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