Application of B-splines to determining eigen-spectrum of Feshbach molecules

TL;DR

This paper demonstrates that B-spline basis sets outperform DVR in calculating the eigen-spectrum of diatomic molecules, especially near dissociation limits, and applies this to Feshbach molecules and universal energy-scattering relations.

Contribution

It introduces a B-spline based numerical method that is more accurate and robust than DVR for studying Feshbach molecules and their energy-scattering relations.

Findings

B-spline method outperforms DVR in eigen-spectrum calculations.

The method accurately describes Feshbach molecules near dissociation.

Numerical validation of quantum-defect theory for 1/R^6 potentials.

Abstract

The B-spline basis set method is applied to determining the rovibrational eigen-spectrum of diatomic molecules. A particular attention is paid to a challenging numerical task of an accurate and efficient description of the vibrational levels near the dissociation limit (halo-state and Feshbach molecules). Advantages of using B-splines are highlighted by comparing the performance of the method with that of the commonly-used discrete variable representation (DVR) approach. Several model cases, including the Morse potential and realistic potentials with 1/R^3 and 1/R^6 long-range dependence of the internuclear separation are studied. We find that the B-spline method is superior to the DVR approach and it is robust enough to properly describe the Feshbach molecules. The developed numerical method is applied to studying the universal relation of the energy of the last bound state to the scattering length. We numerically illustrate the validity of the quantum-defect-theoretic formulation of such a relation for a 1/R^6 potential.