# ⚛  L1 Principle — Time-Dependent Density Functional Theory (TDDFT)

**ID:** `L1-297` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Chemistry — *Excitation spectra and dynamics*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** excitation spectrum
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554113

---

## 🧠 1. Introduction

**Time-Dependent Density Functional Theory (TDDFT)** is a **linear inverse problem** whose unknown lives in **excitation spectrum** space, within the **Excitation spectra and dynamics** sub-domain of **Computational Chemistry**.

Measurements consist of photons collected by an optical detector via a **uv vis spectroscopy** sensing mechanism.

The forward operator applies, in order: E · tddft kernel operator; computes eigen-pairs of a linear operator; O · excitations operator.

Observations are corrupted by additive Gaussian noise. Adiabatic approximation breaks down for double excitations and charge-transfer states.

## ⚙ 2. Forward Model

Physical chain: **x** → E · tddft kernel → O · excitations → **y** (detector).

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

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.tddft_kernel` | E · tddft kernel operator |
| `O.excitations` | O · excitations 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 | Excitation spectra and dynamics |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | excitation_spectrum |
| Noise model | gaussian |
| Integration axis | energy |
| Difficulty delta | 5 |
| L dag | 3.6 |

## 📡 4. Measurement Model

Adiabatic approximation breaks down for double excitations and charge-transfer states.

| Metric | Value |
|---|---|
| Metric | excitation_energy_error_eV |
| Secondary | oscillator_strength_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `time_dependent_dft` · **Forward operator:** `tddft_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Xc | — | B3LYP |
| N atoms | — | 20 |
| N basis | — | 500 |
| N states | — | 20 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Xc | — | B3LYP, CAM-B3LYP, wB97X-D |
| N atoms | — | 2 – 500 |
| N basis | — | 50 – 10000 |
| N states | — | 5 – 500 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** excitation E <= 0.15 eV

| Metric | Range |
|---|---|
| Excitation energy error ev | 0.03 – 1.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **excitation_energy_error_eV**, with κ = `700` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xc0380521533593c72ac2e3d09ca5a4ba111ba0237253bd2f65f470905a6e9f76`
- **Chain tx hash:** `0x8e7aecefa556914651be64a50ce5363c5666627f563d6869d7c3a4a0a092921a`
- **Chain block:** `41554113`

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## File Mapping

This bundle consists of: `L1-297.md`, `L1-297.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-297.md` | Source of truth — edit this | Human or LLM |
| `L1-297.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.
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