Knowledge Bank

In the present paper, the somewhat successful $treatment1$ developed to calculate the rotational energy of the water molecule in the (000) and (010) vibrational states is modified in order to permit the calculation of the rotational energy in the first triad, i.e., in the (100), (020) and (001) vibrational states, which are coupled by strong Fermi-type and Coriolis-type interactions. The approach involves writing first the rovibrational zeroth-order Hamiltonian of the molecule with the help of Radau's coordinates $r1,r2$, and $\theta$, since they lead to a fairly simple expression for this $operator.2-3$ In the next step, the effective Hamiltonian for the interacting (100), (020) and (001) vibrational states is derived. The approximation consists in fixing the two coordinates $r1$ and $r2$, which resemble the two stretching type coordinates, to their average values in these vibrational states in order to obtain, for each of these state, a four-dimensional Hamiltonian corresponding to the large amplitude bending $v2$ mode and to the overall rotation of the $molecule1$. This second step also allows us to obtain expressions for the Coriolis-type coupling operators. In the last step, using vibrational wavefunctions involving Jacobi $polynomials1$, the matrix of the effective Hamiltonian is set-up and diagonalized, and the rovibrational energy is obtained. In the paper, these three steps will be discussed and we hope to have preliminary results concerning analyses of the rotational energy for levels belonging to the first $triad.4$