The magnetic inducibility as a response property

Suppose we have a system that is subject to an external magnetic field \(\require{physics} \vb{B}_{\text{ext}}\) (with gauge origin \(\vb{G}\)) and we would like to describe it with the wave function model \(\ket{\Psi(\vb{p})}\). The optimal wave function parameters \(\vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G})\) should then be determined at the given value of the external field and the related current density is then calculated using these optimal parameters. In a general spinor basis, we can calculate the total (probability) current density as: \[\begin{equation} \vb{j}( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G}), \vb{B}_{\text{ext}}, \vb{G} ) = \sum_{PQ}^M \vb{j}_{PQ}( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) \thinspace D_{PQ}(\vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G})) \thinspace , \end{equation}\] which is a contraction of the 1-DM \(D_{PQ}(\vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G}))\) that depends on the optimal wave function parameters, and the (probability) current density matrix elements \(\vb{j}_{PQ}\): \[\begin{equation} \begin{split} & \vb{j}_{PQ}( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) \\ & \hspace{12pt} = - \frac{i}{2} \Bigg( \phi_P^\dagger( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) \thinspace [ \grad{ \phi_Q( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) } ] - [ \grad{ \phi_P^\dagger( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) } ] \thinspace \phi_Q( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) \\ & \hspace{48pt} + i \phi_P^\dagger( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) \phi_Q( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) \thinspace \vb{B}_{\text{ext}} \cross ( \vb{r} - \vb{G} ) \Bigg) \thinspace . \end{split} \end{equation}\] Since we focus this discussion on the probability current density, we may convert the results to charge current densities by multiplying with a factor \((-1)\).

Note that the spinors are initially considered to be field-dependent, since the underlying scalar basis functions may be London orbitals. The field-free matrix elements are then given by: \[\begin{equation} \vb{j}_{PQ}( \vb{r}; \vb{B}_{\text{ext}, \thinspace 0} ) = - \frac{i}{2} \Bigg( \phi_P^\dagger(\vb{r}) \thinspace [ \grad{ \phi_Q(\vb{r}) } ] - [ \grad{ \phi_P^\dagger(\vb{r}) } ] \thinspace \phi_Q(\vb{r}) \Bigg) \thinspace . \end{equation}\]

The induced (probability) current density \(\vb{j}_{\text{ind}}\) is defined as the (probability) current density that is explicitly induced by the external magnetic field \(\vb{B}_{\text{ext}}\). According to this definition, we should calculate it as the difference of the total (probability) current density (determined at the value of the external field \(\vb{B}_{\text{ext}}\)) and the field-free (probability) current density (determined at zero external field \(\vb{B}_{\text{ext}, \thinspace 0}\)): \[\begin{multline} \vb{j}_{\text{ind}} ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G}), \vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0}), \vb{B}_{\text{ext}}, \vb{G} ) \\ = \vb{j}( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G}), \vb{B}_{\text{ext}}, \vb{G} ) - \vb{j}( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0}), \vb{B}_{\text{ext}, \thinspace 0} ) \thinspace , \end{multline}\] in which the total (probability) current density is calculated using the optimal parameters \(\vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G})\) determined at the value of the external field \(\vb{B}_{\text{ext}}\), and the field-free (probability) current density using the optimal parameters \(\vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0})\) for the field-free situation.

In the spirit of response theory, we do not always want to calculate the total (probability) current density when the system is subject to the external field \(\vb{B}_{\text{ext}}\), rather, we are often only interested in the linear response. The linear response coefficient of the induced (probability) current density \(\vb{j}_{\text{ind}}\) is the (first-order) magnetic inducibility \(\mathcal{J}(\vb{r})\). Accordingly, we should be able to calculate it as: \[\begin{align} \mathcal{J}_{im} ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0}), \vb{G} ) &= \eval{ \dv{ j_{\text{ind}, \thinspace i} ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0}), \vb{B}_{\text{ext}}, \vb{G} ) }{ B_{\text{ext}, m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} \\ &= \eval{ \dv{ j_i ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G}), \vb{B}_{\text{ext}}, \vb{G} ) }{ B_{\text{ext}, m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} \thinspace , \end{align}\] where we were able to switch from using the induced (probability) current density to using the total (probability) current density in the derivative because the field-free case is a field-independent constant term. Working out the chain rule, we may now work out the derivative as: \[\begin{align} & \mathcal{J}_{im} ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0}), \vb{G} ) \notag \\ & \hspace{24pt} = \eval{ \dv{ j_i ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G}), \vb{B}_{\text{ext}}, \vb{G} ) }{ B_{\text{ext}, m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} \\ & \hspace{24pt} = \eval{ \pdv{ j_i ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G}), \vb{B}_{\text{ext}}, \vb{G} ) }{ B_{\text{ext}, m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} + \sum_k^x \qty( \eval{ \pdv{ j_i ( \vb{r}; \vb{p}, \vb{B}_{\text{ext}, \thinspace 0} ) }{p_k} }_{\vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0})} ) x_{k, m} (\vb{G}) \thinspace , \end{align}\] where we have used the short-hand notation \[\begin{equation} x_{k, m} (\vb{G}) = \qty( \eval{ \pdv{ p_k^\star( \vb{B}_{\text{ext}}, \vb{G} ) }{ B_{\text{ext}, \thinspace m} } }_{ \vb{B}_{\text{ext}, \thinspace 0} } ) \end{equation}\] for the linear response.

If we look at the functional dependence of the terms constituting the previous expression for the magnetic inducibility in more detail, we see that we have dropped the gauge origin dependence in the parameter partial derivative, but not in the partial derivative with respect to the external field. We can do so because the derivative \[\begin{equation*} \eval{ \pdv{ j_i ( \vb{r}; \vb{p}, \vb{B}_{\text{ext}, \thinspace 0} ) }{p_k} }_{\vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0})} \end{equation*}\] is always evaluated at zero external field \(\vb{B}_{\text{ext}, \thinspace 0}\), such that the gauge origin \(\vb{G}\) will always disappear. However, if we first apply the partial derivative with respect to the external magnetic field, and only afterwards evaluate at zero external field, such as in \[\begin{equation*} \eval{ \pdv{ j_i ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0}), \vb{B}_{\text{ext}}, \vb{G} ) }{ B_{\text{ext}, m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} \thinspace , \end{equation*}\] a gauge origin dependence might remain, which is ultimately the reason why the (first-order) magnetic inducibility is in general gauge origin-dependent.

The first term in the overall expression for the (first-order) magnetic inducibility is the explicit contribution to the magnetic inducibility and can be calculated as the explicit partial derivative of the total (probability) current density with respect to the external magnetic field \(\vb{B}_{\text{ext}}\): \[\begin{align} \mathcal{J}_{im, \thinspace \text{explicit}} ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0}), \vb{G} ) &= \eval{ \pdv{ j_i ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0}), \vb{B}_{\text{ext}}, \vb{G} ) }{ B_{\text{ext}, m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} \\ &= \sum_{PQ} \qty( \eval{ \pdv{ j_{i, \thinspace PQ} ( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) }{ B_{\text{ext}, m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} ) \thinspace D_{PQ}(\vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0})) \thinspace , \end{align}\] such that, in order to calculate this contribution, we will require the explicit partial derivative of the (probability) current density matrix elements \[\begin{equation} \eval{ \pdv{ j_{i, \thinspace PQ} ( \vb{r}; \vb{B}_{\text{ext}}, \vb{G} ) }{ B_{\text{ext}, m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} \thinspace . \end{equation}\]

The second term in the expression for the magnetic inducibility is an implicit contribution and it represents the contribution due to the optimal parameters that (may) have changed by turning on the external magnetic field: \[\begin{align} \mathcal{J}_{im, \thinspace \text{implicit}} ( \vb{r}; \vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0}), \vb{G} ) &= \sum_k^x \qty( \eval{ \pdv{ j_i ( \vb{r}; \vb{p}, \vb{B}_{\text{ext}, \thinspace 0} ) }{p_k} }_{\vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0})} ) \thinspace x_{k, m} (\vb{G}) \\ &= \sum_{PQ} j_{i, \thinspace PQ} ( \vb{r}; \vb{B}_{\text{ext}, \thinspace 0} ) \thinspace \sum_k^x \qty( \eval{ \pdv{ D_{PQ}(\vb{p}) }{ p_k } }_{\vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0})} ) \thinspace x_{k, m} (\vb{G}) \thinspace , \end{align}\] where the contribution on the right can be recognized as the first-order (response) derivative of the 1-DM: \[\begin{equation} \eval{ \dv{ D_{PQ}(\vb{p}^\star(\vb{B}_{\text{ext}}, \vb{G})) }{ B_{\text{ext}, \thinspace m} } }_{\vb{B}_{\text{ext}, \thinspace 0}} = \sum_k^x \qty( \eval{ \pdv{ D_{PQ}(\vb{p}) }{ p_k } }_{\vb{p}^\star(\vb{B}_{\text{ext}, \thinspace 0})} ) \thinspace x_{k, m} (\vb{G}) \thinspace , \end{equation}\] which is not to be confused with the one-electron response density matrix. The implicit contribution to the magnetic inducibility is thus calculated from the field-free (probability) current density elements and the first-order derivative of the 1-DM.