NMR Overview

Diamagnetic Insulators

By a diamagnetic insulator we mean a system with an electronic gap between its highest occupied state and lowest unoccupied state, and all of the electronic spins are paired, simple examples are diamond, quartz, glycine… This class of materials is the most suitable for calculations as the main NMR interactions, magnetic sielding, J-coupling and electric field gradients can all be computed.

Magnetic Shielding

In a diamagnetic insulator this arises from orbital currents induced by an external magnetic field. This current ${\bf j}({\bf r})$ , produces a non-uniform induced magnetic field in the material, which is given by the Biot-Savart law as

${\bf B}_{\rm in}({\bf r}) =\frac{1}{c}\int d^3r' {\bf j}({\bf r}')\times \frac{{\bf r}-{\bf r}'}{|{\bf r}-{\bf r}'|^3}.$

The shielding tensor is defined as the ratio between this induced field, and the external applied field

${\bf B}_{\rm in}({\bf r})=-{\vec{\sigma}}({\bf r}){\bf B}_{\rm ext}.$

It can thus be seen that the question of computing the shielding tensor is that of computing the induced electronic current.

J-coupling

The J-coupling is a small perturbation to the electronic ground-state of the system and we can identify it as a derivative of the total energy E, of the system

${\bf J}_{\rm KL}= \frac{\hbar\gamma_{\rm K}\gamma_{\rm L}}{2\pi}\frac{\partial^2 E}{\partial {\bf m}_{\rm K} \partial {\bf m}_{\rm L}}$

An equivalent expression arises from considering one nuclear spin (L) as perturbation which creates a magnetic field at a second (receiving) nucleus (K)

${\bf B}^{(1)}_{\rm in}({\bf R}_{{\rm K}}) = \frac{2\pi}{\hbar\gamma_{{\rm K}}\gamma_{{\rm L}}}{\bf J}_{{\rm KL}} \cdot {\bf m}_{{\rm L}}.$

Eqn.~\ref{eq:J} tells us that the question of computing J is essentially that of computing the magnetic field induced indirectly by a nuclear magnetic moment. When spin-orbit coupling is neglected we can consider the field as arising from two, essentially independent, mechanisms. Firstly, the magnetic moment can interact with electronic charge inducing an orbital current ${\bf j}({\bf r})$ , which in turn creates a magnetic field at the other nuclei in the system. This mechanism is similar to the case of magnetic shielding in insulators. The second mechanism arises from the interaction of the magnetic moment with the electronic spin, causing an electronic spin polarisation. By working to first order in these quantities we can write the magnetic field at atom ${\rm K}$ induced by the magnetic moment of atom ${\rm L}$ as

${\bf B}^{(1)}_{\rm in}({\bf R}_{{\rm K}}) = \frac{\mu_{0}}{4\pi}\int {\bf m}^{(1)}({\bf r})\cdot \left[\frac{3{\bf r}_{{\rm K}}{\bf r}_{{\rm K}} - |{\bf r}_{{\rm K}}|^{2}}{|{\bf r}_{{\rm K}}|^{5}}\right]\,{\rm d}^{3}{\bf r}$

$+ \frac{\mu_{0}}{4\pi}\frac{8\pi}{3}\int {\bf m}^{(1)}({\bf r}) \delta({\bf r}_{{\rm K}})\,{\rm d}^{3}{\bf r}$

$+ \frac{\mu_{0}}{4\pi}\int {\bf j}^{(1)}({\bf r})\times \frac{{\bf r}_{{\rm K}}}{|{\bf r}_{{\rm K}}|^{3}}\,{\rm d}^{3}{\bf r}.$

Several quantum chemistry packages provide the ability to compute J coupling tensors in molecular systems (see Ref ¹} for a review of methods). An approach to compute J tensors within the planewave-pseudopotential approach has recently been developed.²

EFG

For a nucleus with spin $>$ ½ the NMR response will include an interaction between the quadrupole moment of the nucleus, Q, and the electric field gradient (EFG) generated by the surrounding electronic structure. The EFG is a second rank, symmetric, traceless tensor $G({\bf r})$ given by

$G_{\alpha\beta}({\bf r}) = \frac{\partial E_{\alpha}({\bf r})}{\partial r_{\beta}} - \frac{1}{3}\delta_{\alpha\beta}\sum_{\gamma}\frac{\partial E_{\gamma}({\bf r})}{\partial r_{\gamma}}$

where $\alpha,\beta,\gamma$ denote the Cartesian coordinates x,y,z and $E_{\alpha}({\bf r})$ is the local electric field at the position ${\bf r}$ , which can be calculated from the charge density $n({\bf r})$ : \begin{equation} E_{\alpha}({\bf r})=\int d^3r \frac{n({\bf r})}{|{\bf r}-{\bf r}'|^3} (r_{\alpha}-r_{\alpha}'). \end{equation} The EFG tensor is then equal to \begin{equation}\label{eq:efg_rs} G_{\alpha\beta}({\bf r}) = \int d^3r \frac{n({\bf r})}{|{\bf r}-{\bf r}'|^3}\left[ \delta_{\alpha\beta} -3 \frac{(r_{\alpha}-r_{\alpha}')(r_{\beta}-r_{\beta}')}{|{\bf r}-{\bf r}'|^2}\right]. \end{equation} The computation of electric field gradient tensors is less demanding than either shielding or J-coupling tensors as it requires only knowledge of the electronic ground state. The LAPW approach in its implementation within the Wien series of codes\cite{blaha90} has been widely used and shown to reliably predict Electric Field Gradient (EFG) tensors³. The equivalent formalism for the planewave/PAW approach is reported in Ref.~\cite{profeta03}.

The quadrupolar coupling constant, C $_Q$ and the asymmetry parameter, $\eta_Q$ can be obtained from the the diagonalized electric field gradient tensor whose eigenvalues are labelled V $_{xx}$ , V $_{yy}$ , V $_{zz}$ , such that $|V_{zz}|>|V_{yy}|>|V_{xx}|$ : \begin{equation} C_{Q}=\frac{eV_{zz}Q}{h}, \end{equation} where h is Planck's constant and \begin{equation} \eta_Q=\frac{V_{xx}-V_{yy}}{V_{zz}}. \end{equation}

Helgaker, T.; Jaszunski, M.; Pecul, M.Progress in Nuclear Magnetic Resonance Spectroscopy2008,53, 249 – 268 ↩
Joyce, S. A.; Yates, J. R.; Pickard, C. J.; Mauri, F.J. Chem. Phys.2007,127, 204107 ↩
Blaha, P.; Sorantin, P.; Ambrosch, C.; Schwarz, K.Hyperfine Interact.1989,51, 917 ↩