Second-order vibrational perturbation theory energy levels

The ab initio calculated energies were obtained at the SCF level, followed by the evaluation of the second-order electronic correlation contribution with the many-body perturbation theory [SCF+MBPT(2)]. These calculations were performed on HF/3-21G(d) optimized geometries and include the zero-point vibrational energy corrections. 

For any work of even greater accuracy, one would definitely have to contend with the deficiencies of the vibrational second-order perturbation theory used for the analysis, and apply a more accurate variational method for rovibronic energy level calculation of molecules (see Vibrational Energy Level Calculations). At present, the applicability of such methods to a system with more than four atoms is rather limited. 

Further improvements can be made in the vibrational-rotational energy level expression for a diatomic by carrying through the next (third) term in the truncation of Equation 9.29 via perturbation theory. This term is quadratic in the displacement coordinate, s. With second order as a chosen point of truncation of the perturbative corrections to the energy, the energy level expression is... 

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