# Changes in metal-ligand bond lengths of coordination compounds

Consider change in length of the metal-ligand bond in the following reaction:

$$\ce{[Cr(CN)6]^3- + e^- -> [Cr(CN)6]^4-}$$

I initially had two theories (based on general electrostatics and d-orbital splitting) that contradicted eachother.

General Electrostatics Theory

Since $$\ce{Cr}$$ is reduced from $$\ce{Cr^3+}$$ to $$\ce{Cr^2+}$$, the electrostatic attractions between the $$\ce{Cr}$$ atom and the $$\ce{CN-}$$ ligand will decrease. Therefore, according to this explanation, the $$\ce{Cr-CN-}$$ bond length would increase.

D-orbital Splitting Theory

I examined the change in the d-orbital splitting diagram. Both the oxidized and reduced forms of the compound are in low-spin states due to the strong cyanide ligand. Since the electron change is from $${d^3}$$ to $${d^4}$$, the diagrams would look like this:

After drawing the diagrams, I concluded that the $$\ce{Cr}$$ ion experiences a net gain of since the energy loss from the extra electron in the bonding $${t_{2g}}$$ orbitals is greater than the energy gain from the paired electrons. Due to this increase in ligand stablization energy, I concluded that the chromium-cyanide bond would decrease in size.

Conclusions

I later saw that my first theory was correct, and that the chromium-cyanide bond increases. However, the answer to this problem also supported the changes in the d-orbital diagrams oultined in the second theory.

Is only the first theory correct, or are both correct with a stronger effect from the first scenario?