  Machine learning models for the potential energy of multi-atomic systems,
such as the deep potential (DP) model, make possible molecular simulations with
the accuracy of quantum mechanical density functional theory, at a cost only
moderately higher than that of empirical force fields. However, the majority of
these models lack explicit long-range interactions and fail to describe
properties that derive from the Coulombic tail of the forces. To overcome this
limitation we extend the DP model by approximating the long-range electrostatic
interaction between ions (nuclei+core electrons) and valence electrons with
that of distributions of spherical Gaussian charges located at ionic and
electronic sites. The latter are rigorously defined in terms of the centers of
the maximally localized Wannier distributions, whose dependence on the local
atomic environment is modeled accurately by a deep neural network. In the deep
potential long-range (DPLR) model, the electrostatic energy of the Gaussian
charge system is added to short-range interactions that are represented as in
the standard DP model. The resulting potential energy surface is smooth and
possesses analytical forces and virial. Missing effects in the standard DP
scheme are recovered, improving on accuracy and predictive power. By including
long-range electrostatics, DPLR correctly extrapolates to large systems the
potential energy surface learned from quantum mechanical calculations on
smaller systems. We illustrate the approach with three examples, the potential
energy profile of the water dimer, the free energy of interaction of a water
molecule with a liquid water slab, and the phonon dispersion curves of the NaCl
crystal.

Chemical Physics;Computational Physics