A Note about MACE

I came about a subtle fact I didn't catch when I first read the MACE paper [1], that is they use the total energy of the system, not the binding or cohesive energy. More specifically, the energy is not:

$$ \begin{equation} E_{\text{MACE}} = E_{\text{system}} - \sum_{i=1} N_i E_i \label{eq:binding_energy} \end{equation} $$

but rather just,

$$ E_{\text{MACE}} = E_{\text{system}}\quad , $$

where $i$ denotes the atom type and $N_i$ is the number of atoms of that type. Here, $E_{\text{system}}$ is the total ground state energy of the system. The energy expansion is with respect to $E_{\text{MACE}} = \sum_{i=1} \mathrm{E}_i(\mathbf{h}_i)$, where $\mathbf{h}_i$ represents the learned node feature embeddings for an atom $i$ and contains all the body-order dependencies through the learned message construction, passing, and updating [2-3].

This realization came about because I was performing some equation of state calculations for certain systems using MACE and couldn't understand why the energy seemed excessively negative compared to other atomistic calculations and experimental values. It's important to note that this energy shift doesn't affect the equilibrium volume or bulk modulus from the fits, as those are related to the shape and derivatives of the curve, not the energy scale/level. Similarly, for MD, this won't affect the dynamics since the forces are again based on $F = -\nabla E$ and the energy is continuous and smooth.

The only quantities affected are thermodynamic ones like enthalpy, free energy, etc., which are related to the energy of the system. Any observed thermodynamic state changes you are investigating will be with respect to this reference energy. However, the interesting aspect of MACE is that the site energies are actually learned! This means an isolated atom in a vacuum has a non-zero energy.

MACE allows you to calculate the energy of an isolated atom, enabling you to subtract it to obtain the binding energy of the system as in eq.~\ref{eq:binding_energy}. Here is an example of the output for the diatomic curve of CO with MACE without the energy shift:

C-O diatomic curve showing the total energy shift

If we calculate the site energies for $\textrm{C}$ and $\textrm{O}$ in this system, we get:

• $\textrm{C}$ = -2.0562 eV
• $\textrm{O}$ = -2.0121 eV

Subtracting the sum of these two site energies from the total energy at the equilibrium bond length gives a binding energy of approximately 10.8 eV, which is quite close to the experimental value of 11.11 eV per molecule.

Therefore, be sure to keep this in mind when analyzing the results of MACE. I need to review if other equivariant models adopt this same approach or not.

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References

[1] I. Batatia, D.P. Kovács, G.N.C. Simm, C. Ortner, G. Csányi, MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields, (2022). https://doi.org/10.48550/arXiv.2206.07697.

[2] B. Sanchez-Lengeling, E. Reif, A. Pearce, A.B. Wiltschko, A Gentle Introduction to Graph Neural Networks, Distill (2021). https://doi.org/10.23915/distill.00033.

[3] D. Grattarola, A practical introduction to GNNs - Part 2, Daniele Grattarola (2021). https://danielegrattarola.github.io/posts/2021-03-12/gnn-lecture-part-2.html (accessed February 28, 2025).

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