Valence Bond Theory (VBT) of Coordination Compounds

Valence bond theory (VBT), applied to coordination compounds by Linus Pauling, explains both the geometry and the magnetic behaviour of a complex by proposing that the central metal ion makes available a specific set of empty atomic orbitals, hybridizes them into an equal number of new, equivalent hybrid orbitals, and then accepts one lone pair of electrons from each ligand into one of these hybrid orbitals to form the coordinate bonds. The particular combination of orbitals chosen for hybridization determines the geometry directly: $sp$ hybridization gives a linear complex, $sp^3$ or $dsp^2$ hybridization gives a tetrahedral or square planar complex respectively (both coordination number 4), and $sp^3d^2$ or $d^2sp^3$ hybridization gives an octahedral complex (coordination number 6).

For an octahedral complex specifically, the theory distinguishes two possibilities depending on which d-orbitals are used. If the metal's own $(n-1)d$ orbitals (the inner, already-occupied d-subshell) are used together with the outer $ns$ and $np$ orbitals, the hybridization is $d^2sp^3$ and the resulting complex is called an inner orbital complex. If, instead, the outer $nd$ orbitals (from the next principal shell out) are used together with $ns$ and $np$, the hybridization is $sp^3d^2$ and the complex is called an outer orbital complex. Crucially, using the inner $(n-1)d$ orbitals for bonding is only possible if enough of them are left vacant — which requires the metal's own d-electrons to first pair up as much as possible within a smaller number of orbitals, freeing the rest for hybridization.

This is precisely why $[\text{Co}(\text{NH}_3)_6]^{3+}$ and $[\text{CoF}_6]^{3-}$ — both containing octahedral $\text{Co}^{3+}$, a $d^6$ ion — show such different magnetic behaviour. $\text{NH}_3$ is a strong-field ligand for $\text{Co}^{3+}$: its approach forces all six d-electrons to pair up within just three of the five $3d$ orbitals, leaving the remaining two $3d$ orbitals empty and available for hybridization. Cobalt therefore uses $d^2sp^3$ hybridization (two inner $3d$ + one $4s$ + three $4p$ orbitals), giving an inner orbital, low-spin, diamagnetic octahedral complex (zero unpaired electrons). $\text{F}^-$, by contrast, is a weak-field ligand: it does not force any extra pairing, so the six d-electrons remain spread across all five $3d$ orbitals in their normal, mostly-unpaired arrangement, leaving none of the inner $3d$ orbitals free. Cobalt is then forced to use its outer $4d$ orbitals instead — $sp^3d^2$ hybridization (one $4s$ + three $4p$ + two outer $4d$ orbitals) — giving an outer orbital, high-spin, paramagnetic octahedral complex, with four unpaired electrons remaining in the original $3d$ set. …