Origin of the strong spin-phonon coupling in transition metal complexes

Spin crossover (SCO) materials are fascinating and raise many interesting questions.
Here I want to address the underlying physics of why there is a large change in bond length (typically 10-20 per cent) when the spin state of the complex changes. Basically, it is because the ligand field splitting Delta changes significantly with bond length. The change in spin state is associated with electrons moving between from the upper d -levels on the metal ion  (e_g in an octahedral complex) to the upper levels (t_g2).

What is the physical origin of this splitting?

How does Delta vary with the distance between the metal (M) and the ligand L?

One can answer the second question theoretically with quantum chemistry computations and experimentally by changing the ligand L, which leads to changes in bond length. The figure below is taken from the book, Ligand Field Theory and its Applications.

The variation of Delta and the bond length with ligand reflects the spectrochemical series.

Quantum chemistry computations show a similar variation for a given complex by varying the M-L distance. For example, see Figure S5 in the Supplementary Material for this paper.

What is the physical origin of this splitting?

A first guess is from "crystal field theory" that associates the energy level splitting with classic electrostatic effects. This gives a value that falls of as the sixth power of the M-L distance, and makes the concrete (and roughly correct ) prediction that for an octahedral complex the e_g levels move up by 3/2 times the amount that the t_2g levels move down. For a tetrahedral complex the opposite happens. However, there are two significant problems with this prediction. First, the predicted splitting is an order of magnitude too small. Second, this model predicts the opposite trend to the spectrochemical series.

A better description is obtained from "ligand field theory" where the splitting arises from covalent bonding between the d-orbitals on the metal and the p-orbitals on the ligand. For an octahedral complex, the t_2g (e_g) orbitals have positive (zero or negative) overlap with the ligand orbitals.

Aside.

It is interesting (and disturbing?) that the authors of the figure above compare the data for Delta vs. R to power laws, 1/R^6 and 1/R^5.  For the data R varies by about 10 per cent. To distinguish between power laws one should be comparing data over several orders of magnitude!

In reality, the data is just as consistent with a linear decay.

What is of interest to me is the magnitude of the decay, G= 1 eV/Angstrom. The next step is to argue why this is "large". The change in bond length with spin crossover will be approximately G/B where B is the elastic constant for the bond.