A study of the local to normal mode transition in pyramidal molecules and their vibrational description in terms of an algebraic model

First a new perspective to study the local to normal mode transition in the series of pyramidal molecules is presented. Then a full study of the vibrational excitations of the series of pyramidal molecules XH${}_3$ with X = P, As and Sb is given in the framework of a polyad-conserving Hamiltonian of a set of interacting Morse oscillators. The model is based on an algebraic representation of the Hamiltonian in terms of $su\left(2\right)$-operators. A simple Hamiltonian including both Fermi and Darling–Dennison interactions is considered. For the PH${}_3$ molecule a fit involving 64 experimental energies up to polyad 8 (energies up to 9054 cm⁻¹) provided an $\mathrm{rms}=1.74{\mathrm{cm}}^{-1}$ using a Hamiltonian with $14+3$(frozen) parameters. For the AsH${}_3$, with 35 experimental energies the deviation obtained was $\mathrm{rms}=2.50{\mathrm{cm}}^{-1}$ with $10+4$ parameters. Lastly, for the SbH${}_3$ molecule, considering 23 experimental energy levels an $\mathrm{rms}=0.86{\mathrm{cm}}^{-1}$ was reached with $9+6$ parameters. In all cases the polyad scheme $P=2({\nu }_1+{\nu }_3)+{\nu }_2+{\nu }_4$ was considered.