Pairing Energy and Magnetic Moments

Magnetic moment measurements remain a routine, cheap, and diagnostic tool in synthetic inorganic and bioinorganic chemistry labs: the Evans NMR method, requiring only a standard NMR spectrometer and no specialized magnetometer, lets a chemist confirm the oxidation state and spin state of a newly synthesized complex in minutes, cross-checking against the same $\Delta_o$-versus-$P$ logic developed here. In bioinorganic chemistry, the spin state of iron in heme proteins is a direct functional switch. Deoxyhemoglobin's iron(II) is five-coordinate and high-spin ($S=2$); on binding O₂, the iron becomes six-coordinate and switches to low-spin ($S=0$), a spin-state change that physically pulls the iron into the porphyrin plane and triggers the conformational signal underlying cooperative oxygen binding — the same electronic competition between $\Delta_o$-like ligand field splitting and pairing energy, now doing physiological work. In molecular magnetism and materials science, spin-crossover complexes are engineered so that $\Delta_o$ and $P$ are deliberately balanced to be nearly equal, so that temperature, pressure, or light can flip the complex between high-spin and low-spin states — the basis of proposed molecular switches, displays, and memory devices, where the same $\Delta_o$ vs $P$ inequality developed here becomes a tunable, externally-triggerable bistability rather than a fixed property of one compound. In catalysis and organometallic chemistry, predicting whether an intermediate is high-spin or low-spin (and hence how many d-orbitals are available for bonding, and how reactive/labile the complex will be) is a standard first diagnostic step before invoking a mechanism — low-spin $d^6$ octahedral complexes, with fully occupied nonbonding $t_{2g}$ orbitals, are characteristically substitution-inert (as flagged in Worked Example 3 for cobalt(III) ammines), while high-spin configurations with occupied $e_g^*$ orbitals are typically far more labile — a link this article shares directly with Crystal Field Stabilization Energy, where the thermodynamic (rather than magnetic) consequences of the same splitting diagram are quantified.

Ask the tutor: For a given metal ion, how would you estimate whether a tetrahedral complex could ever become low-spin, and what ligand field strength would that require relative to $\Delta_o$?

Take a free gaseous transition-metal ion with $n$ electrons to place among five degenerate $d$-orbitals. Hund's rules say: maximize total spin first, then maximize orbital angular momentum. Maximizing spin means spreading electrons across as many of the five orbitals as possible before doubly occupying any of them — this is simply lower in energy because parallel-spin electrons in different orbitals avoid each other more effectively (lower Coulomb repulsion, plus a genuine quantum-mechanical exchange stabilization) than either antiparallel electrons in different orbitals or two electrons forced into the same orbital. Now put that ion in an octahedral field of six ligands. The five-fold degeneracy is broken into a lower $t_{2g}$ set (three orbitals) and an upper $e_g$ set (two orbitals), separated by $\Delta_o$. For $d^1$, $d^2$, $d^3$ there is no dilemma: three electrons fit into three different $t_{2g}$ orbitals with parallel spin, satisfying Hund's rule at zero pairing cost, regardless of how large or small $\Delta_o$ is. The same absence-of-choice happens at the top of the shell: $d^8$, $d^9$, $d^{10}$ have only one way to distribute electrons once you insist on filling $t_{2g}$ first (three doubly-occupied orbitals are unavoidable by $d^6$ already in $t_{2g}$, and $e_g$ is forced to hold the remainder). The interesting cases are $d^4$ through $d^7$. Here the electron count is large enough that Hund's-rule spreading would require promoting an electron across the $t_{2g}$–$e_g$ gap, and small enough that pairing within $t_{2g}$ is a real alternative. The system must decide: pay $\Delta_o$ to promote an electron to $e_g$ and keep it unpaired (high-spin), or pay $P$ to pair it up within $t_2g$ and avoid the promotion (low-spin). This is a direct energetic comparison, and its outcome is read out experimentally through the number of unpaired electrons via the magnetic moment.

The magnetic consequence of all this bookkeeping is that unpaired electrons act as tiny magnetic dipoles due to their spin (and, in principle, their orbital motion). A sample containing many such dipoles is pulled into a magnetic field gradient — it is paramagnetic — with a strength proportional to how many unpaired electrons each ion carries. Complexes with all electrons paired have no net spin dipole and are weakly repelled from a field (diamagnetic), a much smaller effect arising from induced circulation of paired electrons.

The spin-only magnetic moment. For a first-row transition-metal ion where orbital angular momentum is largely quenched by the ligand field (true for most $A$- and $E$-term ground states), the effective magnetic moment is well approximated by counting only spin contributions:

where: $n$ = number of unpaired electrons, $\mu_B$ = the Bohr magneton ($9.274\times10^{-24}\ \mathrm{J\,T^{-1}}$), and $\mu_{\text{eff}}$ is reported in units of $\mu_B$. This comes from $\mu_{\text{eff}} = g_s\sqrt{S(S+1)}\,\mu_B$ with $g_s \approx 2$ (the free-electron $g$-factor) and total spin $S = n/2$, so $S(S+1) = \frac{n}{2}\left(\frac{n}{2}+1\right) = \frac{n(n+2)}{4}$, and $g_s\sqrt{S(S+1)} = 2\sqrt{n(n+2)/4} = \sqrt{n(n+2)}$.

Reading: this single formula converts a count of unpaired electrons — an integer you can read straight off an orbital-filling diagram — into a number with units of $\mu_B$ that is directly comparable to what a Gouy balance or Evans NMR method measures. It says nothing about why electrons paired or didn't; that decision is made separately by comparing $\Delta_o$ and $P$.

The high-spin/low-spin criterion. For $d^4$–$d^7$ octahedral complexes, the configuration adopted minimizes total electronic energy, which reduces to a direct comparison:

where: $\Delta_o$ = octahedral crystal field splitting energy (ligand- and metal-dependent, read from the spectrochemical series and tabulated spectroscopically), and $P$ = the mean pairing energy required to force a second electron into an already-singly-occupied orbital (an intrinsic, largely ligand-independent property of the free ion, though it is somewhat reduced — the nephelauxetic effect — by covalent delocalization onto ligands).

Reading: this is the master inequality of the whole subject. Everything about whether a complex is paramagnetic with a large moment or nearly diamagnetic follows from which side of this inequality a given $d^n$/ligand/geometry combination falls on.

Decomposing the pairing energy. $P$ is not a single elementary quantity; it is a sum of two physically distinct penalties incurred when an electron is forced into an orbital that is already singly occupied rather than into a still-empty orbital of the same energy:

where: $\Pi_c$ = the Coulombic pairing energy, the extra electrostatic repulsion of confining two electrons to the same spatial orbital rather than to two different ones, and $\Pi_e$ = the exchange pairing energy, the stabilization lost because pairing removes one same-spin electron pair (exchange stabilization exists only between electrons of parallel spin in different orbitals, a purely quantum mechanical, Pauli-exclusion-related effect with no classical analogue).

Reading: $\Pi_c$ is repulsive and always opposes pairing (same-orbital electrons are, on average, closer together than electrons in different orbitals, even accounting for correlation). $\Pi_e$ is a stabilization that is lost upon pairing, because reducing the number of unpaired, parallel-spin electrons reduces the number of stabilizing same-spin exchange pairs. Both effects push $P$ in the same direction — both make pairing costly — which is exactly why $P$ is a positive quantity of a few thousand to tens of thousands of cm⁻¹, tabulated per free ion (see Worked Example 3).

Example 1 (foundational): $\text{[Fe(H}_2\text{O)}_6\text{]}^{2+}$, a high-spin $d^6$ ion. Fe²⁺ is $d^6$. Water is a weak-field ligand (spectrochemical series places $\mathrm{H_2O}$ well below $\mathrm{CN^-}$ or $\mathrm{NH_3}$), giving a modest $\Delta_o \approx 10{,}400\ \mathrm{cm^{-1}}$ for this complex (from $d$–$d$ absorption spectroscopy). The pairing energy for free Fe²⁺ is $P \approx 17{,}600\ \mathrm{cm^{-1}}$.

From Step 5 of the first derivation, HS $d^6$ is $t_{2g}^4 e_g^2$ with $n=4$ unpaired electrons. Applying the spin-only formula:

This matches the experimental range for high-spin octahedral Fe(II) complexes, typically $5.1$–$5.7\ \mu_B$ (the small excess above the spin-only value comes from a residual orbital contribution in the $T$ ground term of octahedral $d^6$, discussed under Limiting Cases). The qualitative conclusion — paramagnetic, four unpaired electrons — is exactly right; the magnitude match to within about 15% confirms the spin-only approach is a good zeroth approximation even where orbital contributions are not fully quenched.

Example 2 (applied): $\text{[Fe(CN)}_6\text{]}^{4-}$ versus $\text{[FeF}_6\text{]}^{4-}$, same ion, opposite outcome. Both are Fe²⁺, $d^6$, octahedral, with $P \approx 17{,}600\ \mathrm{cm^{-1}}$ fixed. The ligand field splitting differs enormously: cyanide is a strong-field, $\pi$-accepting ligand giving $\Delta_o(\text{CN}^-) \approx 33{,}000\ \mathrm{cm^{-1}}$ for this complex, while fluoride is weak-field, giving $\Delta_o(\text{F}^-) \approx 13{,}000\ \mathrm{cm^{-1}}$. For $\text{[Fe(CN)}_6\text{]}^{4-}$:

For $\text{[FeF}_6\text{]}^{4-}$:

Experimentally, ferrocyanide salts are indeed diamagnetic and $\text{[FeF}_6\text{]}^{4-}$-type high-spin Fe(II) fluorides show moments near 5.0–5.4 $\mu_B$. This pair is the cleanest textbook demonstration that $\Delta_o$, not $P$, is the variable doing the work when comparing complexes of the same metal ion in different ligand environments — $P$ stays essentially fixed at ~17,600 cm⁻¹ throughout, while $\Delta_o$ swings by a factor of 2.5.

Example 3 (exam/competition level): predicting the moment of $\text{[Co(NH}_3)_5\text{Cl]}^{2+}$, an asymmetric ligand set, and cross-checking against the pairing-energy decomposition. Cobalt here is Co²⁺ (since the complex overall is $2+$ and five neutral NH₃ plus one $\mathrm{Cl^-}$ contribute $-1$, so metal oxidation state $= +2 - (-1) = +2$... more carefully: total charge $2+$ = (metal oxidation state) + 5(0) + (−1), so metal oxidation state $= +3$). Co³⁺ is $d^6$. Take $\Delta_o$ for this mixed-ligand complex as approximately $\Delta_o \approx 23{,}000\ \mathrm{cm^{-1}}$ (intermediate between pure $\text{[Co(NH}_3)_6\text{]}^{3+}$, $\Delta_o \approx 22{,}900\ \mathrm{cm^{-1}}$, and the weaker chloride perturbation on one site, which lowers the average ligand field only slightly since five of six sites remain strong-field ammine). The pairing energy for free Co³⁺ is $P \approx 21{,}000\ \mathrm{cm^{-1}}$ (Co³⁺ pairing energies run higher than Fe²⁺ because of the higher ionic charge, which contracts the $d$-orbitals and increases electron-electron repulsion, hence increases both $\Pi_c$ and $\Pi_e$).

Experimentally, cobalt(III) ammine complexes — with very few exceptions involving weak-field halide-rich coordination spheres — are diamagnetic low-spin $d^6$, consistent with $\mu_{\text{eff}} \approx 0$ measured for $\text{[Co(NH}_3)_5\text{Cl]}^{2+}$ and related pentaamminecobalt(III) salts. As a limiting-case sanity check: if the single ammine ligand were replaced progressively by more chlorides, $\Delta_o$ would fall (Cl⁻ is a weaker-field ligand than NH₃), and once enough substitution pushed $\Delta_o$ below $P \approx 21{,}000\ \mathrm{cm^{-1}}$, the complex would cross over to high spin — consistent with the empirical observation that highly halide-substituted Co(III) complexes are far rarer and less kinetically inert than the ammine-rich ones, partly because the low-spin $d^6$ configuration itself, with all electrons in nonbonding $t_{2g}$, is what confers Co(III) ammines their famous substitution inertness in the first place.

The correct attribution for the specific spin-only formula and the systematic use of magnetic moments to determine electronic structure in coordination compounds belongs jointly to Linus Pauling, who in the 1930s connected valence-bond theory to observed magnetic moments of transition-metal complexes, and to Hans Bethe and John Van Vleck, whose crystal-field and ligand-field theoretical framework (Bethe's 1929 group-theoretical treatment of ions in crystal fields, and Van Vleck's development of magnetochemical theory through the 1930s–1960s) supplied the quantitative machinery — $\Delta_o$, $P$, Racah parameters — used throughout this article. Van Vleck in particular deserves the central credit for magnetochemistry as a predictive, quantitative science. Working largely at Harvard and later at the University of Minnesota and back at Harvard, Van Vleck developed the theory connecting molecular and ionic electronic structure to bulk magnetic susceptibility — what became known as the Van Vleck equation, the general statistical-mechanical formula from which the spin-only limit used throughout ordinary coordination chemistry is recovered as a special case for ions with negligible spin-orbit coupling and no low-lying excited states. He shared the 1977 Nobel Prize in Physics (with Philip Anderson and Nevill Mott) for fundamental theoretical work on the electronic structure of magnetic and disordered systems — work whose everyday chemical payoff is exactly the ability to read $\Delta_o$ vs $P$ off a measured magnetic moment, as this article has done throughout.