What is the symmetry of dxy orbital?

Question

Considering the sign of orbital and assuming the $z$-axis as a principal axis, for me, it looks like that it has two perpendicular $C_2$ axes that penetrate the lobes, so I think it is $C_\mathrm{2v}$. But the book (Inorganic chemistry by Miessler and Tarr) says it is $D_\mathrm{2h}$.

Answer 1

Note that $C_\mathrm{2v}$ is a subgroup of $D_\mathrm{2h}$. This means that you're just overlooking some of the symmetry elements. In addition to the $C_2$ axes penetrating the lobes "end to end", you also have a $C_2$ axis perpendicular to the orbital plane through its center.

Your $C_\mathrm{2v}$ guess is also inconsistent because that point group only has one $C_2$ axis, but you said yourself that you already found two.

Answer 2

Imagine that the $x$ axis and $y$ axis lobes have a different color (phase). Locate the principal axis, that of highest rotational symmetry, if there is more than one just choose one. A $C_2$ or 180 degree rotation will make the orbital indistinguishable from its starting position if this points out of the plane of the orbital (i.e. I choose this to be along $z$ if orbital is in $xy$-plane). Any 'D' point group has a 2-fold, $C_2$ or 180 degree rotation axis, perpendicular to the principal axis so the $\mathrm{d}_{xy}$ belongs to the $D_\mathrm{2h}$ point group. You can check other symmetry elements by looking at the point group table. A $C_\mathrm{2v}$ molecule, i.e. one with symmetry of water molecule, has no 2-fold axis perpendicular to the principal axis. If you do not distinguish $x$ and $y$ with 'color' the orbital will have $D_\mathrm{4h}$ symmetry.