Algorithms

Mathematical details of tricor’s structure-generation pipeline, ordered to match the natural flow of a build: measure the reference crystal, lay down a Voronoi-tiled initial cell, optionally walk grain orientations into a better-aligned basin, clean up grain-boundary overlaps, relax (with the built-in FIRE spring network, or hand off to the MACE-MP0 potential), and optionally synthesise a target g3 distribution against which the supercell can be compared.

• Three-Body Distribution (g3)
  • Definition
  • Reduced coordinates
  • Coordination shell target
• Supercell generation
  • Voronoi grain construction
• Orientation refinement
  • Why orient grains before FIRE
  • Algorithm
  • Pair-distance score
  • Multi-species grains
  • What gets tuned
  • Cost of the search
  • Where to read the code
• Cleanup sweeps
  • bond_relax: attract-to-peak + repel-from-hard-core
  • enforce_hard_core: pure projection
  • Pseudocode
  • Where to read the code
  • See also
• FIRE relaxation
  • Energy
  • Force terms
  • Bond topology
  • Integration
  • Cost and convergence
  • Per-atom cost (trajectory colour scale)
  • See also
• MACE-MP0 potential
  • Energy model
  • Soft-wall regularisation
  • Update strategies
  • Cost
  • See also
• FIRE vs MACE-MP0
  • The two energy models
  • What the spring network can and cannot match
  • Calibrating FIRE weights from a single .cif against MACE
  • See also
• Target g3 Construction
  • Construction pipeline
  • Parameters
  • Usage
  • The g2 pair distribution
• Glossary
  • Disorder regimes
  • Three-body distribution
  • Coordination shell target
  • Grain construction
  • Shell relaxation
  • Orientation refinement
  • Composite shell targets

Stage	Page	What it covers
1. Measure	Three-body distribution	g3 + g2 histogram, reduced coordinates, CoordinationShellTarget extraction.
2. Build	Supercell generation	Voronoi grain construction: seeds → tessellation → master block tiling → cell filling → overlap removal → close-pair push → optional thermal jitter.
3. Align grains	Orientation refinement	Per-grain SO(3) coordinate search that re-runs the full grain assembly per trial and keeps rotations that lower a global pair-distance cost, before the final relaxation (FIRE or MACE) runs.
4. Cleanup	Cleanup sweeps	cKDTree bond_relax + enforce_hard_core sweeps that remove grain-boundary overlaps before the relaxer starts. O(N log N) per iteration.
5. Relax	FIRE relaxation	Bond + angle + repulsion (+ restraint) springs minimised with a simplified FIRE integrator. This is what Supercell.generate() calls internally.
5’. ML relax	MACE-MP0 potential	The machine-learning Hamiltonian and the LBFGS / Langevin update strategies used by the MACE refinement examples.
—	FIRE vs MACE-MP0	Term-by-term comparison of the two energy models, and the strategy for calibrating FIRE’s spring weights against MACE from a single reference crystal.
6. Compare	Target g3 construction	Blur + blend a measured crystalline g3 toward the random limit to produce a comparison target.
Reference	Glossary	Disorder regimes, shell-target fields, force-term nomenclature.