Quantum Basis of NMR and ESR

NMR spectroscopy is the workhorse structure-elucidation technique of organic and inorganic synthetic chemistry, run daily in every research laboratory to confirm connectivity, stereochemistry, and purity from $^1$H, $^{13}$C, $^{19}$F, and $^{31}$P spectra, with two-dimensional experiments (COSY, HSQC, NOESY) extending the same Zeeman-and-coupling framework to full three-dimensional protein and nucleic-acid structure determination. Magnetic resonance imaging (MRI) uses exactly the proton Zeeman resonance condition derived here, spatially encoded with magnetic field gradients so that resonance frequency maps onto position, turning $T_1$/$T_2$ relaxation contrast into anatomical images without ionizing radiation. Solid-state NMR, using magic-angle spinning to average out anisotropic shielding and dipolar interactions, characterizes battery electrode materials, pharmaceutical polymorphs, and zeolite catalysts where diffraction alone cannot resolve local disorder. ESR/EPR spectroscopy is indispensable wherever unpaired electrons appear: characterizing transition-metal centers in metalloenzymes (iron-sulfur clusters, copper proteins, manganese in photosystem II), detecting and quantifying free-radical intermediates in oxidative stress and radiation biology, monitoring radical-mediated polymerization and degradation in materials science, and reading out electron-spin qubits and defect centers (such as the nitrogen-vacancy center in diamond) for quantum sensing and quantum-information applications. Site-directed spin labeling combines ESR with genetically engineered nitroxide radical tags to map protein conformational dynamics and distances via dipolar coupling (DEER/PELDOR), extending the basic hyperfine and Zeeman machinery developed here into a nanometer-scale molecular ruler.

Ask the tutor: How would the ESR spectrum change if the three methyl protons in a radical became magnetically inequivalent — say, through restricted rotation locking one C–H bond in a different orientation?

Every NMR- or ESR-active particle carries an intrinsic magnetic moment tied to its spin angular momentum,

where $\gamma$ is the electron gyromagnetic ratio, $g_N$ the nuclear $g$-factor, and $\mu_N$ the nuclear magneton (a factor of $\sim$1836 smaller than the Bohr magneton $\mu_B$, reflecting the proton-to-electron mass ratio). Placed in an external field $\mathbf{B}_0 = B_0\hat{z}$, the moment's energy is $-\boldsymbol{\mu}\cdot\mathbf{B}_0$. Because $S_z$ (or $I_z$) is quantized in units of $\hbar$ with $2S+1$ (or $2I+1$) allowed projections $m_S$ (or $m_I$), the continuous classical picture of a spinning magnet collapses into a discrete ladder of energy levels — the Zeeman ladder. For a lone electron ($S=\tfrac12$), this ladder has just two rungs, $m_S = \pm\tfrac12$, separated by $g_e\mu_B B_0$ with $g_e \approx 2.0023$. For a proton ($I=\tfrac12$), the two rungs are separated by $g_N\mu_N B_0$, with $g_N(^1\mathrm{H}) \approx 5.586$. Because $\mu_N \ll \mu_B$, nuclear Zeeman gaps are roughly three orders of magnitude smaller than electronic ones at the same field — which is exactly why NMR uses radiofrequency photons (MHz) while ESR uses microwaves (GHz) for comparable, laboratory-achievable magnets. The subtlety that makes resonance spectroscopy a structural tool rather than a mere field-measuring device is that the "local" field felt by a spin is not simply $B_0$. Surrounding electrons respond to $B_0$ by circulating and generating a small induced field that typically opposes $B_0$ at the nuclear position — this is shielding, parametrized by the dimensionless shielding constant $\sigma$, and it is what produces the chemical shift in NMR: chemically inequivalent nuclei, sitting in different electronic environments, resonate at slightly different frequencies. In ESR, the dominant local perturbation is different in character: it is the direct quantum-mechanical contact (or dipolar) interaction between the unpaired electron's spin density and nearby nuclear spins, the hyperfine interaction, which splits each electronic transition into a multiplet whose spacing and pattern map the spatial distribution of the unpaired electron.

The Zeeman Hamiltonian and resonance condition. For a spin-$\tfrac12$ nucleus in field $B_0$,

where: $g_N$ is the nuclear $g$-factor (dimensionless, isotope-specific), $\mu_N = e\hbar/2m_p$ the nuclear magneton, $B_0$ the static field, $m_I$ the magnetic quantum number. Reading: the two levels $m_I=+\tfrac12$ (lower energy, $\alpha$ or "spin-up," since $g_N>0$ for $^1$H means moment aligned with field is lower energy) and $m_I=-\tfrac12$ are split by

where: $\Delta E$ is the energy gap between adjacent Zeeman levels. Resonance absorption of a photon connecting these levels requires

where: $h$ is Planck's constant, $\nu$ the photon frequency, $g_e \approx 2.0023$ the free-electron $g$-factor. Reading: this is the master resonance condition. Everything downstream — chemical shift, $J$-coupling, hyperfine structure — is a small, structure-dependent correction to the field (or to $g$) appearing on the right-hand side.

Chemical shift. The local field at a nucleus is reduced by shielding,

where: $\sigma$ is the (dimensionless, typically $10^{-6}$–$10^{-2}$) shielding constant, itself a sum of local diamagnetic, paramagnetic, and neighboring-group contributions. Since $\sigma$ depends on field only negligibly, shifts are reported relative to a reference compound (TMS for $^1$H/$^{13}$C) as

where: $\delta$ is the chemical shift in parts per million, spectrometer-independent by construction. Reading: $\delta$ isolates the electronic environment term and cancels both $h$ and the absolute field $B_0$, which is why the same molecule gives the same $\delta$ on a 300 MHz or a 900 MHz instrument even though the absolute frequency difference (in Hz) scales with field.

Spin–spin (J) coupling. Two coupled nuclear spins $I$ and $S$ (heteronuclear, weak coupling limit) interact via

where: $J$ is the scalar (indirect, electron-mediated) coupling constant in Hz, transmitted through bonding electrons and independent of $B_0$. This splits each nucleus's resonance into $(2nI+1)$ lines from $n$ equivalent coupling partners of spin $I$, with relative intensities given by the multiplicities of angular-momentum addition (binomial for spin-$\tfrac12$ partners), each line separated by $J$ hertz regardless of field strength.

Hyperfine coupling (ESR). The unpaired electron interacts with a nuclear spin $I$ via the isotropic Fermi contact term plus (in anisotropic media) a dipolar term:

where: $a$ is the isotropic hyperfine coupling constant (Hz or gauss), proportional to $|\psi(0)|^2$, the electron spin density at the nucleus. Reading: because $a\propto|\psi(0)|^2$, hyperfine coupling constants are direct, quantitative probes of where the unpaired electron actually sits — this is the celebrated McConnell relation for aromatic radicals, $a_H = Q\rho_\pi$, linking measured splittings to $\pi$-spin density $\rho_\pi$ at each carbon.

Example 1 — Foundational: proton NMR resonance frequency and Boltzmann polarization at 7.05 T. A common NMR magnet operates at $B_0 = 7.05\ \text{T}$ (chosen so that $^1$H resonates near 300 MHz). Using $g_N(^1\mathrm H) = 5.5857$, $\mu_N = 5.0508\times10^{-27}\ \text{J/T}$, and $h = 6.626\times10^{-34}\ \text{J}\,\cdot\,\text{s}$:

Numerator: $5.5857 \times 5.0508\times10^{-27} = 2.8210\times10^{-26}\ \text{J/T}$; multiplied by $7.05\ \text{T}$ gives $1.9888\times10^{-25}\ \text{J}$.

This matches the design field of a "300 MHz" spectrometer. Now the Boltzmann polarization at $T=298\ \text{K}$:

Numerator: $1.989\times10^{-25}\ \text{J}$. Denominator: $2\times1.381\times10^{-23}\times298 = 8.231\times10^{-21}\ \text{J}$.

Only about 24 nuclei per million show a net excess in the lower level — an explicit, concrete illustration of why NMR is an intrinsically low-sensitivity technique compared to optical spectroscopies, and why high fields and signal averaging matter so much. Limiting case check: as $T\to\infty$, $\Delta N/N \to 0$ — the populations equalize and the NMR signal vanishes, as required, since infinite temperature erases any Boltzmann bias between the two Zeeman levels.

Example 2 — Applied: predicting the ¹H NMR splitting pattern and shift of the CH₂ protons in ethyl acetate's ethoxy group. Ethyl acetate, CH₃–C(=O)–O–CH₂–CH₃, has an ethoxy $\mathrm{OCH_2CH_3}$ fragment. The $\mathrm{OCH_2}$ protons are coupled to the three equivalent $\mathrm{CH_3}$ protons (and negligibly to anything else, by the $n+1$ rule for equivalent, vicinally-coupled spin-$\tfrac12$ nuclei). Multiplicity: with $n=3$ equivalent coupling partners of spin $\tfrac12$, the OCH$_2$ resonance splits into $n+1 = 4$ lines — a quartet — with relative intensities from $\binom{3}{k}$, $k=0,1,2,3$: $1:3:3:1$. This follows directly from Step 4 of the hyperfine derivation above, applied to nuclear–nuclear $J$-coupling instead of electron–nuclear hyperfine coupling; the combinatorics of angular-momentum addition are identical whether the coupled partner is a nucleus or an electron. Line spacing: the vicinal coupling constant for this fragment is experimentally $J_{\text{HH}} \approx 7.1\ \text{Hz}$, and this spacing is the same in Hz at any spectrometer field, since $J$ arises from through-bond electron-mediated coupling and does not depend on $B_0$. Chemical shift: the OCH$_2$ protons resonate near $\delta = 4.12\ \text{ppm}$ (deshielded by the adjacent electronegative oxygen), versus $\delta \approx 1.26\ \text{ppm}$ for the terminal CH$_3$. At a 400 MHz spectrometer ($\nu_{\text{ref}}=400\times10^6\ \text{Hz}$ for $^1$H), the absolute frequency offset from TMS is

This offset scales linearly with spectrometer field (it would be $2 \times 1648 = 3296\ \text{Hz}$ at 800 MHz), while $\delta=4.12\ \text{ppm}$ itself and the 7.1 Hz splitting stay fixed — the diagnostic feature that separates chemical-shift-driven from coupling-driven line positions in a real spectrum. Limiting case check: if the three methyl protons were made chemically and magnetically inequivalent (e.g., by isotopic labeling breaking the threefold symmetry), the $n+1$ rule would break down and a more complex multiplet (up to 8 lines for three inequivalent spin-½ partners) would appear instead of the clean 1:3:3:1 quartet — confirming that the simple rule strictly requires the coupling partners to be equivalent.

Example 3 — Exam/competition level: extracting spin density from the ESR spectrum of the methyl radical (a nine-line 1:8:28:56:70:56:28:8:1 pattern) via the McConnell relation. The methyl radical, $^\bullet$CH$_3$, has its unpaired electron in a $2p_z$-like orbital on carbon (planar, $\pi$-radical). Its ESR spectrum shows hyperfine coupling to the three equivalent $\alpha$-hydrogens, giving (by the $n+1$ rule with $n=3$, spin $I=\tfrac12$) a quartet, with relative intensities $1:3:3:1$, spacing $a_H = 23.0\ \text{G}$ (measured). Step 1 — apply the McConnell relation. For an aromatic or planar $\pi$-radical, the isotropic proton hyperfine coupling constant relates to the $\pi$-spin density on the adjacent carbon via

where $Q \approx -23\ \text{G}$ is an empirically calibrated proportionality constant (from spin polarization of the C–H $\sigma$ bond by the $\pi$ electron) and $\rho_C^\pi$ is the unpaired $\pi$-electron spin density on the attached carbon. Step 2 — solve for spin density. Rearranging,

This confirms — as expected on structural grounds — that essentially the entire unpaired electron resides in the single carbon $2p_z$ orbital of the methyl radical, since there is only one carbon to house it and the sum of $\pi$-spin densities over all atoms of the radical must equal 1. Step 3 — extend to a substituted case with unequal shifts (asymmetric input). Suppose a hypothetical planar allylic-type radical distributes its spin density unequally as $\rho_1 = 0.62$ on carbon 1 and $\rho_3 = 0.38$ on carbon 3 (with node at $C_2$, $\rho_2=0$, as required by symmetry of the nonbonding SOMO in a simple Hückel treatment of an unsymmetrically substituted allyl radical). Using $Q=-23\ \text{G}$, the predicted hyperfine couplings to the terminal protons are

(By convention the magnitude is what is measured spectroscopically; the sign is inferred from theory or ENDOR.) These two inequivalent couplings, together with the small coupling to the two equivalent central-carbon protons (which see essentially zero $\pi$ density and couple mainly by a smaller spin-polarization mechanism), would produce a spectrum splitting first into a doublet of splitting $14.3\ \text{G}$, each line further split into a doublet of splitting $8.74\ \text{G}$, each of those further split by the smaller central coupling — a total of up to 8 lines before any additional equivalence collapses them, illustrating how a full ESR simulation must apply the splitting rule iteratively, coupling-constant by coupling-constant, for each inequivalent set of nuclei. Limiting case check: if the radical were perfectly symmetric (allyl itself, $\rho_1=\rho_3=0.5$), the two terminal couplings would collapse to a single value $a_H = (-23)(0.5)=-11.5\ \text{G}$, restoring the simpler equivalent-nuclei multiplet pattern — showing the asymmetric example above correctly reduces to the familiar symmetric result when the substitution asymmetry is removed.