MACE-MP0 potential¶

The MACE refinement examples relax tricor-generated cells on MACE-MP0, a machine-learning interatomic potential. This page summarises the energy model (the Hamiltonian being minimised) and the update strategies tricor’s pipeline uses on it. tricor does not implement MACE; it consumes the published medium-mpa-0 weights through the mace-torch ASE calculator.

References:

I. Batatia, D. P. Kovács, G. N. C. Simm, C. Ortner, and G. Csányi, MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields, NeurIPS 35 (2022). arXiv:2206.07697

I. Batatia et al., A foundation model for atomistic materials chemistry, arXiv:2401.00096.

R. Drautz, Atomic cluster expansion for accurate and transferable interatomic potentials, Phys. Rev. B 99, 014104 (2019) — the body-ordered basis MACE builds on.

Energy model¶

MACE is a graph neural network over atoms. The total energy is a sum of per-atom contributions read out after \(T = 2\) rounds of message passing:

\[ E \;=\; \sum_i E_i, \qquad E_i \;=\; \sum_{t=1}^{T} \mathcal R_t\!\big(\mathbf h_i^{(t)}\big), \]

where \(\mathbf h_i^{(t)}\) is the feature vector (node state) of atom \(i\) after layer \(t\) and \(\mathcal R_t\) is a learned readout.

Each layer first forms the two-body atomic basis by summing over neighbours within the cutoff \(r_\text{cut} = 6\) Å:

\[ A_i \;=\; \sum_{j \,:\, r_{ij} < r_\text{cut}} R\!\left(r_{ij}\right)\, Y\!\left(\hat{\mathbf r}_{ij}\right) \otimes W\,\mathbf h_j^{(t)}, \]

with learned radial functions \(R\) (Bessel basis with a smooth polynomial envelope that takes the interaction to zero at \(r_\text{cut}\)) and spherical harmonics \(Y\) carrying the angular information. The distinguishing MACE step is the higher-order product basis: symmetrised tensor products of up to \(\nu = 3\) copies of \(A_i\),

\[ B_i \;=\; \big(\underbrace{A_i \otimes A_i \otimes A_i}_{\nu \text{ copies}}\big)_{\text{sym}}, \]

so a single layer builds messages of body order \(\nu + 1 = 4\) (centre atom + three neighbours) — angular and dihedral-like correlations enter directly rather than through deep stacks of two-body messages. Two layers compose these features, giving an effective receptive field of \(2\,r_\text{cut} = 12\) Å and effective body order far beyond four. All features are E(3)-equivariant, so predicted energies are exactly invariant under rotation, translation, and permutation.

Forces and stress come from exact differentiation of the network (autograd), so they are consistent with the energy:

\[ \mathbf F_i = -\frac{\partial E}{\partial \mathbf r_i}, \qquad \sigma = \frac{1}{V}\frac{\partial E}{\partial \varepsilon}. \]

The MP0 / MPA-0 training data¶

The MACE-MP0 family are foundation models: one parameter set covering the periodic table, trained on DFT relaxation trajectories from the Materials Project (the MPtrj dataset, ~1.5 M configurations at the PBE / PBE+U level). The medium-mpa-0 checkpoint used in the examples additionally trains on the Alexandria dataset. Accuracy on bulk inorganic materials is near-DFT for energies and forces; known limitations relevant here are short-range artefacts on far-from-equilibrium geometries (next section).

Soft-wall regularisation¶

Foundation models are trained near equilibrium, so strongly disordered inputs can fall into spurious low-energy basins where atom pairs approach unphysically closely. tricor wraps the MACE calculator with a one-sided per-pair wall (scripts/_wall_calculator.py in tricor-docs):

\[\begin{split} U_\text{wall}(r_{ij}) = \begin{cases} \dfrac{k}{n}\big(r^{\min}_{s_i s_j} - r_{ij}\big)^{n}, & r_{ij} < r^{\min}_{s_i s_j},\\[6pt] 0, & \text{otherwise}, \end{cases} \end{split}\]

with \(k = 1000\) eV/Å\(^n\), \(n = 4\), and the per-species-pair floors \(r^{\min}\) measured from the cleaned pre-MACE geometry (per_pair_min_from_atoms). The wall is exactly zero at and above each floor, so it never perturbs valid physics; it only blocks descent into the near-overlap basins.

Update strategies¶

Two updates are used, chosen per regime:

LBFGS minimisation (all ordered + amorphous regimes). The ASE LBFGS optimiser performs quasi-Newton descent, building a limited-memory approximation of the inverse Hessian from recent \((\Delta\mathbf r, \Delta\mathbf F)\) pairs (D. C. Liu and J. Nocedal, Math. Program. 45, 503 (1989)). Pipeline settings: maxstep = 0.2 Å, 50 steps, with the force threshold set far below reach so the step cap is the deterministic stop — matching the fixed-sweep behaviour of the FIRE relaxation.

Langevin molecular dynamics (liquid regime). A melt is a thermal state, not an energy minimum, so the liquid regime runs NVT Langevin dynamics at the material’s melting point instead of a minimisation:

\[ m_i \ddot{\mathbf r}_i = \mathbf F_i - \gamma m_i \dot{\mathbf r}_i + \sqrt{2 m_i \gamma k_B T}\, \boldsymbol\eta_i(t), \]

with \(T = T_\text{melt}\) (Cu 1358 K, Si 1687 K, SiO₂ 1986 K, SrTiO₃ 2353 K), time step 2 fs, friction \(\gamma = 0.02\) (ASE units), 80 steps, and Maxwell–Boltzmann initial velocities. This is what makes the liquid structurally distinct from the energy-minimised amorphous regime built from the same grain-free start.

Cost¶

One MACE evaluation is \(\mathcal O(N)\) in atom count (fixed neighbour cutoff). Measured on the 40³ Å examples (8 CPU threads): ~4 s/step at ~3 k atoms (Si) and ~9–13 s/step at ~5 k atoms (Cu, SiO₂, SrTiO₃); a GPU is roughly 10–50× faster. Extrapolated by atom count, a 200³ Å cell (~600 k atoms) costs on the order of a day per relaxation on CPU — at those sizes the FIRE spring network (minutes) is the working option.