  In recent years, significant progress has been made in the development of
machine learning potentials (MLPs) for atomistic simulations with applications
in many fields from chemistry to materials science. While most current MLPs are
based on environment-dependent atomic energies, the limitations of this
locality approximation can be overcome, e.g., in fourth-generation MLPs, which
incorporate long-range electrostatic interactions based on an equilibrated
global charge distribution. Apart from the considered interactions, the quality
of MLPs crucially depends on the information available about the system, i.e.,
the descriptors. In this work we show that including -- in addition to
structural information -- the electrostatic potential arising from the charge
distribution in the atomic environments significantly improves the quality and
transferability of the potentials. Moreover, the extended descriptor allows to
overcome current limitations of two- and three-body based feature vectors
regarding artificially degenerate atomic environments. The capabilities of such
an electrostatically embedded fourth-generation high-dimensional neural network
potential (ee4G-HDNNP), which is further augmented by pairwise interactions,
are demonstrated for NaCl as a benchmark system. Employing a data set
containing only neutral and negatively charged NaCl clusters, even small energy
differences between different cluster geometries can be resolved, and the
potential shows an impressive transferability to positively charged clusters as
well as the melt.
