= Solution
An <infrared> fundamental must transform like a component of the <electric dipole moment>; a <Raman spectroscopy> fundamental must transform like a component of the symmetric <polarizability> tensor. For this centrosymmetric <D4h molecular point group>, the resulting classification is:
|| Symmetry
|| Number of modes of this symmetry
|| Infrared
|| Raman
| $A_{1g}$
| 1
| No
| Yes: $x^2+y^2,z^2$
| $B_{1g}$
| 1
| No
| Yes: $x^2-y^2$
| $B_{2g}$
| 1
| No
| Yes: $xy$
| $A_{2u}$
| 1
| \b[Parallel], $z$
| No
| $B_{2u}$
| 1
| No
| No
| $E_u$
| 2, each doubly degenerate
| \b[Perpendicular], $(x,y)$
| No
Here parallel and perpendicular refer to the principal fourfold axis. The table obeys <infrared and Raman mutual exclusion>.
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