Shadow Molecular Dynamics for General Nonlinear Flexible-Charge Models

Machine-learned interatomic potentials have greatly expanded the reach of molecular dynamics simulations, but complex chemical environments often require explicit treatment of long-range electronic response, including charge transfer and polarization. Physics-based flexible-charge models capture these effects but require global nonlinear charge equilibration at every time step to obtain Born-Oppenheimer energies and forces. This process is computationally expensive and can lead to instabilities unless tightly converged. Here, we introduce a shadow molecular dynamics framework for flexible-charge models with arbitrary local nonlinearities and long-range multipolar electrostatics. We replace the exact Born-Oppenheimer potential with a nearby shadow potential obtained by exactly equilibrating a partially linearized electronic energy functional. This provides conservative forces and reduces the global nonlinear equilibration to a quasi-diagonal linear problem that can be solved directly at low cost. Auxiliary dynamical electronic degrees of freedom, propagated through an extended-Lagrangian dynamics, capture the long-range electronic response and define the reference state for the linearization of the energy functional. GPU-accelerated simulations with over 100,000 atoms demonstrate that the shadow potential closely follows the tightly converged Born-Oppenheimer surface, while maintaining long-term stability at a computational cost approaching that of a fixed-charge model. These advances establish an efficient and broadly applicable framework for machine-learned interatomic potentials that explicitly incorporate long-range electrostatic interactions and environment-dependent electronic response.