Knowledge Bank

A two-dimensional kinematic model is proposed for the internal rotation and skeletal bending motion in $AX3YZ$ molecules similar to methanol. The Hamiltonian, which is appropriate for states of zero angular momentum, is expressed in terms of the internal rotation angle $\alpha$ and the supplement of the AYZ angle $\beta$. Included in this vibrational Hamiltonian is the precise dependence of all inertial parameters on the angle $\beta$. The potential surface is that of a $\beta$ -modulated periodic potential in $\alpha$ superimposed upon an anharmonic function in $\beta$ appropriate for ``quasi-linear” molecular frameworks. Thus the skeleton is allowed to flex at the eclipsed configuration, atom Z passes from one minimum to another across saddle points, and the three-fold potential goes to 0 at $\beta =0$ for which a central hump exists. Solution of this two-variable problem is accomplished through numerical techniques for which a type of separation of variables is possible. The differential equation in $\alpha$ is solved at fixed values of $\beta$ and the corresponding eigenvalues, E($\beta$) are combined with the remaining terms in $\beta$ alone to yield an equation which is solved by the method of finite differences. These eigenvalue results reflect a dependence of the torsional levels on the state of the bending motion which becomes severe as the central hump approaches zero. The evidence shows conclusively that the torsional transitions, when fitted to a rigid skeletal model, produce three-fold barriers higher than those required by the flexing model.