[rentj]

Hi, I would like to ask about one of the problems in the 50th IChO, regarding crystal field splitting in a dodecahedral complex. Here is the link to the problems (question 5.4): https://www.icho2021.org/pdf/Theoretical-Problems-50-IChO_final_sol.pdf

Based on the question, the Fe^IIO₈ complex has a dodecahedral arrangement, shown in the picture attached. What I want to ask is how are we supposed to know whether it is high spin or low spin; as in for octahedral complexes, I believe it is clear since there are only two 'types' of d orbitals, the t2g and eg orbitals, where three t2g orbitals are degenerate while 2 eg orbitals are degenerate. Thus, I can say that in high spin complexes, the electron filling is until the eg orbitals and the pairing is done after all 5 orbitals are filled in parallel (Δoct < P). Meanwhile in low spin complexes, the Δoct is larger than the P term, which it prefers pairing first rather than filling the eg orbitals. My first question is how do you define high spin and low spin complex in a dodecahedral complex since there are more than two energy levels of orbitals.

My second question is related to the 5.5 in the picture attached. Does it means the analogous Δoct in dodecahedral complex (Δdod) equals to E1 - E3? If so, why should it be that way? Is there any keyword regarding this topic that I can use to search for reference?

Thank you.

[Corribus]

I'm not really clear on what your question is. In the low spin arrangement, you just fill the electrons into the orbitals from low energy to high energy, two electrons per orbital, and only pairing in degenerate orbitals if necessary. In the high spin arrangement, you fill in the orbitals such that you have the maximum number of unpaired orbitals, and pair electrons in order of lowest energy orbital to highest. The arrangement of orbitals (and therefore the symmetry designation of the complex) does not affect these rules.

For your second question, whether or not the complex exists in high or low spin is determined solely be whether the energy gaps across the available orbitals is exceeded by the electron pairing energy. Since it basically costs energy to pair electrons (since it means they have to be in the same region of space), electrons like to spread out into to different orbitals. If available orbitals are degenerate, electrons will accordingly always maximize their spin since they will spread out among the orbitals as much as possible to minimize pairing energy. But if some available orbitals are higher energy than others, it will cost energy to put electrons into the higher energy orbitals. If the amount of energy it costs to put electrons into higher energy orbitals exceeds the pairing energy, then electrons will pair into lower energy orbitals rather than spread out into higher energy orbitals - a low spin configuration. If the amount of energy it costs to pair electrons exceeds the amount it costs to put them (solely) into higher energy orbitals, then they will occupy higher energy orbitals rather than pair in lower energy orbitals - a high spin configuration. So, the key parameters to determine high spin vs low spin is the energy gap across the available orbitals and the pairing energy.

You may consider the (trivial) case of the Aufbau principle in atomic orbital configurations. You probably know by now that in a multi-electron atom, electrons generally pair into lower energy orbitals and then fill into higher energy orbitals. Atoms are all "low spin" configurations. Consider helium, which has two electrons. The first electron goes into the 1S and the second electron pairs with the first in the 1S: both electrons go into the 1S and the overall spin is 0. We might consider an alternative configuration of the first electron going into the 1S and the second going into the 2S. This configuration would be higher spin (S = 1). But we usually don't observe the high spin configuration under most circumstances (without dumping energy into the system, say, with light) - why? Because the energy gap between the 1S and 2S orbitals greatly exceeds the electron pairing energy: it costs more to promote an electron to the 2S orbital (compared to 1S orbital) than it does to pair two electrons into the 1S orbital. Because atomic orbital energy gaps generally exceed electron pairing energies, lone atoms in the upper echelons are virtually always low spin, hence the Aufbau principle. In transition metal complexes, however, d-orbital splitting energies are generally on the same order as the pairing energy, so both high and low spin configurations become possible. In this case, the field strength imposed by the coordinating ligands tweak the orbital splitting energies, which means that the electron filling order and thus spin character can change quite a pit depending on what the ligands are and how they are arranged around the central metal ion.