10.2 Magnetic Properties of Coordination Compounds

Magnetic Behavior of Coordination Compounds

The magnetic properties of coordination compounds reveal critical information about their electronic structure, particularly the number of unpaired electrons and how d orbitals are occupied. By measuring a compound's magnetic moment, you can distinguish between high-spin and low-spin configurations, confirm oxidation states, and even determine coordination geometry.

Types of Magnetic Behavior

Paramagnetism occurs when unpaired electrons in a complex align with an external magnetic field, producing a weak attraction toward the field. Paramagnetic materials have a positive magnetic susceptibility.

Diamagnetism arises from the interaction of paired electrons with an external magnetic field, causing a weak repulsion. Diamagnetic materials have a negative magnetic susceptibility. Every compound has a diamagnetic contribution, but it's only the dominant effect when all electrons are paired.

Magnetic susceptibility ($\chi$) quantifies how strongly a material responds to an applied magnetic field. It's defined as the ratio of magnetization to the applied field strength, and its sign tells you the type of behavior: positive for paramagnetic, negative for diamagnetic.

Temperature-Dependent Magnetic Phenomena

The Curie law describes how magnetic susceptibility varies with temperature for simple paramagnetic materials:

$\chi = \frac{C}{T}$

• $\chi$ is the molar magnetic susceptibility
• $C$ is the Curie constant (proportional to the square of the effective moment, and thus to the number of unpaired electrons)
• $T$ is the absolute temperature in Kelvin

The key idea: as temperature increases, thermal motion randomizes the alignment of magnetic moments, so susceptibility decreases. A plot of $1/\chi$ vs. $T$ gives a straight line through the origin for an ideal Curie paramagnet.

For materials that show magnetic ordering at low temperatures, the Curie-Weiss law is a better model:

$\chi = \frac{C}{T - \theta}$

Here $\theta$ is the Weiss constant. A positive $\theta$ indicates ferromagnetic interactions, a negative $\theta$ indicates antiferromagnetic interactions, and $\theta = 0$ recovers the simple Curie law. Plotting $1/\chi$ vs. $T$ still gives a straight line, but it intercepts the temperature axis at $\theta$ rather than at the origin.

Temperature-independent paramagnetism (TIP) is a small, constant paramagnetic contribution that doesn't change with temperature. It arises from quantum mechanical mixing of the ground state with excited electronic states (a second-order Zeeman effect). TIP shows up in some transition metal complexes, notably low-spin octahedral $\text{Co}^{3+}$ ($d^6$), which has no unpaired electrons yet still displays a small paramagnetic susceptibility. TIP is usually small but needs to be subtracted from experimental data before calculating the true paramagnetic susceptibility.

Spin States and Magnetic Moments

The Spin-Only Magnetic Moment

The spin-only formula estimates the magnetic moment by considering only the spin angular momentum of unpaired electrons:

$\mu_{s.o.} = \sqrt{n(n+2)} \; \mu_B$

• $n$ = number of unpaired electrons
• $\mu_B$ = Bohr magneton ($9.274 \times 10^{-24} \; \text{J T}^{-1}$)

This gives the following reference values:

You should memorize this table (or be able to quickly compute it). On an exam, comparing a measured $\mu_{\text{eff}}$ to these values is how you determine the number of unpaired electrons and, from that, the spin state.

The spin-only formula works well for most first-row transition metal complexes because the ligand field quenches orbital angular momentum. For second- and third-row metals, and especially for lanthanides and actinides, orbital contributions become significant and the spin-only value underestimates the true moment.

The effective magnetic moment ($\mu_{\text{eff}}$) is what you actually measure experimentally. You extract it from susceptibility data using:

$\mu_{\text{eff}} = 2.828\sqrt{\chi_M T}$

where $\chi_M$ is the molar paramagnetic susceptibility (in cgs-emu units of $\text{cm}^3 \text{ mol}^{-1}$) and $T$ is in Kelvin. Comparing $\mu_{\text{eff}}$ to the spin-only prediction tells you whether orbital contributions matter for a given complex.

Spin Configurations in Octahedral Complexes

Whether a complex adopts a high-spin or low-spin configuration depends on the competition between two energies: the crystal field splitting energy ($\Delta_o$) and the electron pairing energy ($P$).

• High-spin complexes form when $\Delta_o < P$. Electrons spread across all five d orbitals before any pairing occurs, maximizing the number of unpaired electrons. This is typical with weak-field ligands like $\text{Cl}^-$, $\text{F}^-$, and $\text{H}_2\text{O}$.
• Low-spin complexes form when $\Delta_o > P$. Electrons fill the lower-energy $t_{2g}$ set first and pair up before occupying the higher-energy $e_g$ orbitals, minimizing unpaired electrons. Strong-field ligands like $\text{CN}^-$, $\text{CO}$, and $\text{NO}_2^-$ favor this arrangement.

The distinction only matters for $d^4$ through $d^7$ configurations in octahedral geometry. For $d^1$–$d^3$ and $d^8$–$d^{10}$, the electron count is the same regardless of field strength.

Tetrahedral complexes are almost always high-spin because $\Delta_t$ is inherently smaller (roughly $\frac{4}{9}\Delta_o$), so it rarely exceeds the pairing energy. This is worth remembering: if a problem gives you a tetrahedral complex, default to high-spin unless you're explicitly told otherwise.

Here's a concrete example to tie things together. Consider $\text{Fe}^{3+}$ ($d^5$) in two octahedral environments:

• $[\text{FeF}_6]^{3-}$: $\text{F}^-$ is a weak-field ligand, so this is high-spin with 5 unpaired electrons. Predicted $\mu_{s.o.} = 5.92 \; \mu_B$. Experimental value is about $5.9 \; \mu_B$.
• $[\text{Fe(CN)}_6]^{3-}$: $\text{CN}^-$ is a strong-field ligand, so this is low-spin with 1 unpaired electron. Predicted $\mu_{s.o.} = 1.73 \; \mu_B$. Experimental value is about $2.3 \; \mu_B$, higher than predicted because this complex has a $^2T_{2g}$ ground state with an unquenched orbital contribution.

That second example shows exactly when the spin-only formula starts to break down.

Spin crossover is a phenomenon where a complex switches between high-spin and low-spin states in response to changes in temperature, pressure, or irradiation with light. This occurs when $\Delta_o \approx P$, so small perturbations tip the balance. Iron(II) complexes ($d^6$) are the most well-studied examples, and spin crossover materials are actively researched for molecular switching and data storage applications.

Magnetic Ordering

Types of Magnetic Ordering

Most mononuclear coordination compounds are simple paramagnets. But in extended solids and polynuclear complexes, cooperative interactions between magnetic centers produce ordered magnetic states.

Ferromagnetism occurs when neighboring magnetic moments align parallel to each other, producing a strong net magnetization that persists even without an external field. Classic examples include metallic iron (Fe), cobalt (Co), and nickel (Ni). Above the Curie temperature ($T_C$), thermal energy overcomes the cooperative alignment and the material becomes paramagnetic.

Antiferromagnetism occurs when neighboring moments align antiparallel, canceling each other out so the net magnetization is zero in the absence of a field. Manganese oxide (MnO) is a textbook example, with $T_N = 118 \; \text{K}$. Above the Néel temperature ($T_N$), antiferromagnetic ordering breaks down and the material behaves as a paramagnet.

Ferrimagnetism is worth knowing as well. Here, neighboring moments align antiparallel but are unequal in magnitude, so there's a net magnetization (unlike in a pure antiferromagnet). Magnetite ($\text{Fe}_3\text{O}_4$) is the classic example.

Both ferromagnetic and antiferromagnetic ordering arise from exchange interactions between adjacent metal centers. These interactions can be direct (metal–metal overlap) or mediated through bridging ligands (superexchange). In coordination chemistry, superexchange through oxide or halide bridges is the more common pathway. The Goodenough-Kanamori rules predict whether a superexchange pathway will be ferromagnetic or antiferromagnetic based on the geometry and orbital occupancy of the metal-bridge-metal unit.

Advanced Magnetic Properties

Spin-Orbit Coupling

Spin-orbit coupling is the interaction between an electron's spin angular momentum and its orbital angular momentum. It causes the effective magnetic moment to deviate from the spin-only prediction.

For first-row transition metals, spin-orbit coupling is relatively weak, which is why the spin-only formula works reasonably well. However, even here you'll see systematic deviations. The key pattern: complexes with $T$ ground states (such as octahedral $d^1$, $d^2$, high-spin $d^6$, and $d^7$) show larger orbital contributions than those with $A$ or $E$ ground states (like $d^3$, $d^5$ high-spin, $d^8$). The reason is that $T$ states have orbital degeneracy, which means orbital angular momentum is not fully quenched by the ligand field.

For heavier metals (4d, 5d series) and especially for lanthanides and actinides, spin-orbit coupling is strong and you must use the full expression involving the total angular momentum quantum number $J$:

$\mu_{\text{eff}} = g_J\sqrt{J(J+1)} \; \mu_B$

where $g_J$ is the Landé g-factor:

$g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$

For lanthanides, this expression gives excellent agreement with experiment (with the notable exception of $\text{Sm}^{3+}$ and $\text{Eu}^{3+}$, where low-lying excited $J$ states mix into the ground state).

Experimental Measurement: The Gouy Balance

The Gouy balance is a classic method for measuring magnetic susceptibility:

1. A finely powdered sample is packed into a glass tube of uniform cross-section.
2. The tube is suspended from a balance between the poles of an electromagnet, with one end at the center of the field (maximum field strength) and the other end outside the field.
3. The electromagnet is turned on, creating an inhomogeneous field along the length of the sample.
4. A paramagnetic sample is pulled into the stronger field region, increasing its apparent weight. A diamagnetic sample is pushed out, decreasing its apparent weight.
5. The change in apparent weight ($\Delta w$) is measured and used to calculate the volume magnetic susceptibility, which is then converted to molar susceptibility.

The Gouy method works for solids and liquids and remains a straightforward way to determine whether a complex is high-spin or low-spin. More modern techniques like SQUID magnetometry (Superconducting Quantum Interference Device) offer far higher sensitivity and can measure very small samples, making them the standard in research labs. The Evans NMR method is another practical technique you may encounter: it determines $\chi$ in solution by measuring the shift of a reference signal (like TMS) caused by the paramagnetic solute. Regardless of the method, the goal is the same: get $\chi_M$, calculate $\mu_{\text{eff}}$, and compare to spin-only values to determine the electronic structure.