The Vibrational Levels of Linear Symmetrical Triatomic Molecules

Abstract

The theory proposed by E. Fermi of the energy levels of molecules of the C${\mathrm{O}}_2$ type is discussed. It is shown that this assumes a particularly simple form when expressed in the coordinates used by Dennison for describing such molecules. When there exist no integral or nearly integral relations between the vibrational frequencies the first order energy correction $\lambda W_1$ caused by the anharmonic forces vanishes leaving only the second order term ${\lambda }^2{W}_2$. When, however, two of the fundamental frequencies are commensurable, certain of the energy levels coincide thus becoming degenerate. This degeneracy may be removed by the anharmonic forces in which case there appears a first order energy constant $\lambda W_1$ different from zero. The value of $\lambda W_1$ is computed explicitly for certain of the lower energy states and it appears that the only levels which interact under the influence of the resonance are those having the same value of the azimuthal quantum number $l$. From this it follows that the selection rules are not affected by the existence of the resonance. Finally it is shown how the first order term $\lambda W_1$ goes over into the second order term ${\lambda }^2{W}_2$ as the resonance between the frequencies becomes less and less exact. When these results are applied to the C${\mathrm{O}}_2$ spectrum it is found that the resonance between ${\nu }_1$ and $2{\nu }_2$ is almost perfect and consequently the energy levels can only be ordered with the help of the first order term $\lambda W_1$. For C${\mathrm{S}}_2$ on the other hand the difference between ${\nu }_1$ and $2{\nu }_2$ is so large that the effect of the resonance on the positions of the energy levels may be disregarded.