ML/MM

Electrostatic embedding allows a local machine-learning potential to respond to an extended molecular environment without including every environmental atom in its message-passing graph. We construct a fixed-charge machine-learning/molecular-mechanics (ML/MM) interface for the pretrained MACE-POLAR-1-M model. Analytic Gaussian monopole and dipole operators supply the environmental potential, the internal electrostatic features, and the interaction energy. Differentiation through both field-conditioned density updates gives forces on the model fragment and the environment, which are combined with a subtractive classical energy and exact link-atom force redistribution. A reciprocal-space extension treats three-dimensional periodicity at fixed cell. We assess the implementation using fixed-geometry numerical checks and 24 matched QM/MM reference calculations for a 2589-atom rhodopsin model, with 54 atoms in the neural-network graph. The analytic model gives a relative-energy root-mean-square error of 8.62 kJ mol−1 after one constant energy shift, and force-component errors of 0.07345 and 0.00797 eV Å−1 on the 53 real high-level atoms and 2536 environmental atoms, respectively. Analytic operators substantially reduce rotational errors while leaving the QM/MM accuracy essentially unchanged. A nearby water accounts for approximately 74% of the environmental squared force error. Matched charge-density and frozen-density force comparisons identify limitations of both the local multipole representation and its predicted coefficients. The results establish the numerical consistency of the embedding implementation and delimit the single-point accuracy of the transferred pretrained model.