An approach to establish the connection between configuration and su(n + 1) algebraic spaces in molecular physics: application to ammonia

ABSTRACT

An approach aimed to connect configuration and algebraic spaces is discussed. This approach emerges as the need to translate a vibrational description in configuration space to an algebraic representation based on unitary dynamical algebras, where a straightforward connection does not exist e.g. the vibron model case. Our method is based on the mapping of the algebraic to configuration states, a premise that allows arbitrary operators in configuration to be expanded in terms of generators of the dynamical algebra. The coefficients are determined through a minimisation procedure and given in terms of matrix elements defined in configuration space. We apply the general formalism to the Morse potential, representing anharmonic vibrations in a molecule, as a benchmark case where a dynamical symmetry exits, and to the symmetric double-Morse potential, representing vibrations that can tunnel through a potential barrier, as an example in which a dynamical symmetry is not present. We discuss how the tunnelling effect in the double Morse can be described very simply in the su(2) algebraic representation, taking the ammonia inversion vibrational spectrum as an example.

KEYWORDS:

• Second quantisation
• configuration space
• algebraic realisation
• morse potential
• double morse potential
• double minimum potential
• ammonia

Acknowledgments

This work is partially supported by DEGAPA-UNAM, Mexico, under project IN-109113 and IN-227017, by the Spanish Ministerio de Economía y Competitividad under Projects FIS2014-53448-C2-1-P, FPA2013-47327-C2-1-R, and by Junta de Andalucía under group number FQM-160 and Project P11-FQM-7632.

Disclosure statement

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

Notes

1. Please note that to describe a system in equilibrium with a thermal bath of temperature T, we can use with .

Additional information

Funding

Spanish Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía [grant numbers FQM-160, P11-FQM-7632]; and Secretaría de Estado de Investigación, Desarrollo e Innovacióon [grant numbers FIS2014-53448-C2-1-P, FPA2013-47327-C2-1-R]. Mexican DEGAPA-UNAM [grant numbers IN-109113, IN-227017].