Question

How are $d_{xy}$ and $d_{x^{2}-y^{2}}$ orbitals related?

Answer 1

The $d_{xy}$ and  $d_{x^{2} -y^{2} }$ orbitals have same shape and they both lie in the same xy plane. They both differ only in their orientation i.e. ${x^{2} -y^{2} }$  lies along the axes x and y whereas xy lies between these axes and the xy plane rotate 45° relative to ${x^{2} -y^{2} }$.

The two orbitals have the same shape, and they lie in the same plane (the xy plane). They differ in their exact orientation: ${x^{2} -y^{2} }$   lies along the x and y axes, whereas xy lies between these axes (so rotated 45° relative to x²-y²).

Answer 2

The d_{xy}dxy​ and  d_{x^{2} -y^{2} }dx2−y2​ orbitals have same shape and they both lie in the same xy plane. They both differ only in their orientation i.e. {x^{2} -y^{2} }x2−y2  lies along the axes x and y whereas xy lies between these axes and the xy plane rotate 45°relative to {x^{2} -y^{2} }x2−y2 .

The two orbitals have the same shape,and they lie in the same plane (the xy plane). They differ in their exact orientation: {x^{2} -y^{2} }x2−y2   lies along the x and y axes, whereas xy lies between these axes (so rotated 45° relative to x²-y²).