Structured reformulation of many-body dispersion: towards pairwise decomposition and surrogate modeling

Abstract

We present a structured reformulation of the many-body dispersion (MBD) model that enables a physically consistent decomposition of forces into pairwise components. By introducing a many-body correlation matrix that scales dipole-dipole interactions, we derive unified expressions for the MBD energy, force, and Hessian. This reformulation reveals a natural structure for pairwise force decomposition and provides a promising foundation for interpretable analysis and machine learning surrogate modeling of MBD interactions.

Figures (4)

• Figure 1: (a) and (b) show schematic representations of two parallel carbon chains and two parallel carbon rings, respectively, separated by a distance $h$ along the $y$ direction. Each molecule contains $n$ atoms ($N=2n$) with a uniform interatomic spacing of $1.2\,\text{\AA}$, and atoms in the two chains or rings are perfectly aligned along the $y$ axis. These are fixed, fictitious systems, with no consideration of covalent bonding. Atoms in the upper and lower structures have reverse numbering to facilitate heatmap analysis in subsequent sections. (c) shows the $F_y$ force profile along the upper chain or ring computed using different vdW dispersion models for systems with $n=100$ and $h=10\,\text{\AA}$.
• Figure 2: Heatmaps of condensed $\boldsymbol{B}$ and $\nabla\boldsymbol{C}$, see definition in Eq. \ref{['eq:condensed_mat']}, for the two molecular systems (chain and ring) with $n=100$, and $h=10\,\text{\AA}$. The plot axes correspond to atomic indices $i$ and $j$, where the first 100 indices represent atoms in the upper molecule and the second 100 represent atoms in the lower molecule, ordered in reverse (see Fig. \ref{['fig:chain_ring_geo_force']}). Under this convention, the main diagonal blocks correspond to intra-molecular interactions, while the off-diagonal blocks capture inter-molecular interactions.
• Figure 3: Heatmaps of the MBD force decomposition component $f_{i,y,j}$ for the two molecular systems (chain and ring) with $h=10\,\text{\AA}$. Each value represents the contribution from atom $j$ to $f_{y}$ of atom $i$. The same atomic indexing convention is used as in Fig. \ref{['fig:Fmbd_chain_ring_conBdC']}. In (a) and (b), the systems preserve symmetry between the upper and lower molecules with $n=100$. In (c), the last atom of the upper ring is removed ($n_\text{upper}=99$), while the lower ring remains unchanged.
• Figure 4: Roadmap for ML surrogate modeling of MBD. Three types of ML models are considered, corresponding to three different paths. Model (GF) directly predicts MBD forces from the geometric and chemical information of the molecular systems. Model (CB) instead maps matrix $\boldsymbol{C}$ to matrix $\boldsymbol{B}$. The third model (CF) takes both $\boldsymbol{C}$ and $\nabla\boldsymbol{C}$ as inputs and directly outputs MBD forces. The Hessian $\boldsymbol{H}$ could be involved in the loss function to regularize the prediction to the force.