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I wanted to understand mechanism of the formation of $\ce{F3-}$ anion through the molecular orbital theory.

UPDATE - What I really want to know is that when $\ce{F3-}$ is formed from $\ce{F2-}$ and $\ce{F}$, the unpaired electron of $\ce{F}$ enters the anti-bonding orbital $\sigma^*_{\mathrm{2s}}$ then shouldn't the $\ce{F2-}$ ion break up, as all the bonding and anti-bonding orbitals would be filled up, resulting in a bond order of 0?

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I'm not too sure, but let me try. In the case of the F3- ion, the effective bond order is 1 (see attached images), not 2. In other words, the only effective bonding MO holding the entire thing together is a 3-centre 2-electron bond (think linear H3+ and FHF-).

I presume you know how the F2 molecule MO diagram looks like, so I'll just explain what F3- looks like. Using the D2h character table and forming our reducible representations, we get the following: enter image description here Note s-p mixing occurs in the Ag and B1u orbitals, as shown:

enter image description here The MO diagram for F3- would look disastrously messy, so I'll simply show how the molecular orbitals look like and how they're ordered.

enter image description here enter image description here

The first thing you'll probably notice is how the central fluorine isn't allowed to s-p mix anymore; although the symmetry remains D2h the now-presence of a central atom means the s and p orbitals of the central fluorine are no longer of the same symmetry, and no longer interact. That, in addition to the fact that the sole bond that is effectively holding the ion together is a 3-centre 2-electron bond, makes answering your question rather difficult.

I think a way to consider it would be in terms of the formation of a Lewis adduct; in other words, the pz electrons in the F- ion interact with the empty LUMO of the F2 molecule while all the lower-energy orbitals form bonding/antibonding sets with the lower-energy F2 MOs, forming a new HOMO and LUMO from the "HOMO" of F- and LUMO of F2.

Hope this helped.

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