How to Find Transition Metal Cation Electron Configurations

Before diving into crystal field theory, a quick but essential review of transition metal cation electron configurations is in order. The number of d electrons a transition metal ion possesses is central to everything in this lesson — and the next one. Two rules govern this calculation, and getting them right every time is non-negotiable.

Scandium 3+ (Sc³⁺): A d⁰ Ion

Neutral scandium (atomic number 21) has the configuration [Ar] 4s² 3d¹. Forming Sc³⁺ requires removing three electrons. The two 4s electrons go first, then the single 3d electron, leaving a configuration identical to argon. The key point: Sc³⁺ is a d⁰ ion — it has no d electrons at all — which will bear on its optical properties in the next lesson.

Iron 3+ (Fe³⁺): A d⁵ Ion

Iron's neutral configuration is [Ar] 4s² 3d⁶. To form Fe³⁺, remove the two 4s electrons first, then one 3d electron. The result is [Ar] 3d⁵ — five d electrons. Every coordination compound containing Fe³⁺ will have exactly five d electrons to distribute among the crystal field orbitals.

Manganese 2+ (Mn²⁺): Another d⁵ Ion

Manganese is [Ar] 4s² 3d⁵. Removing the two 4s electrons to form Mn²⁺ leaves [Ar] 3d⁵ — again, five d electrons, the same count as Fe³⁺.

2 Periodic Table Exceptions: Chromium and Copper

Chromium and copper — along with some of their congeners — do not follow the standard filling order. Understanding their anomalous configurations is necessary before you can correctly determine the d electron count for their ions.

Chromium 3+ (Cr³⁺): A d³ Ion

Starting from the actual configuration [Ar] 4s¹ 3d⁵, removing three electrons from Cr to form Cr³⁺ means taking the 4s electron first, then two d electrons, yielding [Ar] 3d³ — three d electrons.

Copper 1+ (Cu⁺): A d¹⁰ Ion

Starting from copper's actual configuration [Ar] 4s¹ 3d¹⁰, removing the single 4s electron to form Cu⁺ gives [Ar] 3d¹⁰ — a d¹⁰ ion with a completely full d subshell.

The 5 d Orbitals and Their Orientation in Space

Crystal field theory is fundamentally a theory about how incoming ligands interact differently with different d orbitals. To understand that interaction, you need a clear mental picture of where the five d orbitals point in space.

Names and Shapes of the 5 d Orbitals

The five 3d orbitals are named d_xy, d_yz, d_xz, d_x²−y², and d_z². The first four adopt a four-leaf clover shape with four lobes of electron density.

• d_xy — four lobes in the xy plane, pointing between the x and y axes
• d_yz — four lobes in the yz plane, pointing between the y and z axes
• d_xz — four lobes in the xz plane, pointing between the x and z axes
• d_x²−y² — four lobes in the xy plane, pointing directly along the x and y axes
• d_z² — electron density concentrated along the z axis

2 Groups: On-Axis vs. Between-Axis Orbitals

This geometry creates a natural division into two groups: the between-axis set (d_xy, d_yz, d_xz) and the on-axis set (d_x²−y² and d_z²). This distinction is the geometric foundation of crystal field theory.

Crystal Field Splitting in Octahedral Complexes

In a free transition metal ion, all five d orbitals are degenerate — they have identical energies. When six ligands bind in an octahedral arrangement, that degeneracy breaks and the d orbitals split into two distinct energy levels. This is the core prediction of crystal field theory for octahedral coordination geometry.

The t_2g and e_g Orbital Sets

The lower-energy set — the t_2g orbitals — consists of d_xy, d_yz, and d_xz. These between-axis orbitals are stabilized relative to the average d orbital energy because they avoid the incoming ligands.

The higher-energy set — the e_g orbitals — consists of d_x²−y² and d_z². These on-axis orbitals are destabilized because their lobes of electron density point directly at the approaching ligands, generating electron-electron repulsion.

Crystal Field Splitting Energy (Δ_o)

The energy gap between the t_2g and e_g sets is called the crystal field splitting energy, symbolized Δ_o (delta-o, where the subscript "o" stands for octahedral). You may also encounter the term crystal field stabilization energy (CFSE) — both refer to the same quantity.

The magnitude of Δ_o depends on the specific metal ion and the ligands bound to it. That magnitude — whether it is relatively small or relatively large compared to another energy quantity called the pairing energy — determines how electrons are distributed across the two sets, and therefore whether the complex is high spin or low spin.

High Spin vs. Low Spin in Octahedral Complexes

Pairing Energy and the Filling Decision

When two electrons occupy the same orbital they repel each other. The energy cost of forcing two electrons to share one orbital is called the pairing energy. Every time electrons are filled into d orbitals in a split crystal field, the system faces a competition between two costs:

• The cost of moving an electron up to the higher-energy e_g set (the splitting energy, Δ_o)
• The cost of pairing two electrons together in the same lower-energy t_2g orbital (the pairing energy)

Whichever cost is smaller determines which filling pattern the electrons adopt.

High Spin: Weak Field Ligands and Small Δ_o

When Δ_o is smaller than the pairing energy, it is energetically cheaper to place an electron in the higher e_g set than to pair it with an electron already in t_2g. Electrons therefore spread out across all five d orbitals — one per orbital — before any pairing occurs. This produces the maximum number of unpaired electrons and is called the high spin configuration.

Ligands that create a small Δ_o are called weak field ligands. They do not bind especially tightly to the metal, so the electrostatic perturbation they impose is modest.

Low Spin: Strong Field Ligands and Large Δ_o

When Δ_o is larger than the pairing energy, it is energetically cheaper to pair electrons within the t_2g set than to promote them to the e_g set. Electrons therefore fill all three t_2g orbitals completely — pairing up — before any electron enters the e_g set. This produces the minimum number of unpaired electrons and is called the low spin configuration.

Ligands that create a large Δ_o are called strong field ligands. They bind tightly to the metal and impose a large electrostatic perturbation on the d orbitals.

The Spectrochemical Series: Ranking Ligand Field Strength

The spectrochemical series ranks ligands by their tendency to produce a large or small Δ_o. At the strong-field end of the series sit ligands like CN⁻ (cyanide) and CO (carbon monoxide), which bind tightly and generate a large splitting. At the weak-field end sit ligands like I⁻ (iodide) and SCN⁻ (thiocyanate), which bind loosely and generate a small splitting. A partial ordering looks like this:

An important caveat: the series ranks ligand tendencies, not absolute categories. Whether a given ligand actually produces high spin or low spin depends on the metal ion as well as the ligand. Some metal ions bind all ligands more tightly; others bind them more loosely. The cutoff between weak field and strong field behavior shifts for every metal. Cyanide has a strong tendency to be a strong field ligand, and SCN⁻ has a strong tendency to be weak field — but neither label is guaranteed without knowing the metal.

Identifying High Spin vs. Low Spin by d Electron Count

In practice, you can determine whether a complex is high spin or low spin by measuring its magnetic properties — specifically by counting the number of unpaired electrons. The two filling patterns only produce different numbers of unpaired electrons for certain d electron counts.

Which d Electron Counts Allow You to Distinguish High Spin from Low Spin?

Fill the d orbitals systematically for both high spin and low spin across the full range of d⁰ through d¹⁰:

Crystal Field Splitting in Tetrahedral Complexes

Tetrahedral complexes produce an inverted d orbital splitting pattern relative to octahedral complexes. In a tetrahedral geometry, the four ligands approach the metal from directions that lie between the Cartesian axes — at the alternating corners of a cube, roughly 109.5° apart. This means the ligands now point toward the lobes of d_xy, d_yz, and d_xz, not toward d_x²−y² and d_z².

As a result, the energy ordering is reversed: the former between-axis set (d_xy, d_yz, d_xz) moves to higher energy, and the former on-axis set (d_x²−y² and d_z²) sits at lower energy. The crystal field splitting energy for a tetrahedral complex is labeled Δ_t.

Why All Tetrahedral Complexes Are High Spin

There is a crucial practical consequence: the tetrahedral splitting energy Δ_t is inherently small — for comparable ligands, Δ_t ≈ 4/9 Δ_o, less than half the octahedral value. This small splitting is always less than the pairing energy, so electrons always spread out across the available orbitals before pairing.

Crystal Field Splitting in Square Planar Complexes

Square planar coordination geometry produces a more elaborate d orbital splitting pattern than either octahedral or tetrahedral geometries. The full energy ordering of the five d orbitals in a square planar field is unique and worth memorizing at the advanced level, though the most important practical facts are these:

Square planar complexes are almost exclusively found with metal ions that have eight d electrons (d⁸) — examples include Ni²⁺, Pd²⁺, and Pt²⁺. The large crystal field splitting of square planar geometry stabilizes a configuration in which the highest-energy orbital, d_x²−y², remains empty while all other d orbitals are doubly occupied.

Hybridization of Square Planar Complexes: dsp²

The hybridization model provides a complementary picture. A d⁸ ion has one empty d orbital in its current shell. Four incoming ligands need four empty hybrid orbitals to donate their electron pairs into. The metal uses that one empty d orbital, then borrows the s and two p orbitals from the next shell. Mixing one d + one s + two p orbitals produces four equivalent hybrid orbitals — named after their component orbitals — giving a hybridization of dsp².

2 Hybridizations of Octahedral Complexes: d²sp³ vs. sp³d²

Octahedral complexes require six empty hybrid orbitals to accommodate the six ligand electron pairs. The identity of those six orbitals depends on whether the complex is low spin or high spin — which is why octahedral complexes have two possible hybridizations.

Low Spin Octahedral Hybridization: d²sp³ (Inner Orbital Complex)

In a low spin octahedral complex, the t_2g set is fully or partially occupied but the e_g orbitals are empty. These two vacant inner d orbitals are the lowest-energy available orbitals. The six hybrid orbitals are formed from these two d orbitals plus the s and three p orbitals of the next shell:

2 (inner d) + 1 s + 3 p = d²sp³

Because the d orbitals involved belong to the same shell as the d electrons, this is called an inner orbital complex.

High Spin Octahedral Hybridization: sp³d² (Outer Orbital Complex)

In a high spin octahedral complex, all five d orbitals in the current shell are occupied (none are empty). The six hybrid orbitals must therefore be built entirely from the next shell up: the s orbital, three p orbitals, and two d orbitals from that higher shell:

1 s + 3 p + 2 (outer d) = sp³d²

Because the d orbitals involved come from a higher shell than the d electrons, this is called an outer orbital complex.