Unit 8: Nuclear Magnetic Resonance (NMR) I: Larmor Precession, Relaxation & Proton Chemical Shifts

Atomic nuclei possessing an odd number of protons, an odd number of neutrons, or both, exhibit intrinsic nuclear spin angular momentum \(\vec{I}\). The magnitude of \(\vec{I}\) and its projection along a quantization \(z\)-axis are quantized:

\[ |\vec{I}| = \hbar \sqrt{I(I+1)}, \quad I_z = m_I \hbar \quad (m_I = -I, -I+1, \dots, +I) \]

where \(I\) is the nuclear spin quantum number (\(I = 1/2\) for \(^1\text{H}, ^{13}\text{C}, ^{19}\text{F}, ^{31}\text{P}\); \(I = 1\) for \(^2\text{H}, ^{14}\text{N}\); \(I = 0\) for \(^{12}\text{C}, ^{16}\text{O}\)).

Nuclear Magnetic Dipole Moment

The spinning nuclear charge produces a collinear magnetic dipole moment \(\vec{\mu}_N\):

\[ \vec{\mu}_N = \gamma_N \vec{I} = g_N \mu_N \frac{\vec{I}}{\hbar} \]

where \(\gamma_N\) is the gyromagnetic ratio (a fundamental constant characteristic of each nuclide), \(g_N\) is the nuclear \(g\)-factor, and \(\mu_N = \frac{e\hbar}{2m_p} = 5.05078 \times 10^{-27}\text{ J/T}\) is the nuclear magneton. For the proton (\(^1\text{H}\)), \(\gamma_H = 2.67522 \times 10^8\text{ rad}/(\text{s}\cdot\text{T})\) (\(\frac{\gamma_H}{2\pi} = 42.577\text{ MHz/T}\)).

The Nuclear Zeeman Interaction

Placing the nucleus in a static magnetic field \(\vec{B}_0 = B_0 \hat{z}\) produces the Zeeman Hamiltonian:

\[ \hat{H}_Z = -\vec{\mu}_N \cdot \vec{B}_0 = -\gamma_N \hbar B_0 \hat{I}_z \]

The energy eigenvalues for magnetic quantum numbers \(m_I\) are:

\[ E(m_I) = -\gamma_N \hbar B_0 m_I \]

For a spin \(I = 1/2\) nucleus (\(^1\text{H}\)), the field splits the state into two energy levels:

• Lower state (\(\alpha\), \(m_I = +1/2\)): aligned with the field, \(E_\alpha = -\frac{1}{2}\gamma_N \hbar B_0\)
• Upper state (\(\beta\), \(m_I = -1/2\)): aligned against the field, \(E_\beta = +\frac{1}{2}\gamma_N \hbar B_0\)

The energy separation is:

\[ \Delta E = E_\beta - E_\alpha = \gamma_N \hbar B_0 = h \nu_0 \]

Resonance absorption occurs at the Larmor frequency: \(\nu_0 = \frac{\gamma_N}{2\pi} B_0\). For protons in a \(7.046\text{ T}\) magnet, \(\nu_0 = 300\text{ MHz}\); in a \(14.092\text{ T}\) magnet, \(\nu_0 = 600\text{ MHz}\).

The energetic splitting \(\Delta E = \hbar\omega_0\) in NMR is extraordinarily small compared to thermal energy at room temperature (\(\Delta E \approx 10^{-25}\text{ J} \ll k_B T \approx 4 \times 10^{-21}\text{ J}\)).

Fractional Boltzmann Excess

According to the Maxwell-Boltzmann distribution:

\[ \frac{N_\beta}{N_\alpha} = \exp\left(-\frac{\Delta E}{k_B T}\right) = \exp\left(-\frac{\gamma_N \hbar B_0}{k_B T}\right) \]

Expanding the exponential in a first-order Taylor series (\(e^{-x} \approx 1 - x\)):

\[ \frac{N_\alpha - N_\beta}{N_\alpha + N_\beta} \approx \frac{\gamma_N \hbar B_0}{2 k_B T} \]

For protons at \(300\text{ MHz}\) (\(B_0 = 7.05\text{ T}\)) at \(T = 300\text{ K}\):

\[ \frac{\gamma_N \hbar B_0}{2 k_B T} \approx 2.4 \times 10^{-5} \]

Out of two million proton spins, only about 48 excess spins populate the lower state! This minute population bias generates the macroscopic net magnetization vector \(\vec{M}_0 = \sum \vec{\mu}_i = \frac{N \gamma_N^2 \hbar^2 I(I+1)}{3 k_B T} \vec{B}_0\) and explains why NMR requires high magnetic fields and signal averaging.

Classical Torque & Larmor Precession

Classically, the static magnetic field exerts a torque \(\vec{\tau} = \vec{M} \times \vec{B}_0\) on the macroscopic magnetization vector. Since torque equals the time rate of change of angular momentum \(\vec{L} = \vec{M} / \gamma\):

\[ \frac{d\vec{M}}{dt} = \gamma_N (\vec{M} \times \vec{B}_0) \]

This differential equation describes a steady precession of \(\vec{M}\) around \(\vec{B}_0\) at angular frequency \(\vec{\omega}_0 = -\gamma_N \vec{B}_0\).

In 1946, Felix Bloch formulated the phenomenological differential equations governing the time evolution of magnetization \(\vec{M}(t) = (M_x, M_y, M_z)\) undergoing precession and relaxation back toward thermal equilibrium \(\vec{M}_0 = (0, 0, M_0)\):

\[ \frac{dM_z}{dt} = \gamma_N (\vec{M} \times \vec{B})_z - \frac{M_z - M_0}{T_1} \] \[ \frac{dM_x}{dt} = \gamma_N (\vec{M} \times \vec{B})_x - \frac{M_x}{T_2} \] \[ \frac{dM_y}{dt} = \gamma_N (\vec{M} \times \vec{B})_y - \frac{M_y}{T_2} \]

Longitudinal (Spin-Lattice) Relaxation Time (\(T_1\))

Relaxation along the \(z\)-axis requires exchanging energy between the nuclear spin system and the thermal surrounding environment ('lattice'). Following a \(180^\circ\) inversion pulse (\(M_z(0) = -M_0\)):

\[ M_z(t) = M_0 \left(1 - 2 e^{-t / T_1}\right) \]

\(T_1\) is measured using the inversion-recovery pulse sequence (\(180^\circ - \tau - 90^\circ - \text{acquire}\)).

Transverse (Spin-Spin) Relaxation Time (\(T_2\))

Relaxation in the \(xy\)-plane represents the irreversible loss of phase coherence among precessing spins due to local magnetic field fluctuations, with no net energy exchange with the lattice:

\[ M_{xy}(t) = M_{xy}(0) e^{-t / T_2} \]

The natural linewidth FWHM of an NMR Lorentzian peak is inversely proportional to \(T_2\): \(\Delta \nu_{1/2} = \frac{1}{\pi T_2^*}\), where \(\frac{1}{T_2^*} = \frac{1}{T_2} + \frac{\gamma \Delta B_0}{2}\) accounts for magnet inhomogeneity.

If all protons resonated at the exact same Larmor frequency, NMR would be useless to chemists. Fortunately, nuclei are surrounded by orbiting electron clouds. In an external field \(\vec{B}_0\), the electrons circulate, generating a small local induced magnetic field \(\vec{B}_{\text{ind}} = -\boldsymbol{\sigma} \vec{B}_0\) that opposes the applied field.

The effective magnetic field experienced at the nucleus is:

\[ B_{\text{local}} = B_0(1 - \sigma) \]

where \(\sigma\) is the dimensionless nuclear shielding constant (typically \(\sim 10^{-6}\) for protons, \(\sim 10^{-4}\) for \(^{13}\text{C}\)). The resonance frequency becomes:

\[ \nu = \frac{\gamma_N}{2\pi} B_0 (1 - \sigma) \]

Ramsey's Equation for Shielding

Norman Ramsey derived the quantum mechanical expression for \(\sigma\), separating it into two opposing terms:

\[ \sigma = \sigma_d + \sigma_p \]

1. Diamagnetic Shielding (\(\sigma_d > 0\)): Arises from the circulation of electrons in ground-state orbitals (Lamb formula). It opposes \(B_0\), shielding the nucleus and shifting the resonance upfield. Dominated by s-electron density.
2. Paramagnetic Deshielding (\(\sigma_p < 0\)): Arises from magnetic-field-induced mixing of low-lying excited electronic states with the ground state, creating orbital angular momentum that reinforces \(B_0\). Dominates heavy nuclei (\(^{13}\text{C}, ^{31}\text{P}, ^{19}\text{F}\)) possessing p and d valence electrons.

The Delta (\(\delta\)) Chemical Shift Scale

Because absolute frequency shifts \(\Delta\nu\) scale linearly with applied field \(B_0\), the universal dimensionless chemical shift \(\delta\) (parts per million, \(\text{ppm}\)) is defined relative to tetramethylsilane (\(\text{TMS}\), \(\text{Si(CH}_3)_4\)):

\[ \delta = \frac{\nu_{\text{sample}} - \nu_{\text{TMS}}}{\nu_{\text{spectrometer}}} \times 10^6 = \frac{\sigma_{\text{TMS}} - \sigma_{\text{sample}}}{1 - \sigma_{\text{TMS}}} \approx (\sigma_{\text{TMS}} - \sigma_{\text{sample}}) \times 10^6 \]

The \(\delta\) scale is completely independent of the operating magnetic field strength of the spectrometer.

Proton chemical shifts in organic molecules span a characteristic range of \(0 - 14\text{ ppm}\), determined by the local electronic environment:

1. Electronegativity & Inductive Effects

Electronegative substituents (F, Cl, Br, O, N) withdraw electron density through \(\sigma\)-bonds, reducing diamagnetic shielding (\(\sigma_d\)) around adjacent protons. Less shielded protons experience a stronger effective field and resonate downfield (higher \(\delta\)):

• \(\text{CH}_4\): \(\delta = 0.23\text{ ppm}\)
• \(\text{CH}_3\text{I}\): \(\delta = 2.16\text{ ppm}\)
• \(\text{CH}_3\text{Br}\): \(\delta = 2.68\text{ ppm}\)
• \(\text{CH}_3\text{Cl}\): \(\delta = 3.05\text{ ppm}\)
• \(\text{CH}_3\text{F}\): \(\delta = 4.26\text{ ppm}\)

The inductive deshielding effect attenuates rapidly with distance, becoming negligible beyond three bonds.

2. Carbon Hybridization

Higher s-character increases the electronegativity of the carbon atom: \(sp^3\ (25\% s) < sp^2\ (33\% s) < sp\ (50\% s)\). However, chemical shifts do not follow simple electronegativity because magnetic anisotropy dominates \(sp^2\) and \(sp\) carbons:

• Aliphatic \(sp^3\) (\(\text{-CH}_3, \text{-CH}_2\text{-}\)): \(\delta = 0.8 - 1.8\text{ ppm}\)
• Alkyne acetylenic \(sp\) (\(\text{-C}\equiv\text{C-H}\)): \(\delta = 2.0 - 3.0\text{ ppm}\) (anomalously shielded!)
• Alkene vinylic \(sp^2\) (\(>\text{C}=\text{CH}_2\)): \(\delta = 4.5 - 6.5\text{ ppm}\)
• Aromatic \(sp^2\) (\(\text{Ar-H}\)): \(\delta = 6.5 - 8.5\text{ ppm}\)
• Aldehyde carbonyl (\(\text{-CHO}\)): \(\delta = 9.5 - 10.5\text{ ppm}\)
• Carboxylic acid (\(\text{-COOH}\)): \(\delta = 10.5 - 13.0\text{ ppm}\)

### Advanced Quantum Formalism: Product Operator Formalism for Two-Spin Systems For multi-pulse and multidimensional NMR experiments, vector models fail whenever spin-spin scalar coupling \(J\) or quantum coherence transfer is involved. The rigorous description requires the density matrix and **product operator formalism**. #### 1. Basis Operators for a Two-Spin System (\(I = 1/2, S = 1/2\)) The 16 orthogonal Cartesian product operators spanning Liouville space are: \[ \frac{1}{2}\hat{E}, \quad \hat{I}_x, \hat{I}_y, \hat{I}_z, \quad \hat{S}_x, \hat{S}_y, \hat{S}_z, \quad 2\hat{I}_x\hat{S}_z, 2\hat{I}_y\hat{S}_z, 2\hat{I}_z\hat{S}_x, 2\hat{I}_z\hat{S}_y, \dots, 4\hat{I}_x\hat{S}_x\dots \] Physical significance: - \(\hat{I}_z, \hat{S}_z\): Longitudinal polarization (Zeeman magnetization). - \(\hat{I}_x, \hat{I}_y\): Single-quantum in-phase coherence (observable transverse magnetization). - \(2\hat{I}_x\hat{S}_z, 2\hat{I}_y\hat{S}_z\): Single-quantum anti-phase coherence (multiplet components with opposite phase). - \(2\hat{I}_x\hat{S}_x, 2\hat{I}_x\hat{S}_y\): Multiple-quantum coherences (zero-quantum and double-quantum coherences, strictly invisible to direct detection). #### 2. The Three Fundamental Evolution Rotations The time evolution of the density operator under any interaction Hamiltonian \(\hat{H}\) is governed by the Liouville-von Neumann equation: \[ \hat{\sigma}(t) = \exp\left(-\frac{i \hat{H} t}{\hbar}\right) \hat{\sigma}(0) \exp\left(\frac{i \hat{H} t}{\hbar}\right) \] 1. **Chemical Shift Evolution (\(\hat{H}_{\text{CS}} = \Omega_I \hat{I}_z\)):** \[ \hat{I}_x \xrightarrow{\Omega_I t \hat{I}_z} \hat{I}_x \cos(\Omega_I t) + \hat{I}_y \sin(\Omega_I t) \] \[ \hat{I}_y \xrightarrow{\Omega_I t \hat{I}_z} \hat{I}_y \cos(\Omega_I t) - \hat{I}_x \sin(\Omega_I t) \] 2. **Scalar Coupling Evolution (\(\hat{H}_J = 2\pi J \hat{I}_z \hat{S}_z\)):** \[ \hat{I}_x \xrightarrow{\pi J t 2\hat{I}_z\hat{S}_z} \hat{I}_x \cos(\pi J t) + 2\hat{I}_y\hat{S}_z \sin(\pi J t) \] \[ 2\hat{I}_y\hat{S}_z \xrightarrow{\pi J t 2\hat{I}_z\hat{S}_z} 2\hat{I}_y\hat{S}_z \cos(\pi J t) + \hat{I}_x \sin(\pi J t) \] At \(t = \frac{1}{2J}\) (\(\pi J t = \pi/2\)), total conversion of in-phase into anti-phase coherence occurs: \[ \hat{I}_x \rightarrow 2\hat{I}_y\hat{S}_z \] 3. **Radiofrequency Pulse Rotations (Flip angle \(\beta\) along axis \(x\) or \(y\)):** \[ \hat{I}_z \xrightarrow{\beta \hat{I}_x} \hat{I}_z \cos\beta - \hat{I}_y \sin\beta \] \[ 2\hat{I}_y\hat{S}_z \xrightarrow{(\pi/2)\hat{S}_x} -2\hat{I}_y\hat{S}_y \quad (\text{double-quantum / zero-quantum coherence}) \] This algebra enables exact closed-form tracking of any arbitrary multi-dimensional NMR pulse sequence without numerical matrix integration.

Chemical bonds with non-spherical electron distributions (double bonds, triple bonds, aromatic rings) possess anisotropic magnetic susceptibility (\(\Delta\chi = \chi_\parallel - \chi_\perp \neq 0\)). The secondary field generated by circulating electrons depends dramatically on spatial orientation:

The McConnell equation describes the dipolar shielding shift at distance \(R\) and angle \(\theta\) relative to the symmetry axis:

\[ \Delta\sigma = \frac{\Delta\chi}{12 \pi R^3} (1 - 3\cos^2\theta) \]

Aromatic Ring Currents (Diatropic)

In benzene (\(\text{C}_6\text{H}_6\)), the applied field \(B_0\) perpendicular to the ring plane drives a circulation of the \(6\pi\) aromatic electrons around the ring perimeter (Paulings ring current model). By Lenz's law, the induced magnetic field opposes \(B_0\) in the ring interior, but loops around and reinforces \(B_0\) at the ring periphery where aromatic protons reside!

Consequently, benzene protons experience an additional downfield deshielding of \(\Delta\delta \approx +1.5 - 2.0\text{ ppm}\), shifting them to \(\delta = 7.27\text{ ppm}\). In [18]annulene, outer protons reside in the deshielding zone (\(\delta = 9.3\text{ ppm}\)), while inner protons reside in the intense shielding cone (\(\delta = -3.0\text{ ppm}\)), confirming diatropic ring currents.

Acetylenic Shielding Cone

In terminal alkynes (\(\text{R-C}\equiv\text{C-H}\)), the cylindrical \(\pi\)-electrons circulate freely around the triple bond axis when \(B_0\) is parallel to the bond. The induced field opposes \(B_0\) along the bond axis, placing the acetylenic proton in an intense shielding cone that shifts its resonance upfield to \(\delta \approx 2.0 - 2.5\text{ ppm}\).

Protons attached to heteroatoms (\(\text{-OH}, \text{-NH}_2, \text{-COOH}\)) and molecules undergoing conformational exchange (e.g., chair-chair flip of cyclohexane or amide rotation of dimethylformamide) undergo dynamic chemical exchange between distinct magnetic sites \(A\) and \(B\) with lifetimes \(\tau_A\) and \(\tau_B\).

Timescale Regimes of NMR Exchange

The appearance of the spectrum depends on the exchange rate \(k = \tau^{-1}\) relative to the chemical shift frequency difference \(\Delta\nu = |\nu_A - \nu_B|\) (in Hz):

1. Slow Exchange (\(k \ll \Delta\nu\)): The exchange process is frozen on the NMR timescale. Two distinct, sharp peaks appear at \(\nu_A\) and \(\nu_B\).
2. Intermediate Exchange & Coalescence (\(k \approx \Delta\nu\)): As temperature increases, the two peaks broaden and migrate toward each other, eventually merging into a single flat-topped peak at the coalescence temperature (\(T_c\)). At coalescence, the rate constant for an uncoupled two-site exchange with equal populations is: \[ k_c = \frac{\pi \Delta\nu}{\sqrt{2}} \approx 2.22 \Delta\nu \]
3. Fast Exchange (\(k \gg \Delta\nu\)): The exchange is rapid. A single sharp peak appears at the population-weighted average chemical shift: \(\nu_{\text{avg}} = p_A \nu_A + p_B \nu_B\).

Eyring Activation Barrier Calculation

Measuring the coalescence temperature \(T_c\) and frequency difference \(\Delta\nu\) enables the determination of the Gibbs free energy of activation \(\Delta G^\ddagger\) for the conformational or chemical exchange process using the Eyring equation:

\[ \Delta G^\ddagger = R T_c \left[ 23.76 + \ln\left(\frac{T_c}{k_c}\right) \right] = R T_c \left[ 22.96 + \ln\left(\frac{T_c}{\Delta\nu}\right) \right] \] ### Advanced Research Monograph: Dynamic Nuclear Polarization (DNP) Surface-Enhanced NMR The fundamental bottleneck of NMR spectroscopy is its intrinsically low sensitivity, caused by tiny nuclear Zeeman energy splittings relative to thermal energy (\(\Delta E_{\text{Zeeman}} \ll k_B T\)), which produces Boltzmann population differences of only \(\sim 10^{-5}\). #### 1. Principle of Dynamic Nuclear Polarization (DNP) Because the gyromagnetic ratio of the electron is much larger than that of nuclei (\(|\gamma_e| \approx 658 \times \gamma_{^1\text{H}}\)): - Unpaired electron spins polarize almost completely (\(P_e \approx 100\%\)) at cryogenic temperatures (\(100\text{ K}\)) and high magnetic fields (\(9.4\text{ T}\)). - By introducing stable bis-nitroxide biradicals (e.g., AMUPol, TEKPol) and irradiating the sample with continuous-wave high-power microwave radiation generated by a **gyrotron** (\(\sim 263\text{ GHz}\) at \(9.4\text{ T}\)), polarization is transferred from electron spins to nuclear spins via the Cross Effect (CE): \[ \epsilon = \frac{P_{\text{DNP}}}{P_{\text{Boltzmann}}} \sim 100 - 400 \] #### 2. Signal Gain and Acquisition Speedup Since experimental signal-to-noise ratio scales with \(\sqrt{N_{\text{scans}}}\), a DNP enhancement factor of \(\epsilon = 100\) reduces experimental acquisition time by a factor of: \[ \text{Time Reduction Factor} = \epsilon^2 = (100)^2 = 10000 \] An experiment requiring 3 years of continuous signal averaging on a conventional NMR spectrometer can be recorded in less than 3 hours under DNP conditions! #### 3. DNP-SENS: Surface-Enhanced NMR Spectroscopy of Materials In DNP Surface-Enhanced NMR Spectroscopy (DNP-SENS): - Incipient wetness impregnation wets porous catalysts, metal-organic frameworks (MOFs), or functionalized nanoparticles with a biradical solution without penetrating dense inorganic bulk lattices. - Hyperpolarization originates exclusively at the external liquid-solid interface and propagates into the material surface via \(^1\text{H}-^1\text{H}\) spin diffusion. - This selectively amplifies surface species (\(^{13}\text{C}, ^{15}\text{N}, ^{29}\text{Si}, ^{17}\text{O}, ^{27}\text{Al}\)) by several orders of magnitude, allowing sub-monolayer active catalytic sites to be characterized with structural precision previously restricted to bulk solution NMR.

Enantiomers in an achiral solvent environment possess identical chemical shifts and coupling constants in \(^1\text{H}\) and \(^{13}\text{C}\) NMR spectra because their internal stereochemical relationships are enantiotopic and related by symmetry.

Lanthanide Shift Reagents (LSR)

In 1969, Hinckley discovered that paramagnetic coordination complexes of trivalent lanthanide ions (primarily Europium \(\text{Eu}^{3+}\) and Praseodymium \(\text{Pr}^{3+}\)) with fluorinated \(\beta\)-diketonate ligands (such as \(\text{Eu(fod)}_3\) or \(\text{Eu(dpm)}_3\)) reversibly coordinate to Lewis-basic functional groups (\(-\text{OH}, -\text{NH}_2, >\text{C=O}, -\text{O-}\)):

\[ \text{Substrate} + \text{LSR} \rightleftharpoons [\text{Substrate} \cdot \text{LSR}] \]

Fast chemical exchange on the NMR timescale produces a weighted average chemical shift. The paramagnetic lanthanide induces a massive pseudo-contact (dipolar) shift \(\Delta\delta_{\text{dip}}\) governed by the Bleaney equation:

\[ \Delta\delta_{\text{dip}} = C_J \frac{\mu_B^2}{k_B^2 T^2} \left[ \frac{3\cos^2\theta - 1}{r^3} \right] \]

where \(r\) is the distance from the lanthanide metal center to the nucleus, \(\theta\) is the angle relative to the principal magnetic axis of the complex, and \(C_J\) is a constant characteristic of the lanthanide (\(C_J > 0\) for \(\text{Eu}^{3+}\), causing downfield shifts; \(C_J < 0\) for \(\text{Pr}^{3+}\), causing upfield shifts). Because the shift attenuates as \(r^{-3}\), complex overlapping multiplets are spread across many ppm without line broadening.

Enantiomeric Excess Determination via Chiral Solvating Agents

When a chiral shift reagent (such as chiral tris[3-(heptafluoropropylhydroxymethylene)-(+)-camphorato]europium(III), \(\text{Eu(hfc)}_3\)) is added to a racemic mixture of enantiomers (\(R\) and \(S\)):

\[ (R)\text{-Substrate} + \text{Eu(hfc)}_3 \rightleftharpoons \text{Diastereomeric Complex } [(R) \cdot \text{Eu(hfc)}_3] \] \[ (S)\text{-Substrate} + \text{Eu(hfc)}_3 \rightleftharpoons \text{Diastereomeric Complex } [(S) \cdot \text{Eu(hfc)}_3] \]

The resulting complexes are diastereomers, possessing different thermodynamic stability constants and different spatial geometries. Consequently, the enantiotopic protons become diastereotopic and exhibit distinct chemical shifts (\(\Delta\Delta\delta \sim 0.05 - 0.5\text{ ppm}\)). Direct integration of the separated peak areas determines the enantiomeric excess (ee):

\[ \text{ee} = \frac{|A_R - A_S|}{A_R + A_S} \times 100\% \]