What is the symmetry of dxy orbital?

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 $D\_\mathrm{2h}$.




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.


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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.


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