A multi-channel dipole coupling approach to constructing a pseudoscalar chirality order parameter for finite molecular and atomic systems.
A pseudoscalar cannot be formed from a single vector field. The key insight is that three independent polar vectors, combined via a triple scalar product, naturally yield a pseudoscalar with the correct transformation properties under O(3).
| Structure | N | χTP (Triple Product) | χCM (Chiral Multipole) | χFull (Extended) | SCSM |
|---|---|---|---|---|---|
| CHFClBr (L) | 5 | -4.800 | 0.000 | 0.000 | 1.333 |
| CHFClBr (R) | 5 | +4.800 | 0.000 | 0.000 | 1.333 |
| CH₂F₂ (achiral) | 5 | -0.308 | 0.000 | 0.000 | 1.333 |
| Right helix (12) | 12 | +0.009 | ∼0 | +0.520 | 0.280 |
| Left helix (12) | 12 | -0.009 | ∼0 | -0.520 | 0.286 |
| Planar triangle | 3 | 0.000 | 0.000 | 0.000 | 0.034 |
| Propeller (Δ) | 7 | -0.431 | 0.000 | ∼0 | 0.214 |
| Propeller (Λ) | 7 | +0.330 | 0.000 | ∼0 | 0.124 |
The chiral multipole correctly produces opposite signs for enantiomeric pairs and vanishes for achiral structures. The triple product incorrectly yields χTP = -0.308 for the achiral CH₂F₂. The CSM is always non-negative and cannot distinguish enantiomers.
| Structure | Measure | χref | Max Rel. Error | SO(3) Invariance | O(3)\SO(3) Sign Flip |
|---|---|---|---|---|---|
| CHFClBr (L) | Triple Product | -77.549 | 1.28e-15 | PASS | PASS |
| Chiral Multipole | 0.000 | 4.00e-29 | PASS | PASS | |
| Full Multipole | 0.000 | 6.14e-27 | PASS | PASS | |
| Right Helix | Triple Product | +0.016 | 1.895 | FAIL | FAIL |
| Chiral Multipole | ∼0 | 6.55 | PASS | PASS | |
| Full Multipole | +0.520 | 1.99e-14 | PASS | PASS | |
| Propeller (Δ) | Triple Product | -6.321 | 1.792 | FAIL | FAIL |
| Chiral Multipole | ∼0 | 1.85e-28 | PASS | PASS | |
| Full Multipole | ∼0 | 8.20e-15 | PASS | PASS |
Chirality increases monotonically from zero (flat ring, achiral) as helix pitch grows. The full multipole shows cubic scaling for small pitch.
For a helix with fixed pitch = 2.0, chirality grows superlinearly with the number of atoms in the extended multipole formulation.
Testing whether χ(A ∪ B) ≈ χ(A) + χ(B) for two separated CHFClBr (L) molecules. Exact extensivity corresponds to ratio = 1.
| Separation | χTP Ratio |
|---|---|
| 10 | 0.190 |
| 20 | 0.190 |
| 50 | 0.190 |
| 100 | 0.190 |
| 200 | 0.190 |
| 500 | 0.190 |
| 1000 | 0.190 |
χCM and χFull yield machine-zero for CHFClBr due to channel degeneracy in this symmetric tetrahedron.
Multi-channel measures pass all O(3) tests with errors below 10-14. The triple product fails for structures with identical atomic species.
Correctly assigns opposite signs to L/R enantiomers. Helices: +0.520 (R) vs -0.520 (L). CSM cannot distinguish them.
For highly symmetric systems with identical species, the three weighting channels can become coplanar, yielding zero. Future work: electronic density channels.
The triple product of three independently weighted dipole moments (χ = p₁ · (p₂ × p₃)) is the natural pseudoscalar for structural chirality. For the bulk periodic generalization, each pα would be promoted to a Berry phase, and χ would become a triple product of Berry phases -- the "chiralization" of the crystal.