Nitrogen Organizes Electronic Structure in Diamond Defects: VQE Ensemble Data and Boundary Conditions of σ-Asymmetry Diagnostic

Title: Nitrogen Organizes Electronic Structure in Diamond Defects: VQE Ensemble Data and Boundary Conditions of σ-Asymmetry Diagnostic Authors: Brahmbhatt, Amit (Quantum Clarity LLC) Abstract This dataset reports Variational Quantum Eigensolver (VQE) ensemble results for two complementary investigations: (1) charge-state-dependent orbital organization in nitrogen-vacancy (NV⁻) and carbon-vacancy (C¹³V⁻) defects in diamond, and (2) geometric dissociation of molecular nitrogen (N₂) across four bond lengths spanning equilibrium to the multi-reference regime. Both employ the σ-asymmetry diagnostic — the inter-seed variance of VQE energy ensembles — as a probe of electronic landscape complexity. Together, the datasets establish the upper capability of the diagnostic under a controllable electronic perturbation in diamond defects, and its explicitly characterized boundary conditions under geometric perturbation. All calculations were performed on NVIDIA L40S GPU hardware (48 GB VRAM) using a 20-qubit UCCSD-like ansatz with sector-enforced spin and particle number constraints. Select NV center conditions were independently validated on IBM's 156-qubit Heron-2 processor (IBM_FEZ) at 96.4% circuit fidelity. Goal The overarching goal of this work is twofold. Goal 1 — Diamond Defect Systems: Determine whether nitrogen atoms in diamond vacancy defects function as passive structural substituents or as active electronic organizers — and whether this distinction is detectable via the σ-asymmetry diagnostic. The practical motivation is understanding why NV centers exhibit exceptional quantum coherence properties that make them leading candidates for quantum information hardware. Goal 2 — N₂ Dissociation: Establish the boundary conditions of σ-asymmetry by testing whether the diagnostic extends reliably from electronic topology perturbations (redox, charge, dopant substitution — validated in prior work) to purely geometric perturbations (bond stretch). N₂ was chosen because its dissociation curve spans a well-characterized transition from single-reference to strongly multi-reference character, providing a controlled and theoretically well-understood test case. Methodology All calculations use the Quantum Clarity VQE engine, a custom quantum-classical hybrid framework built on the following stack: Molecular Structure and Hamiltonian Construction PySCF for molecular integral generation, Hartree-Fock reference, and active space partitioning via frozen-core approximation OpenFermion for second-quantized Hamiltonian construction and Jordan-Wigner transformation to qubit operators Active space: 10 electrons / 10 orbitals → 20 qubits for all systems Basis: 6-31G (N₂ and diamond cluster models) Gradient Chunking for GPU Memory Management Hamiltonian measurement is decomposed into manageable Pauli term batches evaluated sequentially, with gradients accumulated across chunks before parameter updates. This reduces peak VRAM from >44 GB to 33–40 GB while preserving mathematically exact variational gradients — enabling 20-qubit strongly correlated calculations on a single 48 GB GPU that would otherwise require HPC cluster resources. Ansatz and Optimization UCCSD-like ansatz with particle-conserving circuit structure Hartree-Fock initialization for all seeds Depth 6 (NV center systems and N₂ RegimeA) and depth 2 (N₂ RegimeB) for ansatz sensitivity comparison Adam-style gradient descent, learning rate 0.02, convergence threshold 10⁻⁶ Ha, patience 30 iterations Sector Enforcement via Penalty Terms Two physical constraints are enforced via penalty Hamiltonians added directly to the qubit operator: Particle number: λ_N · (N̂ − N_target)² with λ_N = 1.0 Ha Spin projection: λ_Sz · (Ŝz − Sz_target)² with λ_Sz = 1.0 Ha This confines optimization to the correct physical sector without requiring post-selection, which is critical for obtaining physically meaningful σ measurements. Energy Bookkeeping The identity offset (frozen core energy + nuclear repulsion) is stripped from the qubit Hamiltonian prior to optimization and restored post-convergence. The validity gate reference is the HF expectation value in qubit space (hf_qubit_energy), not the full molecular HF energy — these differ by tens of thousands of kcal/mol and are not comparable. Active Space Note — NV Center Systems The NV center calculations use a 12-electron / 10-orbital active space for all four systems (NV⁰, NV⁻, C¹³V, C¹³V⁻). The NV⁻ anion charge state is modeled through modification of the external Hamiltonian (effective field change from the additional charge), not by adding an electron directly into the active space. Both NV⁰ and NV⁻ therefore contain 12 active electrons; the electronic landscape difference between them arises from the Hamiltonian change induced by the anion charge environment. This is physically meaningful — it isolates the orbital reorganization effect of nitrogen from a simple electron-count change — but should be noted when comparing active-space electron counts across systems. σ-Asymmetry Protocol Fifteen independent VQE seeds are run per condition, each initialized from a distinct random parameter vector; fifteen was chosen because σ stabilized empirically beyond approximately 12 seeds. The primary metric is: σ_valid = std({E_i : gate_i ∈ {PASS, WARN}}) where PASS = VQE error ≤ 1.0 kcal/mol, WARN = ≤ 5.0 kcal/mol, relative to the qubit-space HF reference. Outlier removal uses robust median-based detection: |E_i − median| > 2.0 kcal/mol. Complementary metrics include inter-basin energy gap analysis, Sarle's bimodality coefficient, trapped fraction (seeds converging above the global minimum), and convergence speed statistics. Note: the σ signal is contingent on sufficient ansatz expressibility. Under-expressive circuits flatten σ across conditions, producing artificially low variance regardless of the underlying electronic complexity. The RegimeA/RegimeB comparison in the N₂ dataset illustrates this explicitly. Hardware Validation (NV Centers) Select NV center conditions were submitted to IBM_FEZ, a 156-qubit Heron-2 processor, via IBM Quantum services. Circuit fidelity was assessed via state tomography. The directional distinction between NV⁻ and C¹³V⁻ charge states — opposite signs of the Z₀Z₁ spin-spin correlator — was independently confirmed at 96.4% circuit fidelity, consistent with and supportive of the computational σ differential. Hardware validation confirms the sector-level physics (charge-state occupancy ordering); σ-asymmetry magnitude itself requires fault-tolerant hardware for direct measurement. Outcome Finding 1 — Directional Inversion in Diamond Defect Systems The σ-asymmetry diagnostic reveals a qualitatively distinct and opposite response to charge state between nitrogen-containing and nitrogen-free diamond vacancy defects. The full four-system σ hierarchy (spin-constrained, 10 seeds per system) is: System σ (kcal/mol) ⟨Sz⟩ Physical State C¹³V⁻ (anion control) 0.6384 +0.5 Vacancy + charge, no N NV⁰ (neutral) 0.3707 +1.0 N present, no extra charge C¹³V (neutral control) 0.3666 0.0 Bare vacancy baseline NV⁻ (anion) 0.3405 +1.0 N present + charge — smoothest The charge-state response is opposite depending on nitrogen presence: System Charge State Δσ (⁰ → ⁻) Interpretation NV center ⁰ → ⁻ −0.0302 kcal/mol σ decreases — N organizes charge C¹³V center ⁰ → ⁻ +0.2718 kcal/mol σ increases — vacancy cannot organize charge Adding negative charge to a nitrogen-free vacancy increases electronic complexity — the intuitive result for a system gaining electrons into partially degenerate orbitals. Adding negative charge to a nitrogen-containing vacancy decreases complexity — the opposite behavior. This directional inversion was reproduced on IBM_FEZ hardware at 96.4% circuit fidelity. The inversion reveals that nitrogen functions as an active orbital organizer: its directional lone-pair orbitals hybridize with vacancy states and break the degeneracy that would otherwise exist, converting a multi-basin optimization landscape into an ordered, low-σ state when charged. This represents a charge-state-controlled modulation of electronic landscape complexity within the diamond lattice — charge state serves as the knob, nitrogen provides the organizing mechanism, and σ is the readout. Finding 2 — Boundary Conditions of σ-Asymmetry under Geometric Perturbation The N₂ dissociation sweep (R = 1.1, 1.6, 2.0, 2.5 Å, RegimeA depth=6, 15 seeds per condition) yields a non-monotonic σ profile: R (Å) σ_valid (kcal/mol) Landscape Character 1.1 (near-equil.) 0.173 π/π* near-degeneracy at equilibrium 1.6 (stretched) 0.035 Single dominant basin — σ dip 2.0 (dissoc.) 0.305 Multi-reference onset — recovery 2.5 (dissoc.) 0.308 Strongly multi-reference plateau The elevated σ at R=1.1 Å reflects π/π* orbital near-degeneracy present at N₂ equilibrium — a well-known feature of its ground-state electronic structure that is distinct from the static correlation onset at R>1.6 Å. The σ dip at R=1.6 Å marks a transient single-reference window between these two correlation regimes. σ alone is therefore non-monotonic across the full dissociation curve. However, inter-basin gap analysis recovers the physical signal: the energy gap between competing basins collapses at R=1.6 Å (gap ratio 6.9×) and re-opens sharply at R=2.0 Å (gap ratio ~45×), consistent with known N₂ multi-reference onset chemistry. The σ plateau at R=2.0–2.5 Å coincides with the well-established onset of static correlation in N₂ dissociation. This establishes that σ alone is reliable for electronic topology perturbations (redox, charge, dopant) but requires the complementary metric ensemble — σ + inter-basin gap ratio + trapped fraction — for geometric perturbation regimes. Significance For Quantum Information and Sensing The directional inversion finding reframes the NV center not as a passive quantum register but as an actively regulated electronic system. Nitrogen's orbital organizing role — converting multi-basin electronic landscapes into ordered single-basin states upon charging — may be a key contributor to NV center quantum coherence. Fewer competing electronic configurations means fewer decoherence channels. This suggests a design principle: diamond defect qubits should be engineered to maximize the orbital organizing influence of adjacent heteroatoms, not merely to place vacancies at controlled locations. More broadly, the finding introduces the concept of a heteroatom-mediated electronic dial — a charge-state-controlled modulation of electronic landscape complexity within a solid-state lattice, achievable without moving parts or external fields beyond charge state modulation. For the σ-Asymmetry Diagnostic Framework The N₂ results complete the validation picture for the framework by explicitly characterizing where it works and where supplementary metrics are needed. A diagnostic method that knows its own limits is more scientifically valuable than one that claims universal applicability. The basin gap analysis developed here — measuring the energy separation between competing optimizer attractors rather than their variance — constitutes a reusable secondary diagnostic applicable to any VQE ensemble study. For Computational Quantum Chemistry The combination of these two datasets demonstrates that statistical VQE ensembles contain latent physical information well beyond ground state energy. The optimization landscape itself — its roughness, its basin structure, its sensitivity to charge state or geometry — is a physically meaningful observable that encodes electronic complexity invisible to single-run calculations. This repositions the VQE not merely as an energy calculator but as a landscape probe. Path Forward This dataset closes the experimental phase of the σ-asymmetry validation program. The path forward operates on three parallel tracks: Track 1 — Publication The companion methods paper (σ-Asymmetry as a Universal Diagnostic for Electronic Complexity in Strongly Correlated Quantum Systems) is in final preparation for submission to a peer-reviewed computational chemistry journal. This Zenodo dataset provides the underlying computational data for the N₂ and diamond defect sections of that paper. Track 2 — Patent The σ-asymmetry diagnostic protocol — specifically the use of inter-seed VQE variance as a complexity probe, the sector-enforced ensemble methodology, and the heteroatom-mediated electronic dial concept — is being prepared for provisional patent filing. The Zenodo timestamp provides prior art documentation. Track 3 — Application The immediate practical application is as a pre-screening tool for materials discovery: run a 15-seed VQE ensemble, compute σ, and use it to identify which candidate materials have the electronic complexity profile associated with superconductivity, catalytic activity, or quantum coherence — before committing to expensive experimental synthesis or deeper computational investigation. Longer term, the heteroatom-mediated electronic dial concept opens a design pathway for solid-state quantum hardware: engineering defect environments to maximize orbital organizing influence, creating controllable complexity switches in diamond and related wide-bandgap semiconductor lattices. Dataset Structure /NV_C13V_defects/ NV_minus/ # NV⁻ anion — 10 seeds, spin-constrained NV_neutral/ # NV⁰ neutral — 10 seeds, spin-constrained C13V_minus/ # C¹³V⁻ anion control — 10 seeds C13V_neutral/ # C¹³V neutral control — 10 seeds /N2_dissociation/ R1.1/ # Near-equilibrium — 15 seeds, RegimeA + RegimeB R1.6/ # Stretched — 15 seeds, RegimeA + RegimeB R2.0/ # Dissociation onset — 15 seeds, RegimeA R2.5/ # Strongly multi-reference — 15 seeds, RegimeA /analysis/ sigma_calculation.py basin_gap_analysis.py phase6_harvest.py Each seed directory contains: VQE stdout/stderr logs, final energy (Ha), full iteration trace, convergence diagnostics (σ_tail, ⟨N⟩, ⟨Sz⟩, dominant bitstring, p_dom), and PASS/WARN/FAIL gate classification. Related Datasets (This Series) Version Date Title DOI 1.0 Jan 24, 2026 VQE Benchmarks: Chatt Cycle Nitrogen Fixation Intermediates 10.5281/zenodo.18356899 1.1 Jan 27, 2026 Chemical Accuracy for High-Spin Tri-Iron Nitrogen Activation 10.5281/zenodo.18382689 1.2 Jan 30, 2026 Redox-Driven Electronic Structure Collapse in Fe₄N₂ Clusters 10.5281/zenodo.18434137 1.3 Feb 14, 2026 Redox-Controlled Modulation of Correlation Energy in TM Oxide Clusters 10.5281/zenodo.18643269 1.0 Feb 17, 2026 Redox-Dependent Charge Redistribution in Mo-Doped Cuprate Clusters 10.5281/zenodo.18674751 1.0 Feb 17, 2026 Electron-Sector Integrity in VQEs — Validation Tools 10.5281/zenodo.18674828 Keywords variational quantum eigensolver, VQE, NV center, diamond defect, nitrogen-vacancy, quantum defects, NV center physics, σ-asymmetry, electronic complexity, electronic structure, static correlation, N₂ dissociation, quantum coherence, multi-reference, optimization landscape, inter-seed variance, sector enforcement, GPU quantum chemistry, IBM quantum hardware, Heron-2, strongly correlated electrons, orbital organization, basin analysis License Creative Commons Attribution 4.0 International (CC BY 4.0) Dataset prepared: March 2026 Quantum Clarity LLC All calculations performed on Linux (NVIDIA L40S 48GB, 92-core Linux server) Hardware validation: IBM_FEZ 156-qubit Heron-2 processor via IBM Quantum services