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 ...


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 ...


Having a half-filled or filled subshell is stabilising, as you say. But for Cl, the difference in energy between 3s and 3p is greater than the additional stabilisation from filling the 3p subshell. 3d and 4s are closer in energy, so for Cr it is favourable to promote a 4s electron and have a 4s1 configuration.

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