D3h Trigonal Prismatic Geometry Links Nuclear Spin Entanglement to Ligand Field Splitting — E8 Intelligence Research

Abstract

FINDING: D3h trigonal prismatic geometry in coordination complexes is linked to nuclear spin entanglement in Posner clusters, with paramagnetic/diamagnetic behavior determined by ligand field splitting and electron pairing. MATH: - D3h point group symmetry: character table includes irreducible representations A1', A2', E', A1'', A2'', E''. - Ligand field splitting: Δ(trigonal prism) ≈ 0.9–1.1 × Δ(octahedral) for d-orbital energies (e.g., d_z², d_x²-y², d_xy, d_xz, d_yz). - Spin states: paramagnetic (unpaired electrons) vs. diamagnetic (paired) determined by Δ vs. pairing energy P. - Posner cluster: Ca₉(PO₄)₆, nuclear spin entanglement via ³¹P (I=1/2) in D3h symmetry, possible quantum coherence at room temperature. CONNECTION: - D3h symmetry is a subgroup of O_h (octahedral) and relates to trigonal prismatic coordination, which appears in hexagonal close-packed (hcp) lattices (c/a ratio ≈ 1.633, close to 1.618 golden ratio). - The 60° rotation axis in D3h links to base-60 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com