# ⚛  L1 Principle — Semi-Empirical Quantum Chemistry (AM1, PM6, DFTB, xTB)

**ID:** `L1-304` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Chemistry — *Parameterized approximate QM*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** semi empirical wavefunction
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41554114

---

## 🧠 1. Introduction

**Semi-Empirical Quantum Chemistry (AM1, PM6, DFTB, xTB)** is a **nonlinear inverse problem** whose unknown lives in **semi empirical wavefunction** space, within the **Parameterized approximate QM** sub-domain of **Computational Chemistry**.

Measurements consist of electrons collected by an electron detector via a **property benchmark** sensing mechanism.

The forward operator applies, in order: E · parameterized fock operator; a fixed-point or gradient iteration on the unknown; O · observables operator.

Observations are corrupted by additive Gaussian noise. Well-posed once parameterized; out-of-domain systems fail silently.

## ⚙ 2. Forward Model

Physical chain: **x** → E · parameterized fock → O · observables → **y** (detector).

```
y = `O.observables` `E.parameterized_fock` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.parameterized_fock` | E · parameterized fock operator |
| `O.observables` | O · observables operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.iter` | A fixed-point or gradient iteration on the unknown |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Chemistry |
| Sub domain | Parameterized approximate QM |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | semi_empirical_wavefunction |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Well-posed once parameterized; out-of-domain systems fail silently.

| Metric | Value |
|---|---|
| Metric | energy_error_mHa |
| Secondary | geometry_RMSD_A |

## 📏 5. Operating Range (Ω)

**Center problem class:** `semiempirical_qc` · **Forward operator:** `semiempirical_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Method | — | GFN2-xTB |
| N atoms | — | 100 |
| N basis | — | 500 |
| Dispersion | — | D4 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Method | — | AM1, PM6, PM7, DFTB3, GFN1-xTB, GFN2-xTB |
| N atoms | — | 2 – 100000 |
| N basis | — | 10 – 500000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** E error <= 3 mHa (~2 kcal/mol)

| Metric | Range |
|---|---|
| Energy error mha | 1.0 – 50 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **energy_error_mHa**, with κ = `200` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x7c26b92831516ade7766f5e9090df0136f213203573091666787806fcfea6cc2`
- **Chain tx hash:** `0xbe564926d5d03d4f5a7299f3dd8ef5cbda7dbd66be0e2d57d7493b86f5136310`
- **Chain block:** `41554114`

---

## File Mapping

This bundle consists of: `L1-304.md`, `L1-304.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-304.md` | Source of truth — edit this | Human or LLM |
| `L1-304.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 `L1-304`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._