I. Complete and orthonormal sets of exponential-type orbitals with noninteger principal quantum numbers

TL;DR

This paper generalizes exponential-type orbitals with non-integer quantum numbers, deriving analytical expressions and orthogonalization methods, and explores their applications in fractional calculus within atomic and molecular physics.

Contribution

It introduces a complete orthonormal set of exponential-type orbitals with non-integer principal quantum numbers and derives analytical formulas for their transformation and orthogonalization.

Findings

Derived analytical expressions for linear combination coefficients.

Developed a closed-form expression for orthogonalized orbitals.

Connected fractional calculus operators with relativistic molecular functions.

Abstract

The definition for the Slater-type orbitals is generalized. Transformation between an orthonormal basis function and the Slater-type orbital with non-integer principal quantum numbers is investigated. Analytical expressions for the linear combination coefficients are derived. In order to test the accuracy of the formulas, the numerical Gram-Schmidt procedure is performed for the non-integer Slater-type orbitals. A closed form expression for the orthogonalized Slater-type orbitals is achieved. It is used to generalize complete orthonormal sets of exponential-type orbitals obtained by Guseinov in [Int. J. Quant. Chem. 90, 114 (2002)] to non-integer values of principal quantum numbers. Riemann-Liouville type fractional calculus operators are considered to be use in atomic and molecular physics. It is shown that the relativistic molecular auxiliary functions and their analytical solutions for positive real values of parameters on arbitrary range are the natural Riemann-Liouville type fractional operators.