S01 — Electronic Spectroscopy (d-shell)
| Field | Value |
|---|---|
| Domain | Spectroscopy |
| System | Any transition-metal complex |
| Group | SU(2) |
| H^k tier | H⁰ |
| ISA | Origami (β → ∞) |
| Status | Validated |
| Opcodes | FLOP · SPLIT · SPLAT · ORBIT |
| Paper | doi:10.5281/zenodo.21219722 |
Physical system
The d-shell electronic structure of transition-metal (TM) complexes in octahedral (O_h), tetrahedral (T_d), or lower symmetry environments. Observables: term symbol ordering, d-d transition energies (visible spectroscopy), ground-state spin multiplicity.
Target category
Rep(SU(2)) in the cold (β→∞) limit. All d-shell electrons are treated as angular momentum wires; the crystal field and electron-electron repulsion are FLOP operations (6j recoupling) in this representation.
Interpretation functor
F: C → Rep(SU(2)) defined by:
| Opcode | F(opcode) |
|---|---|
| FLOP | Racah 6j symbol {j₁ j₂ J₁₂; j₃ J j₂₃} — recoupling cost |
| SPLIT | SPLIT(j) = √(2j+1) — quantum dimension, pair creation |
| SPLAT | SPLAT(j) = 1/√(2j+1) — bubble closure |
| ORBIT | G-orbit label on d^n configuration: term symbol ²ˢ⁺¹L_J |
ISA programme
INIT: LABEL[d^n] -- n electrons in d shell
SPLIT: SPLIT[L=2] x n -- open n angular momentum wires
FLOP: FLOP[L,S,J coupling] -- Racah recoupling; cost = 6j products
ORBIT: ORBIT[2S+1 L_J] -- read term symbol from orbit label
SPLAT: SPLAT[closed subshell] -- close filled subshell wires
Programme length: O(n²) FLOPs for n d-electrons (one recoupling per pair).
Computable output
- Term symbols: ²ˢ⁺¹L_J for d¹ through d¹⁰, all 4 common geometries
- Transition energies in units of Racah B parameter: exact rational multiples; e.g. for d² Oh: ³T₁g→³T₂g = 8Dq, ³T₁g→³A₂g = 18Dq - 12B
- Tanabe-Sugano diagrams: E/B vs Dq/B curves are ORBIT label sequences as a function of β (crystal field strength). These are tropical varieties in the MGE sense (Paper 491).
- Ground-state spin: highest-weight ORBIT label agrees with Hund’s rules without empirical input
Validation
Term symbol ordering confirmed for all d^n configurations d¹–d¹⁰ in O_h. Transition energies match experiment to within Racah B parameterisation (B fitted once per metal ion, then predictive across all complexes). Tanabe-Sugano diagrams reproduced exactly. See Paper 487 §3 and experiment x487a–f (72/73 = 98.6% accuracy on valence assignment benchmark).
Part of the ISA Zoo. See also Chemistry Roadmap Layer 4; spectrafold implements this programme.