DOI: 10.5281/zenodo.18674750
title: Redox-Dependent Charge Redistribution and Correlation Energy in Molybdenum-Doped Cuprate Clusters — VQE Computational Data

description: 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