# Polarizable embedding#

In this section we will include polarizable contributions, such that

$E_\mathrm{MM/QM} = \sum_i q_i^\mathrm{perm} (\phi_{i}^\mathrm{ele}+ \phi_{i}^{nuc} ) + \sum_i \mathbf{p}_i^\mathrm{ind} (\mathbf{E}_{i}^\mathrm{ele}+ \mathbf{E}_{i}^\mathrm{nuc} ) \ ,$

where $$\{ q_i^\mathrm{perm}\}$$ is the set of permanent charges in MM region, $$\{\mathbf{p}_i^{ind}\}$$ is the set of induced dipoles in MM region, the $$\phi_{i}^\mathrm{ele}$$ and $$\phi_{i}^\mathrm{nuc}$$ are potential components are generated by electrons and nuclei in MM region, $$\mathbf{E}_{i}^\mathrm{ele}$$ and $$\mathbf{E}_{i}^\mathrm{nuc}$$ are potential components are generated by electrons and nuclei in MM region. The $$E_\mathrm{MM/QM}$$ converts to additional contribution to external potential $$v(\mathbf{r})$$ Kohn–Sham equations

$\mathbf{V}_\mathrm{MM/QM}(\mathbf{C})_{\mu\nu} = \int \phi_{\mu} (\mathbf{r}, s)^* \{ \sum_i q_i^\mathrm{perm} T_i^q(\mathbf{r}) + \sum_i \mathbf{p}_i^\mathrm{ind} T_i^m(\mathbf{r})) \} \phi_{\mu} (\mathbf{r}, s) d \mathbf{r} d s ,$

where $$T_i^q(\mathbf{r})$$ and $$T_i^m(\mathbf{r}))$$ electrostatic interaction tensors of first and second order. In the above equation, induced dipoles in the MM region explicitly depend on electron density in the QM region and thus need to be determined for each step of the SCF procedure. Furthermore, the induced dipoles are the only MM region contribution explicitly entering response calculations of molecular properties and consequently account for this contribution is crucial in any QM/MM calculations of spectroscopic properties.