Theory

Following an additive QM/MM scheme, the isolated QM Hamiltonian, \(\hat{H}_0\), is modified according to:

\[ \hat{H} = \hat{H}_0 + \hat{H}_\mathrm{QM/MM-elec.} + V_\mathrm{QM/MM-mech.} + V_\mathrm{MM}. \]

The last term in this equation is the potential energy (bonding and non-bonding) of the MM region, i.e. the forcefield energy not including QM atom contributions. The second term is the electrostatic embedding interaction of the QM and MM subsystems:

\[ \hat{H}_\mathrm{QM/MM-elec.} = -\sum_i^{N_\mathrm{elec}} \sum_b^{{N_\mathrm{MM}}^\prime} \frac{q_b}{\left | \hat{\mathbf{r}}_i - \mathbf{R}_b \right |} + \sum_a^{N_\mathrm{QM}} \sum_b^{{N_\mathrm{MM}}^\prime} \frac{Z_a q_b}{\left | \mathbf{R}_a - \mathbf{R}_b \right |}, \]

where the first term captures the Coulomb interaction of the MM partial charges, \(q_b\), and the QM electrons. This is an operator that gets added to the isolated 1-electron Hamiltonian and thus influences the QM wavefunction, i.e., the MM region polarizes the QM region. The second term is the interaction of the MM charges and QM nuclei of atomic number \(Z_a\). The \(^\prime\) marks indicate that electrostatic interactions between QM and directly bonded MM (link) atoms are excluded, to avoid over-polarization errors.

The mechanical embedding potential, \(V_\mathrm{QM/MM-mech.}\), captures bonding and Van der Waals non-bonding interactions between the QM and MM subsystems:

\[ \begin{aligned} V_\mathrm{QM/MM-mech.} &= \sum V_\mathrm{QM/MM-bonds} + \sum V_\mathrm{QM/MM-angles} + \sum V_\mathrm{QM/MM-dihedrals} + \sum V_\mathrm{QM/MM-CMAP} + \\ &\quad \sum_a^{N_\mathrm{QM}} \sum_b^{{N_\mathrm{MM}}^\prime} 4\epsilon_{ab} \left[ \left(\frac{\sigma_{ab}}{R_{ab}}\right)^{12} - \left(\frac{\sigma_{ab}}{R_{ab}}\right)^6 \right], \end{aligned} \]

where the first four terms sum the bond stretch, bond bend, dihedral, and CMAP (if relevant) potentials between the QM and MM atoms. If the QM/MM interface does not cut across any covalent bonds, these terms are all zero. The last term captures the Lennard-Jones interactions between the QM and MM atoms. The \(^\prime\) mark here indicates that the usual 1-4 exclusions/exceptions are applied to the QM/MM LJ interactions.

The parameters for the bonding and non-bonding QM/MM mechanical embedding interactions are unchanged from the topology file. It is therefore important that a reasonable forcefield is used to model both the QM and MM regions of the system. However, since the MM partial charges of the QM region are zeroed and replaced with explicit nuclei and electrons, it is not so important to construct the charge model of the QM region with high accuracy when building the system.