3.8 Nuclear Magnetic Resonance (NMR) Spectroscopy

Principles of NMR Spectroscopy

Fundamentals of NMR

NMR spectroscopy exploits the magnetic properties of certain atomic nuclei to determine the physical and chemical properties of atoms or molecules. Only nuclei with a non-zero spin quantum number are "NMR active." The two most commonly studied nuclei are $^1\text{H}$ (spin $I = \frac{1}{2}$) and $^{13}\text{C}$ (spin $I = \frac{1}{2}$), both of which possess an intrinsic magnetic moment that lets them interact with an external magnetic field.

When placed in an external magnetic field $B_0$, these nuclei split into $2I + 1$ energy levels (two levels for spin-$\frac{1}{2}$ nuclei: aligned with or against the field). The nuclei can absorb and re-emit electromagnetic radiation at a specific resonance frequency that depends on the field strength and the identity of the isotope.

That resonance frequency is given by the Larmor equation:

$\omega_0 = \gamma B_0$

where $\omega_0$ is the Larmor (angular) frequency, $\gamma$ is the gyromagnetic ratio (a constant specific to each isotope), and $B_0$ is the strength of the external magnetic field. In terms of linear frequency: $\nu_0 = \frac{\gamma B_0}{2\pi}$. The energy gap between the two spin states is therefore:

$\Delta E = \hbar \gamma B_0$

This energy gap is small, which means NMR transitions fall in the radiofrequency range (tens to hundreds of MHz at typical field strengths). The small $\Delta E$ also means the population difference between spin states is tiny, which is why NMR is inherently a low-sensitivity technique compared to, say, UV-Vis or IR spectroscopy.

Chemical Environment and Molecular Structure

The electrons surrounding a nucleus partially shield it from the full external field. The nucleus actually experiences an effective field:

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

where $\sigma$ is the shielding constant. Because different nuclei in a molecule have different electronic environments, they resonate at slightly different frequencies. This difference is the chemical shift, reported on the $\delta$ scale in parts per million (ppm) relative to a reference compound (tetramethylsilane, TMS, defined as $\delta = 0$):

$\delta = \frac{\nu_{\text{sample}} - \nu_{\text{TMS}}}{\nu_{\text{TMS}}} \times 10^6$

Chemical shift is influenced by:

• Electron density around the nucleus (more electron density → more shielding → lower ppm)
• Electronegativity of neighboring atoms (electronegative substituents withdraw electron density → deshielding → higher ppm)
• Ring current effects in aromatic systems (aromatic protons are strongly deshielded, appearing at 6.5–8.5 ppm)
• Hydrogen bonding, which can deshield the bonded proton and also broaden the signal

Chemical shift values let you identify functional groups. For example, aldehyde protons appear near 9–10 ppm, aromatic protons near 6.5–8.5 ppm, and alkyl protons near 0.5–2.5 ppm.

Spin-spin coupling arises from the interaction between neighboring NMR-active nuclei transmitted through bonding electrons. This interaction splits NMR signals into multiplets and provides direct information about atom connectivity.

• The n+1 rule: a proton with $n$ equivalent neighboring protons splits into $n + 1$ lines. (This rule assumes all coupling constants to those neighbors are equal, which holds for equivalent neighbors.)
• The intensity ratios of the multiplet lines follow Pascal's triangle: a doublet is 1:1, a triplet is 1:2:1, a quartet is 1:3:3:1, and so on.

The spacing between the lines of a multiplet is the coupling constant $J$, measured in Hz. Unlike chemical shift (in ppm), $J$ is independent of the spectrometer's operating frequency.

Interpreting NMR Spectra

$^1$H NMR Spectroscopy

$^1\text{H}$ NMR is the most widely used form of NMR because hydrogen is abundant in organic molecules and $^1\text{H}$ has high natural abundance (99.98%) and a large gyromagnetic ratio, giving strong signals.

Three pieces of information come from a $^1\text{H}$ spectrum:

1. Number of signals → number of chemically distinct hydrogen environments in the molecule.
2. Integration (relative peak area) → proportional to the number of hydrogen atoms in each environment.
3. Splitting pattern and coupling constants → connectivity to neighboring hydrogens.

Chemical shift trends for $^1\text{H}$:

• Alkyl $\text{C–H}$: 0.5–2.5 ppm (relatively shielded)
• $\text{C–H}$ adjacent to electronegative atoms ($\text{O, N, X}$): shifted downfield, roughly 2.5–4.5 ppm depending on the substituent
• Aromatic $\text{C–H}$: 6.5–8.5 ppm, deshielded by the ring current of the $\pi$ system
• Aldehyde $\text{C–H}$: ~9–10 ppm (strong deshielding from the carbonyl)
• Carboxylic acid $\text{O–H}$: 10–12 ppm (very deshielded; often broad due to exchange)

Spin-spin coupling in $^1\text{H}$ NMR reveals how many hydrogens sit on adjacent atoms. For instance, the $\text{–CH}_3$ group of ethanol ($\text{CH}_3\text{CH}_2\text{OH}$) appears as a triplet because the two neighboring $\text{CH}_2$ protons split it into $2 + 1 = 3$ lines.

$^{13}$C NMR Spectroscopy and Advanced Techniques

$^{13}\text{C}$ NMR provides a map of the carbon skeleton. Its natural abundance is only ~1.1%, so sensitivity is much lower than $^1\text{H}$ NMR. In routine practice, $^{13}\text{C}$ spectra are acquired with broadband proton decoupling, which collapses all $^{13}\text{C–}^1\text{H}$ coupling and gives one sharp singlet per unique carbon environment. This simplifies the spectrum but means you lose multiplicity information (which can be recovered with DEPT experiments).

Chemical shift trends for $^{13}\text{C}$:

• Alkyl carbons: 0–50 ppm
• Carbons bonded to electronegative atoms ($\text{C–O, C–N, C–X}$): 50–90 ppm
• Aromatic and vinyl carbons: ~100–160 ppm
• Carbonyl carbons (aldehydes, ketones, esters, acids): ~160–220 ppm, strongly deshielded by the electron-withdrawing oxygen

Combining $^1\text{H}$ and $^{13}\text{C}$ data with 2D NMR techniques allows full structural elucidation:

• COSY (COrrelation SpectroscopY): identifies pairs of $^1\text{H}$ nuclei that are spin-spin coupled to each other (typically 2–3 bonds apart). You read it by finding off-diagonal cross-peaks that connect two signals on the diagonal.
• HSQC (Heteronuclear Single Quantum Coherence): correlates each $^{13}\text{C}$ signal with the $^1\text{H}$ directly bonded to it (one-bond $^{1}J_{\text{CH}}$). Quaternary carbons produce no cross-peak.
• HMBC (Heteronuclear Multiple Bond Correlation): correlates $^{13}\text{C}$ and $^1\text{H}$ signals that are 2–4 bonds apart. This is especially useful for connecting fragments across quaternary carbons or heteroatoms where direct $\text{C–H}$ bonds are absent.

Factors Affecting NMR Spectra

Chemical Shift and Spin-Spin Coupling

Chemical shift was introduced above; here the focus is on the factors that modulate it and on coupling constants in more detail.

Shielding and deshielding effects:

• Electron-withdrawing groups (halogens, carbonyls, nitro groups) reduce electron density around nearby nuclei → deshielding → downfield shift (higher ppm).
• Electron-donating groups (alkyl groups) increase electron density → shielding → upfield shift (lower ppm).
• These effects are additive and roughly decrease with distance from the substituent.

Coupling constants ($J$) and structural information:

The magnitude of $J$ depends on the number of intervening bonds and the geometry:

• Geminal coupling ($^2J_{\text{HH}}$, two bonds): typically 0–5 Hz for $\text{sp}^3$ carbons, but can be larger (~2–12 Hz) depending on bond angle and substituents.
• Vicinal coupling ($^3J_{\text{HH}}$, three bonds): typically 2–15 Hz, and strongly geometry-dependent.

The Karplus equation quantifies how vicinal coupling depends on the dihedral angle $\phi$ between the two coupled protons:

$^3J = A\cos^2\phi + B\cos\phi + C$

where $A$, $B$, and $C$ are empirical constants (a common parameterization gives roughly $A \approx 7$, $B \approx -1$, $C \approx 5$ Hz for $\text{H–C–C–H}$ fragments, though exact values vary with substitution). The key takeaway: $^3J$ is largest when $\phi = 0°$ or $180°$ (anti-periplanar) and smallest near $\phi = 90°$. This makes $J$ values a direct probe of molecular conformation.

Relaxation Processes

After a radiofrequency pulse excites the spin system, the nuclei must return to thermal equilibrium. This return is governed by two distinct relaxation processes, each with its own time constant.

$T_1$ (spin-lattice / longitudinal relaxation):

• Describes the recovery of the net magnetization along the direction of $B_0$ (the $z$-axis).
• Energy is transferred from the excited spins to the surrounding molecular framework (the "lattice").
• $T_1$ determines how long you must wait between successive scans. If you pulse again before the spins have substantially relaxed, signal intensity is reduced (partial saturation).

$T_2$ (spin-spin / transverse relaxation):

• Describes the decay of magnetization in the plane perpendicular to $B_0$ (the $xy$-plane).
• Caused by loss of phase coherence among the precessing spins due to local field fluctuations from neighboring nuclei.
• $T_2$ directly determines the linewidth of the NMR signal: a shorter $T_2$ gives a broader line. The relationship is $\Delta\nu_{1/2} = \frac{1}{\pi T_2}$, where $\Delta\nu_{1/2}$ is the full width at half maximum.
• $T_2 \leq T_1$ always holds.

Factors influencing relaxation times:

• Molecular size and tumbling rate: Small molecules in low-viscosity solvents tumble rapidly and tend to have longer $T_1$ and $T_2$. Large molecules (e.g., proteins) tumble slowly, leading to efficient relaxation and shorter $T_2$ (broad lines). The relationship between $T_1$ and molecular tumbling is non-monotonic: $T_1$ passes through a minimum when the tumbling rate matches the Larmor frequency.
• Viscosity and temperature: Higher viscosity or lower temperature slows molecular motion, generally shortening $T_2$ and broadening lines. The effect on $T_1$ depends on whether the molecule is in the fast- or slow-tumbling regime.
• Paramagnetic species: Ions or molecules with unpaired electrons (e.g., $\text{Cu}^{2+}$, $\text{Mn}^{2+}$, dissolved $\text{O}_2$) create large fluctuating local fields and dramatically shorten both $T_1$ and $T_2$. This is why NMR samples are often degassed and why paramagnetic relaxation agents are used deliberately in MRI contrast imaging.