General aspects#

In recent years, the hybrid quantum mechanics/molecular mechanics (QM/MM) methods have become a rather widely used approach for modeling molecules’ electronic structure in complex environments. Most QM/MM methods partition the molecular system into a small QM region (up to few hundreds of atoms) and a large MM region (thousands of atoms). Full or partial coupling in intermolecular interactions (electrostatics, induction, dispersion, short-range repulsion) is accounted for between the QM and MM parts of the system, where each interaction acts both ways. Non-polarizable or polarizable force fields typically describe the MM part with parameters (point charges, distributed polarizabilities) determined empirically or using quantum chemistry methods. For bonded systems, the MM part of the system’s residues is often subject to bond capping, and the QM and MM parts interface to approximations like atom in-linking. After partitioning the molecular system into QM and MM regions and selecting interaction mode between these systems, the electronic wave function of the QM system can be determined by solving the wave function model specific equation in the presence of external potential generated by the MM region.

To illustrate this concept, here we consider the QM/MM version of the Kohn–Sham method. The total energy of the molecular system in the KS/MM method is partitioned into three contributions:

\[ E = E_\mathrm{MM} + E_\mathrm{MM/QM} + E_\mathrm{QM} \ , \]

where the first term is the energy of the MM region, the second term is the interaction energy of the QM and MM regions, and the last term is the QM region energy. Assuming a polarizable force field describes MM region, the second term can be expanded as

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

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.