The Water Vapor Molecule

The problem of the vibration rotation spectrum of water vapor is treated by means of the theory of semi-rigid polyatomic molecules developed by Wilson and Howard. The potential energy is expanded as a power series in the normal coordinates and involves three zeroth-order constants, six first-order and six second-order constants. The positions of the band centers are calculated and found to depend upon ten quantities, $X_i$, $X_{\mathrm{ik}}$, and $\gamma$ which are functions of the potential constants. A new feature of the treatment is the recognition of a resonance interaction between certain of the overtone bands which arises from the near equality of ${\nu }_1$ and ${\nu }_3$. Eighteen band centers are known experimentally. These serve to determine the $X_i$, $X_{\mathrm{ik}}$, $\gamma$ and furnish eight self-consistency checks which are very adequately satisfied. There exists no discrepancy between the Raman and infrared spectra as reported earlier. In order to obtain the geometric displacements corresponding to each normal co-ordinate it is necessary to examine the spectrum of ${\mathrm{D}}_2$O. This not only furnishes the required information but also allows two independent checks upon the theory both of which turn out to be in nearly perfect accord. The interaction between vibration and rotation is considered and the effective moments of inertia are calculated. These are functions of the normal frequencies and of the first-order potential constants. It is shown that $\Delta =I_C-I_A-I_B$ depends only upon the normal frequencies and hence may be computed at once. A comparison between the observed and predicted $\Delta$ yields a very satisfactory agreement. The analysis of the rotational structure made by Mecke is supplemented by taking account of the rotational stretching. The resulting molecular constants fix the valence angle to be 104°31′ and the O-H distance to be 0.9580A. From the effective moments of inertia the first-order potential constants may be evaluated and these, together with $X_{\mathrm{ik}}$ determine the second-order potential constants. It is now possible to compute the interaction constant $\gamma$ and a comparison with the observed $\gamma$ again results most satisfactorily.