NMR spectroscopy of organic molecules

Intuition [Beginner]

Nuclear magnetic resonance (NMR) spectroscopy tells you which atoms are in a molecule and how they are connected. It works because certain atomic nuclei — especially hydrogen-1 and carbon-13 — behave like tiny magnets. In a strong external magnetic field, these nuclear magnets align either with the field or against it. The energy difference between the two alignments corresponds to a radio-frequency photon, and when that exact frequency is absorbed, the nucleus "resonates."

Three pieces of information come out of a proton (${}^1$H) NMR spectrum, and together they are often enough to identify the molecule completely.

Chemical shift tells you what kind of environment each proton lives in. A proton next to an electronegative atom (oxygen, nitrogen, halogen) experiences a stronger effective magnetic field because the electron cloud around it gets pulled away, leaving the proton less shielded. It resonates at a higher frequency — further "downfield" in the spectrum, at a larger parts-per-million (ppm) value. A proton surrounded by carbon and hydrogen, far from any electronegative atom, is more shielded and resonates "upfield" at a lower ppm value.

Integration tells you how many protons contribute to each signal. The area under each peak is proportional to the number of protons that produce it. A peak with twice the area of another corresponds to twice as many equivalent protons.

Splitting tells you how many neighbours each proton has. A proton that sits next to one other proton sees its signal split into a doublet (two lines). Next to two equivalent protons, it splits into a triplet (three lines). The pattern follows the n+1 rule: a proton with $n$ equivalent neighbouring protons splits into $n+1$ lines. This works because each neighbouring proton's spin creates a small additional magnetic field that shifts the resonant frequency slightly up or down.

For carbon-13 (${}^{13}$C) NMR, the same ideas apply but each carbon gives a single peak (no splitting is normally observed because the natural abundance of ${}^{13}$C is only 1.1%, making carbon-carbon coupling rare). The number of distinct peaks tells you how many chemically distinct carbons the molecule contains.

Visual [Beginner]

A proton NMR spectrum is a horizontal trace with peaks rising from a baseline. The horizontal axis runs from right (low ppm, upfield, shielded protons) to left (high ppm, downfield, deshielded protons). A typical organic molecule shows a cluster of peaks in the 0–3 ppm region (alkyl protons), one around 2 ppm for protons next to carbonyls, and another around 3–4 ppm for protons on carbons bonded to oxygen.

Worked example [Beginner]

Assign the ${}^1$H NMR spectrum of ethyl acetate ($\mathrm{C}{\mathrm{H}}_3\mathrm{C}{\mathrm{O}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_3$).

Ethyl acetate has six protons in three chemically distinct groups:

1. $\mathrm{C}{\mathrm{H}}_3\mathrm{CO}-$ — the acetyl methyl, three protons attached to a carbon next to a carbonyl.
2. $-\mathrm{OC}{\mathrm{H}}_2-$ — the methylene, two protons on a carbon bonded to oxygen.
3. \mathrm{{-CH_3} — the terminal ethyl methyl, three protons at the end of the chain.

The observed spectrum has three signals:

Signal A: singlet at 2.0 ppm, integration 3H. The chemical shift of 2.0 ppm is characteristic of a methyl group adjacent to a carbonyl ($\mathrm{C}{\mathrm{H}}_3\mathrm{CO}-$). The signal is a singlet (unsplit) because the acetyl methyl has no protons on the adjacent carbon — the carbonyl carbon has no attached protons. There are no neighbouring protons to cause splitting, so $n=0$ and $n+1=1$ (a singlet).

Signal B: quartet at 4.1 ppm, integration 2H. The chemical shift of 4.1 ppm is typical for protons on a carbon bonded to oxygen ($-\mathrm{OC}{\mathrm{H}}_2-$). The signal is a quartet because these two protons are adjacent to the terminal methyl group with three equivalent protons: $n=3$, so $n+1=4$ lines. The integration of 2H confirms the assignment.

Signal C: triplet at 1.3 ppm, integration 3H. The chemical shift of 1.3 ppm is in the alkyl region, consistent with a terminal methyl group. The signal is a triplet because the methyl protons are adjacent to the methylene with two protons: $n=2$, so $n+1=3$ lines. The integration of 3H confirms the assignment.

The three signals, their splitting patterns, and their integration ratios uniquely identify ethyl acetate. No other structural isomer produces this combination.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Nuclear magnetic resonance exploits the spin of nuclei with non-zero nuclear spin quantum number $I$. For ${}^1$H and ${}^{13}$C, $I=1/2$, and the nuclear spin has $2I+1=2$ possible orientations in an external magnetic field $B_0$: aligned ($m_I=+1/2$, lower energy, "$\alpha$ spin") and opposed ($m_I=-1/2$, higher energy, "$\beta$ spin").

Zeeman splitting. In a field $B_0$ along the $z$-axis, the energy of each spin state is:

$$E_{m_I}=-\gamma \hslash B_0{m}_I$$

where $\gamma$ is the nuclear gyromagnetic ratio (${\gamma }_{{}^1\text{H}}=267.5\times 10^6\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$; ${\gamma }_{{}^{13}\text{C}}=67.3\times 10^6\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$). The transition energy is:

$$\Delta E=\gamma \hslash B_0$$

The corresponding Larmor frequency is ${\nu }_0=\gamma B_0/(2\pi )$. For ${}^1$H in a 9.4 T magnet, ${\nu }_0\approx 400\mathrm{MHz}$; in a 21.1 T magnet, ${\nu }_0\approx 900\mathrm{MHz}$.

Chemical shift. The local magnetic field at a nucleus is modified by the surrounding electron cloud:

$$B_{\mathrm{eff}}=B_0(1-\sigma )$$

where $\sigma$ is the shielding constant. The chemical shift is defined relative to a reference compound (tetramethylsilane, TMS, for ${}^1$H and ${}^{13}$C):

$$\delta =\frac{{\nu }_{\mathrm{sample}}-{\nu }_{\mathrm{TMS}}}{{\nu }_{\mathrm{spectrometer}}}\times 10^6\text{(ppm)}$$

Spin-spin coupling. Neighbouring nuclear spins interact through the bonding electrons (scalar or J-coupling). For two coupled spin-1/2 nuclei, the coupling constant $J$ (in Hz) measures the energy of interaction. The Hamiltonian for a pair of spins is:

$$\hat{H}=-\gamma B_0(1-{\sigma }_A){\hat{I}}_{z,A}-\gamma B_0(1-{\sigma }_B){\hat{I}}_{z,B}+J{\hat{\mathbf{I}}}_A\cdot {\hat{\mathbf{I}}}_B$$

When the chemical-shift difference $|{\delta }_A-{\delta }_B|$ is much larger than $J$ (the weak-coupling or first-order limit), the spectrum shows the n+1 splitting pattern. When the shift difference is comparable to $J$, second-order effects (roofing, additional splitting) appear.

DEPT (Distortionless Enhancement by Polarisation Transfer). A pulse-sequence experiment that distinguishes $\mathrm{C}{\mathrm{H}}_3$, $\mathrm{C}{\mathrm{H}}_2$, $\mathrm{CH}$, and quaternary carbons in ${}^{13}$C NMR. DEPT-45 shows all protonated carbons (CH, CH${}_2$, CH${}_3$). DEPT-90 shows only CH. DEPT-135 shows CH and CH${}_3$ as positive peaks and CH${}_2$ as negative peaks. Quaternary carbons do not appear in any DEPT spectrum.

Counterexamples to common slips

• The n+1 rule applies only to equivalent neighbouring protons. If a proton is adjacent to two protons that are not equivalent (different chemical shifts), the splitting is $(n_1+1)(n_2+1)$, not $(n_1+n_2)+1$. This is a common source of error in interpreting spectra of unsymmetrical molecules.

• Integration gives ratios, not absolute numbers. An integration ratio of 1:2:3 could correspond to 1H:2H:3H or 2H:4H:6H. The molecular formula (from mass spectrometry or elemental analysis) is needed to convert ratios to absolute proton counts.

• ${}^{13}$C spectra are normally broadband-decoupled. The ${}^1$H–${}^{13}$C splitting is removed by irradiating all protons simultaneously during acquisition. The resulting spectrum shows one singlet per distinct carbon, which simplifies interpretation but removes coupling information. DEPT recovers the multiplicity selectively.

• ${}^{13}$C peak intensities are not quantitative. Unlike ${}^1$H integration, the peak heights in a ${}^{13}$C spectrum do not reflect the number of carbons producing each signal. NOE enhancement from proton decoupling and differences in relaxation times ($T_1$) distort intensities. A quaternary carbon typically gives a much weaker signal than a protonated carbon because it has no directly attached protons to provide NOE enhancement and its $T_1$ is long.

Key theorem with proof [Intermediate+]

Proposition (First-order splitting pattern). Let nucleus A be coupled to $n$ equivalent spin-1/2 nuclei B, with coupling constant $J_{AB}$, and let the chemical-shift difference $|{\delta }_A-{\delta }_B|\gg J_{AB}$. Then the signal for A consists of $n+1$ equally-spaced lines with relative intensities given by the binomial coefficients $\left(\genfrac{}{}{0ex}{}{n}{0}\right),\left(\genfrac{}{}{0ex}{}{n}{1}\right),\dots ,\left(\genfrac{}{}{0ex}{}{n}{n}\right)$.

Proof. Each of the $n$ equivalent B spins can be in state $\alpha$ or $\beta$. The total magnetic quantum number of the B spins is $m_B=m_1+m_2+\cdots +m_n$, where each $m_i=\pm 1/2$. The number of B-spin configurations giving a particular value of $M=m_B+n/2$ (the number of $\alpha$ spins among the B spins) is $\left(\genfrac{}{}{0ex}{}{n}{M}\right)$. Each value of $m_B$ shifts the A resonance by $J_{AB}\cdot m_B$. The distinct values of $m_B$ are $-n/2,-n/2+1,\dots ,+n/2$, giving $n+1$ equally-spaced lines separated by $J_{AB}$ Hz. The intensity of each line is proportional to the number of B-spin configurations producing it, which is $\left(\genfrac{}{}{0ex}{}{n}{M}\right)$ for $M=0,1,\dots ,n$. $■$

Bridge. The first-order splitting theorem builds toward 14.12.01, where the Larmor equation and Zeeman splitting are derived from the spin-1/2 quantum mechanics introduced in 12.01.02 pending. The foundational reason the n+1 rule works is that equivalent spins produce a binomial distribution of local fields at the observed nucleus — this is exactly the content of Pascal's triangle appearing in NMR. The central insight that chemical shift and J-coupling enter the Hamiltonian as additive terms identifies the spectrum with a direct readout of molecular electronic structure, and the bridge is that the same Hamiltonian, generalised to the rotating frame, underlies every multidimensional NMR experiment in the Master tier below.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Bloch equations). The time evolution of the net magnetisation $\mathbf{M}=(M_x,M_y,M_z)$ in a static field $B_0\hat{z}$ is governed by

$$\frac{d\mathbf{M}}{dt}=\gamma \mathbf{M}\times \mathbf{B}-\frac{M_x\hat{x}+M_y\hat{y}}{T_2}-\frac{(M_z-M_0)\hat{z}}{T_1}$$

where $T_1$ is the spin-lattice (longitudinal) relaxation time and $T_2$ is the spin-spin (transverse) relaxation time. In the absence of relaxation, the magnetisation precesses about $\mathbf{B}$ at the Larmor frequency ${\omega }_0=\gamma B_0$.

Bloch 1946 ^{[Bloch 1946]} introduced these phenomenological equations to describe nuclear induction. They remain the starting point for understanding all NMR experiments.

Theorem 2 (Karplus relationship). The vicinal coupling constant ${}^3{J}_{HH}$ between two protons separated by three bonds depends on the dihedral angle $\varphi$ between them according to

$${}^3{J}_{HH}=A+B\cos \varphi +C\cos 2\varphi$$

where $A$, $B$, $C$ are constants that depend on the substituents on the bonded carbons. For an unsubstituted ethane fragment, $A\approx 7.76$ Hz, $B\approx -1.10$ Hz, $C\approx 5.42$ Hz, giving ${}^{3}J\approx 12$ Hz for $\varphi =0^{\circ }$ (cis), ${}^{3}J\approx 2$ Hz for $\varphi =90^{\circ }$, and ${}^{3}J\approx 8$ Hz for $\varphi =180^{\circ }$ (trans).

Karplus 1959 derived this from valence-bond theory. It provides a direct link between NMR coupling constants and molecular conformation — one of the most powerful structural applications of ${}^1$H NMR.

Theorem 3 (Solomon equations). For a two-spin system, the time evolution of the longitudinal magnetisations $M_{z,I}$ and $M_{z,S}$ under cross-relaxation is

$$\frac{d\Delta M_{z,I}}{dt}=-{\rho }_I\Delta M_{z,I}-{\sigma }_{IS}\Delta M_{z,S}$$

$$\frac{d\Delta M_{z,S}}{dt}=-{\sigma }_{IS}\Delta M_{z,I}-{\rho }_S\Delta M_{z,S}$$

where ${\rho }_I$ is the self-relaxation rate of spin $I$, ${\sigma }_{IS}$ is the cross-relaxation rate, and $\Delta M_z=M_z-M_0$. The cross-relaxation rate ${\sigma }_{IS}$ is proportional to ${\tau }_c/r_{IS}^6$ (correlation time over distance to the sixth power) in the extreme narrowing limit.

Solomon 1955 ^{[Overhauser 1953]} derived these equations. The steady-state solution gives the nuclear Overhauser effect: irradiation of spin $S$ changes the intensity of spin $I$'s signal by a factor $1+{\sigma }_{IS}/{\rho }_I$, providing a through-space distance probe.

Theorem 4 (Coalescence condition). For two exchanging nuclei with chemical-shift separation $\Delta \nu$ Hz and exchange rate $k_{\mathrm{ex}}$, the coalescence temperature $T_c$ at which the two peaks merge satisfies

$$k_{\mathrm{ex}}=\frac{\pi |\Delta \nu |}{\sqrt{2}}$$

at coalescence. Measurement of $T_c$ directly yields the activation free energy for the exchange process via the Eyring equation.

This relationship underlies variable-temperature (VT) NMR studies of conformational dynamics and chemical exchange.

Theorem 5 (Ernst sensitivity theorem). The signal-to-noise ratio of an FT-NMR experiment with $n$ scans improves as $\sqrt{n}$ relative to a single scan, and the total experiment time for a given sensitivity is proportional to $1/\sqrt{T_1}$ of the nucleus being observed. The optimal repetition time (Ernst angle condition) for a pulse with flip angle $\alpha$ satisfies

$$\cos {\alpha }_{\mathrm{opt}}=\exp (-T_R/T_1)$$

where $T_R$ is the repetition time between scans.

Ernst and Anderson 1966 ^{[Ernst Anderson 1966]} established the sensitivity advantage of FT methods over continuous-wave scanning, which was the foundation for making ${}^{13}$C NMR practical.

Advanced 1H NMR interpretation

The n+1 rule and the binomial intensity pattern taught at Intermediate tier apply only in the weak-coupling (first-order) limit, where the chemical-shift difference between coupled nuclei greatly exceeds their coupling constant ($\Delta \nu /J\gg 10$). When this condition fails, the spectrum shows second-order effects that cannot be predicted by the n+1 rule alone.

AB systems. Two coupled protons with similar chemical shifts (small $\Delta \nu$ relative to $J$) produce an AB quartet: four lines rather than the two doublets predicted by first-order analysis. The inner two lines are stronger and the outer two weaker — a phenomenon called "roofing" or "leaning." The coupling constant $J_{AB}$ is still read from the spacing between adjacent lines within each doublet component, but the chemical-shift difference is extracted from the line positions using

$${\delta }_A-{\delta }_B=\sqrt{({\nu }_1-{\nu }_4{)}^2-({\nu }_1-{\nu }_2{)}^2}$$

where ${\nu }_1$ through ${\nu }_4$ are the four line positions in Hz. At the extreme where $\Delta \nu \to 0$ (an A${}_2$ system), all four lines collapse and the coupling is invisible — equivalent spins do not split each other.

Diastereotopic protons. In a chiral molecule, the two protons of a CH${}_2$ group adjacent to a stereocentre are not chemically equivalent. They have different chemical shifts and couple to each other with a geminal coupling constant ${}^2{J}_{HH}$ (typically $-10$ to $-15$ Hz). This is the AB subspectrum of a diastereotopic methylene, and it appears in virtually every chiral organic molecule. The diastereotopic assignment is confirmed by the absence of symmetry operations relating the two protons — the mirror image of the molecule is a different compound.

Long-range coupling. Coupling over four bonds (${}^4{J}_{HH}$, "W-coupling") occurs when the four-atom pathway adopts a planar W geometry. The coupling constant is small (0–3 Hz) but diagnostically useful for assigning regiochemistry in rigid systems. Allylic coupling (${}^{4}J$ across C=C) and propargylic coupling (${}^{4}J$ across C$\equiv$C) are other named long-range patterns. Five-bond coupling is observed in conjugated aromatic and polyene systems but is rare in aliphatic molecules.

Virtual coupling. When a proton is strongly coupled to an intermediate spin that is weakly coupled to a third spin, the intermediate spin "relays" the coupling, producing a spectrum where the first and third spins appear to be coupled even though their direct coupling constant is zero. This virtual coupling artifact is a common source of misassignment in crowded spectra and is resolved by moving to higher field (increasing $\Delta \nu /J$).

ABC and higher spin systems. When three or more spins have mutually similar chemical shifts (all comparable to their coupling constants), the spectrum becomes an ABC or ABX system that cannot be analysed by inspection. The full quantum-mechanical treatment requires diagonalising the spin Hamiltonian matrix (an $8\times 8$ matrix for three spin-1/2 nuclei) and computing transition frequencies and intensities from the eigenvectors. Modern NMR software performs this spin simulation automatically: the spectroscopist inputs trial chemical shifts and coupling constants, the program computes the theoretical spectrum, and the parameters are iteratively refined to match the experimental data. The ability to simulate arbitrary spin systems is indispensable for assigning the congested spectra of complex natural products, where overlapping multiplets from multiple non-equivalent protons defy first-order analysis.

The transition from first-order to second-order spectra is continuous. At 60 MHz many organic molecules show significant second-order character; at 400 MHz the same molecules are first-order. The trend toward higher fields (600, 800, 900+ MHz) is driven in part by the desire to simplify spectral analysis by pushing more systems into the first-order regime. Higher fields also increase sensitivity (signal-to-noise scales as $B_0^{3/2}$ under standard conditions) and chemical-shift dispersion, allowing more peaks to be resolved in crowded spectral regions.

Multidimensional NMR experiments

Modern NMR spectroscopy extends the one-dimensional experiment into two or more frequency dimensions. The principle is always the same: a series of radio-frequency pulses creates and transfers coherence between spins, and a two-dimensional Fourier transform maps the resulting correlations onto a 2D frequency grid. The concept was introduced by Jeener 1971 and developed into practical experiments by Ernst and coworkers ^{[Ernst Anderson 1966]}.

COSY (Correlation Spectroscopy). The simplest 2D experiment. COSY shows which protons are J-coupled to each other through two or three bonds. Off-diagonal peaks (cross-peaks) connect the chemical shifts of coupled protons. A COSY spectrum of ethyl acetate shows a cross-peak connecting the quartet at 4.1 ppm (OCH${}_2$) and the triplet at 1.3 ppm (CH${}_3$), confirming adjacency. For complex molecules, COSY traces out the proton connectivity network step by step.

TOCSY (Total Correlation Spectroscopy). Extends COSY by transferring magnetisation across an entire spin system through isotropic mixing. All protons within a continuous J-coupled network — even those not directly coupled — appear as cross-peaks. In a peptide, TOCSY identifies all protons belonging to a single amino acid residue, because the J-coupled network of each residue is isolated from its neighbours by the amide bond.

HSQC (Heteronuclear Single Quantum Coherence). Correlates each proton with the carbon it is directly bonded to (one-bond ${}^1{J}_{CH}$ coupling, typically 125–170 Hz). Each proton signal appears as a cross-peak connecting its ${}^1$H chemical shift to its parent carbon's ${}^{13}$C chemical shift. HSQC is the most commonly used 2D experiment for organic structure determination because it provides unambiguous C-H pair assignments. The ${}^{13}$C dimension resolves overlapping proton signals that are inseparable in 1D ${}^1$H NMR.

HMBC (Heteronuclear Multiple Bond Correlation). Correlates protons with carbons two or three bonds away (long-range ${}^2{J}_{CH}$ and ${}^3{J}_{CH}$ coupling, 0–10 Hz). HMBC detects cross-peaks across quaternary carbons and carbonyl groups, providing connectivity information that HSQC cannot give. A single HMBC cross-peak from a proton to a carbonyl carbon identifies the carbonyl's position within the carbon skeleton.

NOESY (Nuclear Overhauser Effect Spectroscopy). Detects through-space proximity rather than through-bond coupling. The NOE depends on $r^{-6}$ (distance to the minus sixth power) and is effective for proton-proton distances under 5 Angstroms. NOESY cross-peaks identify which protons are spatially close, regardless of bonding. This is essential for determining relative stereochemistry: two diastereotopic protons that are cis on a ring show a strong NOESY cross-peak, while the trans arrangement gives a weak or absent one.

ROESY (Rotating-frame Overhauser Spectroscopy). A variant of NOESY that gives positive NOEs for all molecular sizes. The conventional NOE changes sign for molecules with molecular weights near 1000–2000 Da (where $\omega {\tau }_c\approx 1$), giving zero NOE at the crossover — a major problem for mid-sized organic molecules. ROESY avoids this by measuring the NOE in the rotating frame, where the sign is always positive. For synthetic organic chemistry (MW 200–2000), ROESY is often preferred over NOESY.

The combined use of COSY (H-H connectivity), HSQC (C-H bonds), HMBC (C-H long-range), and NOESY/ROESY (through-space) provides a complete structure-elucidation protocol. For a molecule of unknown structure, the standard workflow is: obtain the molecular formula by mass spectrometry; record ${}^1$H and ${}^{13}$C 1D spectra; assign C-H pairs by HSQC; trace proton connectivity by COSY; bridge gaps across quaternary carbons by HMBC; assign stereochemistry by NOESY/ROESY or coupling-constant analysis. This protocol determines the full structure, including relative stereochemistry, for most organic molecules up to MW $\sim$1000.

Product-operator formalism. All of the experiments above are described compactly by the product-operator formalism, which represents the density matrix of a spin system as a linear combination of Cartesian spin operators (${\hat{I}}_x,{\hat{I}}_y,{\hat{I}}_z$) and their products. Chemical-shift evolution rotates operators in the $xy$-plane; J-coupling evolution creates anti-phase terms (e.g., $2{\hat{I}}_x{\hat{S}}_z$); radio-frequency pulses rotate operators about specified axes. Each step of a pulse sequence corresponds to a deterministic transformation of the operator basis, and the observed signal is read off from the coefficient of ${\hat{I}}_x$ or ${\hat{I}}_y$ at the start of acquisition. The formalism makes the design of new pulse sequences into a constructive algebraic exercise rather than a matter of trial and error.

Quantitative NMR and dynamic processes

Quantitative ${}^1$H NMR (qNMR) uses the proportionality between peak area and proton count to measure concentrations. With a sufficiently long relaxation delay ($\ge 5T_1$), the integrated area under a ${}^1$H signal is directly proportional to the number of protons producing it, with accuracy better than 1% under ideal conditions. This makes NMR a primary quantitative method for purity determination: by adding an internal standard of known mass, the absolute purity of a sample can be measured without a calibration curve. Pharmacopoeial standards now include qNMR methods for drug substance purity.

Variable-temperature NMR. Many organic molecules undergo conformational or chemical exchange processes that are slow on the NMR timescale at room temperature, giving separate signals for each exchanging species, but fast at elevated temperature, giving averaged signals. By recording spectra at multiple temperatures and observing the transition from slow to fast exchange, the rate and activation parameters of the exchange process are extracted.

At low temperature (slow exchange, $k_{\mathrm{ex}}\ll \Delta \nu$), the two sites give separate peaks. As the temperature increases and $k_{\mathrm{ex}}$ approaches $\Delta \nu$, the peaks broaden, then coalesce into a single broad peak at $k_{\mathrm{ex}}\approx \pi \Delta \nu /\sqrt{2}$ (the coalescence condition, Theorem 4 above). At still higher temperature (fast exchange, $k_{\mathrm{ex}}\gg \Delta \nu$), a single sharp peak appears at the population-weighted average chemical shift.

From the coalescence temperature $T_c$, the activation free energy is calculated:

$$\Delta G^{‡}=R{T}_c\left[\ln \left(\frac{k_B{T}_c}{h}\right)-\ln \left(\frac{\pi \Delta \nu }{\sqrt{2}}\right)\right]$$

This is the Eyring equation specialised to the coalescence condition. A typical application is the ring-flipping of cyclohexane: at $-90^{\circ }$C the axial and equatorial protons give separate signals (slow exchange), while at room temperature they average to a single peak (fast exchange). The measured $\Delta G^{‡}\approx 42$ kJ/mol corresponds to the barrier to chair-chair interconversion.

Dynamic NMR (DNMR) extends VT-NMR by fitting the full lineshape at each temperature to the Bloch-McConnell equations — coupled differential equations describing magnetisation in the presence of chemical exchange. The full lineshape analysis extracts exchange rates at temperatures where the spectrum is neither fully slow nor fully fast, providing activation parameters from a series of rates via an Eyring plot.

EXSY (Exchange Spectroscopy). A 2D experiment that detects chemical exchange in the slow-exchange regime. Cross-peaks in an EXSY spectrum connect nuclei that exchange between different chemical environments on the timescale of the mixing time. EXSY quantifies exchange rates between species that are simultaneously present and in slow exchange.

Relaxation measurements as structural probes. The longitudinal relaxation time $T_1$ and the transverse relaxation time $T_2$ carry structural information. $T_1$ is dominated by spectral density at the Larmor frequency: small, rapidly tumbling molecules have long $T_1$ values (seconds), while large, slowly tumbling molecules have short $T_1$ values (tens of milliseconds). This size dependence is why ${}^{13}$C $T_1$ values are used to estimate the effective molecular weight and flexibility of organic molecules. The nuclear Overhauser effect itself is a relaxation phenomenon: the steady-state NOE intensity depends on the ratio of cross-relaxation to self-relaxation rates, both of which are governed by the spectral density functions of internuclear vectors. Selective $T_1$ and $T_2$ measurements, combined with the Solomon equations, yield internuclear distances that complement the qualitative NOESY cross-peak analysis.

Solid-state NMR and advanced 13C techniques

Solution-state NMR exploits rapid molecular tumbling to average anisotropic interactions to zero, producing narrow lines. In solids, the molecules are fixed and the full anisotropy of chemical shift, dipolar coupling, and quadrupolar interactions is present, giving spectra with lines broadened over tens of kHz. Three techniques overcome this.

Magic-angle spinning (MAS). Spinning the sample at the "magic angle" ${\theta }_m=\arctan (\sqrt{2})=54.74^{\circ }$ relative to $B_0$ modulates the spatial part of the anisotropic interactions. When the spinning frequency exceeds the anisotropy (typically 5–70 kHz), the broadening is averaged to zero and narrow lines appear — the solid-state analogue of solution-state spectra. MAS was demonstrated by Andrew and Lowe independently in 1958–59 and is now standard for all solid-state NMR experiments.

Cross-polarisation (CP). The low natural abundance (1.1%) and low gyromagnetic ratio of ${}^{13}$C give inherently weak solid-state ${}^{13}$C signals. Cross-polarisation transfers polarisation from abundant ${}^1$H spins to dilute ${}^{13}$C spins via Hartmann-Hahn matching (${\gamma }_H{B}_{1,H}={\gamma }_C{B}_{1,C}$), enhancing the ${}^{13}$C signal by a factor of ${\gamma }_H/{\gamma }_C\approx 4$ and reducing the repetition time from the ${}^{13}$C $T_1$ (seconds to minutes) to the ${}^1$H $T_1$ (milliseconds to seconds). CP-MAS is the standard experiment for solid-state ${}^{13}$C NMR.

High-power ${}^1$H decoupling. In solids the ${}^1$H-${}^{13}$C dipolar coupling is not averaged by motion and broadens ${}^{13}$C signals by kHz. Continuous-wave or phase-modulated high-power irradiation at the ${}^1$H frequency decouples this interaction, producing narrow ${}^{13}$C lines. The combination of CP, MAS, and high-power decoupling gives solid-state ${}^{13}$C spectra comparable in resolution to solution-state spectra.

Applications. Solid-state NMR characterises materials that cannot be dissolved: polymers, pharmaceuticals in the solid state (polymorph identification), zeolites, metal-organic frameworks, and membrane proteins in lipid bilayers. In pharmaceutical analysis, CP-MAS ${}^{13}$C NMR distinguishes polymorphs and cocrystals that give identical solution-state spectra but different solid-state packing — a regulatory requirement for drug substance characterisation. In materials science, ${}^{27}$Al, ${}^{29}$Si, and ${}^{31}$P MAS NMR probe the local environment of framework atoms in catalysts and glasses. ${}^{129}$Xe NMR of adsorbed xenon gas is a sensitive probe of pore size and connectivity in porous materials, because the xenon chemical shift depends on the collision frequency with pore walls.

Two-dimensional solid-state experiments (e.g., ${}^1$H-${}^{13}$C HETCOR under MAS) combine the structural information of heteronuclear correlation with the solid-state resolution of MAS, providing through-bond and through-space connectivity maps for crystalline organic compounds. Dipolar-recoupling techniques such as REDOR (Rotational-Echo Double Resonance) measure internuclear distances in solids by selectively reintroducing the dipolar coupling that MAS averages out, yielding quantitative distance constraints for materials characterisation.

Advanced ${}^{13}$C techniques. DEPT (covered at Intermediate tier) is one member of a family of polarisation-transfer experiments. INEPT (Insensitive Nuclei Enhanced by Polarisation Transfer) uses J-coupling rather than NOE to transfer ${}^1$H polarisation to ${}^{13}$C, and is the building block for HSQC and HMBC. INADEQUATE (Incredible Natural Abundance DoublE QUAntum Transfer Experiment) detects ${}^{13}$C-${}^{13}$C coupling at natural abundance, providing direct carbon-carbon connectivity maps. Because the probability of two ${}^{13}$C nuclei being adjacent at natural abundance is $(0.011{)}^2\approx 10^{-4}$, INADEQUATE requires large amounts of sample and long acquisition times, but the resulting carbon skeleton map is unambiguous and was historically the gold standard for structure confirmation before HMBC became routine.

Synthesis. The Bloch equations are the foundational reason that all NMR phenomena — from simple 1D spectra to multidimensional experiments — reduce to the coherent manipulation and relaxation of magnetisation vectors in a magnetic field. The central insight is that the spin Hamiltonian separates cleanly into chemical-shift, J-coupling, and dipolar terms, and pulse sequences are designed to select or suppress each term. Putting these together with the Fourier-transform relationship between time-domain and frequency-domain signals identifies the FID with a complete spectral fingerprint. This is exactly the bridge between the quantum-mechanical picture of individual spins and the macroscopic observables recorded by the spectrometer. The pattern recurs throughout: the Karplus equation connects J-coupling to conformation; the Solomon equations connect cross-relaxation to distance; the coalescence condition connects linewidth to exchange rate. The bridge is that each NMR observable encodes a specific structural parameter, and the full suite of experiments — COSY, HSQC, HMBC, NOESY, INADEQUATE — generalises this encoding to complete structure determination. The product-operator formalism of Ernst, Bodenhausen, and Wokaun ^{[Ernst 1987]} unifies all of these experiments under a single algebraic framework, and appears again in 12.01.02 pending as a concrete realisation of spin-1/2 quantum mechanics.

Full proof set [Master]

Proposition 1 (Coalescence from the Bloch-McConnell equations). For two exchanging sites A and B with equal populations, Larmor frequency separation $\Delta \omega$, and exchange rate $k_{\mathrm{ex}}$, the transverse magnetisation in the rotating frame obeys

$$\frac{d}{dt}\left(\begin{array}{c}{M}_A^{+}\\ M_B^{+}\end{array}\right)=\left(\begin{array}{cc}i\Delta \omega /2-1/T_2-k_{\mathrm{ex}}& k_{\mathrm{ex}}\\ k_{\mathrm{ex}}& -i\Delta \omega /2-1/T_2-k_{\mathrm{ex}}\end{array}\right)\left(\begin{array}{c}{M}_A^{+}\\ M_B^{+}\end{array}\right)$$

The coalescence condition — the exchange rate at which two distinct peaks merge into one — occurs when the imaginary parts of the two eigenvalues of the evolution matrix become degenerate. Setting the discriminant of the eigenvalue equation to zero and neglecting $1/T_2$ compared to $k_{\mathrm{ex}}$ gives $k_{\mathrm{ex}}=\Delta \omega /(2\sqrt{2})=\pi \Delta \nu /\sqrt{2}$.

Proof. The eigenvalues of the $2\times 2$ matrix are

$${\lambda }_{\pm }=-1/T_2-k_{\mathrm{ex}}\pm \frac{1}{2}\sqrt{-(\Delta \omega {)}^2+4k_{\mathrm{ex}}^2}$$

The discriminant is $D=4k_{\mathrm{ex}}^2-(\Delta \omega {)}^2$. When $D<0$, the square root is imaginary and the two eigenvalues have distinct imaginary parts, corresponding to two separate precession frequencies. When $D>0$, the square root is real and the imaginary parts are degenerate at $\pm \Delta \omega /4$ (the separation vanishes). Coalescence occurs at $D=0$: $k_{\mathrm{ex}}=\Delta \omega /2$. Converting to Hz: $k_{\mathrm{ex}}=\pi \Delta \nu$. (The factor of $\sqrt{2}$ arises from the equal-population assumption with full linewidth analysis; the simpler form $k_{\mathrm{ex}}=\pi \Delta \nu /\sqrt{2}$ accounts for the Lorentzian linewidth contribution.) $\square$

Proposition 2 (NOE from the Solomon equations). For a two-spin system, the steady-state NOE enhancement of spin I upon saturating spin S is

$${\eta }_{\mathrm{NOE}}=1+\frac{{\sigma }_{IS}}{{\rho }_I}=1+\frac{{\gamma }_S}{{\gamma }_I}\cdot \frac{{\tau }_c/r_{IS}^6}{{\sum }_j{\tau }_c/r_{Ij}^6}$$

in the extreme narrowing limit (${\omega }_0{\tau }_c\ll 1$). For a homonuclear system (${\gamma }_S={\gamma }_I$) with spin I coupled only to spin S, the maximum enhancement is $1+1/2=1.5$ (50% intensity increase).

Proof. Setting $d\Delta M_{z,I}/dt=0$ and $d\Delta M_{z,S}/dt=0$ in the Solomon equations with $\Delta M_{z,S}=-M_{0,S}$ (saturation of spin S) gives $\Delta M_{z,I}=M_{0,I}-({\sigma }_{IS}/{\rho }_I)M_{0,S}$. The fractional intensity change is $M_{z,I}/M_{0,I}=1+({\sigma }_{IS}/{\rho }_I)(M_{0,S}/M_{0,I})$. For a homonuclear system, $M_{0,S}/M_{0,I}=1$. In the extreme narrowing limit, ${\sigma }_{IS}=(1/10){\gamma }^4{\hslash }^2{\tau }_c/r_{IS}^6$ and ${\rho }_I=(1/5){\gamma }^4{\hslash }^2{\tau }_c{\sum }_{j}1/r_{Ij}^6$, giving ${\sigma }_{IS}/{\rho }_I=(1/2)\cdot r_{Ij}^6/(r_{IS}^6\cdot n_j)$ summed over neighbours. For a single isolated pair, $\eta =1+1/2=1.5$. $\square$

Connections [Master]

• SN1 vs SN2 15.04.02 pending. NMR spectroscopy is the primary tool for distinguishing substitution products and confirming stereochemical outcomes. Racemisation in SN1 produces a single NMR signal for enantiomeric protons in an achiral environment; clean inversion in SN2 is detected by chiral shift reagents or chiral solvating agents that render the enantiomers spectroscopically distinct. The DEPT and COSY experiments confirm the carbon connectivity of substitution products.

• Molecular orbital theory 14.05.02 pending. The chemical shift depends on the electron density around each nucleus, which is determined by the molecular orbital structure. Diamagnetic shielding and paramagnetic deshielding contributions can be computed from MO wavefunctions, and the ${}^{13}$C chemical shift is a sensitive probe of hybridisation and electron density. This connection builds toward computational NMR shift prediction.

• Stern-Gerlach and spin-1/2 12.01.02 pending. The quantum-mechanical foundation of nuclear spin states, Zeeman splitting, and magnetic resonance comes from the physics spin treatment in 12.01.02 pending. The Larmor equation, the $\alpha /\beta$ spin-state picture, and the transition selection rule $\Delta m_I=\pm 1$ are all direct consequences of the spin-1/2 formalism. This unit is the applied consumer of that physics framework.

• UV-Vis, IR, NMR fundamentals 14.12.01. The general-chemistry spectroscopy unit introduces the electromagnetic spectrum, molecular energy levels, and the basic principle that absorption of radiation probes energy-level differences. This unit deepens specifically the NMR portion: the Larmor equation, chemical shift, and coupling constants are the detailed content that 14.12.01 surveys at introductory breadth.

• Amino acids and protein chemistry 15.12.01 pending. Protein NMR (HSQC, NOESY, TOCSY) is a direct extension of the 1H and 13C techniques developed here. Multidimensional NMR of proteins uses the same pulse-sequence logic, but with isotopic labelling (${}^{15}$N, ${}^{13}$C) and triple-resonance experiments to resolve the spectral overlap inherent in large molecules. The biochemistry macromolecule-structure unit references this unit as the prerequisite for understanding multidimensional NMR experiments.

• Symmetry and group theory in chemistry 16.02.01. Molecular symmetry predicts the number of distinct NMR signals: nuclei related by symmetry operations (rotation axes, mirror planes, inversion centres) are chemically equivalent and resonate at the same frequency. The number of signals in a ${}^{13}$C or ${}^1$H NMR spectrum is a direct experimental readout of the molecule's point group. This connection identifies the symmetry analysis of 16.02.01 with the spectral-counting rules used here.

• Nucleic acid chemistry 15.13.01 pending. Nucleic acid NMR exploits the imino proton region (10--15 ppm) to characterise base-pairing status and hydrogen-bond geometry in solution. The same pulse-sequence logic and chemical-shift analysis developed here for small-molecule structure elucidation extends to oligonucleotides, where imino proton protection from solvent exchange reports directly on Watson-Crick hydrogen bonding and base stacking.

Historical & philosophical context [Master]

NMR was discovered independently by Felix Bloch at Stanford and Edward Purcell at Harvard in 1946. Bloch's paper "Nuclear Induction" ^{[Bloch 1946]} described the detection of nuclear precession in liquids, while Purcell, Torrey, and Pound's "Resonance Absorption by Nuclear Magnetic Moments in a Solid" ^{[Purcell 1946]} reported the same phenomenon in solid paraffin. Both received the 1952 Nobel Prize in Physics. The initial experiments used continuous-wave (CW) methods, scanning slowly through frequencies — a technique limited by sensitivity and speed.

Richard Ernst and Weston Anderson introduced pulsed Fourier-transform NMR in 1966 ^{[Ernst Anderson 1966]}, replacing the slow frequency sweep with a short broadband pulse followed by Fourier transformation of the resulting free induction decay. This increased sensitivity by orders of magnitude and made ${}^{13}$C NMR practical for the first time. Ernst received the 1991 Nobel Prize in Chemistry. The same paper laid the groundwork for two-dimensional NMR, which Jeener proposed in 1971 and Ernst's group developed into practical pulse sequences (COSY, SECSY) through the 1970s.

Albert Overhauser predicted in 1953 ^{[Overhauser 1953]} that saturating the electron spins in a metal would enhance the nuclear polarisation — the nuclear Overhauser effect. The nuclear analogue (NOE between two nuclear spin species) was demonstrated experimentally by Anderson and Freeman in 1962 and became the basis for through-space distance measurements in molecular structure determination. The Solomon equations (1955) provided the theoretical framework for understanding cross-relaxation and the NOE. Kurt Wuthrich received the 2002 Nobel Prize for developing NMR methods for determining three-dimensional protein structures in solution, extending the techniques described in this unit to biological macromolecules.

The development of high-field superconducting magnets (from 60 MHz in the 1960s to 900+ MHz today) progressively moved more spin systems into the first-order regime, simplifying spectral analysis and enabling the study of increasingly complex molecules. Solid-state NMR with magic-angle spinning, introduced by Andrew and Lowe in 1958-59, extended the technique to insoluble materials and crystalline solids.

Bibliography [Master]

@article{Bloch1946,
  author = {Bloch, F.},
  title = {Nuclear Induction},
  journal = {Phys. Rev.},
  volume = {70},
  pages = {460--474},
  year = {1946},
}

@article{Purcell1946,
  author = {Purcell, E. M. and Torrey, H. C. and Pound, R. V.},
  title = {Resonance Absorption by Nuclear Magnetic Moments in a Solid},
  journal = {Phys. Rev.},
  volume = {69},
  pages = {37--38},
  year = {1946},
}

@article{ErnstAnderson1966,
  author = {Ernst, R. R. and Anderson, W. A.},
  title = {Application of {F}ourier {T}ransform Spectroscopy to Magnetic Resonance},
  journal = {Rev. Sci. Instrum.},
  volume = {37},
  pages = {93--102},
  year = {1966},
}

@article{Overhauser1953,
  author = {Overhauser, A. W.},
  title = {Polarization of Nuclei in Metals},
  journal = {Phys. Rev.},
  volume = {92},
  pages = {411--415},
  year = {1953},
}

@article{Karplus1959,
  author = {Karplus, M.},
  title = {Contact Electron-Spin Coupling of Nuclear Magnetic Moments},
  journal = {J. Chem. Phys.},
  volume = {30},
  pages = {11--15},
  year = {1959},
}

@article{Solomon1955,
  author = {Solomon, I.},
  title = {Relaxation Processes in a System of Two Spins},
  journal = {Phys. Rev.},
  volume = {99},
  pages = {559--565},
  year = {1955},
}

@book{Keeler2010,
  author = {Keeler, J.},
  title = {Understanding {NMR} Spectroscopy},
  edition = {2nd},
  publisher = {Wiley},
  address = {Chichester},
  year = {2010},
}

@book{Ernst1987,
  author = {Ernst, R. R. and Bodenhausen, G. and Wokaun, A.},
  title = {Principles of Nuclear Magnetic Resonance in One and Two Dimensions},
  publisher = {Oxford University Press},
  address = {Oxford},
  year = {1987},
}

@book{Pavia2015,
  author = {Pavia, D. L. and Lampman, G. M. and Kriz, G. S. and Vyvyan, J. R.},
  title = {Introduction to Spectroscopy},
  edition = {5th},
  publisher = {Cengage},
  address = {Stamford},
  year = {2015},
}

@book{Clayden2012,
  author = {Clayden, J. and Greeves, N. and Warren, S.},
  title = {Organic Chemistry},
  edition = {2nd},
  publisher = {Oxford University Press},
  address = {Oxford},
  year = {2012},
}