2a. Platinum tetrachloride, [PtCl4], belongs to the point group Dah. Determine how the s, p, and d orbitals of the Pt ion would behave when operated on by all of the D4h symmetry elements. Use the attached D4h character table to assign a symmetry label to each of these orbitals. 2 Cl Pt Cl Orbital 2C2 (z)(x. v) (dihedral) Label Platinum s Platinum px Platinum p Platinum pz Platinum dz2 Platinum dx2-y2 Platinum d Platinum dxz Platinum dyz LGO #1 LGO #2 2 0 0 0 -210|2|0|0 2 0 0 0 2 0 200
The Correct Answer and Explanation is :
To determine the symmetry labels (irreducible representations) for the orbitals of platinum in PtCl₄, which belongs to the D₄h point group, we use the D₄h character table. The goal is to analyze how each orbital transforms under the symmetry operations of D₄h.
Step 1: Understanding D₄h Point Group
- The D₄h group includes symmetry operations such as the identity (E), rotations (C₄, C₂), reflections (σh, σv, σd), and inversion (i).
- The symmetry-adapted basis functions (such as atomic orbitals) transform according to irreducible representations listed in the character table.
Step 2: Assigning Symmetry Labels
Each orbital of the platinum atom behaves according to the irreducible representations in D₄h.
| Orbital | Transformation Behavior in D₄h | Irreducible Representation |
|---|---|---|
| Platinum s | Symmetric under all operations | A₁g |
| Platinum pₓ | Changes sign under σv(xz), remains under C₄ | B₁u |
| Platinum pᵧ | Changes sign under σv(yz), similar to pₓ | B₂u |
| Platinum p𝓏 | Symmetric under C₄, antisymmetric under i | A₂u |
| Platinum d𝓏² | Symmetric under C₄, C₂, σh | A₁g |
| Platinum dₓ²₋y² | Changes sign under C₄, symmetric under σh | B₁g |
| Platinum dₓʸ | Changes sign under C₂ and σh | B₂g |
| Platinum dₓ𝓏 | Changes under σh but remains in-plane | E₁u |
| Platinum dᵧ𝓏 | Similar to dₓ𝓏 | E₁u |
Step 3: Explanation of the Symmetry Labels
- s-Orbital (A₁g): The spherical s-orbital is totally symmetric and remains unchanged under all symmetry operations.
- p-Orbitals:
- p𝓏 (A₂u): Oriented along the principal C₄ axis, it remains unchanged under rotation but inverts under inversion.
- pₓ, pᵧ (B₁u, B₂u): These orbitals are in-plane and transform differently under reflection through σv(xz) and σv(yz).
- d-Orbitals:
- d𝓏² (A₁g): Similar to s, it is symmetric under all operations.
- dₓ²₋y² (B₁g): Changes sign under a C₄ rotation.
- dₓʸ (B₂g): Behaves similarly but with a different transformation under mirror reflections.
- dₓ𝓏, dᵧ𝓏 (E₁u): These two orbitals form a degenerate pair.
Final Answer
- s: A₁g
- pₓ: B₁u
- pᵧ: B₂u
- p𝓏: A₂u
- d𝓏²: A₁g
- dₓ²₋y²: B₁g
- dₓʸ: B₂g
- dₓ𝓏, dᵧ𝓏: E₁u
These assignments help in understanding bonding and electronic properties in square-planar complexes like PtCl₄.