The alternating short and long bonds preserve fourfold, but not eightfold, rotations. Including the molecular plane and inversion symmetry, the planar structure belongs to $D_{4 h}$ , the D4h molecular point group.

Use the printed axes: the short bonds are

1

–

2

3

–

4

5

–

6

and

7

–

8

, and the

x

and

y

axes pass through the midpoints of opposite short bonds. Interchanging short and long bonds is not a molecular symmetry operation of this geometry.

ii

Solution

For the

p_{z}

atomic orbital basis, only orbitals whose centres remain fixed contribute to a character. Each such orbital contributes

+ 1

if its

z

direction is preserved and

- 1

if it is reversed. In the standard D4h molecular point group class order,

χ_{p_{z}} E 8 2 C_{4} 0 C_{2} 0 2 C_{2}^{'} 0 2 C_{2}^{''} 0 i 0 2 S_{4} 0 σ_{h} - 8 2 σ_{v} 0 2 σ_{d} 0

No nonidentity operation other than

σ_{h}

leaves any carbon site fixed. Reflection in the molecular plane fixes all eight sites but reverses every

p_{z}

Γ_{p_{z}} = A_{1 u} + A_{2 u} + B_{1 u} + B_{2 u} + 2 E_{g} .

1 + 1 + 1 + 1 + 2 \cdot 2 = 8

. The two copies of

E_{g}

are separate equivalent irreducible representations which may mix with one another; they do not force all four functions to have a common energy.

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