inZOR-ND: Active Space Selection for Cr₂ — 30M Search Space, 0.005% Coverage

Abstract

Cr₂ is one of the most challenging molecules in quantum chemistry: its bond order is disputed (1 to 6), multi-reference character is extreme, and the optimal active space remains debated. Using inZOR-ND with no chemical priors, we search for the optimal CASSCF(8,8) active space from C(36,8) = 30,260,340 candidates. inZOR-ND evaluates only 1,572 spaces (0.0052%) and discovers the optimal active space containing Cr 3d/4s orbitals, capturing 0.559 Eh of correlation energy. 10 degenerate optima are found. The search survives a near-extinction event (4 active candidates at step 14) and recovers to full capacity.

30.3M

Total C(36,8)

1,572

Evaluated

0.005%

Coverage

Degenerate optima

−0.559

Correlation (Eh)

51.7m

Wall time (14 cores)

1. Why Cr₂?

The chromium dimer is notorious in computational chemistry. Its ground state has a formal sextuple bond (3dσ + 3dπ×2 + 3dδ×2 + 4sσ), but the potential energy curve is extremely shallow and difficult to reproduce. Active space selection requires expert knowledge of transition metal d-orbital ordering — knowledge that inZOR-ND does not have.

Molecule	Basis	n_MO	k	Search Space	Coverage
N₂ (companion)	6-31G	18	6	18,564	2.91%
Cr₂ (this work)	STO-3G	36	8	30,260,340	0.005%

2. Results

2.1 Optimal Active Space

Property	Value
Best MOs	[8, 20, 21, 22, 24, 27, 33, 35]
E(CASSCF)	−2064.595189 Eh
E(HF)	−2064.036056 Eh
Correlation captured	−0.559 Eh
Converged	Yes
Active electrons	8
Active orbitals	8 (MOs 20–27 are 3d/4s region)

inZOR-ND finds the Cr 3d/4s active space from 30.26 million candidates, evaluating only 0.005%. Brute-force would require ~30 million CASSCF evaluations × ~5s each = ~5 years on 14 cores. inZOR-ND completes in 51.7 minutes.

Figure 1. Left: Cache growth over 150 steps — 1,572 unique CASSCF evaluations out of 30.26 million. Right: Population dynamics — near-extinction at step 14 (4 candidates surviving the hostile landscape) followed by full recovery to 40 candidates with saturated fitness.

2.2 Degeneracy — 10 Equivalent Optima

As with N₂, inZOR-ND discovers multiple degenerate optimal spaces — all achieving identical CASSCF energy. MOs 20, 22, 24, 27, 33, 35 appear in most sets (3d/4s manifold). The remaining 2 orbitals vary — symmetry-equivalent substitutions.

Figure 2. Left: Top 10 active spaces — degenerate optima with identical fitness. Right: Orbital frequency across top 10 solutions. MOs 20, 22, 24, 27, 33, 35 appear in most sets (core 3d/4s manifold), while remaining orbitals vary.

2.3 Search Dynamics

Near-extinction and recovery: At step 14, only 4 active candidates survived — most were eliminated because Cr₂'s hostile landscape (many non-converging CASSCF evaluations) exhausted their viability. The surviving candidates were well-tuned for this harsh environment, and by step 60 the search had fully recovered to 40 active candidates with saturated fitness. Natural selection at work.

3. Comparison: N₂ vs Cr₂

Property	N₂ (6-31G)	Cr₂ (STO-3G)
Search space	18,564	30,260,340
Evaluated	541 (2.91%)	1,572 (0.005%)
Correlation captured	0.464 Eh	0.559 Eh
Degenerate optima	10	10
Near-extinction event	No (stable growth)	Yes (4 candidates at step 14)
Wall time (14 cores)	20.7 min	51.7 min
inZOR-ND engine modified	No	No

4. Reproducibility

All results are fully reproducible from scratch.

Seed: 7 — fully deterministic
inZOR-ND engine: used without modification
Dependencies: PySCF 2.x, NumPy, Python 3.10+
Hardware: 14-core CPU, Linux

inZOR-ND: Active Space Selection for Cr₂ — The Most Controversial Diatomic