C06 — C/T Skeleton Active-Space Pre-screening
| Field | Value |
|---|---|
| Domain | Chemistry |
| System | Any molecule with MP2 natural orbitals |
| Group | U(n) × U(n) (orbital rotation group) |
| H^k tier | H⁰ / H¹ / H² |
| ISA | Forge (β ≈ β*) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · BIND · LABEL |
| Papers | doi:10.5281/zenodo.21300668 (Paper 588), Paper 596 |
Physical system
Given a molecule, run a cheap MP2 calculation to obtain natural orbital occupation numbers (NOONs). Each bonding/antibonding pair (nᵢ, 2−nᵢ) is classified into one of three strata by two ISA descriptors:
- θ_G = 2·arccos(√(n/2)) — Grassmannian angle; measures departure from a frozen (ionic/covalent) pair
- c₂ = min(n, 2−n, ñ, 2−ñ) — Weyl chamber coordinate; measures entanglement between bonding and antibonding partners
This produces a C/T skeleton (sCeleTon): a bipartite graph over GF(2) where C-boxes (frozen pairs, θ_G < δ) are labelled 0 and T-arrows (active pairs, θ_G ≥ δ) are labelled 1. The T-arrow set is the recommended CASSCF active space.
Target category
Vect(ℝ) with graded H^k structure — the category of real vector spaces stratified by cohomological degree:
- H⁰: C-box pairs (NOON near 0 or 2) — tropical fixed points, handled by DFT
- H¹: T-arrow pairs with c₂ < δ* — Clifford/CCSD regime
- H²: T-arrow pairs with c₂ ≥ δ* — magic/CASSCF regime
Interpretation functor
F: C → Vect(ℝ) defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Eigenvalue of occupation number operator; output = NOON nᵢ ∈ [0,2] |
| TWIST | Berry phase between bonding/antibonding pair; captured by θ_G |
| BIND | Non-abelian holonomy of multi-pair entanglement; triggers CASSCF |
| LABEL | Stratum assignment: H⁰ (C-box) / H¹ / H² (T-arrow) |
ISA programme
NOON: ORBIT[nᵢ for all pairs] -- MP2 natural orbital occupations
ANGLE: LABEL[θ_G = 2·arccos(√(n/2))] -- Grassmannian angle per pair
WEYL: LABEL[c₂ = min(n,2-n,ñ,2-ñ)] -- Weyl coordinate per pair
ROUTE: LABEL[stratum | θ_G, c₂, δ*] -- classify C-box vs T-arrow
BETA: LABEL[β_eff = Δ_HL/(σ·(2S+1))]-- MGE inverse temperature
ACTIVE: ORBIT[T-arrow indices] -- recommended active space
METHOD: TWIST or BIND -- route to CCSD (H¹) or CASSCF (H²)
Routing threshold: δ* = (π/4)√α where α = β_eff · ε (correlation energy per pair). This is derived from the MGE cost-accuracy tradeoff, not fitted to data.
Computable output
- Active space (n_elec, n_orb): list of T-arrow orbital indices and electron count for CASSCF input. Reduces to 7–36× smaller active space than full valence (Paper 588, x588g: r = −0.979 correlation with CASSCF NOONs).
- Weyl c₂: DFT failure predictor with r = 0.990 correlation to multi-reference correlation energy (Paper 596, x596a).
- Speedup: 7–36× wall-time reduction over manual active-space selection, with no loss in accuracy for the validated benchmark set.
Validation
- x588g (SHA 45dbc10): Weak Lifting Theorem confirmed numerically — 10/10 molecules, r = −0.979 between C/T stratum and CASSCF NOON deviation.
- x596a–e (Papers 596): Weyl c₂ predicts DFT error r = 0.990 across N₂, Fe²⁺, and 6-molecule benchmark; MGE soft router p continuous 0.024→0.911 along N₂ dissociation curve.
- alchemi (PyPI): reference implementation at github.com/roguetrainer/alchemi.