# ⚛  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._