Configuration localised states from orthogonal polynomials for effective potentials in 3D systems vs. algebraic DVR approaches

Abstract

Configuration localised states (CLS) based on the zeroes of orthogonal polynomials were introduced some time ago for 1D systems [Phys. Rev. A 61, 042504 (2000)]. Recently two algebraic approaches based on a discrete variable representation were proposed to describe 3D systems: the 3D harmonic oscillator discrete variable representation (HO-DVR) approach and the U(4)-UGA [Mol. Phys. 118, e1662959 (2020)]. Both of them together with the CLS are based on the harmonic oscillator basis. A comparison between the three ways of generating a DVR is timely. For that purpose, here, first the CLS method based on zeroes of orthogonal polynomials is generalised so as to be applied to 3D systems and, then, its results are compared with the algebraic DVR methods. It is shown that the CLS states obtained from the zeroes of orthogonal polynomials are equivalent to the localised states of the HO-DVR approach, while the U(4)-DVR method does not provide localised states. As applications, the ro-vibrational spectra of O molecules and berylium dimers, Be, are analysed. Excellent results are obtained with relatively small CLS and HO-DVR basis size.