Knowledge Bank

The two nearly coincident vibration frequencies $\omega 1$ and $\omega 3$ and the overlapping bands $\omega 2$ and $\omega 4$ (Herzberg's notation) of $XH3$ molecules are coupled, respectively, by Coriolis interactions. It is possible to show that in the case of $\omega 1$ and $\omega 3$ the interaction is so slight that it may satisfactorily be taken into account in the effective moments of $inertia.1$ Standard methods, using combination relations, may therefore be employed in the analysis of these bands. The Coriolis coupling between $\omega 2$ and $\omega 4$ is not small, however, and the interaction must be calculated by the use of degenerate perturbation $methods.2$ It may be shown, however, that for values of the quantum number $K=J$ and $K=J-I$ (i.e., where the rotation of the molecule classically is almost entirely about the axis of symmetry) the perturbation is small enough so that the corresponding term values in the first excited state of $\omega 4$ may be obtained by an expansion of the secular determinant. One may show by this method that the frequency interval between two lines, and is equal to $\Delta \nu =2[(1-\zeta 4)C+2\zeta 2.42B2/\Delta ]$ where $B=h/8\pi Ixxc$ and $C=h/8\pi 2Izzc,Ixx$ and $Izz$ being the two moments of inertia, $\zeta 4$ is the degenerate Coriolis coefficient coupling the incipient angular momentum of $\omega 3$ and $\omega 4$ to the rotation of the molecule and where $\Delta =\omega 4-\omega 3$. Similarly one may show that the frequency interval between two lines and is equal to $\Delta v\prime =2[(1-\zeta 4)C-4\zeta 2.42B2/\Delta$]. A direct means of determining the magnitude of the perturbation from the experimental data is therefore available if the above lines can be identified. The term values of $\omega 3$ where $K=O$ (i.e., where the molecule is tumbling) may also readily be obtained from the secular determinant. Here also the frequency interval between two lines in the P branch were $K=O$ gives a measure of $\zeta 2.42B2/\Delta$. When applied to the molecules $PH3,3AsH3$, and $SbH3$ and recalling that $\zeta 2+\zeta 4=(B/2C)-1$ the following values for the H-X-H angle, $\alpha$, and the XH distance $r\circ$, are obtained: for $PH3,\alpha =93\circ 50\prime ,r\circ =1.424$ {\AA}; for $Ash3,\alpha =91\circ 35\prime ,r\circ =1.523$ {\AA}; and for $SbH3,\alpha =91\circ 30\prime ,r\circ =1.711$ {\AA}. These values agree within experimental error with the values given by Loomis and $Strandberg4$ from microwave data.