BF3 D3h Symmetry and Spectroscopic Activity

Understanding the infrared (IR) and Raman activity of molecular transitions requires analyzing the symmetry of the molecule and the transition using character tables. For a fundamental transition from the ground state (which is typically the totally symmetric representation, $\rm A_{1}^{'}$ in D3h) to an excited state with a specific irreducible representation, the transition is active if the irreducible representation of the excited state transforms as a dipole moment component (for IR activity) or a polarizability component (for Raman activity).

Determining IR and Raman Activity from the Character Table

The D3h character table provided gives information on how different functions transform under the symmetry operations. The last column indicates the transformation properties relevant to spectroscopy:

• IR Activity: A transition is IR active if the excited state's irreducible representation transforms as one of the dipole moment vectors (x, y, z). In the D3h table, (x, y) transforms as $\rm E^{'}$ and z transforms as $\rm A_{2}^{''}$.
• Raman Activity: A transition is Raman active if the excited state's irreducible representation transforms as one of the components of the polarizability tensor ($x^2, y^2, z^2, xy, xz, yz$). In the D3h table, ($x^2 + y^2, z^2$) transforms as $\rm A_{1}^{'}$, ($x^2 - y^2, xy$) transforms as $\rm E^{'}$, and (xz, yz) transforms as $\rm E^{''}$.

Analyzing the Character Table for D3h

Evaluating the Options

We are considering fundamental transitions from the ground state ($\rm A_{1}^{'}$) to various excited states. The activity depends directly on the symmetry of the excited state.

1. Transition to $\rm A_{1}^{'}$ state: The $\rm A_{1}^{'}$ representation transforms as ($x^2 + y^2, z^2$), which are polarizability components. This state is Raman active. It does not transform as x, y, or z, so it is not IR active. Thus, option 1 is incorrect.
2. Transition to $\rm A_{2}^{'}$ state: The $\rm A_{2}^{'}$ representation is listed with $R_z$ (rotation about z-axis). It does not appear in the last column as a dipole moment vector (x, y, z) or a polarizability component ($x^2, y^2, z^2, xy, xz, yz$). Therefore, a transition to the $\rm A_{2}^{'}$ state is neither IR active nor Raman active. This aligns with the stated correct answer.
3. Transition to $\rm A_{2}^{''}$ state: The $\rm A_{2}^{''}$ representation transforms as z, which is a dipole moment vector. This state is IR active. It does not transform as any polarizability component listed. Thus, option 3 is incorrect.
4. Transition to $\rm E^{''}$ state: The $\rm E^{''}$ representation transforms as (xz, yz), which are polarizability components. This state is Raman active. It does not transform as x, y, or z (dipole moment vectors). Thus, option 4 is incorrect.

Based on the analysis of the D3h character table, a fundamental transition to the $\rm A_{2}^{'}$ state does not transform as a dipole moment or polarizability component, making it neither IR nor Raman active.