Effect of Symmetry Operations on Non-Degenerate Vibrations

For a given non-degenerate normal vibration $\nu_r$, there is only one possible way of displacement coordinates of the atoms. As the symmetry operation acts simultaneously on all atom displacements, it can only either simultaneously change signs of all displacement coordinates $u_i$, or remain them unchanged. Thus, a non-degenerate vibration can only be symmetric, or antisymmetric with respect to any symmetry operation which is permitted by the symmetry of the molecule. The former means that $Q_r' = Q_r$ and the latter means that $Q_r' = -Q_r$, where the primed coordinates are the ones after the symmetry operation.

For example, for the formaldehyde molecule, $H_2CO$ the reflection over the plane $xz$ perpendicular to the molecular plane leaves the normal modes $\nu_1$, $\nu_2$, $\nu_3$, and $\nu_6$ the same, whereas inverts directions of all displacement vectors for the modes $\nu_4$ and $\nu_5$. In a similar way it can be seen that all vibrations but $\nu_6$ are symmetric with respect to the molecular plane, while $\nu_6$ is antisymmetric. Finally, the vibrations $\nu_4$, $\nu_5$, and $\nu_6$ are antisymmetric with respect to the rotation by $\pi$ about the two-fold $XY$ axis.

Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.