Hund's Rule from Golden Angle Interference

We demonstrate that Hund's rule—the empirical observation that electrons preferentially occupy different orbitals with parallel spins—emerges naturally from wave interference when electron phases are quantised in golden angle units. KEY RESULTS: 1. GOLDEN ANGLE PHASE STRUCTURE The golden angle θ = 2π/φ² ≈ 137.5° creates specific interference: • Adjacent orbitals (1 golden angle apart): cos(2π/φ²) = −0.7374 (DESTRUCTIVE) • Opposite orbitals (2 golden angles apart): cos(4π/φ²) = +0.0874 (CONSTRUCTIVE) 2. REMARKABLE IDENTITY cos(2π/φ) = cos(2πφ) = cos(2π/φ²) — all equal due to golden ratio identities 3. NULL EIGENVALUE (EXACT) The 3×3 Hamiltonian matrix has determinant EXACTLY zero. This follows analytically from the double-angle identity t₂ = 2t² − 1. A protected null mode exists in the orbital system. 4. HUND'S RULE EMERGENCE Electrons spread across orbitals to AVOID destructive interference. Adjacent orbital coupling is 8× stronger than opposite orbital coupling. This asymmetry naturally enforces electron spreading. PHYSICAL INTERPRETATION: The golden angle is the unique angle producing maximum irrationality and self-similarity under golden ratio scaling. If electrons are waves on an icosahedral structure, golden angle phase relationships are inevitable, and Hund's rule follows as a geometric consequence. This paper complements "Atomic Orbitals as Icosahedral Interference Patterns" by providing a Fourier/Hamiltonian perspective on the same icosahedral framework.