SCIENCE · INVESTIGATE AND EXPLAIN

Compare splitting and pairing while preserving a degenerate boundary

An orbital energy model can constrain spin occupation without uniquely predicting observed color.

Learning outcome: Compare d4–d7 at weak, equal and strong splitting. Retain orbital occupations, tied-energy unpaired ranges and the gap wavelength without claiming that every matching photon is an allowed transition.

Starting recap

Refresh state-function boundaries, absolute temperature, dimensionless logarithms, written reaction/electron equivalents, conserved charge and formal orbital/identity bookkeeping.

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Try

For d6 and pairing cost2 eV, splitting3 eV favors t2g6 eg0 with no unpaired electrons; splitting1 eV favors a high-unpaired configuration.

Open the experiment

Predict

What happens when the simplified splitting and pairing costs are exactly equal?

Your prediction and reason

Experiment

Compare d4–d7 at weak, equal and strong splitting. Retain orbital occupations, tied-energy unpaired ranges and the gap wavelength without claiming that every matching photon is an allowed transition.

Compare orbital splitting and pairing without hiding a degeneracy

Worked teaching model · synthetic values · Idealized octahedral five-orbital model: three t2g levels at−.4Δ, two eg at+.6Δ, pairing cost P2 eV per doubly occupied orbital. Enumerate occupations0–2 per orbital and minimize energy; choose maximum unpaired within equal-energy states, but report full unpaired range at degeneracy. No exchange, spin-orbit, vibronic or selection-rule model; Δ photon wavelength is a possible gap scale, not predicted observed color.

d-electron count: 6 electrons Octahedral splitting: 3 eV

Reset keeps your prediction, captures and practice history.

Minimum electronic energy versus splitting: solid

Axes rescale. A multiplicity distribution, inverse-temperature relation and charge-budget graph have different meanings. Read each axis and do not treat a computed curve as additional measured evidence.

All calculated plot coordinates

Minimum electronic energy versus splitting
octahedral splitting (eV)	minimum model electronic energy (eV)
1	1.6
1.033333	1.586667
1.066667	1.573333
1.1	1.56
1.133333	1.546667
1.166667	1.533333
1.2	1.52
1.233333	1.506667
1.266667	1.493333
1.3	1.48
1.333333	1.466667
1.366667	1.453333
1.4	1.44
1.433333	1.426667
1.466667	1.413333
1.5	1.4
1.533333	1.386667
1.566667	1.373333
1.6	1.36
1.633333	1.346667
1.666667	1.333333
1.7	1.32
1.733333	1.306667
1.766667	1.293333
1.8	1.28
1.833333	1.266667
1.866667	1.253333
1.9	1.24
1.933333	1.226667
1.966667	1.213333
2	1.2
2.033333	1.12
2.066667	1.04
2.1	0.96
2.133333	0.88
2.166667	0.8
2.2	0.72
2.233333	0.64
2.266667	0.56
2.3	0.48
2.333333	0.4
2.366667	0.32
2.4	0.24
2.433333	0.16
2.466667	0.08
2.5	0
2.533333	-0.08
2.566667	-0.16
2.6	-0.24
2.633333	-0.32
2.666667	-0.4
2.7	-0.48
2.733333	-0.56
2.766667	-0.64
2.8	-0.72
2.833333	-0.8
2.866667	-0.88
2.9	-0.96
2.933333	-1.04
2.966667	-1.12
3	-1.2

Calculated under the declared model
Quantity	Value	Unit
Minimum model electronic energy	-1.2	eV
Selected t2g occupation	6	electrons
Selected eg occupation	0	electrons
Minimum unpaired count among tied states	0	electrons
Maximum unpaired count among tied states	0	electrons
Tied orbital occupation patterns	1	count
Photon wavelength matching splitting	413.2807	nm

The model minimum has all d electrons paired.

Three lower t2g orbitals and two upper eg orbitals share the same barycenter: energies−.4Δ and+.6Δ per electron. Pairing costs compete with promotion. Distinct degenerate orbital patterns are not automatically distinct spin multiplicities; exchange and other interactions are omitted. Unpaired electrons support paramagnetic response, but wavelength matching a gap does not establish an allowed/intense transition or a unique observed color; charge-transfer bands can also matter.

These calculated packets are not performed chemical measurements or separation procedures. Keep feasible inventories, activity conventions, fitting observations and independent checks distinct.

Selected minimum occupation; tied-state range remains in the ledger

t2g 1

↑↓

t2g 2

↑↓

t2g 3

↑↓

eg 1

empty

eg 2

empty

Three lower and two upper orbitals; each box holds at most two electrons. This displayed representative does not erase other equal-energy occupations.

The model minimum has all d electrons paired.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Split five orbitals without losing the reference

In the idealized octahedral model, three t2g orbitals lie at−.4Delta per electron and two eg at+.6Delta. Their orbital-weighted barycenter remains zero. Putting a second electron in an orbital adds a supplied pairing cost P=2 eV. The model enumerates occupations up to two per orbital and compares total splitting-plus-pairing energy.

At d6 and Delta3 eV, occupying all six electrons in the lower three orbitals gives t2g6 eg0 and no unpaired d electrons. At Delta1 eV, promoting electrons can avoid enough pairing cost to favor four unpaired electrons. These are model predictions under explicit omitted interactions, not measurements of a named compound.

Do not hide the equality boundary

When Delta=P in this simplified account, states with different unpaired counts can tie. The display reports the full tied range rather than claiming one uniquely selected spin state. Orbital occupation-pattern degeneracy is not automatically identical to spin multiplicity; exchange, spin-orbit coupling and distortion are not represented.

Unpaired electrons support paramagnetic response, while all-paired arrangements are diamagnetic in the simple account. Real observed magnetism can include additional contributions and interactions.

Transfer and check

A photon matching a2 eV gap has wavelength hc/E≈619.921 nm. This is an energy scale, not proof of an allowed intense absorption or a unique observed solution color. Transition selection rules, band widths and charge-transfer absorption matter. Explain why observing color alone cannot uniquely establish oxidation state, geometry and spin occupation without independent evidence.

Source and evidence boundary

Conceptual reference checked 30 October 2026: OpenStax Chemistry 2e. All prose, cases, graphs and questions are original. These supplied thermodynamic, electrical, nuclear or orbital models are not actual measurements, apparatus instructions, synthesis protocols or practical certification. Retain conventions, independent constraints, changed-case reasoning and limits. No heating, wiring, electrolysis, source handling or chemical manipulation is requested.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which omitted exchange, covalency, selection-rule or charge-transfer effect could matter in a real spectrum?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Separate formal charge, donor count and geometric possibilities Next: Design an independent constraint before spending the sample

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