This question concerns the following exercise from an old exam:
The vibrational motion of a linear diatomic molecule can be approximated as simple harmonic motion.
A CO molecule has a bond with force constant $k = 1900 \, Nm^{-1}$. What frequency of radiation would excite transitions between the different vibrational enery levels? (C and O have masses of $12 \, m_u$ and $16\, m_u$ respecitvely, where $m_u$ is the atomic mass unit.)
Explain why electromagnetic radiation can not excite vibrational transitions in an $O_2$ or $N_2$ molecule. How is this important for the heating of the earth's atmosphere by the sun (the greenhouse effect)?
For the first part I know the energy levels of a harmonic oscillator (in center of mass reference frame)
$$E_n = \hbar\omega(n+1/2),\quad \text{where }\omega = \sqrt{\frac{k}{\mu}}, \quad \mu = \frac{m_Om_C}{m_O + m_C}$$
and the energy of a photon of frequency $\nu$ is given by $E_{ph} = h\nu$. The photon energy must be the energy difference between two states: $E_{ph} = E_n - E_m = \hbar \omega (n-m)$. This gives me the frequency
$$\nu = \frac{(n-m)}{2\pi}\sqrt{\frac{k}{\mu}}$$
Or minimum frequency $\nu_0 = \frac1{2\pi}\sqrt{\frac{k}{\mu}}$, right?
However, now to my actual question: I don't know why we cannot do the exact same thing for the other two molecules? What is different for $O_2$ and $N_2$?
I'd appreciate some help, thanks! =)
