Why can't fluorine be the central atom in inter-halogen compounds?
First off, fluorine can be at the "center of things." Examples would include the strongly hydrogen bonded hydrofluoric acid
and the very relevant example of the trifluoride anion
$$\ce{[F-F-F]^-}$$
The trifluoride anion example is critical as it demonstrates that fluorine can be hypercoordinated (hypervalent). Of course, in this last example we've "tricked" the fluorine into occupying the middle position, so your question remains (slightly altered), why doesn't fluorine occupy a central position more often?
In order to answer this question, two general cases need to be considered,
- The anionic situation $\ce{[X-F-X]^-}$ vs. $\ce{[X-X-F]^-}$
- The cationic situation $\ce{[X-F-X]^+}$ vs. $\ce{[X-X-F]^+}$
where X is Cl, Br or I.
In the case of the anion, Lewis dot structures suggest a negative charge on the central atom.
However, computational analysis of the electron distribution in the triiodide anion (see p. 156) suggests that the central atom is positively charged (+0.27 electron), while the terminal atoms are negatively charged (-0.63 electrons on each terminal atom). Assuming that the same general trend would be applicable to mixed trihalide anions, then in the case of $\ce{[X-F-X]^-}$ we would be placing positive charge on highly electronegative fluorine atom instead of the larger, more polarizable (e.g. more capable of bearing a positive charge) bromine, chlorine or iodine atom.
In the case of the cation, similar arguments come into play. Again, calculations for a variety of trihalide cations (mixed and homogenous) suggest substantial positive charge on the central atom (this time in agreement with Lewis Dot structures). The same arguments made above would operate here to destabilize structures with a central fluorine atom. This analysis also found extremely long (less stabilizing) bonds resulted when fluorine (or the most electronegative of the 3 atoms) occupied the central position.
Arguments that don't explain the infrequent occurrence of fluorine at a central position in inter-halogen compounds include:
- the lack of "d-orbitals"; as other's have commented, this is an old, invalid argument; the existence of $\ce{F3^-}$ demonstrates that fluorine is capable of hypercoordinate bonding without the need for d-orbital involvement
- inefficient 2p-3p overlap; $\ce{FCl}$ has a higher bond strength and lower heat of formation than $\ce{Cl2}$