L1 ⚛ L1-294 ⊙ Testnet

Kohn-Sham Density Functional Theory

Computational Chemistry · Ground-state electronic structure

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⬇ L1-294.md

# ⚛ L1 Principle — Kohn-Sham Density Functional Theory **ID:** `L1-294` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Computational Chemistry — *Ground-state electronic structure* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** kohn sham orbitals > **📡 Carrier:** electron · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554113 --- ## 🧠 1. Introduction **Kohn-Sham Density Functional Theory** is a **nonlinear inverse problem** whose unknown lives in **kohn sham orbitals** space, within the **Ground-state electronic structure** sub-domain of **Computational Chemistry**. Measurements consist of electrons collected by an electron detector via a **quantum chemistry benchmark** sensing mechanism. The forward operator applies, in order: E · potential ext operator; E · hartree xc operator; computes eigen-pairs of a linear operator; O · density energy operator. Observations are corrupted by additive Gaussian noise. Well-posed (Hohenberg-Kohn theorem); XC functional approximation is dominant error source. ## ⚙ 2. Forward Model Physical chain: **x** → E · potential ext → E · hartree xc → O · density energy → **y** (detector). ``` y = `O.density_energy` `E.hartree_xc` `E.potential_ext` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.potential_ext` | E · potential ext operator | | `E.hartree_xc` | E · hartree xc operator | | `O.density_energy` | O · density energy operator | **🛠 Solver components** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `E.eigensolve` | Computes eigen-pairs of a linear operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Computational Chemistry | | Sub domain | Ground-state electronic structure | | Carrier | electron | | Problem class | nonlinear_inverse | | Solution space | kohn_sham_orbitals | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 5 | | L dag | 3.3 | ## 📡 4. Measurement Model Well-posed (Hohenberg-Kohn theorem); XC functional approximation is dominant error source. | Metric | Value | |---|---| | Metric | total_energy_error_mHa | | Secondary | force_L2_error_eV_per_A | ## 📏 5. Operating Range (Ω) **Center problem class:** `density_functional_theory` · **Forward operator:** `dft_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | Xc | — | PBE | | Ecut ev | — | 400 | | N basis | — | 1000 | | N kpoints | — | 100 | | N electrons | — | 20 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | Xc | — | LDA, PBE, B3LYP, HSE06 | | Ecut ev | — | 100 – 2000 | | N basis | — | 100 – 1000000 | | N kpoints | — | 1 – 100000 | | N electrons | — | 2 – 10000 | ## 🎯 6. Tolerance (ε) **Center tolerance:** E error <= 1.5 mHa (~kJ/mol) | Metric | Range | |---|---| | Total energy error mha | 0.5 – 50 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **total_energy_error_mHa**, with κ = `600` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x1f70b327655c0d3a0cd23c77b104714cdf889ce9bfd329bd69f9536723feb083` - **Chain tx hash:** `0x8c1b577419fc5db696288a5e0dde41d695671830b20c74919b2097a994344112` - **Chain block:** `41554113` --- ## File Mapping This bundle consists of: `L1-294.md`, `L1-294.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-294.md` | Source of truth — edit this | Human or LLM | | `L1-294.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-294`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-294.md).
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📋 JSON metadata

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  "chain_block": 41554113,
  "chain_hash": "0x1f70b327655c0d3a0cd23c77b104714cdf889ce9bfd329bd69f9536723feb083",
  "chain_tx_hash": "0x8c1b577419fc5db696288a5e0dde41d695671830b20c74919b2097a994344112",
  "domain": "Computational Chemistry",
  "hardness_fn": {
    "delta": 5,
    "kappa": 600,
    "metric": "total_energy_error_mHa",
    "type": "epsilon_fn"
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      "name": "primary",
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  "observable_profile": {
    "metric": "total_energy_error_mHa",
    "regime": "Well-posed (Hohenberg-Kohn theorem); XC functional approximation is dominant error source.",
    "secondary": "force_L2_error_eV_per_A"
  },
  "physics_fingerprint": {
    "L_DAG": 3.3,
    "carrier": "electron",
    "difficulty_delta": 5,
    "domain": "Computational Chemistry",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "E.potential_ext",
      "E.hartree_xc",
      "E.eigensolve",
      "O.density_energy"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "quantum_chemistry_benchmark",
    "solution_space": "kohn_sham_orbitals",
    "sub_domain": "Ground-state electronic structure",
    "title": "Kohn-Sham Density Functional Theory"
  },
  "size_tiers": {
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      "dft_forward"
    ],
    "allowed_omega_dimensions": [
      "N_electrons",
      "N_basis",
      "Ecut_eV",
      "xc",
      "N_kpoints"
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    "allowed_problem_classes": [
      "density_functional_theory"
    ],
    "center_spec": {
      "epsilon_fn_center": "E error \u003c= 1.5 mHa (~kJ/mol)",
      "forward_operator": "dft_forward",
      "input_format": "measurement_only",
      "omega": {
        "Ecut_eV": 400,
        "N_basis": 1000,
        "N_electrons": 20,
        "N_kpoints": 100,
        "xc": "PBE"
      },
      "problem_class": "density_functional_theory"
    },
    "epsilon_bounds": {
      "total_energy_error_mHa": [
        0.5,
        50
      ]
    },
    "omega_bounds": {
      "Ecut_eV": [
        100,
        2000
      ],
      "N_basis": [
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        1000000
      ],
      "N_electrons": [
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        10000
      ],
      "N_kpoints": [
        1,
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      ],
      "xc": [
        "LDA",
        "PBE",
        "B3LYP",
        "HSE06"
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Ground-state electronic structure",
  "title": "Kohn-Sham Density Functional Theory"
}

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