Orgel Diagrams for d², d³, d⁷ and d⁸

Introduction

Unlike a D ground term, an F ground term produces three octahedral components. Moreover, the same d configurations also possess a higher P term of the same spin multiplicity. The F Orgel diagram therefore predicts three principal spin-allowed transitions for high-spin d², d³, d⁷ and d⁸ cases. It explains why a spectrum with several d–d bands cannot be described by a single t₂g-to-e g arrow or by one number Δₒ alone.

Core explanation

Free-ion d² and d⁸ have ^3F ground and ^3P excited terms; d³ and d⁷ have ^4F ground and ^4P excited terms. Under O h, F gives A₂g+T₁g+T₂g, whereas P gives T₁g. Spin multiplicity is retained for this first treatment. Thus a d² triplet system contains two different ^3T₁g states, one F-derived and one P-derived. The d³ quartet system likewise contains ^4T₁g(F) and ^4T₁g(P). Parenthetical F and P indicate ancestry and help distinguish equal-symmetry states.

For octahedral d³ and d⁸ in the usual weak-field picture, the ground state is ^4A₂g(F) or ^3A₂g(F). The three principal spin-allowed targets are T₂g(F), T₁g(F) and T₁g(P), each with the matching spin superscript. For octahedral d² and high-spin d⁷, the ground state is T₁g(F), and the three principal spin-allowed targets are T₂g(F), A₂g(F) and T₁g(P). The energy order of the first two excited targets and the observed visibility of the high-energy P-derived band must be read from the appropriate diagram and spectrum rather than assumed solely from this list.

The two T₁g states can interact because they have the same symmetry and spin. As field strength changes and their unperturbed energies approach, their actual energy curves repel and avoid crossing. Their wavefunctions exchange some F- and P-parent character. This is why a simple straight-line extrapolation can place an upper band inaccurately. The F–P free-ion separation involves the Racah B parameter, so the third band's position carries electron-repulsion information in addition to ligand-field splitting.

Orgel diagrams remain qualitative and high-spin. They display spin-allowed term trajectories without providing a reliable quantitative E/B scale or all spin-forbidden states. For d² and d⁷, an apparently simple first transition need not equal Δₒ exactly because term interactions shift state energies. For extracting Δₒ and B from measured bands, Tanabe–Sugano analysis is more appropriate. A charge-transfer band may obscure the highest d–d band, so “three predicted” does not guarantee three isolated experimental peaks.

Tetrahedral relatives use similar F/P parent terms but omit g/u labels and reverse the effective ordering relative to the octahedral counterpart in the Orgel correlation. For example, octahedral d³/d⁸ and tetrahedral d²/d⁷ occupy corresponding sides of the standard F diagram, while the other geometry/count pair occupies the opposite side. Rather than memorising a figure's left and right, derive geometry, d count, parent multiplicity and ground-state symmetry before assigning arrows.

Every octahedral F-parent d–d arrow is g→g and therefore Laporte-forbidden in the ideal electric-dipole model even when spin-allowed. Vibronic borrowing makes the bands observable. Tetrahedral analogues can be stronger because they lack inversion. Band intensities are still modified by geometry, covalency and mixing, so they should corroborate, not replace, state assignments.

Step-by-step reasoning

Count d electrons and establish geometry and spin state. Choose the ^3F/^3P pair for d² or d⁸ and ^4F/^4P for d³ or d⁷. Split F into A₂, T₁ and T₂, and P into another T₁. Identify A₂ ground for octahedral d³/d⁸ or T₁(F) ground for octahedral d²/high-spin d⁷. Draw arrows only to excited states of the same multiplicity. Note possible T₁ mixing and avoid treating all arrow energies as equal to Δₒ.

Visual explanation

Draw an F line and a higher P line at zero field. Let F fan into A₂g, T₁g(F) and T₂g while P becomes T₁g(P). Show the two T₁g curves bending apart where they approach. In an octahedral d³ inset place A₂g lowest and draw three upward arrows to T₂g, T₁g(F) and T₁g(P), with the same spin superscript on every label.

Real-world analogy

Several runners start from one team while one runner starts from another. A new course separates the original team into different lanes; two runners entering lanes of the same type can influence one another's routes. The F and P parents similarly create several symmetry states, and equal-symmetry T₁ states mix rather than retain perfectly independent identities.

Real-world example

Octahedral Cr³⁺ is d³ and has a ^4A₂g(F) ground state. Its principal spin-allowed absorptions can be assigned to ^4T₂g(F), ^4T₁g(F) and ^4T₁g(P). The first two may lie in visible or near-visible regions depending on ligands; the third may be in the ultraviolet and could overlap a charge-transfer feature. A measured two-band visible spectrum therefore need not contradict a three-transition term scheme.

Why?

Why do F configurations have three spin-allowed targets rather than two? Their ground F parent splits into three O h states, and a distinct same-spin P parent contributes another T₁g state. One F-derived component is ground, leaving two F-derived targets plus the P-derived target.

Common misconception

“Two T₁g states are duplicate labels for one level.” The F and P parent terms supply distinct states with the same O h symmetry. Their equal labels permit mixing, but they remain two energy levels and can support separate transitions from an appropriate ground state.

Worked example

Predict the principal spin-allowed transitions of ideal octahedral Ni²⁺. Ni²⁺ is d⁸, whose free-ion ground term is ^3F and same-spin excited parent is ^3P. The octahedral ground is ^3A₂g(F). The three targets are ^3T₂g(F), ^3T₁g(F) and ^3T₁g(P). Write arrows from ^3A₂g(F) to each. All preserve S=1 but are g→g, so expect relatively weak d–d bands; quantitative energies need a fuller term diagram.

Quick check

1. What is the F-ground state of octahedral d³ in the usual weak-field case? Answer: ^4A₂g(F), because d³ has quartet F parentage and the A₂g component is lowest in that octahedral correlation.

Exam focus

Keep the d³/d⁸ A₂g-ground family separate from the d²/d⁷ T₁g-ground family. Label parent terms in parentheses when two T₁g states appear. Orgel predicts possible spin-allowed arrows qualitatively; do not claim exact Δₒ or B from a sketch alone.

Advanced insight

The avoided crossing of equal-symmetry T₁ states is an example of configuration interaction. Their wavefunctions are linear combinations of the basis states that correlate with the free-ion parents. The parent label remains useful near weak field, but becomes less exact where mixing is strong. This is one reason diagram branches curve.

Summary

High-spin d², d³, d⁷ and d⁸ have F ground and same-spin P excited parent terms. The F splitting plus P-derived T₁ state provides three principal spin-allowed targets. Their energies reflect ligand field and electron repulsion, and equal-symmetry T₁ states can mix.

Practice questions

1. List the three principal octahedral d³ spin-allowed target states from ^4A₂g(F). Answer: ^4T₂g(F), ^4T₁g(F) and ^4T₁g(P). The shared quartet superscript shows that each arrow preserves spin. 2. Why might an octahedral d⁸ spectrum show fewer than three resolved d–d maxima? Answer: A transition can overlap another broad band or intense charge transfer, and the highest transition may fall outside the measured wavelength window. Resolution and vibronic width also affect visible maxima. 3. Does the first d² transition necessarily equal Δₒ? Answer: No. It connects many-electron terms whose energies include interelectronic repulsion and configuration interaction. A quantitative ligand-field analysis is needed to extract Δₒ reliably. 4. Why are there two T₁g trajectories in the F Orgel diagram? Answer: F and P parent terms each yield a T₁g component of the same spin multiplicity; the trajectories can mix and avoid crossing.