  The accurate description of electrostatic interactions remains a challenging
problem for fitted potential-energy functions. The commonly used fixed
partial-charge approximation fails to reproduce the electrostatic potential at
short range due to its insensitivity to conformational changes and anisotropic
effects. At the same time, possibly more accurate machine-learned (ML)
potentials struggle with the long-range behaviour due to their inherent
locality ansatz. Employing a multipole expansion offers in principle an exact
treatment of the electrostatic potential such that the long-range and
short-range electrostatic interaction can be treated simultaneously with high
accuracy. However, such an expansion requires the calculation of the electron
density using computationally expensive quantum-mechanical (QM) methods. Here,
we introduce an equivariant graph neural network (GNN) to address this issue.
The proposed model predicts atomic multipoles up to the quadrupole,
circumventing the need of expensive QM computations. By using an equivariant
architecture, the model enforces the correct symmetry by design without relying
on local reference frames. The GNN reproduces the electrostatic potential of
various systems with high fidelity. Possible use cases for such an approach
include the separate treatment of long-range interactions in ML potentials, the
analysis of electrostatic potential surfaces, and for the static multipoles in
polarizable force fields.
