Oh Molecular Orbital Diagram

Understanding the Oh Molecular Orbital Diagram Without Losing Your Mind

The Oh molecular orbital diagram is one of those things professors draw on the board like it's simple, then expect you to memorize the energy ordering. It's not magic. You just need to know which orbitals actually overlap and which ones don't. I don't use crystal field theory anymore for anything beyond a quick mental model. It gives you the right colors but the wrong mechanism. The Oh molecular orbital diagram tells you what's actually happening: which ligand orbitals mix with which metal orbitals, and how much the bonding versus antibonding character shifts the energies. That's what matters when you're trying to rationalize redox potentials or interpret UV-vis spectra of octahedral complexes. The basic framework starts with the octahedral point group. You have six ligands arranged along the x, y, and z axes. Each ligand contributes a sigma donor orbital pointing at the metal center. You work out the symmetry species of those six ligand orbitals using a character table, and you get a1g + t1u + eg. Those three sets can only interact with metal orbitals of the same symmetry. That's the whole constraint. Metal s is a1g, metal p is t1u, and metal d splits into eg and t2g. The t2g set has no sigma-symmetry match, so in a pure sigma-only model it stays non-bonding.

Building the Diagram Step by Step

I always start from the metal first, then add the ligands. Put the metal orbitals in the middle column. The s orbital goes at the bottom since it's usually the lowest energy valence orbital. The three p orbitals sit above it as the t1u set. Then the five d orbitals split: the eg pair (dz² and dx²-y²) higher than the t2g triplet (dxy, dxz, dyz) in terms of raw metal orbital energy, though I should note that's not always a clean separation — it depends on which d-block row you're in. On the left, draw the six ligand sigma orbitals as a1g, t1u, and eg groups. On the right, do the same but label them as antibonding combinations. Draw lines connecting orbitals of matching symmetry. Where two orbitals of the same symmetry meet, they mix and push each other apart. The bonding combination drops below the original ligand energy. The antibonding combination rises above the original metal energy. Non-matching orbitals just stay where they are. The result is a set of occupied bonding orbitals at the bottom (mostly ligand character), a middle region containing the non-bonding t2g and possibly some metal-ligand pi interactions, and a set of high-energy antibonding orbitals — the ones that determine your d-electron count and spin state.

The Pi-Bonding Complication That Nobody Warns You About

Here's where it gets messy. If your ligands have pi orbitals — halides, water, oxide, anything with lone pairs not pointing at the metal — those pi orbitals also have symmetry matches in Oh. The t2g set, which I just told you was "non-bonding" in the sigma-only model, now interacts with ligand pi orbitals of t2g symmetry. Pi-donor ligands push the t2g orbitals up in energy because you're mixing filled ligand pi orbitals with filled metal t2g orbitals. That's repulsive. It shrinks the gap between t2g and eg* and changes your predicted spin state. Conversely, pi-acceptor ligands like CO or CN- pull the t2g orbitals down. Back-bonding from metal t2g into empty ligand pi* orbitals stabilizes that set and widens the splitting. This is the real reason spectrochemical series exists. Crystal field theory hand-waves this as "strong field" and "weak field" without explaining what's actually moving electrons around. I spent an afternoon once trying to reconcile the magnetic moment of [CrF6]³ with a textbook Oh molecular orbital diagram that only showed sigma bonding. The prediction came out wrong every time. The issue was fluoride's pi-donor effect on t2g. I had to add the pi-interaction manually, treating the F- 2p orbitals as additional basis functions in the t2g symmetry block. Once I did that, the diagram correctly showed a smaller o and the high-spin configuration matched the experimental data. It took me about forty-five minutes to fix what should have been in the textbook in the first place.

Counting Electrons Correctly

This is where most people mess up. You don't just dump the metal's d-electron count into the diagram. You count all valence electrons: metal d electrons plus the electrons donated by each ligand through sigma bonds. In an octahedral complex with six sigma-only ligands, that's metal d count plus twelve ligand electrons. Fill the orbitals from the bottom up. The bonding orbitals (a1g, t1u bonding, eg bonding) take priority. Then the non-bonding or pi-modified t2g set. Then the antibonding eg* orbitals last. For d³ Cr(III) with six water ligands, you get 3 + 12 = 15 electrons to place. The first twelve fill the three bonding sets. The remaining three go into the t2g set. That's why [Cr(H2O)6]³ is paramagnetic with three unpaired electrons. It's not a coincidence. The diagram gets it right.

Where This Approach Actually Breaks Down

The Oh molecular orbital diagram assumes you can treat each interaction independently and that the symmetry labeling holds. That's fine for ideal octahedral geometry. But real complexes distort. Jahn-Teller distortions in d Cu(II) complexes, for example, lower the symmetry from Oh to D4h and split the eg and t2g sets further. The diagram you drew for perfect octahedral symmetry no longer applies without modification. You'd need to rebuild it with the lower symmetry group, which is more work and the qualitative predictions don't necessarily get clearer. Another limitation: the diagram is qualitative. It tells you ordering and bonding character, but it doesn't give you numerical energy values. If you need actual orbital energies for quantitative work — say, calculating excitation energies or fitting electrochemical data — you're going to need DFT anyway. I use the diagram as a planning tool before running calculations, not as a replacement for them. It usually saves me about twenty minutes of guessing which orbitals to look at in the output. The biggest practical problem I run into is when ligands aren't all the same. Mixed-ligand octahedral complexes like [Co(NH3)4Cl2]+ break the Oh symmetry entirely. The diagram becomes messy because you can't cleanly assign symmetry labels to the ligand group orbitals anymore. People sometimes still draw an Oh diagram and pretend it works, but the energy levels they read off it are essentially guesses. In those cases, I skip straight to a computational approach or use group theory for the actual point group of the molecule, which for trans-[Co(NH3)4Cl2]+ is D4h and for the cis isomer is C2v.

A Quick Reference for the Standard Oh MO Ordering

For a typical d-block metal with sigma-only ligands in Oh symmetry, the orbital energy ordering from lowest to highest is: a1g (bonding, mostly ligand), t1u (bonding, mixed ligand and metal p), t2g (non-bonding, metal d), eg (bonding, mixed ligand and metal d), then the antibonding counterparts eg* (metal d character, this is what you count d electrons into), t1u* (antibonding), and a1g* (antibonding). The t2g remains in the middle unless pi-bonding is present. That middle position is why t2g electrons are the ones involved in ligand field transitions and why they determine magnetism. If you're looking for a visual reference, the diagram is standard enough that any inorganic chemistry textbook will have one. Cotton and Wilkinson is the classic source. Online, the WebMO tutorial page and the Chemistry LibreTexts coordination chemistry section both have decent labeled versions you can work from. I don't link directly because those pages change URL occasionally, but searching for "octahedral molecular orbital diagram free energy level" will bring up usable versions within the first few results. The thing I wish someone had told me clearly when I first learned this: the diagram isn't a puzzle to solve. It's a bookkeeping method for tracking where electrons live and how they move. If you understand which orbitals overlap and why the t2g set is special, you don't need to memorize the diagram. You can reconstruct it in about three minutes whenever you need it.