Sturmians molecular

The problem of evaluating matrix elements of the interelectron repulsion part of the potential between many-electron molecular Sturmian basis functions has the degree of difficulty which is familiar in quantum chemistry. It is not more difficult than usual, but neither is it less difficult. Both in the present method and in the usual SCF-CI approach, the calculations refer to exponential-type orbitals, but for the purpose of calculating many-center Coulomb and exchange integrals, it is convenient to expand the ETO s in terms of a Cartesian Gaussian basis set. Work to implement this procedure is in progress in our laboratory. 

A short presentation of the unpublished Research Report No 217, by Osvaldo Goscinski, from the Quantum Chemistry Group at Uppsala University is made. The Report has the title Conjugate Eigenvalue Problems and the Theory of Upper and Lower Bounds . Some justification of its verbatim inclusion in this Volume in honour of Per-Olov Lowdin is made. It is essentially motivated by the attention that the theory presented there has received in the field of Generalized and Molecular Sturmians. Current work by John Avery and collaborators is alluded to. It is included as an Appendix. 

A new development is the notion of Molecular Sturmians, recently published /6/. It is stated there that ... 

Interelectron Repulsion Integrals for Molecular Sturmians from... 

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. 

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. 

If we wish to malce accurate molecular calculations with many-electron Sturmians, we must pay careful attention to the question of normalization Suppose that we wish to find the normalization constant M for the basis function ... 

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

Fig. 2 This figure shows the Sturmian molecular orbital corresponding to the ground state of the Hj ion, withS = 6, k = 1.16885, and/J = 5.13325 Bohrs...

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... 

Avery JS (2003) Sturmians. In Wilson S (ed) Handbook of Molecular Physics and Quantum Chemistry. Wiley, Chichester... 

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... 

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]. 

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