Vibrational energy levels Dunham expansion

Figure 4. Vibrogram of C2HD calculated with = 2000 cm-1 from all the vibrational energy levels predicted by the Dunham expansion corresponding to the Hamiltonian (3.12) obtained by Herman and co-workers by fitting to high-resolution spectra [112], The periods of the bulk periodic orbits of Table I obtained numerically for the classical Hamiltonian (3.12) are superimposed as circles. On the right-hand side, the main labels (n4,ns) of the periodic orbits are given.

At higher levels of excitation anharmonicity has to be included to obtain accurate energy levels. Perturbation theory has been used to derive the following expression, often called a Dunham expansion (Hirst, 1985), for polyatomic anharmonic vibrational energy levels, which is similar to the Morse energy level expression Eq. (2.59), for a diatomic molecule ... 

Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

The next step beyond pictorial patterns is a simple algebraic representation, which is based on the observed regularity of the pattern rather than any physical model. One expects that the energy levels Evj sampled in a band spectrum should be well represented by a simple, rapidly convergent, polynomial function of the rotational and vibrational quantum numbers, J and v. The Dunham expansion (1932),... 

As the molecule is vibrationally excited, couplings among these harmonic modes become important. These couplings are taken into account by the Dunham expansion. The Dunham expansion for the vibration-rotation energy of a linear polyatomic molecule above the zero-point level is given by... 

---