Theoretical Background

This section explains the theory behind Magesty.jl — the spin-cluster expansion (SCE) — and connects each concept to the module that implements it. The pages are meant to be read in order; each builds on the previous one.

• Overview
• • The problem
  • The spin-cluster expansion
  • Pipeline and modules
  • References
• Real spherical harmonics
• • Why a spherical-harmonic basis
  • Real (tesseral) harmonics
  • Implementation
• Angular-momentum coupling
• • Why couple
  • Coupling scheme
  • Implementation
• Symmetry adaptation (SALC)
• • Clusters and orbits
  • From coupled bases to SALCs
  • Implementation
• Orbit clusters
• • What is stored
  • How it is built
  • Why pre-enumerate
  • Invariants worth knowing
• Folded tensor
• • Definition
  • Why fold
  • Storage
  • Numerical note
• Spherical-harmonic cache
• • What is stored
  • Why precompute
  • Threading and lifetime
  • Hot-path simplification
  • Numerical note
• Cluster-major torque kernel
• • The per-atom layout, and what it wastes
  • Cluster-major reverse-mode
  • The accumulation buffer
  • Threading
  • Inference path
  • Numerical note
• Design matrix and fitting
• • The design feature
  • Building the design matrix
  • Fitting the coefficients
  • Result types
  • Cross-validation diagnostics
  • References
• Technical notes
• • Conversion between the lowest SCE model and conventional spin model