The Rotation and Vibration of Linear Triatomic Molecules

The rotation and vibration of a linear molecule cannot be treated by the method of rotating axes if it contains more than three atoms. The definition of the axes for the triatomic case may be based on the fact that the three atoms always determine a plane. The relation of this definition to that for the nonlinear molecules is discussed, and the appropriate normal coordinates are introduced. The wave equation of the problem is derived and its approximate solution is discussed. The fact that the Eulerian angles do not enter into the zero‐ and first‐order terms of the Hamiltonian makes it possible to calculate some of the second order terms of the expression for the energy values with complete generality. These are the terms that arise from the rotation and from the coupling between the rotation and vibration. They are found to be $${\hslash }^2{\text{[J(J+1)−l}}^2\text{]/2A+C,}$$ where A is the value of the moment of inertia at equilibrium, and $${\mathrm{C=[\hslash }}^2{\mathrm{(V}}_2{\text{+1)/2A]{s}}^2{\mathrm{(V}}_1+\frac{1}{2}{\mathrm{)[(\omega }}_1{\mathrm{/\omega }}_2{\mathrm{)+(\omega }}_2{\mathrm{/\omega }}_1{\mathrm{)]\; +c}}^2{\mathrm{(V}}_3+\frac{1}{2}{\mathrm{)[(\omega }}_3{\mathrm{/\omega }}_2{\mathrm{)+(\omega }}_2{\mathrm{/\omega }}_3{\mathrm{)]\}-\hslash }}^2\text{/2A.}$$ In this expression, the V's are the vibrational quantum numbers and the ω's are the natural frequencies. The index 2 refers to the perpendicular vibration. The quantities c and s are determined by the potential energy function and are related by c²+s²=1. For symmetric molecules, X – Y – X, their values are c=1, s=0. If the natural frequencies are commensurable, the derivation ceases to be valid, but it appears likely that the result is essentially the same in this case.