Sturmian expansions

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. 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. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

Table 5.2. The highest principal quantum number for an eigenstate of hydrogen whose eigenvalue is approximated within the relative-error tolerance 5 by a Sturmian expansion of dimension M (adapted from Bray, Konovalov and McCarthy, 1991a)...

Since the hydrogenlike Sturmian basis functions form a complete set, the term %i,o,o(xy-R) can be represented as a single-center expansion in terms of functions localized at the origin ... 

There have been several papers published on a(co) and y(co) for the hydrogen atom[85]-[90]. Shelton[89] used an expansion in Sturmian functions to obtain y values for Kerr, ESHG, THG and DFWM at a number of frequencies. A more straightforward and simpler method is to use the SOS approach and a pseudo spectral series based on the wavefunctions formed by the linear combinations ... 

In other words, if a Coulomb Sturmian located on one center is expanded in terms of Coulomb Sturmians located on another center, the expansion coefficients are Shibuya-Wulfman integrals. It should be noted, however, that this expansion is... 

The series in (171) terminates and the expansion is exact. The coefficients -ifl form a large but very sparse matrix that can be precalculated and stored. What we have done here is to expand a product of two Coulomb Sturmians in terms of a single Coulomb Sturmian with double the k value. When this is done, the exponential part is automatically correct, and only the polynomial parts need to be taken care of. Hence, the sparseness of Cpt p p. Then... 

In this expansion, the coefficients r nJj and a, are universals that can be calculated once and for all, and that never have to be recalculated. When the basis functions scale with changing values of k, the expansion scales automatically too. Because the coefficients are universals, we can use many terms in the expansion and thus obtain especially good accuracy. The fact that the interelectron repulsion integrals divided by k are independent of k can be shown by arguments similar to those shown in (42)-(47). When divided by k, the interelectron repulsion integrals are pure functions of the parameters s = kx and Sa = kXa. Therefore, they scale automatically with changes of scale of the basis functions. The independence from k also implies that the molecular-Sturmian-based interelectron repulsion integrals can be pre-evaluated and stored. 

R. Szmytkowski. The Dirac-Coulomb Sturmians and the Series Expansion of the Dirac-Coulomb Green Functions Application to the Relativistic Polarizability of the Hydrogen Like Atoms. /. Phys. B At. Mol Opt. Phys., 30 (1997) 825-861. 

---