Force Constants of Triatomic Molecules

In the general expression of the potential energy of the triatomic molecule XYZ there are six force constants c₁, c₁′ (valence bond), c₂ (angle deformation), c₃, c₃′ (angle bond interaction), and c₄ (bond‐bond interaction), of which only three can be calculated from the three experimentally determined fundamental frequencies. For the isosceles triatomic molecule YX₂ the number of force constants reduces to four (c₁, c₂, c₃, and c₄). From a study of all the real values of these four force constants, it is seen that they are related to one another by ellipses and straight lines. The graphs of these ellipses indicate the possibility of choosing a certain singular point (near one end of the major axis) which gives the following relation between the force constants: $${\mathrm{dc}}_4{\mathrm{/dc}}_2\text{=0\hspace{0.17em}\hspace{0.17em}}\mathrm{or}{\mathrm{\hspace{0.17em}\hspace{0.17em}dc}}_4{\mathrm{/dc}}_3\text{=0.}$$ This condition and the equations usually obtained from the theory of small vibrations permit the calculation of a singular set of four force constants of the isosceles triatomic molecule. When these force constants are applied to isotopic molecules, satisfactory checks between the calculated and experimentally determined frequencies are obtained. The ellipses showing the relation between the force constants indicate clearly why the simple central force field (c₁, c₂, c₃) or the simple valence force field (c₁, c₂, c₃), containing one interaction constant or crossterm, must yield imaginary values for c₂ and c₃ in certain cases.