Energy levels and partition functions of internal rotation: Analytical approximations

A simple numerical method that has considerably simplified the calculations of torsional energy levels is proposed for one-dimensional barriers of arbitrary shape. The internal rotation constant is also permitted to vary as a function of torsional angle. In two limiting cases of high and low barriers, analytical expressions for hindered rotor state energies have been obtained. The partition function of internal rotation is then evaluated by eigenvalue summation. The derived analytical formula for the partition function of free internal rotation gives reasonable values down to the lowest temperatures. Analytical approaches give a better insight into the physical nature of the problem solvable by a rigorous method.