Variation of angle in GeX2 molecules where X is an halogen

Question

I'm wondering what is the exact reasoning to explain why in $\ce{GeX_2}$ the angle $\widehat{XGeX}$ is smaller when $\ce{X}=\ce{F}$ than when $\ce{X}=\ce{I}$.

I ask this question because I had an exams some days ago and I failed in this question and for me my reasoning is not false.

What I said is that

From $\ce{F}$ to $\ce{I}$ the difference of electronegativity between the germanium and the halogen is decreasing. Then the electrons in the $\ce{Ge-X}$ bond are more near $\ce{X}$ when $\ce{X}=\ce{F}$ than when $\ce{X}=\ce{I}$. Then the two bonds repulse themselves more for $\ce{GeI_2}$ than for $\ce{GeF_2}$ and so the angle increase from $\ce{F}$ to $\ce{I}$.

Why my reasoning is absolutely false, and what could be the perfect one?

Answer 1

As for all atoms of period 3 and higher, we can consider germanium to be essentially unhybridised and the bonds being formed entirely by $\unicode[Times]{x3c3}_\mathrm{p-p}$ overlaps. Therefore, we expect a perfect $90^\circ$ angle for all three compounds.

However, while fluorine is small, the size of the halogens increases from fluorine to iodine. The larger the halogen the more space it requires. Since the central atom remains the same size (and, let’s approximate, is roughly the size of bromine) it cannot offer more space for a strict $90^\circ$ angle without lengthening (= weakening) the bonds which would be unfavourable. It is less unfavourable to widen the angle to generate more space. This requires the s-orbital to hybridise slightly (less favourable) but keeps the desired shorter $\ce{Ge-X}$ bond length (more favourable than anything else in this consideration). Not the bonding electrons repulse each other, but the steric stress of the halogen’s lone pairs causes them to move apart. The same effect can be observed in $\ce{NH3}$ versus $\ce{PH3}$ or $\ce{PH3}$ versus $\ce{PF3}$.

Note the following: If you have two bonds from a central atom which are each described by a bonding and an antibonding orbital, then these two sets of orbitals will always be orthogonal to each other (the integral of overlap is $0$). Thus they will be indifferent to each other, not repulsive.