Boron trifluoride, BF3, belongs to the point group D3h.
i) Draw the symmetry elements belonging to BF3.
ii) Determine the symmetry of the vibrations for BF3.
The Correct Answer and Explanation is :
Solution for BF₃ Symmetry Analysis
(i) Symmetry Elements of BF₃
Boron trifluoride (BF₃) is a trigonal planar molecule, and it belongs to the D₃h point group. The symmetry elements present in this point group are:
- Principal rotation axis (C₃) – A threefold axis passing through the boron atom, allowing 120° rotations.
- Three perpendicular C₂ axes – These pass through the boron atom and the midpoints of opposite B–F bonds.
- Horizontal mirror plane (σₕ) – Lies in the plane of the molecule.
- Three vertical mirror planes (σᵥ) – Each plane contains one B–F bond and the C₃ axis.
- Improper rotation axis (S₃) – A threefold improper rotation (rotation by 120° followed by reflection through σₕ).
- Identity element (E) – The molecule remains unchanged under identity operation.
These symmetry elements define the D₃h point group.
(ii) Determining the Symmetry of Vibrations for BF₃
To determine the vibrational symmetry of BF₃, we follow these steps:
- Find the number of vibrational modes using the formula:
[
\Gamma_{\text{vib}} = 3N – 6
]
where ( N ) is the number of atoms.
- For BF₃, ( N = 4 ) (1 B + 3 F), so:
[
3(4) – 6 = 6 \text{ vibrational modes}
]
- Character Table for D₃h Point Group:
The D₃h group has the following irreducible representations: D₃h E 2C₃ 3C₂ σₕ 2S₃ 3σᵥ A₁’ 1 1 1 1 1 1 A₂’ 1 1 -1 1 1 -1 E’ 2 -1 0 2 -1 0 A₁” 1 1 1 -1 -1 -1 A₂” 1 1 -1 -1 -1 1 E” 2 -1 0 -2 1 0 - BF₃ Vibrational Modes:
The 6 normal modes transform according to: [
\Gamma_{\text{vib}} = 2A’₁ + E’ + 2E’ + A”₂
]
- 2 A₁’: Symmetric stretching and symmetric bending (Raman active).
- E’ (double degenerate): Asymmetric stretching (IR and Raman active).
- E’ (double degenerate): Asymmetric bending (IR and Raman active).
- A₂”: Out-of-plane bending (IR active).
Thus, the vibrational modes of BF₃ are:
- IR Active: E’ (asymmetric stretch), E’ (asymmetric bend), A₂” (out-of-plane bend).
- Raman Active: A₁’ (symmetric stretch), E’ (asymmetric stretch and bend).
These symmetry assignments help in understanding BF₃’s spectroscopic behavior in IR and Raman spectroscopy.