Redox-Dependent Charge Redistribution and Correlation Energy in Molybdenum-Doped Cuprate Clusters — VQE Computational Data Project HELIOS — Quantum Clarity LLC Principal Investigator: Amit Brahmbhatt Date: February 2026 Companion Paper: "Redox-Dependent Charge Redistribution and Correlation Energy in Molybdenum-Doped Cuprate Clusters: A VQE Study" (manuscript submitted) Methods Paper: "Electron-Sector Validation for Variational Quantum Eigensolvers" Overview 1️⃣ The Scientific Problem Understanding how redox state reshapes electronic structure in transition-metal oxides remains a central problem in materials chemistry. In Mo-doped cuprate clusters, oxidation is often interpreted through simple ionic models in which charge removal is localized primarily on the nominal redox-active center. However, in strongly correlated, multi-metal systems, charge redistribution may instead involve collective reorganization of d-orbital occupations across the cluster. The specific question addressed here is: Does oxidation in Cu₅MoO₁₂ correspond to simple charge depletion at Mo, or does it induce non-local redistribution within the correlated d-manifold? Resolving this requires an electronic-structure treatment capable of capturing redox-dependent correlation effects within a controlled active space. 2️⃣ What We Did We performed active-space VQE simulations of Cu₅MoO₁₂ across three charge states (anion, neutral, cation) using: 10 spatial orbitals (20 qubits) UCCSD-like ansatz (depth 6) Jordan–Wigner mapping Statevector simulation on GPU Consistent basis and active space across charge states To ensure comparability of redox states, all solutions were validated for electron-sector purity (P(N=target) > 99%) using Hamming-weight analysis. The cation calculation employed a quadratic number penalty to enforce sector integrity. The penalty expectation value (⟨λ(N̂−13)²⟩ = 0.0008 Ha ≈ 0.49 kcal/mol) is well below chemical accuracy; reported energies reflect the electronic Hamiltonian to within numerical precision (methodology detailed in companion methods paper). This framework enables internally consistent comparison of: Correlation energy scaling Mulliken population trends Natural orbital occupation numbers (NOONs) Optimization landscape diagnostics 3️⃣ Principal Achievement We find that oxidation of Cu₅MoO₁₂ does not behave as a simple ionic removal of electron density from Mo-centered orbitals. Instead: Mo formal charge increases by +1.43 Mo d-electron count increases by +0.27 Charge density redistributes from Cu-dominated orbitals toward Mo-centered d orbitals Correlation energy magnitude increases strongly with oxidation This combination indicates that oxidation induces intra-manifold redistribution within the correlated d-system, rather than localized electron depletion. Within this system and active space, correlation magnitude tracks charge distribution geometry more closely than total d-electron count. In practical terms: These results indicate Mo acts as a charge-redistribution node within the cluster, rather than as an isolated redox site — a distinction with implications for interpreting Mo-doped cuprate superconductors. All reported trends are derived from sector-validated VQE solutions, ensuring that comparisons reflect physical redox behavior rather than electron-sector artifacts. System Molecule: Cu₅MoO₁₂ — a molybdenum-doped cuprate clusterCharge states: Anion (Cu₅MoO₁₂⁻), Neutral (Cu₅MoO₁₂), Cation (Cu₅MoO₁₂⁺)Geometry: 18 atoms — 1 Mo, 5 Cu, 12 OBasis set: LANL2DZ ECP for Mo and Cu; 6-31G for OActive space: 12 electrons in 10 spatial orbitals (20 qubits) for anion and neutral; 13 electrons for cation (sector-constrained)Frozen orbitals: 96 doubly-occupied core orbitals Method Algorithm: Variational Quantum Eigensolver (VQE)Ansatz: UCCSD-like, depth 6, ~534 variational parametersHamiltonian: Jordan-Wigner transformation, ~3100 Pauli terms after threshold filteringOptimizer: Adam, learning rate = 0.02, convergence criterion: 30 iterations without improvement (max 300)Mapping: Jordan-Wigner with parity adjustmentHardware: NVIDIA L40S GPU (48 GB VRAM), statevector simulationQuantum chemistry backend: PySCF (HF reference, Mulliken population analysis) Cation active space note: The cation VQE calculation uses a quadratic number penalty H′ = H + λ(N̂ − N_t)² with λ = 0.5 Ha and N_t = 13 to enforce electron-sector purity. The expectation value of the penalty term is ⟨λ(N̂−13)²⟩ = 0.0008 Ha (≈ 0.49 kcal/mol), well below chemical accuracy. Reported energies reflect the electronic Hamiltonian to within numerical precision. Full methodology is described in the companion methods paper (Paper B). Files Statevectors File Charge State Active e⁻ Sector Purity Description anion_statevector.npz Anion (Cu₅MoO₁₂⁻) 12 P(N=12) = 99.69% Converged VQE statevector, sector-pure neutral_statevector.npz Neutral (Cu₅MoO₁₂) 12 P(N=12) = 99.85% Converged VQE statevector, Basin A, sector-pure vqe_Cu5MoO12_cation_penalty_N13_lambda0.5_statevector.npz Cation (Cu₅MoO₁₂⁺) 13 P(N=13) = 99.84% Converged VQE statevector, sector-constrained (λ=0.5 Ha) All statevectors stored as complex128 NumPy arrays of dimension 2²⁰ = 1,048,576, normalized to unit norm. Key: statevector. Natural Orbital Occupation Numbers (NOONs) File Charge State Fractional NOONs Description anion_noons.npz Anion 1/10 spatial Spatial and spin-orbital NOONs from 1-RDM neutral_noons.npz Neutral (Basin A) 0/10 spatial Spatial and spin-orbital NOONs from 1-RDM cation_penalty_N13_lambda0.5_noons.npz Cation (constrained) 2/10 spatial Spatial and spin-orbital NOONs from 1-RDM NOON files contain: noons_spatial (10 values, range 0–2), noons_spinorb (20 values, range 0–1), nelec_trace (1-RDM trace = active electron count). Summary Data File Description vqe_diagnostics_summary.csv All key energetics, sector metrics, and Mulliken populations in tabular form Figures File Figure Description Figure2_Correlation_vs_Geometry.pdf Figure 2 Correlation energy vs Mo charge geometry scatter (PDF, publication quality) Figure2_Correlation_vs_Geometry.png Figure 2 Correlation energy vs Mo charge geometry scatter (PNG, 300 dpi) figure3_charge_redistribution.png Figure 3 Charge redistribution flow diagram across charge states (PNG, 300 dpi) Key Results Energetics (Sector-Pure Solutions) Charge State VQE Energy (Ha) HF Energy (Ha) Correlation (kcal/mol) σ_tail (kcal/mol) P(N=target) Anion (N=12) −1935.9091 −1935.9083 −0.18 ± 0.20 0.43 99.69% Neutral (N=12) −1935.9296 −1935.9062 −14.64 6.39 99.85% Cation (N=13)* −1935.8270 −1935.5969 −144.45† 36.0‡ 99.84% *Sector-constrained via quadratic number penalty (λ = 0.5 Ha); see companion methods paper.†Penalty expectation ⟨λ(N̂−13)²⟩ = 0.0008 Ha (≈ 0.49 kcal/mol); reported energy reflects the electronic Hamiltonian to within numerical precision.‡σ_tail evaluated from unconstrained trajectory to reflect intrinsic Hamiltonian landscape topology; sector enforcement does not materially alter the underlying electronic Hamiltonian curvature. Mulliken Population Analysis Absolute values are basis-set dependent; trends across charge states within the same basis are robust and constitute the primary evidence. All charge states were computed within an identical basis set (LANL2DZ/6-31G) and active-space protocol (10 spatial orbitals), enabling internally consistent cross-charge-state comparison. Active space dimension is identical across charge states, ensuring comparability of correlation energy trends. Charge State Mo Charge Mo d-electrons Cu Total Charge O Total Charge Anion −0.177 5.364 +3.250 −4.072 Neutral −0.281 4.305 +4.393 −4.112 Cation +1.148 4.570 +3.993 −4.141 Charge Redistribution (Δ vs Neutral) Process ΔMo Charge ΔMo d-electrons ΔCu Total ΔO Total Oxidation (→Cation) +1.429 +0.265 −0.400 −0.029 Reduction (→Anion) +0.104 +1.059 −1.143 +0.040 Key finding: Oxidation increases Mo d-electron density (+0.265), inconsistent with simple ionic oxidation. This suggests oxidation redistributes electron density within the active space rather than removing charge directly from Mo-centered orbitals — charge redistribution from Cu-dominated orbitals toward Mo-centered d-orbitals, with Mo becoming formally positive while retaining d-electron density. NOON Summary Charge State Fractional Spatial NOONs Character Anion 1/10 Near single-reference Neutral (Basin A) 0/10 Single-reference Cation (penalty-constrained) 2/10 Open-shell singlet Reproducing Results Loading a Statevector import numpy as np data = np.load('anion_statevector.npz', allow_pickle=True) psi = data['statevector'] # shape: (1, 1048576) complex128 psi = psi[0] / np.linalg.norm(psi[0]) # flatten and normalize Loading NOONs import numpy as np data = np.load('anion_noons.npz', allow_pickle=True) noons_spatial = data['noons_spatial'] # shape: (10,), range 0–2 noons_spinorb = data['noons_spinorb'] # shape: (20,), range 0–1 nelec = data['nelec_trace'] # float, active electron count Loading Summary CSV import pandas as pd df = pd.read_csv('vqe_diagnostics_summary.csv') Verifying Sector Purity (Hamming-Weight Distribution) import numpy as np def compute_sector_purity(psi, n_qubits=20): """Compute P(N) Hamming-weight distribution from statevector.""" probs = (psi.conj() * psi).real idx = np.arange(probs.size, dtype=np.uint32) # Fast popcount x = idx.copy() x = x - ((x >> 1) & 0x55555555) x = (x & 0x33333333) + ((x >> 2) & 0x33333333) popcount = (((x + (x >> 4)) & 0x0F0F0F0F) * 0x01010101) >> 24 return np.bincount(popcount.astype(np.int32), weights=probs, minlength=n_qubits+1) data = np.load('anion_statevector.npz', allow_pickle=True) psi = data['statevector'][0] psi = psi / np.linalg.norm(psi) P_N = compute_sector_purity(psi) print(f"P(N=12) = {P_N[12]:.4f}") # Expected: >0.99 Sector Validation Summary All three charge states used in chemistry comparisons are confirmed sector-pure: System Target N P(N=target) ⟨N⟩ (1-RDM) Purity Proxy Classification Anion 12 99.69% 11.999 0.006 SECTOR-PURE ✓ Neutral 12 99.85% 12.000 0.003 SECTOR-PURE ✓ Cation (penalty) 13 99.84% 13.000 0.003 SECTOR-PURE ✓ Purity proxy = 1 − Σ_N P(N)²; values < 0.01 indicate sector-pure solutions.Full sector validation methodology described in companion methods paper (Paper B). Dependencies Python ≥ 3.8 NumPy ≥ 1.20 PySCF (for Mulliken analysis reproduction) Pandas (for CSV loading) Citation If you use this dataset, please cite: This dataset:Brahmbhatt, A. (2026). Dataset: Redox-Dependent Charge Redistribution in Molybdenum-Doped Cuprate Clusters — VQE Computational Data [Data set]. Zenodo. https://doi.org/10.5281/zenodo.18674751 Companion methods dataset:Brahmbhatt, A. (2026). Electron-Sector Integrity in Variational Quantum Eigensolvers — Computational Data and Validation Tools [Data set]. Zenodo. https://doi.org/10.5281/zenodo.18674828 License Creative Commons Attribution 4.0 International (CC BY 4.0).You are free to share and adapt this material for any purpose, provided appropriate credit is given. Contact Amit BrahmbhattQuantum Clarity LLCquantum-clarity.com Dataset generated February 2026 | Project HELIOS | Version 1.0