⚛️ Charge Redistribution in Molybdenum-Doped Cuprate Clusters
Removing an electron from Cu₅MoO₁₂ causes charge density to redistribute toward Mo despite making it more positive. Our sector-validated VQE simulations indicate that Mo functions as a charge-redistribution node mediating electron flow across the cluster geometry.
Two Discoveries, One System: Chemistry + Methodology
🔬 Discovery A: Mo as Charge-Redistribution Node
System: Cu₅MoO₁₂ (anion, neutral, cation) | 20 qubits | Sector-validated VQE
Oxidation and Non-Local Charge Redistribution
Simple ionic oxidation models predict that oxidizing a transition metal complex removes electrons from the metal center, decreasing its d-orbital occupancy. For Mo-doped cuprate clusters, our sector-validated VQE simulations indicate non-local charge redistribution inconsistent with a localized ionic model.
| Charge State | Mo Charge | Mo d-electrons | Interpretation |
|---|---|---|---|
| Anion (Cu₅MoO₁₂⁻) | -0.18 | 5.36 | Reduced, closed-shell |
| Neutral (Cu₅MoO₁₂) | -0.28 | 4.31 | Reference state |
| Cation (Cu₅MoO₁₂⁺) | +1.15 | 4.57 | More positive, MORE d-electrons ⭐ |
Charge Redistribution Mechanism
Oxidation (neutral → cation):
Mo formal charge: +1.43 (becomes strongly positive)
Mo d-electron density: +0.27 electrons gained
Cu total charge: -0.40 (Cu donates electrons)
Mechanism: Cu → Mo charge donation compensates Mo electrostatic deficit arising from oxidation. Mo functions as a charge-redistribution node mediating electron flow across the cluster geometry, inconsistent with a simple localized ionic oxidation model.
Correlation Energy Tracks Charge Polarization and Open-Shell Character
Within this system and active space, correlation magnitude tracks charge distribution geometry and open-shell character more closely than total d-electron count.
| Charge State | Mo d-electrons | Mo Charge | Correlation Energy |
|---|---|---|---|
| Anion | 5.36 | -0.18 | -0.18 kcal/mol |
| Neutral | 4.31 | -0.28 | -14.64 kcal/mol |
| Cation | 4.57 | +1.15 | -144.45 kcal/mol |
The anion has the most d-electrons but the least correlation. The cation has intermediate d-electrons but correlation energy of −144.45 kcal/mol compared to −0.18 kcal/mol for the anion. The distinguishing factor is Mo charge polarization (+1.15 vs -0.18): strongly positive Mo creates highly asymmetric charge topology that drives open-shell multi-reference character.
Optimization Landscape Roughness: 84-Fold Amplification
Optimization landscape roughness (σ_tail) shows systematic correlation with HOMO-LUMO gap reduction, Mo charge polarization, and correlation energy across all three charge states.
| System | σ_tail (kcal/mol) | HOMO-LUMO Gap (eV) | Character |
|---|---|---|---|
| Anion | 0.43 | 4.37 | Ultra-smooth, closed-shell |
| Neutral | 6.39 | 4.29 | Smooth, moderate |
| Cation | 36.0 | 3.84 | Rough, open-shell singlet |
Implications for Cuprate Dopant Screening
Mo's charge-redistribution behavior is not predicted by simple ionic models of oxidation state change. This framework — combining VQE energetics, Mulliken population analysis, and landscape diagnostics — provides a template for systematic dopant screening in cuprate superconductors.
Key takeaway: For transition metal dopants in ionic cluster environments, charge distribution geometry — rather than occupation numbers alone — governs the accessibility of correlated electronic states. These results are specific to the selected cluster model and active-space protocol, but provide internally consistent insight into redox-driven electronic restructuring in Mo-doped cuprate motifs.
⚙️ Discovery B: VQE Sector Validation Framework
Methodology contribution: System-agnostic detect→fix→verify workflow
A Common but Under-Quantified Failure Mode
Non-number-conserving VQE ansätze (e.g., UCCSD-type parameterizations on parity-transformed Hamiltonians) can converge to solutions exhibiting electron-sector mixing without triggering standard convergence diagnostics. Convergence criteria — energy tolerance, gradient norms — are blind to sector composition.
Case Study: Cu₅MoO₁₂ Cation (Unconstrained)
Target electron count: N = 13
Actual sector composition: P(N=13) = 86.69%
Adjacent sector contamination: P(N=12) = 7.34%, P(N=14) = 5.67%
Classification: SECTOR-MIXED ✗ (not representative of the intended electron-number sector)
Energy bias: -2.91 kcal/mol non-physical variational lowering (exceeds chemical accuracy)
Detect → Fix → Verify Workflow
Hamming-Weight Distribution
Compute P(N) — probability distribution over electron numbers — from final statevector. O(2ⁿ) scan, <0.1% VQE runtime overhead.
P(N=target) > 99%? → PURE ✓
P(N=target) < 90%? → MIXED ✗
Quadratic Number Penalty
Augment Hamiltonian:
H′ = H + λ(N̂ - N_target)²
Penalty strength: λ = 0.5 Ha
P(N=13): 86.69% → 99.84%
Purity: 80× improvement
Post-Penalty Validation
Re-run sector metrics + NOON analysis. Penalty expectation ⟨λ(N̂−13)²⟩ = 0.0008 Ha (0.49 kcal/mol) — below chemical accuracy.
Fractional NOONs: 4 → 2
(2 were sector artifacts)
Sector Validation Checklist for Publication-Quality VQE
| Metric | Threshold | Classification |
|---|---|---|
| P(N=target) | > 99% | SECTOR-PURE ✓ |
| P(N=target) | 90–99% | N-DOMINANT ⚠ |
| P(N=target) | < 90% | SECTOR-MIXED ✗ |
| Purity proxy (1 − P(N)) | < 0.01 | Essentially pure |
| ⟨N⟩(1-RDM) vs ⟨N⟩(Hamming) | Δ < 0.01 | Consistent electron count |
Proposed Standard
Hamming-weight distribution analysis is computationally inexpensive (<0.1% of VQE runtime). We recommend it as a low-overhead best practice for non-number-conserving VQE prior to publication of energetics or orbital diagnostics. The detect→fix→verify workflow is system-agnostic and applicable to any VQE implementation using parity-transformed Hamiltonians, Jordan-Wigner qubit reduction, or hardware-efficient ansätze.
⚙️ Technical Infrastructure
GPU-Accelerated VQE Implementation
These discoveries required building custom VQE infrastructure specifically designed for transition metal oxide clusters with strong correlation and open-shell character.
1. Statevector-Based Sector Monitoring
Real-time Hamming-weight monitoring during optimization. Automatic detection of sector drift before energetic comparisons are made. This is integrated into the VQE loop, not a post-processing afterthought.
2. Adaptive Number-Penalty Hamiltonian Construction
Dynamic λ selection based on sector drift magnitude. Efficient sparse Hamiltonian augmentation that preserves gradient flow. Penalty expectation tracking ensures reported energies reflect the physical Hamiltonian.
3. Multi-Seed Robustness Validation
Every charge state validated across 5 independent random seeds. Sector purity, NOON structure, and energy convergence tracked systematically. No single-run claims — all results statistically validated.
4. CASCI Benchmarking Integration
All VQE energies benchmarked against CASCI (Complete Active Space Configuration Interaction) within the same active space. Chemical accuracy validation (≤1 kcal/mol) confirmed before publication.
Applicability Beyond This System
This infrastructure enables systematic exploration and validation of quantum complexity in chemically realistic transition metal systems: CO₂ reduction catalysts, water oxidation clusters, organometallic complexes, battery cathode materials. The methodology is applicable to any strongly correlated system requiring sector-validated VQE solutions.
📊 Campaign Metrics
📖 Access Complete Research Data
Two published datasets + validation scripts (open data)
Status: Manuscripts currently in preparation and submission.