Fluorescence

Fluorescence is built from the same EHT sTDA singlets as the UV-Vis absorption spectrum. It is a vertical emission model with a spontaneous-emission lineshape and an Einstein A radiative rate. It does not optimize the S1 geometry and it does not report a quantum yield.

Selection of the emitting state

Excitations are the occupied→virtual gaps of the EHT orbitals, including dark states. They are ordered by energy.

1. If the lowest singlet has oscillator strength (f) at or above dark_threshold (default (10^{-3})), it is the emitter. That is Kasha's rule, and kasha is true.
2. If that state is dark, the lowest brighter singlet is used and kasha is false. The note records the departure.
3. If every singlet in the window is dark, the call fails. No band is invented.

Lineshape

Absorption weights each band by (f). Emission does not reuse that envelope. The relative intensity is

[ I(E) \propto n^2 E^3 |\mu|^2 g(E - E_\text{em}) ]

with (|\mu|^2 \propto f / \Delta E_\text{abs}) from (f = \tfrac{2}{3}\Delta E|\mu|^2). (g) is the same Gaussian or Lorentzian used for UV-Vis. The curve is scaled so its maximum is 1.

The emission energy is the vertical gap of the emitter minus stokes_ev. The default is 0: vertical emission, no excited-state relaxation. A positive stokes_ev is a shift you supply. It is not computed from an S1 optimization.

Radiative rate

The Einstein A coefficient, with (\tilde\nu) in cm⁻¹, is

[ A = 0.667025, n^2 \tilde\nu^2 f \quad (\mathrm{s}^{-1}) ]

radiative_lifetime_ns is (10^9 / A). For (f = 1), (E = 3.1,\mathrm{eV}) and (n = 1), the lifetime is about 2.4 ns. The fluorescence quantum yield is not reported, because non-radiative decay is not computed.

API

Rust:

use sci_form::{compute_fluorescence, spectroscopy::FluorescenceConfig};

let mut config = FluorescenceConfig::default();
config.stokes_ev = 0.2;
let spec = compute_fluorescence(&elements, &positions, config)?;

Python: fluorescence_spectrum(elements, coords, stokes_ev=0.2).

WASM: compute_fluorescence(elements_json, coords_json, sigma, e_min, e_max, n_points, stokes_ev, refractive_index, broadening).

What an organometallic spectrum is

Passing a 3D organometallic structure returns a spectrum when EHT can build the orbitals. That spectrum is a screening estimate.

• UV-Vis and fluorescence use minimal-basis EHT. Literature parameters cover the d-block already in the table. Every other metal through Z = 103 uses a hydrogenic fallback: H_ii is the first ionization potential and ζ = sqrt(IP / 13.6 eV). Lanthanides and actinides keep the f shell in the core (valence ns/np/(n−1)d), because the overlap is s/p/d. Shells with n = 6 and n = 7 reuse the STO-3G contraction of n = 5, scaled by (5/n)². Odd-electron molecules keep the SOMO occupied. None of this is TD-DFT.
• NMR shifts exist for the NMR-active isotopes in the nucleus registry, including the metal catalog. Cerium is omitted: every natural Ce isotope has (I = 0). Shifts other than ¹H/¹³C are relative screening values.
• IR frequencies use atomic masses through (Z = 103). The Hessian still needs a method that can evaluate the energy of those elements (eht, pm3, xtb, or uff).