The d-dimensional hydrogen atom: hyperspherical harmonics as momentum space orbitals and alternative Sturmian basis sets

Abstract

Hydrogenoid orbitals, i.e. the solutions to the Schrödinger equation for a central Coulomb field, are considered in mathematical dimensions d = 2 and d > 3 different from the physical case, d = 3. Extending known results for d = 3, Sturmian basis sets in configuration (or direct) space — corresponding to variable separation in parabolic coordinates — are introduced as alternatives to the ordinary ones in spherical coordinates: extensions of Fock stereographic projections allow us to establish the relationships between the corresponding momentum (or reciprocal) space orbitals and the alternative forms of hyperspherical harmonics. Properties of the latter and multi-dimensional Fourier integral transforms are exploited to obtain the matrix elements connecting the alternative basis sets explicitly in terms of Wigner's rotation matrix elements for d = 2 and generalized vector coupling (or Hahn) coefficients for d > 3. The use of these orbitals as complete and orthonormal expansion basis sets for atomic and molecular problems is briefly commented.