Alternative Sturmian bases and momentum space orbitals: An application to the hydrogen molecular ion

The relationship between alternative separable solutions of the Coulomb problem both in configuration and in momentum space is exploited in order to obtain Sturmian orbitals of use as expansion basis sets in atomic and molecular problems. The usual spherical basis is obtained by separation in polar coordinates. The mathematical properties are explored for a type of basis set for quantum mechanical problems with axial symmetry, examples being diatomic molecules, or atoms under the influence of a uniform electric field. Because of its appropriateness for treatment of the Stark effect in atomic physics, this alternative basis set is called Stark basis. The Stark basis corresponds to separation in parabolic coordinates in configuration space and in cylindrical coordinates in momentum space. Fock's projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. As an application, the Shibuya and Wulfman analysis of momentum space molecular orbitals is reformulated and monoelectronic multicenter integrals are calculated by angular momentum algebra. Properties of the Stark basis, which exhibits less coupling at long range, are illustrated with reference to the case of the hydrogen molecular ion in the fixed nuclei approximation.