@@ -606,7 +606,7 @@ Consider carbon dioxide (C0$_2$), which is a linear, triatomic molecule shown be
Write down Newton's equations of motion for the atoms, you may assume that the spring constant is the same for both bonds.
3. Consider the *symmetric* vibrational mode, for which the displacements of the oxygen atoms are equal in magnitude and have an opposite direction. Write down the eigenvector (no calculations) of this mode and find its eigenfrequency. Discuss the eigenfrequency in relation to that for a single mass on a spring connected to a wall.
4. Now consider the *antisymmetric* mode, in which both oxygen atoms move in phase and have the same displacement. Considering that the center of mass of the molecule should remain at rest, write down the eigenvector describing this motion (no calculations).
5. Compute the eigenfrequency of the antisymmetric mode. Discuss if the eigenfrequency behaves as you expect for $M \rightarrow \infty$.
5. Compute the eigenfrequency of the antisymmetric mode. Discuss if the eigenfrequency behaves as you expect when you would let the mass of the center atom $M$ go to infinity: $M \rightarrow \infty$.
6. The vibrational modes of molecules such as CO$_2$ can absorb light when the frequency of the light matches the eigenfrequency of the mode AND the atomic motion mode leads to a net, oscillating electric dipole. Do you expect the non-vibrating molecule to have an electric dipole moment? Do you expect any inhomogeneous electron distribution on the molecule? Argue if you would expect the symmetric or the antisymmetric mode to absorb light.
