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GATE 2023 PH – Question 36

Atomic and Molecular Physics · Raman effect and basics of Raman spectroscopy · 2 marks · Multiple choice

It is given that the electronic ground state of a diatomic molecule X2 has even parity and the nuclear spin of X is 0. Which one of the following is the CORRECT statement with regard to the rotational Raman spectrum (J is the rotational quantum number) of this molecule?

  1. Lines of all J values are present
  2. Lines have alternating intensity in the ratio of 3: 1
  3. Lines of only even J values are present
  4. Lines of only odd J values are present

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Correct answer: (C) Lines of only even J values are present

Explanation

For a homonuclear diatomic molecule $X_2$, the total wave function must have the right symmetry under exchange of the two identical nuclei. The nuclei have spin 0, so they are **bosons**, and the nuclear spin part has only one state (symmetric).

The electronic ground state has even parity (a $\Sigma_g^+$ state), which is symmetric under inversion. The overall symmetry of the rotational part is then fixed:
- For $\Sigma_g^+$ states the rotational wave function with quantum number $J$ has parity $(-1)^J$ under nuclear exchange.
- The total wave function must be symmetric (bosons), and the other factors are symmetric, so $(-1)^J=+1$.

Hence **only even $J$ levels** exist. In the rotational Raman spectrum (which follows $\Delta J=\pm2$ from even $J$) only lines starting from even $J$ levels appear. Odd $J$ levels are absent (they would give the alternating 3:1 pattern for spin-½ nuclei, but here they are missing altogether).

Answer **lines of only even $J$ values are present** (C).