Unitary group approach for effective molecular potentials: 1D systems

ABSTRACT

An algebraic approach based on the algebra is proposed to describe 1D systems. Our approach starts adding a scalar boson to the boson associated with the 1D harmonic oscillator, whose bilinear products constitute elements of the generators of the algebra. As a next step the identification of coordinates and momenta in the algebraic space is established using the approach recently proposed [Mol. Phys. (2017) doi:10.1080/00268976.2017.1358829]. As a result, the three bases provided by the dynamical group are identified with energy, coordinate and momentum representations. This result provides power tools to obtain the algebraic representation of a 1D Hamiltonian for any potential through the transformation brackets connecting the different bases. As examples of our approach, several potentials are considered: Morse potential as a benchmark case where analytical solutions are known, double Morse potential where exact solutions are known for only restricted subspaces (semi-integral system), and the Harmonic-Gaussian potential where no analytical solutions are known.

GRAPHICAL ABSTRACT

KEYWORDS:

• U(2) unitary algebra
• molecular effective potentials
• Morse potential
• Double Morse potential
• Harmonic-Gaussian potential

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Notice that to describe the system in equilibrium with thermal bath at temperature T, we can use with .

Additional information

Funding

This work is partially supported by Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México DEGAPA-UNAM, Mexico, under project IN-227017. Useful discussions with Prof. Octavio Castaños are appreciated.