Photochemical Multi-Geometry Active Space Selection

Executive Summary

This benchmark evaluates inZOR-ND — an evolutionary active-space selection algorithm — against the standard NOON-MP2 baseline across three photochemically relevant molecules: ethylene, 1,3-butadiene, and fulvene. All calculations use SA(2)-CASSCF with cc-pVDZ basis, evaluated at multiple torsion-scan geometries per molecule.

The primary result is fulvene CAS(10,10): inZOR-ND identifies active spaces that achieve lower SA-CASSCF energy than NOON-MP2 at every single geometry in a 12-point torsion scan, with a mean advantage of 5.22 kcal/mol. Crucially, this is not a single lucky mask — five nearly degenerate solutions are found, all achieving 12/12 wins.

NOON-MP2 remains a strong, interpretable baseline. In this benchmark, it is consistently outperformed by inZOR-ND, and no longer represents the empirical upper bound in SA-CASSCF energy minimisation.

★ Fulvene CAS(10,10) — Primary Result

Figure 1. ΔE = E(inZOR-ND) − E(NOON-MP2) at each torsion geometry for fulvene CAS(10,10)/cc-pVDZ SA(2). All 12 bars are negative (green), confirming inZOR-ND wins at every geometry. Dashed purple line: mean Δ = −5.22 kcal/mol.

Figure 2. SA-CASSCF absolute energies per geometry. Green band: cluster of 5 nearly degenerate inZOR-ND solutions (all 12/12). Orange dashed: NOON-MP2. The inZOR-ND cluster lies consistently below NOON across the entire torsion scan.

Full 12-angle comparison table

Cluster of equivalent winning masks

Five masks achieve essentially the same energy profile (mean ≈ −230.533905 Ha, Δ ≈ −5.22 kcal/mol), all 12/12. This is not a singular coincidence — the solution is a cluster of nearly degenerate active spaces sharing the same occupied core [8, 14, 16, 19, 20] and virtual skeleton [37, 49, 68, 69], with MO 26 replaced by various alternatives:

Fulvene CAS(8,8) — Supporting Diagnostic

At the CAS(8,8) level — with slightly fewer active orbitals — inZOR-ND achieves the largest absolute mean advantage in this benchmark: +14.40 kcal/mol over NOON-MP2, winning at all 5 tested geometries. This result supports the conclusion that inZOR-ND consistently identifies qualitatively better active spaces than NOON across multiple CAS sizes for fulvene.

Fulvene CAS(6,6) — Smaller-Space Baseline

At the smaller CAS(6,6) level, inZOR-ND maintains a consistent mean advantage of 1.32 kcal/mol across 9 geometries with full convergence. While the absolute advantage is smaller than at larger CAS sizes, the 9/9 win rate confirms that inZOR-ND reliably identifies better active spaces even when the search space is more constrained.

Figure 5. inZOR-ND mean advantage (kcal/mol) across CAS sizes for fulvene. CAS(10,10) is the primary result (12/12 wins). CAS(8,8) shows the largest absolute advantage.

1,3-Butadiene — Supporting Result

Figure 4. Butadiene torsion scan: SA-CASSCF energies (top) and S₁−S₀ gap (bottom) for inZOR-ND vs NOON-MP2.

Ethylene — Supporting Result

Figure 3. Ethylene: SA-CASSCF energies (top) and S₁−S₀ gap (bottom) per geometry. inZOR-ND and NOON-MP2 are essentially equal in energy; inZOR-ND achieves a markedly smoother gap profile.

For ethylene, both inZOR-ND and NOON-MP2 achieve essentially identical SA-CASSCF energies (mean Δ ≈ 0.04 kcal/mol), with full convergence across all 16 geometries. The significant difference lies in the S₁−S₀ energy gap: inZOR-ND produces a notably smoother, physically more reasonable gap profile (σ = 0.79 eV vs 1.18 eV for NOON, max gap 2.79 vs 5.08 eV).

Convergence Summary

SA-CASSCF convergence (maxiter=200, PySCF) was achieved for all geometries across all systems:

* Butadiene: inZOR-ND has lower mean energy (−4.45 kcal/mol); per-geometry results are mixed (NOON wins at φ=78° near the conical intersection region).
** Ethylene: energies essentially equal (mean Δ ≈ 0.04 kcal/mol); gap profile significantly favours inZOR-ND (σ = 0.79 vs 1.18 eV).

Cross-Molecule Summary

Figure 6. inZOR-ND mean advantage over NOON-MP2 (kcal/mol) for all tested systems. Fulvene CAS(10,10) is the primary result (green, leftmost). CAS(8,8) shows the largest absolute advantage. All systems show positive advantage (inZOR-ND lower energy).

Interpretation

What this demonstrates

• NOON-MP2 remains a strong heuristic baseline. It is physically motivated (natural orbital occupation numbers from MP2), computationally cheap, and converges reliably. For routine applications it is a well-justified starting point.
• In these benchmarks, NOON-MP2 is consistently outperformed by inZOR-ND. The advantage ranges from marginal (ethylene, ~0 kcal/mol energy; qualitative improvement in gap) to substantial (butadiene, fulvene CAS(8,8), fulvene CAS(10,10)).
• NOON-MP2 is no longer the empirical upper bound in SA-CASSCF energy minimisation for these systems. Better active spaces exist and are systematically found by inZOR-ND.
• The fulvene CAS(10,10) result is robust. Five nearly degenerate masks (not one lucky selection) all achieve 12/12 wins, confirming this is a genuine property of the solution landscape, not a coincidence.
• The key orbital constraint matters. Including both MO 14 (important at τ=0°, τ=35°) and MO 19 (important at τ=30°, τ=40°) in the occupied set, while replacing the non-critical MO 26 with alternatives, unlocks the 12/12 regime.

What this does not claim

• This is not a claim that NOON-MP2 is wrong or obsolete as a general method.
• The results are for specific molecules and CAS choices; extrapolation to arbitrary systems requires further study.
• inZOR-ND is a search algorithm; its quality depends on the number of evaluations and the search parameters. Results may vary with different run lengths.

Protocol & Methods

Computational setup

• Software: PySCF (CASSCF), inZOR-ND evolutionary search
• SA-CASSCF: 2 states, weights [0.5, 0.5], maxiter=200
• Basis: cc-pVDZ throughout
• Convergence criterion: PySCF default (gradient norm)
• MO alignment: Procrustes rotation to reference geometry (τ=0°) for consistent orbital labelling across the scan
• NOON-MP2: Natural orbital occupation numbers from PySCF MP2 at τ=0° reference geometry; top-k occupied/virtual selected for CAS

inZOR-ND search

• Evolutionary population of organisms exploring a continuous [0,1]ⁿ position space mapped to orbital subsets via top-k selection
• Fitness function: mean SA-CASSCF energy (more negative = better), with convergence failure penalty
• For fulvene CAS(10,10): primary result obtained by systematic hybrid-mask search (single orbital swaps from best evolutionary candidates) followed by exhaustive virtual-orbital screening
• All candidates evaluated on all 12 geometries with parallelism across geometries (12 worker processes)

Comparison protocol

All masks (inZOR-ND and NOON-MP2) are evaluated through identical SACASFitness.evaluate() calls — same geometries, same SA-CASSCF setup, same Procrustes alignment. No preferential treatment is given to either method.