Molecular Rotational energies and frequencies of diatomic molecule

Question

I am studying Raman spectroscopy (I am still in the introduction though), and I wish to find the rotational and vibrational energies and frequencies for a diatomic molecule, say $0_2$. From the handbook of chemistry and physics I found the following data for $0_2$: $R_0=1.2074 (Angstrom)$, $k=11.77(Nm)$ Here $R_0$ is the distance between atoms and $k$ is the "spring" constant that is used when we treat the diatomic molecule as an oscillator.

I know that I can find the vibrational frequency and energy from the quantum harmonic oscillator formulation:

$$E_{\nu}=h\nu(n+\frac{1}{2})$$ here, $n$ is the vibrational quantum number and $\nu$ is the vibrational frequency given by:

$$\nu=\frac{1}{2\pi}\sqrt{\frac{k}{\mu}}$$ so that part is relatively simple. But I don't know how to find the rotational energy and its frequencies, since the book that I am supposed to find the information on (Introduction to Raman Spectroscopy) doesn't mention it. I have thought about treating it classically, but I just don't know how to find the rotational frequency with which I could find its energy. I also thought about calculating its kinetic energy and considering that to be the rotational energy, but I don't know if it also moving, so I don't know if that works.

What could I do?

Answer 1

You can treat this classically if you consider your diatomic molecule as a rigid rotor. The classical rotational energy is expressed in terms of the angular momentum $J$

$$ E_{rot} = \frac{1}{2} \frac{J^2}{I}, $$

where $I$ is your moment of inertia. Now you can ''quantize'' your angular momentum by writing $J^2 = J(J+1) \hbar^2$, and your rotational energy is

$$ E_{rot}(J) = \frac{1}{2} \frac{J(J+1) \hbar^2}{MR_e^2}, $$

where the moment of inertia was expressed in terms of the equilibirum distance of the two nuclei $R_e$ and $M = \frac{m_A m_B}{m_A+m_B}$ for the two masses $m_A$ and $m_B$ of the two atoms in the diatomic molecule.

For corrections to the expressionon with the equilibrium distance you can include centrifugal widening of the internuclear spacing.