Rotational Spectroscopy Overview

Rotational spectroscopy is used to study the rotational states of molecules. Typically, the rotational constant $B$ falls within the range of 0.1 to 10 cm${}^{-1}$, which corresponds to the microwave region of the electromagnetic spectrum.

Key Formulae and Concepts

• Moment of Inertia: $I=\mu R^2$ where $\mu$ is the reduced mass and $R$ is the bond length.
• Energy Levels: The energy of rotational states is given by $E_J=J(J+1)hc\tilde{B}$ where $J$ is the rotational quantum number and $\tilde{B}$ is the rotational constant in wavenumbers.
• Selection Rule: For microwave spectroscopy, the rotational transitions follow $\mathrm{\Delta }J=\pm 1$, meaning the rotational quantum number changes by one unit.

Distribution of Populated Levels

• The distribution of molecules across different rotational levels follows the Boltzmann distribution. The most populated rotational level, $J_{max}$, is determined by the temperature and rotational constant:$$J_{max}=\sqrt{\frac{k_{B}T}{2hc\tilde{B}}}-\frac{1}{2}$$

Rotational Constant (Wavenumbers)

• $E_J=J(J+1)hc\tilde{B}$
• $\tilde{B}=\frac{h}{8{\pi }^{2}cI}$

Rotational Constant (Frequency)

• The frequency equivalent of the rotational constant $B$ is $B=c\tilde{B}$, where $\tilde{B}$ is in cm${}^{-1}$ and $c$ is the speed of light in cm/s.

Microwave Absorption Spectroscopy

Microwave spectroscopy measures pure rotational transitions, typically for polar molecules with permanent dipole moments. The selection rule for microwave absorption is $\mathrm{\Delta }J=\pm 1$.

• Key Aspects:

  • Narrow peak widths due to long lifetimes of excited states, as described by the Heisenberg uncertainty principle.
  • Doppler broadening, a form of thermal doppler broadening, is observed because molecules move at different velocities, affecting the frequency recorded.

• Spectral Spacing:

  • The spacing between successive peaks is $2\tilde{B}$ or $2B$ in frequency units.
  • 

Example Calculation:

Given a rotational constant $\tilde{B}$ in cm${}^{-1}$:

1. Calculate the energy levels using $E_J=J(J+1)hc\tilde{B}$.
2. Find the most populated level, $J_{max}$, using the formula above.

Rotational Raman Spectroscopy

Raman spectroscopy probes rotational transitions, but with different selection rules: $\mathrm{\Delta }J=\pm 2$.

• Stokes lines (energy gain): $\tilde{\nu }(J+2\leftarrow J)={\tilde{\nu }}_i-2\tilde{B}(2J+3)$
• Anti-Stokes lines (energy loss): $\tilde{\nu }(J-2\leftarrow J)={\tilde{\nu }}_i+2\tilde{B}(2J-1)$
• Lowest state: $J=2$, with lines appearing at $6\tilde{B},10\tilde{B},14\tilde{B},\dots$

Selection Rules in Rotational Spectroscopy

• Microwave Spectroscopy:
  • Requires a permanent dipole moment.
  • Selection rule: $\mathrm{\Delta }J=\pm 1$.
• Rotational Raman Spectroscopy:
  • Can detect transitions in non-polar molecules.
  • Selection rule: $\mathrm{\Delta }J=\pm 2$.

Boltzmann Distribution

The relative population of molecules in a given rotational state $J$ follows the Boltzmann distribution:

• The most populated rotational state is determined by both $J$ and temperature, with $J_{max}$ calculated by the formula provided earlier.

Key Points in Spectroscopy

• Degeneracy: Each rotational level $J$ has a degeneracy factor of $2J+1$.
• Line Intensity: The intensity of each line is proportional to the square of the transition dipole moment. For a transition from $J\to J+1$, the intensity is:

where ${\mu }_0$ is the magnitude of the permanent dipole moment.

Thermal Doppler Broadening

• Arises due to the motion of molecules at different velocities.
• The Doppler effect shifts the frequency of the rotational transitions, and the degree of broadening is related to temperature.

Further Exploration

• You can watch this YouTube video for more in-depth understanding: https://youtu.be/qiTbuKE55Dk
• Wesley's video on rotational Raman spectroscopy: Wesley Video

References

• Peter Atkins Physical Chemistry QMC - Chapter 42 for detailed reading.