A comparative study of the vibrational spectra of HCN in exact vibron model and in mean field approximation

Abstract

The aim of this work is to study the highly excited vibrational states of hydrogen cyanide HCN in the exact vibron model and with mean field approximation in the vibron model. Considering the U(4) ⊗ U(4) spectrum-generating algebra for linear triatomic molecules, the standard Hamiltonian is constructed using the linear and quadratic combination of Casimir operators. For higher order corrections, the quadratic contributions of Casimir operators are used to construct the Hamiltonian. Using this Hamiltonian the higher excited vibrational levels of HCN are calculated in the local mode approximation. The energy levels are observed as a function of vibron number N. The best fit is obtained for N = 184 (N1 = 139, N2 = 45) with root mean square (r.m.s.) deviation 5.598 cm−1. The intermodal coupling within the same polyad is studied and addressed properly by introducing Majorana operator. The r.m.s. deviation is then reduced to 4.755 cm−1. The modification is negligible, which indicates the local nature of HCN. In this work, 35 experimental levels are taken for fit, out of which only two sets of levels are accidentally degenerate. The Fermi resonances of the accidentally degenerate levels are studied using the Fermi operator and r.m.s. deviation becomes 4.835. The coefficient of Majorana and Fermi coupling for different levels are obtained by diagonalzing the Majorana and Fermi matrices for each polyad. The Majorana and Fermi matrices for each polyad are diagonalized with the MATRIX CALCULATOR program. The algebraic parameters are evaluated by a least square fit against the experimental data using MATLAB R2015a. Using this model, a set of energy levels is predicted up to 30 000 cm−1, with very good accuracy. HCN is chosen for this study, because, its vibrational states can be fairly described without any modification due to Fermi resonance. The fundamental vibrational levels of HCN are again calculated, using mean field approximation and compared to those obtained using the vibron model. A good agreement is observed.