Induced molecular current densities
The law of Biot-Savart expresses how an induced (charge) current density of the electrons in turn creates an induced magnetic field: \[\begin{equation} \require{physics} \vb{B}_{\text{ind}} ( \vb{R}_K; \vb{B}_{\text{ext}} ) = \frac{\mu_0}{4 \pi} \int \dd{\vb{r}} \thinspace \frac{ \vb{j}_{\text{ind}} ( \vb{r}; \vb{B}_{\text{ext}} ) \cross (\vb{R}_K - \vb{r}) }{ \norm{ \vb{R}_K - \vb{r} }^3 } \thinspace , \end{equation}\] where \(\vb{R}_K\) is the reference position at which the induced magnetic field is to be calculated.
The induced current density can be expanded with respect to the external magnetic field as \[\begin{equation} \begin{split} j_{\text{ind}, \thinspace i} ( \vb{r}; \vb{B}_{\text{ext}} ) = &\sum_m \eval{ \dv{ j_{\text{ind}, \thinspace i} ( \vb{r}; \vb{B}_{\text{ext}} ) }{ B_{\text{ext}, \thinspace m} } }_{ \vb{B}_{\text{ext}, \thinspace 0} } B_{\text{ext}, \thinspace m} \\ &+ \frac{1}{2!} \sum_{mn} \eval{ \frac{ \dd{^2}{ j_{\text{ind}, \thinspace i} ( \vb{r}; \vb{B}_{\text{ext}} ) } }{ \dd{ B_{\text{ext}, \thinspace m} } \dd{ B_{\text{ext}, \thinspace n} } } }_{ \vb{B}_{\text{ext}, \thinspace 0} } B_{\text{ext}, \thinspace m} B_{\text{ext}, \thinspace n} \thinspace , \end{split} \end{equation}\] where we have omitted the zero-th order term because we are examining an current density by the external magnetic field. We can write this expression in a linear fashion as: \[\begin{equation} \vb{j}_{\text{ind}} ( \vb{r}; \vb{B}_{\text{ext}} ) = \boldsymbol{\mathcal{I}}(\vb{r}) \thinspace \vb{B}_{\text{ext}} \thinspace , \end{equation}\] in which we will call the matrix \(\boldsymbol{\mathcal{I}}(\vb{r})\) the magnetic inducibility: it describes, linearly, how a molecular current density is induced by an external magnetic field.
Up to first order, the induced current density can then be written as \[\begin{align} \vb{j}^{(1)}_{\text{ind}} ( \vb{r}; \vb{B}_{\text{ext}} ) &= \boldsymbol{\mathcal{I}}^{(1)}(\vb{r}) \thinspace \vb{B}_{\text{ext}} \\ &= \boldsymbol{\mathcal{J}}(\vb{r}) \thinspace \vb{B}_{\text{ext}} \thinspace , \end{align}\] reminiscent of the interaction energy of a permanent dipole moment with an external field. Here, we have introduced a new symbol for the first-order coefficient in the expansion of the induced current density: \[\begin{equation} \mathcal{J}_{im}(\vb{r}) = \mathcal{I}^{(1)}_{im}(\vb{r}) = \eval{ \dv{ j_{\text{ind}, \thinspace i} ( \vb{r}; \vb{B}_{\text{ext}} ) }{ B_{\text{ext}, m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} \thinspace . \end{equation}\] We will accordingly call it the first-order magnetic inducibility, although in literature (Lazzeretti, Malagoli, and Zanasi 1994) (Cuesta et al. 2009) (Acke et al. 2018) it is more oftenly called the current density tensor. The (first-order) magnetic inducibility can be calculated through response theory.
Since the total magnetic inducibility \(\boldsymbol{\mathcal{I}}(\vb{r})\) relates the external magnetic field \(\vb{B}_{\text{ext}}\) to the total induced current density \(\vb{j}_{\text{ind}}(\vb{r}; \vb{B}_{\text{ext}})\) in a linear fashion, we must allow for non-linear effects to be described by the total magnetic inducibility. Similarly to how the electric polarizability can be introduced with respect to the electric dipole moment, we can introduce the magnetic inducibility magnetizability \(\mathcal{K}_{imn}(\vb{r})\) as the quadratic response coefficient of the induced current \(j_{\text{ind}, \thinspace i}(\vb{r}; \vb{B}_{\text{ext}})\): \[\begin{equation} \mathcal{K}_{imn}(\vb{r}) = \eval{ \frac{ \dd{^2}{ j_{\text{ind}, \thinspace i} ( \vb{r}; \vb{B}_{\text{ext}} ) } }{ \dd{B_{\text{ext}, m}} \dd{B_{\text{ext}, n}} } }_{\vb{B}_{\text{ext}, \thinspace 0}} \thinspace , \end{equation}\] or the linear response coefficient of the magnetic inducibility: \[\begin{equation} \mathcal{K}_{imn}(\vb{r}) = \eval{ \dv{ \mathcal{I}_{im} (\vb{r}) }{ B_{\text{ext}, n} } }_{\vb{B}_{\text{ext}, \thinspace 0}} \thinspace . \end{equation}\] Up to second order, we can write the induced current density as: \[\begin{equation} j_{\text{ind}, \thinspace i} ( \vb{r}; \vb{B}_{\text{ext}} ) = \sum_m \mathcal{J}_{im}(\vb{r}) \thinspace B_{\text{ext}, \thinspace m} + \frac{1}{2!} \sum_{mn} \mathcal{K}_{imn}(\vb{r}) \thinspace B_{\text{ext}, \thinspace m} B_{\text{ext}, \thinspace n} \thinspace , \end{equation}\] so the first-order induced response current density \(\vb{j}^{(1)}_{\text{ind}}\) can be calculated as \[\begin{equation} j^{(1)}_{\text{ind}, \thinspace i} ( \vb{r}; \vb{B}_{\text{ext}} ) = \sum_m \mathcal{J}_{im}(\vb{r}) \thinspace B_{\text{ext}, \thinspace m} \end{equation}\] and the second-order induced response current density \(\vb{j}^{(2)}_{\text{ind}}\) as \[\begin{equation} j^{(2)}_{\text{ind}, \thinspace i} ( \vb{r}; \vb{B}_{\text{ext}} ) = \frac{1}{2!} \sum_{mn} \mathcal{K}_{imn}(\vb{r}) \thinspace B_{\text{ext}, \thinspace m} B_{\text{ext}, \thinspace n} \thinspace . \end{equation}\]