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HI
How can I explain that F- has a bigger field than I- (Δ bigger) using the ligand field theory ( so using the MO diagram and not the electrostatic theory)

Here is a MO diagram:

https://en.wikipedia.org/wiki/Ligand_field_theory#/media/File:LFTi(III).png



2 case:
M(I)₆  ^n-
MF₆  ^n-


Now looking at  the MO diagram when I pass from I^- to F^- the electronegativity increases...so the SALC for the 6 "s" F orbitals (the ones on the right in the diagram) are lower in energy than the Iodine one


WIth fluorine so there is a bigger difference between the 6 F salc s  and the metal transition orbitals...so a bigger energy gap means  a weaker interaction /overlap and so the stabilization energy of bonding MO is less and also the destabilization energy for the e*_g  is less .(they are lower in energy)

SO for fluorine I should have  a small Δ because e*_g is lower in energy (in comparision with iodine) and the t2g non bonding are also at the same energy...


BUt I know that in the spectrochemical series I have an opposite trend 

[Orcio_87]

F^- and I^- are σ and π donors. π because two p filled orbitals (2p in F^- and 5p in I^-) share electrons with empty d metal orbitals (orbitals d_xy, d_xz and d_yz). But they are still σ donors because one of they p orbital will be directed towards d_x²-y² or d_z² orbitals.

F^- is small electronegative ion so σ bonding will be stronger (I think that size of 2p orbitals is also important).

In turn I^- gives good π bonding and that is why gap between d_xy, d_xz, d_yz and d_x²-y², d_z² is smaller.