Orgel Diagrams for d¹, d⁴, d⁶ and d⁹

Introduction

The D-ground-term Orgel diagram is the simplest many-electron term diagram. It covers free-ion d¹, d⁴, d⁶ and d⁹ ground terms in the weak-field, high-spin setting. The D term becomes one E and one T₂ component in a cubic field, so the elementary diagram predicts one spin-allowed ligand-field transition between them. Learning which component is lower for each electron count makes the diagram useful rather than a pair of unlabeled lines.

Core explanation

The free-ion ground terms are ^2D for d¹ and d⁹, and ^5D for d⁴ and d⁶. Under octahedral symmetry a D term splits into E g+T₂g, preserving multiplicity. In the simplified octahedral sequence, d¹ has lower ^2T₂g and upper ^2E g. High-spin d⁶ likewise has lower ^5T₂g and upper ^5E g. Their ground states are triply orbital-degenerate T states. For high-spin d⁴ and d⁹, the order is reversed: ^5E g or ^2E g is ground and the corresponding T₂g state is excited. This reverses which side of a D Orgel plot applies.

The pattern can be related to electron occupancy. Octahedral d¹ is t₂g¹; exciting its electron toward e g gives the upper E-like state. High-spin d⁶ is t₂g⁴e g², with a comparable T-ground weak-field pattern. High-spin d⁴ is t₂g³e g¹ and d⁹ is t₂g⁶e g³; their E-ground states reflect the way the D parent splits for those occupancies. The many-electron term labels, rather than a single box arrow, remain the precise transition labels. For idealised high-spin D cases, the separation of the two components is closely associated with the relevant ligand-field splitting scale, so one spin-allowed band can provide an estimate of Δ for a simple first analysis.

Orgel diagrams are qualitative correlation diagrams for weak-field/high-spin cases. Their vertical axis is energy but usually has no quantitative calibrated scale; their horizontal axis represents field strength and may display octahedral and tetrahedral sides. They show spin-allowed transitions from the ground term, not all singlet/triplet or other spin-forbidden levels, and they cannot describe a low-spin ground-state crossover. For quantitative Δ/B analysis or strong-field spectra, Tanabe–Sugano diagrams are more suitable.

The tetrahedral side reverses the one-electron ordering compared with octahedral t₂g/e g, and tetrahedral states omit g because there is no inversion centre. One must first choose geometry and d count, then identify which branch of the diagram is ground. Merely reading “left” or “right” from a reproduced figure is risky because diagrams may be mirrored or use different horizontal conventions. The state labels and occupancy provide a safer guide.

The predicted one spin-allowed transition does not imply that a real spectrum has only one visible feature. Vibronic structure, Jahn–Teller distortion, spin-forbidden lines, charge transfer, and low-symmetry splitting can add features or broaden the basic band. d⁹ Cu²⁺ in particular often undergoes substantial tetragonal distortion, so its broad visible absorption may not fit an exact cubic two-level scheme. The Orgel diagram remains a useful parent-state model, not a complete line-shape calculation.

Selection rules determine intensity. In ideal octahedral symmetry both E g and T₂g states are gerade, so their d–d transition is Laporte-forbidden even though spin is conserved. Vibronic distortion gives it weak intensity. In tetrahedral symmetry there is no strict g→g prohibition, so analogous spin-allowed bands can be stronger. This intensity prediction is independent of whether the E or T component lies lower.

Step-by-step reasoning

Determine oxidation state and d count. Confirm that the case is one of d¹, high-spin d⁴, high-spin d⁶ or d⁹ and choose octahedral or tetrahedral symmetry. Write the D parent with its spin multiplicity, split it into E and T₂ components, and use the correct ground ordering. Draw one arrow from the ground to the other component. Then test parity and consider whether distortion or strong-field conditions limit the simple interpretation.

Visual explanation

Draw a central horizontal free-ion D line at zero field. To one octahedral side let its components separate upward and downward into T₂g ground and E g excited, labelled d¹/d⁶; on the other show E g ground and T₂g excited, labelled d⁴/d⁹. Keep each pair's spin superscript. Place one upward transition arrow between its two states and a note that actual energy scale is qualitative.

Real-world analogy

Two teams can start tied before the rules of a game are changed. A new rule splits their scores; which team wins depends on the players present. The D parent always splits into E and T₂ symmetry groups, but d count determines which of those groups is lower. Knowing only that a split occurs is not enough to choose the absorption arrow.

Real-world example

In an approximately octahedral Ti³⁺ aqua ion, d¹ leads to ^2T₂g ground and ^2E g excited states, so the principal spin-allowed d–d band is ^2T₂g→^2E g. For high-spin Fe²⁺ in a sufficiently weak octahedral field, d⁶ gives the analogous ^5T₂g→^5E g transition. Their band energies need not be the same because ligand identity, metal charge and bonding affect Δₒ.

Why?

Why are d⁴ and d⁶ placed on opposite ordering sides even though both have ^5D free-ion ground terms? Their different electron occupancies interact with the octahedral orbital splitting differently. Correlation preserves the same E g and T₂g symmetry types, but not which component receives the lower energy.

Common misconception

“One Orgel arrow means exactly one observed peak and nothing else.” The diagram retains only the main spin-allowed D-parent correlation in an idealised field. Distortion, vibrations, charge transfer or other electronic states can produce extra measured features.

Worked example

Consider a high-spin octahedral Mn³⁺ ion, which is d⁴. Its free-ion ground term is ^5D, so the octahedral components are ^5E g and ^5T₂g. The D Orgel ordering for octahedral d⁴ places ^5E g below ^5T₂g. The spin-allowed ligand-field transition is ^5E g→^5T₂g. Both labels carry g parity, so ideal electric-dipole absorption is Laporte-forbidden and must borrow intensity; possible Jahn–Teller distortion can further affect the observed band.

Quick check

1. What is the octahedral d⁹ D-term transition in the cubic model? Answer: ^2E g→^2T₂g, because the ^2E g component is ground for octahedral d⁹.

Exam focus

Label the ground state explicitly and include spin superscripts and g where appropriate. State “high spin” for d⁴ and d⁶. Use the diagram for a qualitative number and identity of spin-allowed bands, then mention limitations if a distorted d⁹ complex or strong-field crossover is involved.

Advanced insight

The separation of D components can look deceptively like a one-electron Δ. Term energies are many-electron energies, and the approximation works cleanly here because only two spin-allowed components of the same D parent are being compared. In more complex F-parent diagrams, additional terms and configuration interaction make band energies depend on electron-repulsion parameters as well as field strength.

Summary

The D-ground configurations d¹, high-spin d⁴, high-spin d⁶ and d⁹ each give E and T₂ components and one principal spin-allowed Orgel transition. Octahedral d¹/d⁶ have T₂g ground; d⁴/d⁹ have E g ground. The diagram is qualitative and does not account for all real spectral features.

Practice questions

1. State the expected principal octahedral transition for high-spin Fe²⁺. Answer: Fe²⁺ is d⁶; its weak-field ground is ^5T₂g and the principal D-parent spin-allowed transition is ^5T₂g→^5E g. 2. Why can the d⁹ Cu²⁺ band be wider than a simple D Orgel diagram suggests? Answer: d⁹ often undergoes Jahn–Teller tetragonal distortion. The lower symmetry can split cubic states and vibrations can merge several contributions into a broad envelope. 3. Compare the parity restriction for the principal octahedral and tetrahedral D-term bands. Answer: Octahedral d–d states are g, so the ideal electric-dipole transition is Laporte-forbidden. Tetrahedral complexes lack inversion and therefore do not have that strict g→g prohibition. 4. Can a D Orgel diagram alone predict a low-spin d⁶ spectrum? Answer: No. It is a high-spin weak-field diagram; low-spin d⁶ has a different ground state and requires a more complete term-energy treatment.