list the canonical hydrogenic Sturmian sets and the orthogonal transformations connecting them. The latter enjoy useful three-term recurrence relationships that allow their efficient calculations even for large strings. We also consider modifications needed when the conservation of the symmetry of Sturmians with respect to parity. Finally, we discuss some properties of elliptic hydrogenic Sturmians and their relations with canonical Sturmians. Because elliptic Sturmians cannot be expressed in closed form, it is important to find expansions in a suitable basis set and calculate the transformation coefficients, We derive three-term recursion relationships fulfilled by the coefficients of the transformation between elliptic Sturmians and canonical Sturmians. A concluding discussion on the connections between configuration space and momentum space hydrogenic Sturmians completes this article. (C) 2003 Wiley Periodicals, Inc.