Reading Tanabe–Sugano Diagrams for d²

Introduction

The octahedral V³⁺ aqua ion is a useful d² case because two electrons already produce several many-electron terms. A one-electron t₂g²→t₂g¹e g¹ picture suggests a promotion but cannot by itself assign all bands. A d² Tanabe–Sugano diagram retains the free-ion ^3F and ^3P ancestry, the octahedral state labels, and the effect of electron repulsion. Reading it requires a ground state, a set of candidate target curves and an explicit check of measured band ratios.

Core explanation

Each H₂O ligand is neutral, so the complex [V(H₂O)₆]³⁺ has V³⁺. Vanadium(III) is d². In octahedral symmetry its weak-to-moderate-field ground state is ^3T₁g(F), descending from the free-ion ^3F term. This is the E/B=0 baseline of the d² Tanabe–Sugano diagram. Its two electrons occupy the lower t₂g-derived manifold in a simple orbital picture, but the uppercase T label refers to the symmetry of the entire two-electron state.

Three important same-spin target curves are ^3T₂g(F), ^3A₂g(F) and ^3T₁g(P). The ^3F parent contributes the first two as well as the ground ^3T₁g(F); the ^3P parent contributes the other ^3T₁g. Excitations from the triplet ground state to these triplets obey ΔS=0, yet all are g→g and are Laporte-forbidden in ideal O h. Their measured bands are therefore relatively weak d–d features compared with a strongly allowed charge-transfer absorption. Several singlet curves also exist on a complete diagram, but ground-triplet to singlet arrows are spin-forbidden and should usually be weaker.

Read the actual vertical order of curves at the selected Δₒ/B, not merely the ancestry labels. The two ^3T₁g states can mix, and ^3A₂g(F) and the P-derived triplet can change their relative order in the plotted region. Some textbook diagrams arrange target curves in an order that differs from a rough Orgel sketch. Thus a numbered first, second or third observed band should be assigned only after checking its energy, relative intensity and consistency with a single horizontal position on the proper diagram.

Begin with wavelengths, convert them to wavenumbers and sort by increasing energy. A band ratio such as ν̃₂/ν̃₁ is independent of B; compare it with the ratio of candidate ordinates at various x=Δₒ/B positions. Once an x gives a plausible match, read the first ordinate y₁=E₁/B and calculate B=ν̃₁/y₁. Then Δₒ=xB, and every other assigned curve predicts a band at y iB. If a third observed feature badly disagrees, revisit its assignment: it may be charge transfer, an overlapping shoulder, or a sign that the simplified model and assumed C/B are inadequate.

The first d² spin-allowed band must not be set equal to Δₒ by habit. Its energy is between interacting many-electron states, and configuration interaction modifies the term separation. The diagram supplies the relationship between that energy and Δₒ/B. In some ranges the two are numerically similar, but equality is not a general derivation. This is exactly why d² is an instructive step beyond the d¹ orbital picture.

Spectral conditions matter. An aqueous V³⁺ sample must really contain predominantly the hexa-aqua species; hydrolysis or oxidation would change the absorber. A broad visible band may hide two transitions, and an ultraviolet charge-transfer tail can interfere with a high-energy d–d assignment. Treat the diagram as a constrained interpretation of a chemically characterised sample, not as a machine that labels every peak automatically.

Step-by-step reasoning

Find V oxidation state and d count, then choose the octahedral d² diagram. Mark ^3T₁g(F) as baseline and identify the three triplet target curves. Convert observed λ to ν̃=10⁷/λ(nm), order the bands, and test ratios against ordinates at one x. Derive B and Δₒ only after a consistent match. Check remaining bands, weak singlet candidates and possible charge-transfer overlap.

Visual explanation

Sketch the d² plot with the ^3T₁g(F) baseline and three triplet curves above it. Draw a vertical guideline at a hypothetical x. Draw upward arrows from zero to its three intersection heights and label their ordinates y₁,y₂,y₃. Beside the sketch put a small spectral trace whose three marked energies can be compared with y₁B,y₂B,y₃B; leave the high-energy assignment provisional if a charge-transfer band overlaps it.

Real-world analogy

Matching a constellation requires more than finding one bright star: the relative positions of several stars must fit the same sky pattern. A single d² peak can be fitted by many x and B choices; the ratios among multiple peaks constrain one coherent position on the term diagram.

Real-world example

An aqueous V³⁺ solution can show more than one ligand-field absorption because its d² triplet ground state has several triplet targets. If a strong ultraviolet band towers over weaker visible features, intensity suggests checking for charge transfer before assigning it to the highest d–d state. The oxide, pH and sample history should be checked to ensure the ion has not changed oxidation state.

Why?

Why is ^3T₁g(P) a separate target from ^3T₁g(F) despite identical point-group symbols? They descend from distinct free-ion terms. The ground and excited T₁g states are different energy eigenstates that can mix; the parent label tracks their weak-field origin.

Common misconception

“The first d² band directly gives Δₒ because one electron moves from t₂g to e g.” The observed transition connects interacting two-electron terms. Electron repulsion and state mixing alter its energy, so Δₒ should be extracted from a diagram or a justified approximation.

Worked example

Suppose two assigned V³⁺ triplet bands are at 16,000 and 24,000 cm⁻¹. Their ratio is 1.500. On a hypothetical, correctly labelled d² diagram, assume the selected x gives corresponding ordinates y₁=20.0 and y₂=30.0. Then B=16,000/20.0=800 cm⁻¹, and the second band is predicted at 30.0×800=24,000 cm⁻¹. If the selected x is 22.0, Δₒ=22.0×800=17,600 cm⁻¹. These ordinates are an illustrative diagram reading; the actual sample requires measured peaks and the specific published diagram.

Quick check

1. What is the d count of V³⁺, and what is the d² octahedral triplet ground label? Answer: V³⁺ is d²; the ordinary octahedral ground term is ^3T₁g(F).

Exam focus

Label the F/P parents where duplicate T₁g states occur. Use ratios to locate Δₒ/B, then one absolute energy to obtain B. Mark any high-energy band assignment as tentative when charge transfer or overlap is possible, and avoid equating the lowest band with Δₒ without analysis.

Advanced insight

The two same-symmetry ^3T₁g branches exhibit avoided crossing as Δₒ/B changes. At strong mixing, “F-derived” and “P-derived” become approximate ancestry descriptors rather than pure wavefunction identities. A fitted B is also an effective parameter of the complex and may differ from free V³⁺ because of nephelauxetic covalency.

Summary

[V(H₂O)₆]³⁺ is an octahedral d² system with ^3T₁g(F) ground state and several triplet target states. Tanabe–Sugano band ratios constrain Δₒ/B; an absolute energy then yields B and Δₒ. Multiple bands and intensity evidence are needed for a defensible assignment.

Practice questions

1. Name three spin-allowed target symmetries for octahedral d² from ^3T₁g(F). Answer: ^3T₂g(F), ^3A₂g(F) and ^3T₁g(P); their order at a particular field strength should be read from that diagram. 2. If two assigned bands have energies 15,000 and 22,500 cm⁻¹, what ratio should the matching diagram ordinates have? Answer: Their ordinate ratio must be 22,500/15,000=1.50, since the common B factor cancels. 3. A third very intense ultraviolet band fails the triplet-curve prediction. Give one chemically plausible alternative. Answer: It may be ligand-to-metal or metal-to-ligand charge transfer rather than the missing d–d target; check its much higher molar absorptivity, sample speciation and predicted term energy. 4. Why are its triplet-to-triplet d–d bands still relatively weak in ideal O h? Answer: Spin is conserved, but all d-derived initial and final states are gerade, so electric-dipole g→g absorption is Laporte-forbidden and borrows weak intensity.