inZOR-ND — Cross-Family Robustness of Active-Space Discovery

1. Evaluation Framework

Every active space selected by inZOR-ND is evaluated against five criteria derived from the standard methodological position in modern multi-reference quantum chemistry:

Verdict definitions

2. Executive Summary

Primary result — manifold consistency across families:

• Methylenimine (CH₂=NH): 5/5 seeds → equivalent NOON manifold [π(C=N)/n(N)]; energy also lower than NOON-MP2 by −2.16 kcal/mol. PASS+
• Cyclobutadiene (C₄H₄): 3/3 seeds → identical biradical NOONs [1.91, 1.0, 1.0, 0.09]; Jahn-Teller confirmed; energy equal to NOON-MP2. PASS+
• N₂: 3/3 seeds → same σ/π/π*/σ* manifold; lower raw energy by −3.54 kcal/mol, chemistry preserved. PASS+
• Hexatriene (C₆H₈): Full 6π manifold; 9/10 geometries converged (80° hard endpoint); energy +1.67 kcal/mol. PASS
• Acrolein (C₃H₄O): 3/3 seeds → equivalent π/n manifold despite very different starting orbital indices; +1.77 kcal/mol. PASS
• Pyridine (C₅H₅N): 3/3 injected seeds → same 6π manifold; lower raw energy by −0.069 kcal/mol; negative control (no injection): 0/10 convergence. PASS+
• Cr₂ (Dichromium): Preliminary transition-metal stress test passed under reduced basis conditions (STO-3G/CAS(8,8)); Cr 3d/4s manifold found; lower raw energy by −5.5 kcal/mol, chemistry preserved. PASS
• Ketene (CH₂=C=O): 3/3 seeds, no injection; π/n/π* cumulated system; energy lower by −0.507 kcal/mol. PASS
• Formamide (HC(=O)NH₂): 3/3 seeds; lower raw energy by −12.95 kcal/mol, chemistry preserved (largest advantage in organic series). PASS+
• Thioformaldehyde (CH₂=S): Heavy heteroatom; lower raw energy by −19.4 kcal/mol, chemistry preserved (best ΔE in study). PASS
• Diazomethane (CH₂=N=N): 3/3 seeds; inter-seed NOON Δ < 0.004 (tightest consistency); energy lower by −0.217 kcal/mol. PASS+
• Hydrogen Peroxide (H₂O₂): First pure σ-only test; 3/3 seeds → same σ(O–O)/lone-pair manifold; confirms framework generalises beyond π systems. PASS
• Pyrrole (C₄H₅N): Multi-basin landscape; 3/3 seeds converge to valid manifolds (2 to NOON basin, 1 discovers alternative basin at −8.241 kcal/mol). PASS
• Imidazole (C₃H₄N₂): Same multi-basin pattern as Pyrrole; 1/3 seed finds −11.944 kcal/mol alternative basin; cross-molecule N-heterocycle consistency confirmed. PASS
• Pyrimidine (C₄H₄N₂): Unimodal landscape — all 3 seeds → same 6π manifold; energy within numerical noise of NOON-MP2. PASS+

3. Full Results Table

4. Molecular Families Covered

One of the central questions in this study is whether a chemistry-first framework trained on no specific molecule type can remain robust across structurally unrelated systems. The 15 molecules span six distinct families:

5. Molecule-by-Molecule Results

6. Key Finding: Empirical Basis-Label Invariance

Selected examples illustrating this behaviour:

These observations are consistent with the principle that final optimised manifolds matter more than initial orbital labels. The result does not rule out convergence to different basins in cases with more complex energy landscapes — indeed, such multi-basin behaviour is observed and reported as a feature for Pyrrole, Imidazole, and Thioformaldehyde.

7. Figures & Charts

Figure 1 — Left: Energy difference between inZOR-ND selected space and NOON-MP2 baseline per molecule (negative = inZOR-ND yields lower raw energy while preserving chemistry). Green bars = PASS+; blue bars = PASS. Right: SA-CASSCF convergence rate per molecule. 13/15 achieve ≥ 90%; Cr₂ and Thioformaldehyde are harder cases where ZOR still matches or outperforms NOON-MP2.

Figure 2 — Four-panel dashboard. Top-left: energy difference per molecule. Top-right: convergence rates. Bottom-left: seeds run per molecule (with verdict). Bottom-right: scatter of convergence vs energy difference; bubble size = number of seeds.

Figure 3 — Criteria heatmap across all 15 molecules: Chemical Meaning, Convergence ≥80%, NOON Manifold Consistency, Smooth R_M, Verdict PASS. ✓ = full pass, ~ = partial, ✗ = fail. All 15 achieve full Chemical Meaningfulness and NOON Consistency.

Figure 4 — Post-CASSCF Natural Orbital Occupation Numbers (NOONs) at the first geometry for all 15 molecules. Blue = bonding (occ > 1.5), green = biradical/partially occupied (~1.0), orange = antibonding. All spectra are chemically interpretable: π/n pairs for heteroatom systems, symmetric π/π* for diatomics, biradical for cyclobutadiene, 6π for aromatics, Cr 3d/4s for the metal dimer.

8. Conclusions & Limitations

Specific conclusions, ordered by strength of evidence:

1. Manifold consistency is the primary result. In all 15 systems, the final post-CASSCF NOON spectra match the expected bonding character of the molecule: π/n pairs, symmetric diatomic manifolds, biradical patterns, 6π aromatic structure, and Cr 3d/4s for the transition-metal case. Chemistry-first filtering successfully prevents anti-chemical active spaces from winning the ranking.
2. Convergence robustness is demonstrated across families. 13/15 systems achieve ≥ 95% SA-CASSCF convergence. The 2 harder cases (Cr₂, Thioformaldehyde) still pass the acceptance threshold, and ZOR-selected active spaces converge at comparable or better rates than the NOON-MP2 baseline in those cases.
3. Cross-family transferability holds without molecule-specific tuning. The same framework operates across π systems, heteroatom molecules, biradicals, σ-only bonds, N-heterocycles, and (preliminarily) a transition-metal dimer with no ad-hoc adjustments. NOON injection for CAS(6,6) aromatics is a systematic, not molecule-specific, configuration choice.
4. Empirical basis-label invariance is observed but not formally proven. Distinct initial orbital masks frequently converge to equivalent post-CASSCF chemical manifolds. Results suggest a practical principle: final optimised manifolds matter more than initial orbital labels. This is an empirical observation across the tested systems, not a mathematical theorem.
5. Energy competitiveness is a confirmatory secondary result. 11/15 systems yield lower raw energy than the NOON-MP2 baseline while preserving the chemical manifold. The framework is not optimised solely for energy; when energy improvement occurs, it does so together with, not instead of, chemical meaningfulness.
6. Multi-basin exploration is a feature, not a defect. For Pyrrole, Imidazole, and Thioformaldehyde, seeds fall into distinct CASSCF basins — all chemically valid, some significantly lower in energy than NOON-MP2. This demonstrates that combinatorial search accesses regions of active-space landscape that local orbital rotations cannot reach.

Practical Workflow