Sturmian base

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. 

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... 

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. 

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 ... 

Molecular Orbitals Based on Sturmians 5.1 The One-Electron Secular Equation... 

We have just seen that the treatment of a single electron moving in the field of several nuclei has been developed by a number of authors. Let us now turn to the question of whether molecular orbitals based on Coulomb Sturmians can be used to treat /V-electron molecules. To answer this question, let us consider a Slater determinant of the form... 

By comparing experimental or accurate theoretical results with others based on approximate models, it is possible to determine which among those models offers the best approximate constants of the motion and quantum numbers to describe particular states. This approach is used to evaluate and compare the extent of validity of independent-particle, Hartree-Fock and collective, molecule-like descriptions of atoms with two valence electrons. The comparisons are made on the basis of overlaps, oscillator strengths, momentum correlation and quadrupole moments. The criterion for each evaluation is the extent of agreement with results obtained from well-converged Sturmian Configuration Interaction wave functions. 

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