Knowledge Bank

Due to the 1:2 Fermi resonance, the vibrational levels of molecule like $CS2$ or $CO2$ cannot be obtained by a simple Dunham exponsion, but require instead the diagonallsation of on Homiltonian matrix:$$(v_{1}\cdot v_{2}|H|v_{1}\cdot v_{2})=\Sigma w_{i}(v_{i}+d_{i}/2)\Sigma x_{ij}(v_{i}+d_{i}/2)(v_{j}+d_{j}/2)$$$$(v_{1}\cdot v_{2}|H|v_{1}-1\cdot v_{2}-2)=kv_{1}^{1/2}(v_{2}+1)+\ldots$$ The dynamics of the classical counterpart of this Hamiltonian has been studied recently, leading to analytical expressions for the action integrals and classical frequencies, as well as to an exhaustive desceiption of the geometry of the phase space (1,2). These results have been applied to three problems of interest. First, the observed vibrational levels of $CS2$ up to $10000cm-1$ energy have been assigned by replacing the good quantum number, which is destroyed by the fermi resonance, by the remaining semiclassical quantum number obtained from Einstein-Brillouin-Keller(EBK) quantization conditions(3). In addition. EBK quantization enables one to describe the anticrossing of quantum levels as arising classically from the dynamical tunnelling between quantizing torl(4). Also, the first 44 levels arising from combinaations and overtones of the stretching and bending vibrations of $CS2$ were reproduced up to $5000cm-1$ vibrational energy using Berry and Taboors Trace Formula(5).