Hydrogen Sturmian

Weatherford CA Scaled hydrogenic sturmians as ETOs. In Weatherford and Jones [40], pp. 29-34... 

The momentum-space orthonormality relation for hydrogenlike Sturmian basis sets, equation) 17), can be shown to be closely related to the orthonormality relation for hyperspherical harmonics in a 4-dimensional space. This relationship follows from the results of Fock [5], who was able to solve the Schrodinger equation for the hydrogen atom in reciprocal space by projecting 3-dimensional p-space onto the surface of a 4-dimensional hypersphere with the mapping ... 

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. 

The Sturmian eigenfunctions in momentum space in spherical coordinates are, apart from a weight factor, a standard hyperspherical harmonic, as can be seen in the famous Fock treatment of the hydrogen atom in which the tridimensional space is projected onto the 3-sphere, i.e. a hypersphere embedded in a four dimensional space. The essentials of Fock analysis of relevance here are briefly sketched now. 

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 ... 

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)...

The configuration-interaction representation of the lower-energy states of an atom is the IV-electron analogue of the Sturmians in the hydrogen-atom problem. We choose an orbital basis of dimension P, form from them a subset of all possible A/ -electron determinants pk),k = 0,Mp, and use these determinants as a basis for diagonalising the IV-electron Hamiltonian. It may be convenient first to form symmetry configurations kfe) from the pfe). 

V. Aquilanti, A. Caligiana, S. Cavalli, and C. Coletti, Hydrogenic orbitals in momentum space and hyperspherical harmonics elliptic sturmian basis sets. Int. J. Quant. Chem., 92 212-228, 2003. 

Aquilanti V, Cavalli S, Coletti C, Grossi G (1996) Alternative sturmian bases and momentum space orbitals an application to the hydrogen molecular ion. Chem Phys 209 405... 

L-spinors, [87], are related to Dirac hydrogenic functions in much the same way as Sturmian functions [93,94] are related to Schrodinger hydrogenic functions. The radial parts are given by [87, III]... 

The alternative on which we focus here is the study of Coulomb Sturmian orbitals and their generalizations, functions of a type which have so far received limited attention in quantum theory. Coulomb Stiumians are formally similar to the well-known hydrogenic functions, but they are isoenergetic and their use has important and peculiar implications that are worthwhile investigating. 

Sturmians are not simply an alternative orbital set closely related to hydrogenic functions and to Slater-type orbitals. In the last decade the connection between their use and the ever-increasing exploitation of hyperspherical methods to treat the A -body problem in quantum mechanics was established, for both bound-state and scattering problems in nuclear, atomic and molecular physics and this became an important source of progress, as recorded in a recent book [5]. 

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. 

Alternative ETOs would be Slater type orbitals and B-functions with their simple Fourier transforms. Strictly, they should be combined as linear combinations to form hydrogen-like or, better, Sturmian basis sets prior to use. 

---