Post 5: Crystal Field Splitting and Band Gaps in 3d Transition Metal Chalcogenides

With the descriptor set in place, this is the first real correlation post: does crystal field splitting Δ, combined with d-electron count, actually predict band gap behavior across the dataset — and does it do so in a way that matches the crystal field theory I'd reach for at a whiteboard?

The expected pattern

For an octahedral d⁰ or d¹⁰ configuration, there's no orbital degeneracy to split into a gap-determining transition — these tend toward wide gaps or are dominated by the M–X charge-transfer gap instead. For intermediate fillings, the t₂g/eg splitting itself can open a gap if Δ is large enough relative to the bandwidth, but for small Δ or partially filled, degenerate orbitals near E_F, the compound trends metallic. The d⁵ case is special: high-spin d⁵ has one electron in every orbital, which under strong correlation tends to favor a Mott gap regardless of Δ — this is the regime where the Zaanen-Sawatzky-Allen classification (Mott-Hubbard vs charge-transfer) starts to matter more than Δ alone.

Application: band gap regime map

The chart below plots Δ against d-electron count and colors each region by expected gap regime — wide gap (coral), moderate/correlation-driven gap (purple), and metallic/narrow (teal). Hover or click a point to see the reasoning for that combination. This is the kind of map I want the actual dataset points to fall onto, or to clearly deviate from in informative ways.

Where I expect surprises

The map above is deliberately the textbook version. The places I expect the real dataset to deviate — and where the interesting physics lives — are: tellurides, where the larger, more covalent M–Te bond reduces the effective U and can push d⁵ compounds from the correlation-driven regime toward metallic, against the naive d-count expectation; and compounds near the ZSA Mott-Hubbard/charge-transfer boundary, where small structural changes (and therefore small Δ shifts) might flip the dominant gap mechanism entirely, producing band gaps that don't vary smoothly with Δ.

Next post stays on the electronic side but shifts from Δ to bond length and coordination number directly, looking specifically at the metal-insulator transition as a function of structural distortion.