Spin Crossover Points on Tanabe–Sugano Diagrams

Introduction

Some octahedral d counts can rearrange their electrons when ligand-field splitting becomes large. For d⁴–d⁷, a high-spin state can be lowest in a weak field while a lower-spin state becomes lowest in a sufficiently strong field. Their Tanabe–Sugano diagrams are often divided by a vertical line. Understanding this line prevents the mistake of reading all curves from one unchanging ground state across the entire plot.

Core explanation

In an octahedral field, electrons fill lower t₂g and upper e g orbitals while also paying an energy cost for pairing. For d⁴, a weak-field arrangement t₂g³e g¹ has S=2, whereas sufficiently strong field favours t₂g⁴ with S=1. For d⁵, t₂g³e g² has S=5/2 and low spin t₂g⁵ has S=1/2. For d⁶, high-spin t₂g⁴e g² has S=2 and low-spin t₂g⁶ has S=0. For d⁷, high-spin t₂g⁵e g² has S=3/2 and low-spin t₂g⁶e g¹ has S=1/2. These orbital counts summarise the spin competition; exact term energies also involve interelectronic repulsion and symmetry.

The diagram's vertical separator marks the field strength where competing ground-state term energies become equal in the model. On the weak-field side, the high-spin term supplies the E/B=0 baseline; on the strong-field side, a lower-spin term does. Because the plotted ordinate is an excitation energy relative to whichever state is ground, curves can appear to jump or change their vertical relation at the separator. It is more precise to say that the reference state changes than that every physical level jumps abruptly as one tunes a continuous field parameter.

At the crossing, states of different total spin can cross without the same avoided-crossing restriction that applies to equal-symmetry, same-spin states under a spin-independent Hamiltonian. Spin–orbit coupling can weakly mix them in real systems, so the idealised sharp boundary can be softened. Whether a real complex undergoes a thermal spin transition is a further question: lattice energy, entropy, solvation and metal–ligand bond length all contribute to the free-energy balance. A static Tanabe–Sugano x coordinate is an electronic-energy model, not a complete phase diagram.

The spectroscopic consequence is substantial. A high-spin d⁶ Fe²⁺ complex has a quintet ground term and a small set of quintet-allowed target states; a low-spin d⁶ complex has a singlet ^1A₁g ground state and a different set of singlet-allowed transitions. Using a high-spin band assignment on the low-spin side can produce numerically plausible but chemically false B and Δₒ. Magnetic susceptibility is particularly helpful because the spin-only moments of S=2 and S=0 are dramatically different.

The dividing x value is not universal across all diagrams or all materials. It depends on d count and parameters used in generating the plot, including electron-repulsion ratios. It is also not simply “Δₒ equals a single pairing energy” for every dⁿ configuration. That phrase is a useful elementary energy-balance mnemonic, but detailed many-electron terms set the precise crossing. In an exam, read the particular diagram supplied and state the ground-term label on each side before calculating transitions.

No analogous high-/low-spin choice exists for d¹–d³ under the usual octahedral occupancy because electrons can occupy different t₂g orbitals without needing to promote to e g or pair prematurely. For d⁸/d⁹ the standard octahedral diagrams also lack the d⁴–d⁷ type of spin crossover, although other geometries and strong covalency can still produce different electronic states.

Step-by-step reasoning

Count d electrons, identify whether d⁴–d⁷ supports competing octahedral spin arrangements, and locate the vertical crossover marker on that d-count diagram. Read the ground-state term on the chosen side. Select target curves of the same multiplicity for strong spin-allowed bands; check weaker differently spin-labelled curves separately. Use magnetism and ligand strength to decide which side describes the sample before extracting B or Δₒ.

Visual explanation

Sketch two absolute-energy curves, high-spin and low-spin, crossing as Δₒ increases. Under them draw the Tanabe–Sugano version with the baseline reset to zero at the lower curve on each side and a vertical divider at the crossing. Add electron-box sketches for high-spin d⁶ t₂g⁴e g² and low-spin d⁶ t₂g⁶ to connect the term picture to occupancy.

Real-world analogy

Two routes to a destination can exchange which is shortest as a toll changes. If a chart always reports extra distance above the currently shortest route, its baseline switches routes at the equality point. The roads do not teleport; the comparison reference changes. A ground-state crossover causes the same re-referencing of plotted excitation energies.

Real-world example

High-spin aqueous Fe²⁺ and low-spin hexacyanoferrate(II) are both d⁶ but have very different spin states and spectra. H₂O and CN⁻ produce different ligand fields and metal–ligand covalency, so the samples lie in distinct electronic regimes. A d⁶ diagram must be read from the quintet side for the high-spin case and the singlet side for the low-spin one.

Why?

Why are d⁴–d⁷ especially susceptible to this competition? Their electron counts permit a choice between occupying higher e g orbitals to keep spins parallel and pairing more electrons in lower t₂g orbitals. The preferred choice changes when the ligand-field benefit outweighs the pairing and exchange balance.

Common misconception

“The vertical crossover line is a band that the spectrometer observes.” It is a calculated boundary between alternative ground-state regimes on a parameter plot. Individual observed absorptions are vertical energy gaps at the sample's chosen x position, not the divider itself.

Worked example

An octahedral d⁶ complex is found to be diamagnetic. The simple high-spin arrangement t₂g⁴e g² would contain four unpaired electrons and be paramagnetic. The low-spin arrangement t₂g⁶ has S=0, consistent with diamagnetism. Therefore analyse its bands from the low-spin ^1A₁g baseline on the strong-field side of the d⁶ Tanabe–Sugano plot. A quintet-ground Orgel diagram would be inappropriate even if it could fit one visible band by chance.

Quick check

1. What are the high- and low-spin t₂g/e g occupancies for octahedral d⁵? Answer: High spin is t₂g³e g² with five unpaired electrons; low spin is t₂g⁵e g⁰ with one unpaired electron.

Exam focus

Mark high-spin and low-spin sides before reading ordinates. Use the actual diagram's separator, not a memorised universal x value. Corroborate the selected side with magnetic data and spin multiplicity; then assign only transitions from the correct ground state.

Advanced insight

The diagram compares electronic energies at fixed model parameters. Real thermal spin crossover depends on Gibbs free energy, including entropy from spin and vibrational degrees of freedom and cooperative interactions in solids. Thus a compound can change spin population with temperature even when a simple zero-temperature term crossing picture gives only one preferred electronic state at a fixed geometry.

Summary

The d⁴–d⁷ octahedral Tanabe–Sugano diagrams can change from a high-spin to a lower-spin ground state as Δₒ/B increases. Their divider marks a change of baseline and allowed transition set. Magnetism, geometry and ligand identity help choose the right regime for an actual complex.

Practice questions

1. Contrast the spin-only behaviour of high- and low-spin octahedral d⁶. Answer: High-spin t₂g⁴e g² has four unpaired electrons and S=2, so it is strongly paramagnetic; low-spin t₂g⁶ has S=0 and is diamagnetic apart from small temperature-independent effects. 2. Why should a band ratio not be fitted across a d⁶ diagram's two sides without checking spin state? Answer: The ground-state baseline and permissible same-spin target curves differ across the crossing. Mixing one side's ground with the other's curves gives a chemically inconsistent ratio. 3. Does a calculated crossing guarantee abrupt thermal spin transition in a solution? Answer: No. Thermal populations depend on free energies, entropy, solvent and structural relaxation; the electronic-energy crossing alone does not determine whether a measurable thermal transition is sharp, gradual or absent. 4. Does every energy eigenvalue physically jump when a diagram shows a ground-state discontinuity? Answer: No. The plot measures excitation energy from the current lowest state. The zero-energy reference changes at the ground-state crossing, creating apparent discontinuities in relative curves.