11.11.2 Vibrational Perturbation Theory

Vibrational perturbation theory has been historically popular for calculating molecular spectroscopy. Nevertheless, it is notorious for the inability of dealing with resonance cases. In addition, the non-standard formulas for various symmetries of molecules forces the users to modify inputs on a case-by-case basis,^{31, 642, 178} which narrows the accessibility of this method. VPT applies perturbation treatments on the same Hamiltonian as in Eq. (4.1), but divides it into an unperturbed part, $\widehat{U}$,

$$\widehat{U}=\sum _i^m\left(-\frac{1}{2}\frac{{\partial }^2}{\partial q_i^2}+\frac{\omega _i{}^2}{2}q_i{}^2\right)$$

(11.54)

and a perturbed part, $\widehat{V}$:

$$\widehat{V}=\frac{1}{6}\sum _{ijk=1}^m{\eta }_{ijk}{q}_i{q}_j{q}_k+\frac{1}{24}\sum _{ijkl=1}^m{\eta }_{ijkl}{q}_i{q}_j{q}_k{q}_l$$

(11.55)

One can then apply second-order perturbation theory to get the $i$th excited state energy:

$$E^{(i)}={\widehat{U}}^{(i)}+⟨{\mathrm{\Psi }}^{(i)}|\widehat{V}|{\mathrm{\Psi }}^{(i)}⟩+\sum _{j\ne i}\frac{{|⟨{\mathrm{\Psi }}^{(i)}|\widehat{V}|{\mathrm{\Psi }}^{(j)}⟩|}^2}{{\widehat{U}}^{(i)}-{\widehat{U}}^{(j)}}$$

(11.56)

The denominator in Eq. (11.56) can be zero either because of symmetry or accidental degeneracy. Various solutions, which depend on the type of degeneracy that occurs, have been developed which ignore the zero-denominator elements from the Hamiltonian.^{672, 31, 642, 178} An alternative solution has been proposed by Barone,⁵⁸ which can be applied to all molecules by changing the masses of one or more nuclei in degenerate cases. The disadvantage of this method is that it will break the degeneracy which results in fundamental frequencies no longer retaining their correct symmetry. He proposed

$$E_i^{\mathrm{VPT2}}=\sum _j{\omega }_j(n_j+1/2)+\sum _{i\le j}{x}_{ij}(n_i+1/2)(n_j+1/2)$$

(11.57)

where, if rotational coupling is ignored, the anharmonic constants $x_{ij}$ are given by

$$x_{ij}=\frac{1}{4{\omega }_i{\omega }_j}\left({\eta }_{iijj}-\sum _k^m\frac{{\eta }_{iik}{\eta }_{jjk}}{{\omega }_k^2}+\sum _k^m\frac{2({\omega }_i^2+{\omega }_j^2-{\omega }_k^2){\eta }_{ijk}^2}{\left[{({\omega }_i+{\omega }_j)}^2-{\omega }_k^2\right]\left[{({\omega }_i-{\omega }_j)}^2-{\omega }_k^2\right]}\right)$$

(11.58)