FIRE vs MACE-MP0

The FIRE relaxation and the MACE-MP0 potential are the two relaxers used in the example galleries. They sit at opposite ends of a cost–accuracy trade-off, and the goal of the ongoing calibration work is to make FIRE’s spring weights — derived from a single reference .cif — reproduce MACE-relaxed structures as closely as the harmonic form allows.

The two energy models

FIRE spring network

MACE-MP0

functional form

harmonic bond + angle springs, analytic repulsion walls

learned equivariant message-passing network

body order

2-body (bonds, repulsion) + 3-body (angles), fixed

4-body messages per layer × 2 layers; effectively many-body

interaction range

first shell (bonds/angles), \(1.5\,r^\star\) (push)

6 Å per layer, 12 Å receptive field

parameters from

one reference crystal (.cif) + 3 user weights

~1.5 M DFT configurations (MPtrj + Alexandria)

bond topology

explicit K-nearest graph, rebuilt every 10 steps

none — smooth cutoff functions

energy scale

arbitrary (spring units); only ratios matter

eV, transferable across materials

forces

analytic per term

autograd of the network

update

simplified FIRE integrator

LBFGS (or Langevin MD for liquid)

cost / step at 40³ Å

~0.05–0.1 s

~4–13 s (8-thread CPU)

practical size limit

200³ Å+

~40³ Å on CPU

What the spring network can and cannot match

Because every FIRE term is harmonic about a single target, the model can in principle reproduce:

  • bond-length and angle distribution positions — the targets \(r^\star\), \(\phi^\star\) are measured from the same crystal MACE relaxes toward;

  • distribution widths — set by the ratios of bond_weight : angle_weight : repulsion_weight, which is exactly what the calibration below adjusts;

  • first/second-shell separation — via the hard-core and non-bonded-push walls.

It cannot match:

  • anharmonicity — real bond-energy curves are softer under stretch than compression, so MACE-relaxed distributions are skewed while harmonic ones are symmetric;

  • chemistry changes — coordination defects, over/under-bonded sites, and bond rearrangement energetics have no representation in a fixed-topology spring network;

  • absolute energies — the spring cost is a fit metric, not a potential energy; regime stability ladders (eV/atom) require MACE (or DFT) single points.

Calibrating FIRE weights from a single .cif against MACE

The spring targets (\(r^\star\), \(\phi^\star\), hard-core radii) already come from the reference crystal. CoordinationShellTarget.calibrate_to_mace() measures the stiffnesses from MACE’s curvature around that same crystal, so both models share one input file:

shell = tc.CoordinationShellTarget.from_atoms(ref_atoms, phi_num_bins=90)
shell = shell.calibrate_to_mace()        # optional; needs mace-torch

The calibration runs in seconds per material (the reference cell is tiny) and fills in:

  • pair_k_bond (eV/Ų) — interatomic-Hessian projection: displace the representative neighbour \(j\) by \(\pm\delta\) along the bond axis and read the force change on \(i\), \(k = \hat{\mathbf u} \cdot (\mathbf F_i^+ - \mathbf F_i^-) / 2\delta\). This isolates the \(i\)–\(j\) constant from the other springs attached to either atom. Negative bare constants (ionic spectator pairs such as Sr–O, where lattice stability comes from the other terms) are clamped to zero with the raw value kept in mace_calibration.

  • pair_morse_D/a/r — Morse fit \(D(1 - e^{-a(r - r_e)})^2\) of a rigid bond scan, restricted to an energy window (default 4 eV) around the minimum.

  • pair_hard_min_mace (Å) — where the compressive branch rises 1 eV above the bond minimum.

  • angle_k (eV/rad²) — rigid-bend scan of the representative triplet, rotating the endpoint with the fewest bonds, with the moving atom’s bond-stretch contribution \(\sum_n k_{an} (\mathbf v \cdot \hat{\mathbf u}_{an})^2\) subtracted in closed form. Couplings to other angles sharing the moving atom are not removed and remain a known overestimate (worst for high-coordination centres, e.g. Ti–O–Ti in SrTiO₃).

Because tricor’s force definitions make bond_weight ≡ \(k_\text{bond}\) (energy/Ų) and angle_weight ≡ \(k_\text{angle}\) (energy/rad²), the calibrated ratio plugs in directly. Measured with medium-mpa-0:

material

k_bond (eV/Ų)

k_angle (eV/rad²)

angle/bond

preset ratio

Si

5.8

20.3

3.5

~0.45

SiO₂ (Si–O)

26.3

14.2 / 34.0

0.92

~0.8

Cu

1.8

0.6

0.34

0

SrTiO₃ (Ti–O)

1.7

20.9 / 70.3

—

~0.7

The Si row is the headline: MACE says silicon’s angle springs are ~7× stiffer relative to its bond springs than the hand-tuned presets use. Validation protocol: relax identical Voronoi starts with calibrated FIRE and with MACE; compare bond/angle distribution widths and g₂/g₃, and score the FIRE-relaxed structure with a MACE single point. The per-atom energy gap to the MACE-relaxed structure is the figure of merit.

Validated outcome (Si nanocrystalline, 40³ Å). Transplanting the literal stiffness ratio into the quench over-constrains angles (angle σ collapses to 12.9° vs MACE’s 16.2°) and widens the energy gap; a thermal distribution’s width is not set by curvature alone, and the fixed-step integrator under-resolves the stiffened modes. What does transfer:

FIRE configuration

gap to MACE-relaxed (eV/atom)

hand-tuned weights

+0.105

+ literal MACE stiffness ratio

+0.179

+ Morse anharmonicity, MACE wall, per-pair k at the hand-tuned effective scale

+0.081

The winning configuration — used by the Fast FIRE Refinement examples — keeps the proven effective force scale and takes the calibration’s Morse anharmonicity, per-pair stiffness ratios between different species pairs, and hard-core wall.

Composite targets (sp²/sp³ carbon) are calibrated per sub-target before from_targets composes them. Consuming the per-pair stiffnesses and Morse parameters inside shell_relax is the next implementation step.

See also