Ideal gaseous phosphorus pentachloride is a trigonal bipyramid with D3h molecular point group symmetry. There are five chlorine sites and three 3p orbitals per site, hence a fifteen-dimensional basis. A moved atom contributes zero to the character; each fixed chlorine contributes the trace of the three-dimensional polar-vector matrix acting on its local p orbitals.

class fixed Cl sites tr D_{x yz} χ_{15 p} E 5315 2 C_{3} 200 3 C_{2}^{'} 1 - 1 - 1 σ_{h} 313 2 S_{3} 000 3 σ_{v} 313

For example, a

C_{2}^{'}

axis through one equatorial chlorine fixes that site but rotates its p-vector basis with trace

1 - 1 - 1 = - 1

. Counting that atom as

+ 1

would be incorrect.

Apply molecular character reduction with class sizes

(1, 2, 3, 1, 2, 3)

and group order twelve:

n_{λ} = \frac{1}{12} \sum_{C} ∣ C ∣ χ_{15 p} (C) χ_{λ} (C)^{*} .

(2)

For

A_{1}^{'}

, this gives

(15 - 3 + 3 + 9) /12 = 2

; for

E^{'}

(30 + 6) /12 = 3

. The full result is

Γ_{15 p} = 2 A_{1}^{'} \oplus A_{2}^{'} \oplus 3 E^{'} \oplus 2 A_{2}^{''} \oplus 2 E^{''} .

(3)

The dimension check is

2 + 1 + 6 + 2 + 4 = 15

. As an additional transparent separation, the two axial chlorines supply

A_{1}^{'} + E^{'} + A_{2}^{''} + E^{''}

, while the three equatorial ones supply

A_{1}^{'} + A_{2}^{'} + 2 E^{'} + A_{2}^{''} + E^{''}

. These are all chlorine 3p orbitals, including both radial and tangential functions; a five-orbital ligand-sigma basis would answer a different question.

Solution

 0  0

The highest expected orbital degeneracy is five. Ideal buckminsterfullerene has Ih molecular point group symmetry. Its irreducible orbital representations include five-dimensional

H_{g}

and

H_{u}

representations; no single-valued orbital irrep has higher dimension.

This counts distinct spatial molecular orbitals. It does not multiply by electron spin, and it excludes accidental coincidences between unrelated irreps. In a real environment, symmetry lowering can split these degeneracies.^[ref]