Theoretical Interpretation of Force Constants

The quantum‐mechanical definition of the force constant k of a diatomic molecule is given. The three main approaches to the calculation of force constants—the perturbation approach, the virial theorem approach, and the strict Hellmann—Feynman approach—are discussed. Several new expressions for k are derived. In particular the Hellmann—Feynman theorem leads to a general expression for k in terms of the charge density ρ and of the change δρ/δR with internuclear distance R. It is also possible to relate the force constant to the quadrupole coupling constant (eqQ) of either nucleus by the formula $${\mathrm{k=Z}}_{\mathrm{A}}{\mathrm{[q}}_{\mathrm{A}}+\frac{4}{3}\mathrm{\pi \rho (}\mathrm{A}\mathrm{)-}{\displaystyle \int }{\mathrm{\hspace{0.17em}(\partial \rho /\partial X}}_{\mathrm{A}}\mathrm{)\hspace{0.17em}(}\cos {\theta }_{\mathrm{A}}{\mathrm{/r}}_{\mathrm{A}}^2\mathrm{)\; d\tau ].}$$ The experimentally observed relation k≅q_H for diatomic hydrides is then interpreted in terms of the properties of the charge density of these molecules.