Complete and Orthonormal Sets of Exponential-type Orbitals with non-integer quantum numbers. On the results for many-electron atoms using Roothaan's LCAO method

TL;DR

This paper introduces a new basis set of exponential-type orbitals with non-integer quantum numbers for atomic calculations, demonstrating improved energy results and discussing the limitations of treating fractional quantum numbers as variational parameters.

Contribution

It proposes a method to construct accurate, computationally efficient basis sets using non-integer quantum number orbitals and analyzes their physical and mathematical properties.

Findings

Lower ground-state energies for Be- and Ne- like atoms compared to previous basis sets.

Non-integer quantum numbers cannot be solely treated as variational parameters due to unphysical basis sets.

Parameter alpha is not an observable and unsuitable as a variational parameter.

Abstract

Complete orthonormal sets of exponential-type orbitals with non-integer principal quantum numbers are discussed as basis functions in non-relativistic Hartree-Fock-Roothaan electronic structure calculations of atoms. A method is proposed to construct accurate and computationally efficient basis sets using these orbitals. It is demonstrated that principal quantum numbers of fractional order cannot be treated solely as variational parameters, since such a procedure may lead to unphysical basis sets (in particular, linearly dependent Slater-type functions). Ground-state total energies for the Be- and Ne- isoelectronic series are calculated. The results obtained are lower than those reported using other published basis sets. However, the energies obtained using Slater-type functions with non-integer principal quantum numbers are omitted from the comparison. These "orbitals" have no physical interpretation (except the "1s", which coincides with a hydrogenlike eigenfunction). In general linear independence of such Slater-type orbitals is not guaranteed. The results confirm that the parameter alpha, used to represent the complete orthonormal exponential-type orbitals in the weighted Hilbert space, is neither observable nor suitable to be considered as a variational parameter, despite its treatment as such in some prior work.