Simple analytical relation between vibrational frequencies of linear XY_2-type molecules

We have considered vibrational normal modes of linear \(XY_2\)-type molecules having the symmetry \(D_{\infty h}\) (for example, the molecule \(\hbox {CO}_2\)). We derived a formula of the type \(f(\omega _1,\omega _2,\omega _3,m,M,q,Q)=0\), where \(\omega _{1,2,3}\) are all three vibrational frequencies of the molecule, M, m and Q, q are the masses and the charges of nuclei Y and X, respectively. The derivation relies on the Taylor expansion of the total electronic energy of the molecule around the united atom configuration, which is adapted to the following symmetries: the indistinguishability of identical nuclei, homogeneity of space, isotropy of space and space inversion. In spite of the simplicity and crudeness of such an expansion, the formula is in reasonable agreement with experiments and might be useful in the interpretation of vibrational spectra of such molecules.