Lie Algebraic approach to molecular spectroscopy: Diatomic to polyatomic molecules

Interacting dipole ($p$) bosons along with scalar ($s$) bosons, based on the ideas drawn from the interacting boson model of atomic nuclei, led to the development of the vibron model based on $U(4)$ spectrum generating algebra for diatomic molecules. The $U(4) \supset SO(4) \supset SO(3)$ algebra generates rotation-vibration spectra. Extending this to two coupled $SO(4)$ algebras and three $SO(4)$ algebras describe triatomic and four-atomic molecules respectively. Similarly, appropriately coupled $U(2) \supset SO(2)$ algebras will describe the stretching vibrations, with proper point group symmetries, in polyatomic molecules. In addition, coupled $U(3)$ algebras describe coupled benders. The Lie algebraic approach to molecular spectroscopy is briefly described along with a list giving future directions and presented in three appendices results for: (i) $U(3)$ algebra for bending vibrations and coupled benders; (ii) symmetry mixing Hamiltonians generating regular spectra; (iii) partition functions for diatomic and triatomic molecules.