Bromine pentafluoride, BrF5, has a square pyramidal geometry and belongs to the C4v point group. A character table is shown below: A. Draw a 3D formula for BrF5 and identify and illustrate all symmetry elements associated with the molecule. B. What is the order of the C4v group? C. Determine which irreducible representations each of the following atomic orbitals of the central atom, Br belong to: (a) s (b) px and py (c) pz (d) dz2 (e) dxy (f) dxz and dyz (g) dx2-y2
The Correct Answer and Explanation is:
A. Geometry and Symmetry Elements of BrF₅
Structure:
Bromine pentafluoride (BrF₅) has a square pyramidal geometry, where the Br atom is at the center. Four F atoms occupy the equatorial positions of a square base, and one F is axial above the plane. There is one lone pair on Br, which pushes the structure into a square pyramid (from an ideal octahedral base).
Symmetry Elements in C₄v point group:
- E (Identity)
- C₄ and C₄³ (90° and 270° rotation around the z-axis)
- C₂ (C₄², 180° rotation about z-axis)
- σv(xz) and σv’(yz) (two vertical mirror planes containing the z-axis and bisecting the square)
- σd, σd’ (two diagonal vertical mirror planes)
B. Order of the C₄v Group
The order of a point group is the number of symmetry operations in the group.
- C₄v has:
E, C₄, C₄², C₄³, σv(xz), σv’(yz), σd, σd’ → 8 operations total
✅ Order = 8
C. Symmetry and Irreducible Representations of Br Atomic Orbitals
| Orbital | Symmetry in C₄v | Representation |
|---|---|---|
| (a) s | Fully symmetric | A₁ |
| (b) px, py | Transform as x, y | E |
| (c) pz | Along principal axis (z) | A₁ |
| (d) dz² | Symmetric about z-axis | A₁ |
| (e) dxy | Rotates in x-y plane (diagonals) | B₂ |
| (f) dxz, dyz | Contain z and x or y | E |
| (g) dx²−y² | Lies in x-y plane, changes under 90° rotation | B₁ |
Explanation
Bromine pentafluoride (BrF₅) has a square pyramidal molecular geometry, derived from an octahedral electron-pair geometry where one position is occupied by a lone pair. The central atom, Br, is bonded to five fluorine atoms: four in a square plane (equatorial) and one above (axial). The resulting molecular symmetry belongs to the C₄v point group, characterized by a principal C₄ axis, four vertical mirror planes (σv), and its identity operation.
To analyze the orbital symmetries, we match the transformation properties of atomic orbitals under C₄v symmetry operations with those listed in its character table. The s orbital is spherically symmetric and transforms as the A₁ irreducible representation. The pz orbital, aligned along the principal axis, is symmetric under rotation and mirror planes containing the axis, so it also belongs to A₁.
The px and py orbitals lie in the molecular (xy) plane and transform together as a two-dimensional representation under rotation; hence, they belong to the E representation. The same applies to dxz and dyz, which contain z and either x or y.
The dz² orbital, being symmetric around the z-axis, transforms as A₁, while dx²–y², lying in the xy plane and changing sign under 90° rotation, corresponds to B₁. The dxy orbital, oriented between x and y axes in the plane, transforms under the B₂ representation.
Understanding these symmetries is crucial for constructing molecular orbitals and predicting IR/Raman activity, bonding interactions, and electronic transitions in molecular spectroscopy and quantum chemistry
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