Shadow Molecular Dynamics for Flexible Multipole Models

TL;DR

This work extends shadow extended-Lagrangian Born–Oppenheimer MD to flexible multipole models, enabling stable and efficient simulation of long-range electrostatics with both monopoles and dipoles. By decomposing the Coulomb interaction into short-range and long-range parts and introducing shadow energy functions with extended charge variables, the authors achieve accurate BO-like dynamics without tight convergence in each step, while leveraging low-rank Krylov updates for the kernel. The framework is demonstrated on solvated systems, verifying energy conservation, consistent IR spectra, and realistic dipole dynamics, and is shown to work for both fully flexible multipoles and fixed monopole/flexible-dipole variants. The approach is designed to be compatible with AI/ML parameterizations for environment-dependent atomic properties, enabling transferable, high-fidelity simulations across diverse molecular systems.

Abstract

Shadow molecular dynamics provide an efficient and stable atomistic simulation framework for flexible charge models with long-range electrostatic interactions. While previous implementations have been limited to atomic monopole charge distributions, we extend this approach to flexible multipole models. We derive detailed expressions for the shadow energy functions, potentials, and force terms, explicitly incorporating monopole-monopole, dipole-monopole, and dipole-dipole interactions. In our formulation, both atomic monopoles and atomic dipoles are treated as extended dynamical variables alongside the propagation of the nuclear degrees of freedom. We demonstrate that introducing the additional dipole degrees of freedom preserves the stability and accuracy previously seen in monopole-only shadow molecular dynamics simulations. Additionally, we present a shadow molecular dynamics scheme where the monopole charges are held fixed while the dipoles remain flexible. Our extended shadow dynamics provide a framework for stable, computationally efficient, and versatile molecular dynamics simulations involving long-range interactions between flexible multipoles. This is of particular interest in combination with modern artificial intelligence and machine learning techniques, which are increasingly used to develop physics-informed and data-driven foundation models for atomistic simulations. These models aim to provide transferable, high-accuracy representations of atomic interactions that are applicable across diverse sets of molecular systems, which requires accurate treatment of long-range charge interactions.

Figures (15)

• Figure 1: Conceptual picture of how the shadow MD framework can provide accurate and stable atomistic simulations using physics-informed and data-driven foundation models, including long-range flexible multipole interactions, where the interatomic potential is based on coarse-grained conceptual DFT and is parameterized using AI/ML trained on first-principles reference data.
• Figure 2: Model atomic system of eight atoms on the corners of a box with a single atom zoomed in to show details of the model. Blue "${\bf +}$" and magenta "${\bf -}$" symbols represent positive and negative monopole partial charges and arrows represent dipoles. The dotted curve illustrates the charge-independent potential energy surface, $V({\bf R})$, and the solid curve illustrates the Gaussian monopole charge distribution.
• Figure 3: Description of the three tested molecular systems including the total number of atoms and molecules in each system.
• Figure 4: Comparison of electrostatic potential energy values using exact and shadow potentials as a single atom of acetamide in vacuum is displaced (hydrogen in the left panel, oxygen in the right panel). The shadow energy function was expanded around the exact solution at a displacement $D = 0.5$ a.u. (black dot) from the equilibrium bond distance. Around the expansion point ($D \in [0,1.5]$ a.u.) the exact and shadow potentials agree closely, but begin to diverge as the displacement changes ($D < 0$).
• Figure 5: The electrostatic potential energy fluctuations (the Born-Oppenheimer potential without $V({\bf R})$) for acetamide in water along a shadow MD trajectory generated with the flexible multipole model. The figure compares the exact regular electrostatic potential (black line) with the shadow potential (red dots). Simulations were performed using a time step of $\delta t = 0.4$ fs.