# Low-spin / high-spin arguments: why overlooking pairing energy?

There are many arguments in inorganic chemistry based on a transition from a high-spin to a low-spin configuration, for example:

• $K_{f,3} > K_{f,2}$ for $\ce{Fe(bipy)3}$ complex, because of "a considerable increase in LFSE" as the complex becomes low-spin (from Shriver and Atkins)

However, as the complex becomes low-spin there is also an increase in the pairing energy (or a decrease in the number of Fermi holes), so I don't see why the assumption above that low-spin complexes are always more stable than high-spin is necessarily true (it can just reflect a subtle balance between $\Delta$ and P with not indication of the overall change in energy).

Based on my knowledge, successive bipy ligands would only further enhance the mixing of $\mathrm{t_{2g}}$ SALCs and that is where increasing stabilization is coming from. However, I would predict that this factor would get weaker with each ligand substitution, as the energy match of the occupied $\mathrm{t_{2g}}$ and the π* of an incoming bipy becomes worse.

Perhaps, there is some non-linear dependence of the way the pairing energy is reduced as the orbitals get "smeared" over larger space (i.e. over both the metal and π-acceptors) on the number of $\ce{bipy}$ ligands?

• Is $\ce{bipy}$ $\ce{bpy}$, i.e. 2,2'-bispyridyl? – Jan Feb 12 '17 at 19:03
• @Jan yes, sorry for the confusion! – GingerBadger Feb 12 '17 at 19:28

Regarding your question about the non-linear change with ligands, I suspect this isn't that easy of a question to answer, but the data is probably out there somewhere.

Regarding the quote and question in the title, when these kinds of arguments are made, they aren't ignoring the pairing energy, but they also aren't really making any predictions.

Essentially, Ligand Field Theory (LFT) lays out a simple way that one can rationalize the geometry of a particular transition metal complex based on the energy of the d orbitals. I can see that you know this. What this means though is that when they observe whatever change happens for the iron complex, LFT says that this must correspond to some change in the energy of the electrons associated with shifting changes in the energy of occupying orbitals in a specific way.

That is, in LFT the only two terms that describe the energy of a particular electronic arrangement are the ones describing the potential energy of the electrons while ignoring their interactions, and the increase in energy from pairing electrons in orbitals.

So, when any geometrical or similar observation is made, there's really no other explanation than to say the a high-spin complex became low-spin because the orbitals split more ("considerbale change in LFSE") which guarantees this decrease in potential is larger than the cost of pairing electrons.

I agree that it's quite naive to always explain things in this way, but that's why crystal field theory and ligand field theory are not meant as real computational or predictive tools these days. They are useful for connecting geometry, spin, and energy as well as giving a first-order explanation of observations.

It's like when you'll sometimes hear inorganic chemistry students describe the behavior of a catalyst as being due to a "combination of electronic and steric effects." That's true... but those are also the only possibilities, so it doesn't tell you much.

• Thank you for your answer! But in this case, I'm surprised that such arguments work at all, and do consistently! – GingerBadger Feb 13 '17 at 23:10
• Well they aren't really arguments so much as posterior explanations. That's what I'm trying to say in the answer. If you really wanted to be predictive the model you use to has to be much more detailed than CFT or LFT because you don't know the splitting or pairing energies without experimental measurement. And by the time you've made an experimental measurement, you already know the answer. That's why there are other methods for explaining these things from first principles. – jheindel Feb 13 '17 at 23:39