Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section A / 28 / a

For the character of a localized orbital basis, take the trace of the transformation matrix: both permutation of centres and signs of atomic orbitals matter. The two metal centres lie on the principal axis and remain fixed under rotation about $z$. At either metal the $(p_x,p_y)$ basis has the matrixThe full four-dimensional group representation is block diagonal with two such blocks. Consequently $\chi (R)=4\cos \theta$, and for the $60^{\circ }$ rotation of the D6h molecular point group, $\chi (C_6)=4\cos (\pi /3)=2$.