Harmonic and Anharmonic Vibrational Energy Levels

The partition functions thus far have been assumed to be calculated using the harmonic approximation. However, real vibrations contain higher-order force constants and cross terms between the harmonic normal modes, and they are coupled to rotations. If the cross terms and couplings are neglected, each of the vibrational degrees of freedom is bound by an anharmonic potential given by 

One type of anharmonic motion is a hindered internal rotation, or torsion, which can differ substantially from a harmonic normal mode motion. Unlike many other anharmonic motions, torsions can be readily accounted for even in large systems. It has been shown that a vibrational partition function that includes a torsion can be written as 

The frequency, the effective moment of inertia, and the barrier height W, are related to one another by the expression  

The reduced moment of inertia for internal rotation is given by 

The R scheme does not require that the axis of rotation be chosen a priori, but it relies on the generalized normal mode eigenvector of the mode corresponding to the torsion to determine the axis. The equations for I in this scheme are given elsewhere.  

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