Figure 1.

Schematic structure of QM/MM-NN for a system containing N atoms in the QM subsystem. Here r_QM and r_MM are respectively the Cartesian coordinates of the atoms in the QM and MM subsystem, {$G_i^k$} is the symmetry function that depends on r_QM, i denotes the QM atoms, k denotes different radial and angular functions with different hyperparameters, and {$v_i^{\text{ext}}$} is the external electrostatic potential at QM grids close to atom i as a function of r_QM and r_MM. After semiempirical QM/MM calculations with r_QM and r_MM, the total potential energy $E_{\text{QM/MM}}^{\text{L}}$ and the corresponding forces ${\mathbf{F}}_{\text{QM/MM}}^{\text{L}}$ on all QM and MM atoms at the low level are known. Then the energy difference between two levels as ΔE is predicted with QM/MM-NN, and the derivative of ΔE with respect to r_QM and r_MM is calculated analytically. Finally, the total potential energy $E_{\text{QM/MM}}^{\text{H}}$ and the corresponding forces ${\mathbf{F}}_{\text{QM/MM}}^{\text{H}}$ on all QM and MM atoms at the high level is obtained. The differences on the input variables of NN compared to our previous work⁴² are highlighted in red.