Algebraic evaluation of matrix elements in the Laguerre function basis

TL;DR

This paper develops an algebraic framework for Laguerre functions, enabling analytic calculation of matrix elements crucial for atomic, molecular, and nuclear physics computations.

Contribution

It constructs the SU(1,1)xSO(3) algebra and derives shift operators, providing new analytic formulas for matrix elements in the Laguerre basis.

Findings

Derived explicit formulas for radial operator matrix elements.

Established algebraic methods for constructing matrix elements of tensor operators.

Demonstrated applicability to complex spherical tensor operators.

Abstract

The Laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron-structure theory, with applications in hadronic and nuclear theory as well. While similar in form to the Coulomb bound-state eigenfunctions (from the Schroedinger eigenproblem) or the Coulomb-Sturmian functions (from a related Sturm-Liouville problem), the Laguerre functions, unlike these former functions, constitute a complete, discrete, orthonormal set for square-integrable functions in three dimensions. We construct the SU(1,1)xSO(3) dynamical algebra for the Laguerre functions and apply the ideas of factorization (or supersymmetric quantum mechanics) to derive shift operators for these functions. We use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators (involving powers of the radial coordinate and radial derivative) in the Laguerre function basis. We illustrate how matrix elements for more general spherical tensor operators in three dimensional space, such as the gradient, may then be constructed from these radial matrix elements.