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${}_2$ molecules and berylium dimers, Be${}_2$, are analysed. Excellent results are obtained with relatively small CLS and HO-DVR basis size.

GRAPHICAL ABSTRACT

Keywords:

• Effective 3D potentials
• pseudo-states based on orthogonal polynomials
• algebraic DVR approach

Acknowledgements

Authors want to thank Micha Lesiuk to provide us with the original program to reproduce the potential for Be${}_2$.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by DGAPA-UNAM, México, under project IN-212020, by Spanish Consejería de Economía, Inovación, Ciencia y Empleo, Junta de Andalucía [grant number FQM-160], by the Consejería de Conocimiento, Investigación y Universidad, Junta de Andalucía and European Regional Development Fund (ERDF), ref. SOMM17/6105/UGR, and by the Spanish Ministerio de Economía y Competitividad and FEDER funds under projects US-1380840, and P20-01247, and Spanish National Research, Development, and Innovation plan under the projects PID2019-104002GB-C22 and PID2020-114687GB-I00. First author is also grateful for the scholarship (Posgrado en Ciencia e Ingeniería de Materiales) provided by CONACyT, México.