Paramagnetism and Diamagnetism in Complexes

Introduction

A metal complex’s d-electron diagram makes a testable prediction. If one or more electrons remain unpaired, their spin magnetic moments usually produce paramagnetism. If all electrons are paired, the complex is generally diamagnetic in the elementary model. Magnetic measurements therefore help distinguish high- and low-spin assignments, but interpreting a real sample requires care about temperature, coupling and other contributions.

Core explanation

An unpaired electron has a magnetic moment related to its spin. In an applied field, the moments of a paramagnetic sample can align to some extent, causing a net attraction into a region of stronger field. Thermal motion resists alignment, so the response often depends on temperature. A diamagnetic substance has no permanent unpaired spin moment in the simple picture; an applied field induces a small opposing response, so it is weakly repelled from a stronger-field region.

For a mononuclear complex, count unpaired electrons after establishing oxidation state, d count, geometry and spin state. High-spin octahedral Fe²⁺ is d⁶, t₂g⁴e g², with four unpaired electrons and is paramagnetic. Low-spin [Fe(CN)₆]⁴⁻ is also Fe²⁺ d⁶, but t₂g⁶ has no unpaired electrons and is diamagnetic. The shared formal d count makes their contrasting magnetic response strong evidence for a ligand-field difference.

The qualitative rule is not “all transition metals are paramagnetic.” Zn²⁺ is d¹⁰ and has no unpaired d electrons. Low-spin d⁶ and many square-planar d⁸ complexes are also diamagnetic. Conversely, not every ion with an unpaired electron has the same susceptibility. The number of unpaired electrons, the effective moment per centre, interactions between centres and temperature all influence the measured response.

A Gouy or Evans-type measurement can estimate magnetic susceptibility. A sample’s apparent weight can change in a field gradient, or a solution’s NMR signal can shift because paramagnetic solute changes the local field. These methods require calibration and subtraction of background diamagnetism from ligands, counterions and solvent. They do not simply display “three unpaired electrons” on an instrument screen. An effective magnetic moment is inferred from susceptibility under a model.

Multinuclear complexes and solids can complicate interpretation. Two paramagnetic metal centres may couple antiferromagnetically so their moments partly cancel, while ferromagnetic coupling can enhance collective response. A small bulk moment therefore does not automatically prove every metal centre is individually paired. Spin–orbit coupling can also add an orbital contribution, causing measured moments to deviate from the spin-only formula.

The magnetic response provides structural evidence when paired with chemistry. A four-coordinate d⁸ Ni²⁺ ion with roughly two unpaired electrons supports a tetrahedral description; a diamagnetic d⁸ complex supports a square-planar description. This is an inference rather than a unique theorem because an unusual ligand field, excited state or magnetic coupling can alter the observation. Spectroscopy and crystallography strengthen the assignment.

Step-by-step reasoning

Determine oxidation state and d count. Choose a geometry and fill the split orbitals, considering Δ versus P where a spin choice exists. Count arrows without opposite-spin partners. Predict paramagnetism for n>0 and diamagnetism for n=0, then check whether magnetic coupling, orbital contribution or temperature could affect a measured value.

Visual explanation

Draw one box with two opposite arrows for a paired orbital and one box with a single arrow for an unpaired orbital. Beside them show a magnetic field gradient: an unpaired-moment sample moves toward the stronger field, while a paired sample has only a much weaker opposing response.

Real-world analogy

Imagine a crowd carrying small compasses. If many compasses are free, an external field can orient them and create a collective direction. If every compass is tied to an equal opposite partner, little permanent direction remains. Thermal jostling prevents perfect alignment even for the free compasses.

Real-world example

The near-diamagnetic behaviour of ferrocyanide, [Fe(CN)₆]⁴⁻, supports low-spin Fe²⁺ d⁶ with t₂g⁶. Aqueous Fe²⁺ has a much larger paramagnetic response because water’s weaker field permits t₂g⁴e g² and four unpaired d electrons.

Why?

Why does a paired orbital not contribute the same permanent spin moment as two separate unpaired electrons? The two electrons in one orbital have opposite spins, so their spin angular momenta cancel in the elementary picture. An applied field still induces a small diamagnetic response.

Common misconception

“Diamagnetic means absolutely no interaction with a magnetic field.” Diamagnetic samples weakly oppose an applied field. It means no permanent unpaired-spin contribution dominates, not zero response.

Worked example

For [Co(NH₃)₆]³⁺, ammonia is neutral, so cobalt is +3 and d⁶. If the complex is low spin, fill t₂g⁶e g⁰. All three t₂g orbitals contain pairs, so n=0 and the simple prediction is diamagnetism. If it were high-spin d⁶, t₂g⁴e g² would have n=4 and be strongly paramagnetic; magnetic data can distinguish the two assignments.

Quick check

1. What magnetic class is expected for a mononuclear complex with two unpaired electrons? Answer: Paramagnetic, because it has permanent electron spin moments. 2. Can a bulk sample with a small moment contain unpaired electrons on individual metal centres? Answer: Yes. Antiferromagnetic coupling between centres can partly cancel their moments.

Exam focus

Count electrons from a justified d configuration, not from the number of ligands. Use magnetic evidence together with geometry and ligand-field reasoning, and distinguish a qualitative class from a precise measured moment.

Advanced insight

Susceptibility is a macroscopic response averaged over many molecules and thermal states. Converting it into an effective moment assumes a magnetic model; spin–orbit effects, coupled centres or a temperature-dependent spin equilibrium can make a simple n-electron interpretation inaccurate.

Summary

Unpaired electrons generally make a complex paramagnetic, while fully paired electronic configurations are diamagnetic in the simple model. Measurements test spin and geometry assignments but require corrections and attention to coupling.

Practice questions

1. Classify ideal octahedral high-spin d⁵ and low-spin d⁶. Answer: High-spin d⁵ has five unpaired electrons and is paramagnetic. Low-spin d⁶ has t₂g⁶, no unpaired electrons and is diamagnetic. 2. Why should ligand and solvent background be corrected in a susceptibility experiment? Answer: Those parts of the sample also respond to the field, often diamagnetically. Their contribution must be removed to infer the metal-centre response. 3. A four-coordinate Ni²⁺ complex is diamagnetic. What geometry is supported, and what else should be checked? Answer: Square-planar d⁸ is supported; spectroscopy and structural evidence should be checked because magnetic data alone are an inference, not a complete structure determination.