Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 1 / a

Infrared spectroscopy measures absorption as the incident photon frequency is swept through molecular normal mode frequencies. In the electric-dipole approximation the interaction is $-\mathbf{p}\cdot \mathbf{E}$. Expanding the molecular electric dipole moment in a normal coordinate $Q$ givesThe fundamental one-quantum vibrational transition dipole moment is proportional to the derivative multiplying $Q$. Thusis the criterion for vibrational infrared activity; the derivative must also have a component along the incident polarization of an electromagnetic wave. A permanent molecular electric dipole moment is not necessary. Absorption occurs near $\hslash \omega =E_{v+1}-E_v$, with broadened lines reflecting finite lifetimes and the molecular environment.

Use the indicated out-of-plane displacements and the equivalence of the four oxygen sites and of the two carbon sites in the isolated planar oxalate ion. In $V_1$, the two oxygen atoms on one side move oppositely to those on the other side, and the two carbon atoms also move oppositely. Their dipole changes cancel. The mode is even under molecular inversion symmetry, which independently forbids an electric-dipole transition.

In $V_2$, all four oxygen atoms move together out of the plane while both carbon atoms move in the opposite direction. The positive and negative charge distributions acquire a relative out-of-plane displacement, producing a nonzero derivative of the molecular electric dipole moment.

In $V_3$, two oxygen atoms move out and two move in, in a checkerboard arrangement, while the carbon atoms are stationary. The equivalent oxygen contributions cancel. This mode illustrates that being odd under inversion symmetry is necessary but not sufficient for vibrational infrared activity: its full symmetry does not transform like any component of an electric dipole moment.These conclusions concern the isolated-molecule displacements drawn in the source; a different crystal environment can alter selection rules through additional symmetry breaking.

There are six atoms in the nonlinear oxalate ion, hence $3\cdot 6=18$ positional degrees of freedom. Subtract three rigid translations and three rigid rotations to obtain $3N-6=12$ vibrational normal modes. Planarity does not remove out-of-plane motions. Since three modes have been displayed,