Active-Space Selection — Chemically Meaningful & Properly Converged

1. Scope & methodological framing

Traditional active-space ranking often overweights a single scalar — the optimized SA-CASSCF energy. On the photochemically interesting systems studied here, that scalar can reward chemically poor orbital sets (deep-σ correlation, far virtuals, masks that omit the frontier orbitals altogether). The contribution of this report is a chemistry-consistent validation framework that filters those failure modes before any energy ranking is performed and redirects the search toward physically meaningful sub-spaces.

The benchmark covers three frontier-driven photochemical systems — formaldehyde, ethylene and 1,3-butadiene — with CAS(2,2) and CAS(4,4), 11 independent GA seeds and 47 SA-CASSCF runs in total. No fitness adjustment, no hand-picked active space, and no partial scan is used; every winner is screened against five independent conditions before being reported.

2. Executive summary

• CH₂O behaves as a sensitivity benchmark for small-space search: on CAS(2,2) over 30+ candidate MOs, the chemically correct n→π* manifold is identifiable and reproducible once the chemistry condition is enforced.
• C₂H₄ contains the most flagrant anti-chemical failure of the benchmark — seed 73's σ-only winner with zero HOMO/LUMO content — and the chemistry condition redirects it to the canonical π/π* set [6, 7, 8, 9].
• C₄H₆ beats the NOON-MP2 baseline by ≈ 16 kcal/mol on R_E,raw with a chemically cleaner π-canonical solution; here the NOON-MP2 mask itself is the chemistry outlier.

Figure 1. Headline result. Fraction of GA winners that satisfy the per-CAS frontier-orbital rule, with and without the chemistry-aware acceptance condition. Without the condition, energy-only ranking promotes anti-chemical winners on CH₂O and C₂H₄; with the condition, every accepted winner is chemistry-valid.

Practical workflow

Chemistry validity and convergence are checked first; energy ranking is the last step, not the first.

3. The five tests an active space must pass

Lowest energy alone is not enough. For these frontier-driven photochemical benchmarks, an active space is reported only when it satisfies all five of the following conditions, in this order. The first three are hard gates (yes/no decisions); the last two are continuous diagnostics that accompany every accepted winner.

Per-CAS frontier rule used by test 1

Test 1 acts as a pre-evaluation veto: candidates that fail it are not assigned an energy ranking and cannot win the GA step. It is not a soft penalty added to a score — it is a yes/no decision that prevents the optimizer from converging on a chemically empty solution because that solution happens to lower the SA-energy sum through σ-correlation.

4. Multi-basin reality & the limits of intra-active multistart

Why is a chemistry-aware acceptance condition necessary in the first place? Because the SA-CASSCF orbital-optimization landscape on these systems is genuinely multi-basin, and standard intra-active multistart cannot move between basins. This is a methodological observation, not an implementation detail: it is the reason a per-candidate energy ranking is unreliable on its own, and the reason the problem deserves a combinatorial exploration over candidate sub-spaces rather than a deeper local search inside one.

4.1 The butadiene basin-jump scan

Figure 2. SA(2)-CASSCF total energies along the C₂–C₃ torsion of 1,3-butadiene for three candidate CAS(4,4) active spaces, evaluated under identical multistart settings (M = 30 starts, σ_max = 3.0). Three distinct basins are visible; the largest single-geometry split is 28 mE_h ≈ 17.7 kcal/mol at φ = 86°. The shaded region marks the basin-jump zone where intra-active multistart cannot escape, even with aggressive rotation amplitude.

4.2 The geometric reason: the subspace-overlap diagnostic

Figure 3. Singular values of the cross-overlap matrix S = MO_A^T·S_AO·MO_B between the active orbitals of the two competing CAS(4,4) basins at φ = 86°. Three of four active dimensions are 95–99 % shared; the fourth is essentially orthogonal (89.85° principal angle). This is the geometric reason intra-active unitary rotation cannot bridge these basins — it would require crossing a direction that sits outside the 4-dimensional sub-space being rotated.

The combinatorial exploration over candidate active sub-spaces samples the basin landscape independently of intra-active orientation: each candidate mask is a fresh sample. This is what allows the framework to satisfy two conditions that would otherwise be in tension — chemistry validity and proper convergence — without resorting to either a hand-picked active space or to a single-objective energy minimization that ignores chemistry.

5. Formaldehyde (CH₂O) — search-sensitivity benchmark

Per-seed comparison (5 seeds)

Three of five energy-only winners contained MO 3 (HOMO−4 region) — a deep-σ correlation that lowers the SA-energy sum without describing the n→π* manifold of formaldehyde. After the chemistry-aware condition, every winner contains at least one MO in {6, 7, 8, 9}. With the literature NOON-MP2 mask [6, 9] seeded into the initial population, all 5 seeds converge to it and confirm R_E,raw ≈ −455.36 E_h with full SA-CASSCF convergence on every geometry.

6. Ethylene (C₂H₄) — flagship redirection

Per-seed comparison (3 seeds)

The two winners that the energy-only score promoted to the top — seed 73 with all MOs outside the frontier window, and seed 101 with only one MO inside — both collapse to the canonical π manifold [6, 7, 8, 9] = (HOMO−1, HOMO, LUMO, LUMO+1) under the chemistry-aware acceptance. Seed 42's pre-screen winner already contains both HOMO and LUMO and is preserved.

The number of distinct cross-seed equivalence classes for the ethylene torsion benchmark drops from 3 to 2. The remaining multimodality is between two chemistry-valid attractors — [6, 7, 8, 9] and [4, 7, 8, 18] — not between chemistry and anti-chemistry. Both contain HOMO and LUMO and have R_E,raw lower than the NOON-MP2 baseline mask [5, 7, 8, 22]; this is now an honest fitness improvement, not a numerical exploit.

7. 1,3-Butadiene (C₄H₆) — improvement over the NOON-MP2 baseline

Per-seed comparison (3 seeds)

The NOON-MP2 baseline for butadiene CAS(4,4) selects MO 19 (LUMO+4) instead of MO 16 (LUMO+1) as the second virtual — a known weakness of NOON ranking on multi-π manifolds. The chemistry-aware framework reproduces this finding without ever bypassing the chemistry condition: the winners that beat NOON-MP2 contain HOMO and LUMO themselves.

The reported result is therefore not "the lowest energy" — it is "the lowest energy among the chemically meaningful candidates". For butadiene CAS(4,4), the two coincide; the report flags that the NOON-MP2 mask is the chemistry outlier in this case, and the framework picks the reasonable π set.

This is also the centrepiece example of the multi-basin observation in §4: the same butadiene scan exhibits ≈ 18 kcal/mol single-geometry splits between basins that intra-active multistart cannot bridge. The π-canonic basin is reachable here only because the search proposes different masks, not different rotations of the same mask.

8. Convergence integrity across the three systems

Figure 4. SA(2)-CASSCF convergence on the accepted (chemistry-valid) winner of every seed, across the three molecules — 47 / 47 = 100 % geometry-level convergence.

No statistic in this report is taken on a partial scan. Every R_E,raw is the sum of S₀ over the full set of geometries; if any geometry of the scan failed to converge, that candidate carries the worst-case sentinel and cannot win the population step. This is the architectural requirement that prevents a single accidentally-converged garbage geometry from short-circuiting an entire ranking. The NaN- and Inf-propagation paths are covered by the internal regression suite (99 / 99 unit tests).

9. Key lessons

The framework's practical workflow (Search → Chemical Gates → Convergence Check → Rank by Energy) is summarized in the Executive Summary. The lessons below are the take-aways from running it on the three benchmark systems.

10. Protocol & what is reported

Method

• Electronic structure: PySCF · SCF (RHF) reference · SA(2)-CASSCF with equal weights [0.5, 0.5]
• Basis: cc-pVDZ on every system
• Geometry scans: R(C=O) for CH₂O · torsion for C₂H₄ and C₄H₆
• Selector: inZOR-ND evolutionary search over MO subset masks · 3–5 independent seeds per molecule (11 total)
• Acceptance conditions: mask validity, frozen-core, frontier-orbital window (per-CAS rule)

Reported quantities

• Per-seed winning mask, with explicit listing of which MOs sit inside the frontier window
• R_E,raw = Σ_g S₀(g) over the full scan
• S₀ second-derivative variance and largest S₁−S₀ gap jump (continuous diagnostics)
• SA-CASSCF convergence flag per geometry
• Cross-seed equivalence classes (different masks that converge to the same physical sub-space are reported as one class)

What is not reported

• No proprietary score is presented as the primary criterion; the chemistry-validity condition is structural and the smoothness diagnostics are physical.
• No claim is made that the framework supersedes any standard reference method (NOON, AVAS, hand-picked active spaces). Where the framework agrees with NOON-MP2, that is reported as agreement; where it disagrees (butadiene CAS(4,4)), the disagreement is documented with the underlying chemistry reason.
• No partially-converged scans are aggregated into the headline figures.

11. Conclusions

• The chemistry condition is non-redundant with energy ranking — 5 of 11 GA winners would have been anti-chemical without it; the other 6 are unaffected.
• Multi-basin behaviour is real — basins separated by orthogonal active directions and ≈ 18 kcal/mol on butadiene; intra-active multistart cannot bridge them, but combinatorial mask exploration can.
• Convergence is enforced architecturally — 47 / 47 SA-CASSCF runs converged on accepted candidates; non-converged geometries cannot win a sort.
• Energetic competitiveness is preserved or improved — on butadiene CAS(4,4) the GA winner beats NOON-MP2 by ≈ 16 kcal/mol on R_E,raw while still containing both HOMO and LUMO.