Relating nuclear shieldings with induced current densities
Filling in equation \(\eqref{eq:B_ind_shielding}\) in equation \(\eqref{eq:B_ind_Biot-Savart}\), applying the partial derivative with respect to one component of the external magnetic field \(\require{physics} \vb{B}_{\text{ext}}\), evaluating at the field-free case \(\vb{B}_{\text{ext}, \thinspace 0}\) and writing the vector cross product using the Levi-Civita symbol (and its implied summation), we find Acke et al. (2018): \[\begin{equation} \sigma^{(0)}_{im}(\vb{R}_K) % = - \frac{\mu_0}{4 \pi} \int \dd{\vb{r}} % \thinspace % \frac{ % \epsilon_{ijk} \thinspace \mathcal{J}_{jm}(\vb{r}) % \thinspace % \qty[ \vb{R}_K - \vb{r} ]_k % }{ % \norm{ \vb{R}_K - \vb{r} }^3 } % \thinspace , \end{equation}\] but (Acke et al. 2018) seems to be missing a minus sign due to the change from \((\vb{R}_K - \vb{r})\) to \((\vb{r} - \vb{R}_K)\).
The NICS-value (nucleus-independent chemical shift) is the scaled negative trace of the (permanent) shielding tensor: \[\begin{equation} \text{NICS}(\vb{R}_K) % = - \frac{1}{3} \qty[ % \sigma^{(0)}_{xx}(\vb{R}_K) % + \sigma^{(0)}_{yy}(\vb{R}_K) % + \sigma^{(0)}_{zz}(\vb{R}_K) % ] % \thinspace . \end{equation}\]
The related `density’ (i.e. what is integrated over) to the nuclear shielding (tensor field) is called the (Jameson-Buckingham) nuclear shielding (tensor field) density Acke et al. (2018): \[\begin{equation} \Sigma_{im}(\vb{r}, \vb{R}_K) % = - \frac{\mu_0}{4 \pi} \thinspace % \frac{ % \epsilon_{ijk} \thinspace \mathcal{J}_{jm}(\vb{r}) % \qty[ \vb{R}_K - \vb{r} ]_k % }{ % \norm{ \vb{R}_K - \vb{r} }^3 } % \thinspace . \end{equation}\] The corresponding density to the NICS is then called the NICS-density, or NICSD: \[\begin{equation} \text{NICSD}(\vb{r}, \vb{R}_K) % = - \frac{1}{3} \qty[ % \Sigma_{xx}(\vb{r}, \vb{R}_K) % + \Sigma_{yy}(\vb{r}, \vb{R}_K) % + \Sigma_{zz}(\vb{r}, \vb{R}_K) % ] % \thinspace , \end{equation}\] whose \(zz\)-component can be calculated as: \[\begin{equation} \text{NICSD}_{zz}(\vb{r}, \vb{R}_K) % = \frac{\mu_0}{12 \pi} % \thinspace % \frac{ % \mathcal{J}_{xz} ( % \vb{r}; % \vb{B}_{\text{ext}} % ) % \thinspace % \qty[ \vb{R}_K - \vb{r} ]_y % % - \mathcal{J}_{yz} ( % \vb{r}; % \vb{B}_{\text{ext}} % ) % \thinspace % \qty[ \vb{R}_K - \vb{r} ]_x % }{ % \norm{ \vb{R}_K - \vb{r} }^3 } % \thinspace . \end{equation}\]