The Quantum Mechanical Hamiltonian for the Linear Polyatomic Molecule Treated as a Limiting Case of the Non-Linear Polyatomic Molecule

The quantum mechanical Hamiltonian for a general triatomic molecule is derived by the method of Wilson and Howard. It is pointed out that the corresponding wave equation does not lend itself to an exact solution and that the actual Hamiltonian must be replaced by its expansion. When the Hamiltonian is expanded on the basis that the displacement of the particles from equilibrium is small compared to the equilibrium positions of the particles (as is legitimate in the non-linear case) the results of Shaffer and Nielsen are obtained. In the case of linear X${\mathrm{Y}}_2$ molecules the equilibrium value of the moment of inertia $I_{\mathrm{zz}}^{\mathrm{}\left(e\right)\mathrm{}}=0\mathrm{}$ so that the displacements of the particles normal to the axis $z$ must be considered large compared to the values of equilibrium coordinates $x_i^{}{}_{}{}^0$ and $y_i^{}{}_{}{}^0$ of the nuclei. If the molecule is considered as very nearly linear indeed and the Hamiltonian is expanded on the hypothesis that $\mu q^2\gg I_{\mathrm{zz}}^{\mathrm{}\left(e\right)\mathrm{}}$, $\mu$ being the reduced mass and $q$ being the normal coordinate, the results of Dennison for the linear X${\mathrm{Y}}_2$ molecular model are approached asymptotically as $I_{\mathrm{zz}}^{\mathrm{}\left(e\right)\mathrm{}}$ approaches zero.