UV-Vis Spectroscopy (sTDA)

sci-form implements a simplified Tamm-Dancoff approximation (sTDA) for UV-Vis absorption spectra using EHT molecular orbital transitions with oscillator strength weighting and configurable broadening.

Overview

sTDA Pipeline

Theory

Molecular Orbital Transitions

Starting from a converged EHT calculation with $N_{\text{occ}}$ occupied and $N_{\text{virt}}$ virtual orbitals, all single-excitation transitions $i\to a$ are enumerated:

$$\mathrm{\Delta }{E}_{ia}={\epsilon }_a-{\epsilon }_i(\text{HOMO-LUMO and all sub-gap excitations})$$

The energy is converted to wavelength:

$${\lambda }_{ia}=\frac{hc}{\mathrm{\Delta }{E}_{ia}}=\frac{1239.84}{\mathrm{\Delta }{E}_{ia}[\text{eV}]}\text{nm}$$

Oscillator Strength

The oscillator strength $f_{ia}$ accounts for the transition probability via the electric dipole matrix element:

$$f_{ia}=\frac{2m_e\mathrm{\Delta }{E}_{ia}}{3{\hslash }^2}|⟨{\psi }_i|\mathbf{r}|{\psi }_a⟩{|}^2$$

In the sTDA approximation, the transition dipole moment is evaluated as a sum over orbital coefficients:

$$|{\mu }_{ia}{|}^2={|\sum _{\mu \nu }{C}_{\mu i}{C}_{\nu a}⟨{\varphi }_{\mu }|\mathbf{r}|{\varphi }_{\nu }⟩|}^2$$

Broadened Absorption Spectrum

The molar extinction coefficient $\epsilon (\lambda )$ is built by summing Gaussian or Lorentzian line shapes centred at each transition:

Gaussian broadening (default):

$$\epsilon (\lambda )=\sum _{ia}{f}_{ia}\cdot G(\lambda -{\lambda }_{ia},\sigma )$$$$G(x,\sigma )=\frac{1}{\sigma \sqrt{2\pi }}\exp (-\frac{x^2}{2{\sigma }^2})$$

Lorentzian broadening:

$$\epsilon (\lambda )=\sum _{ia}{f}_{ia}\cdot L(\lambda -{\lambda }_{ia},\gamma )$$$$L(x,\gamma )=\frac{\gamma /\pi }{x^2+{\gamma }^2}$$

Broadening Functions Compared

Parameters

Parameter	Type	Default	Description
sigma	f64	0.3	Broadening width in eV (Gaussian σ or Lorentzian γ)
e_min	f64	0.5	Minimum transition energy in eV
e_max	f64	12.0	Maximum transition energy in eV
n_points	usize	500	Number of grid points in the energy axis
broadening	BroadeningType	Gaussian	BroadeningType::Gaussian or BroadeningType::Lorentzian

Output Types

StdaUvVisSpectrum

Field	Type	Description
wavelengths_nm	Vec<f64>	Wavelength axis (nm)
intensities	Vec<f64>	ε(λ) — broadened extinction (arb. units)
excitations	Vec<StdaExcitation>	Individual sTDA excitations
homo_energy	f64	HOMO energy (eV)
lumo_energy	f64	LUMO energy (eV)
gap	f64	HOMO-LUMO gap (eV)
n_transitions	usize	Number of MO→MO transitions used
broadening_type	BroadeningType	Applied broadening type
notes	Vec<String>	Method caveats

StdaExcitation

Field	Type	Description
energy_ev	f64	Vertical excitation energy (eV)
wavelength_nm	f64	Corresponding wavelength (nm)
oscillator_strength	f64	Dimensionless oscillator strength $f$
from_mo	usize	Occupied MO index
to_mo	usize	Virtual MO index

API

Rust

use sci_form::{embed, compute_stda_uvvis};
use sci_form::reactivity::BroadeningType;

let conf = embed("c1ccc(cc1)C=O", 42);
let pos: Vec<[f64;3]> = conf.coords.chunks(3).map(|c| [c[0],c[1],c[2]]).collect();

let spectrum = compute_stda_uvvis(
    &conf.elements,
    &pos,
    0.3,             // sigma in eV
    1.0,             // e_min eV
    8.0,             // e_max eV
    500,             // n_points
    BroadeningType::Gaussian,
).unwrap();

println!("HOMO-LUMO gap: {:.2} eV", spectrum.gap);
println!("Strongest transition: {:.1} nm, f={:.4}",
    spectrum.excitations.iter()
        .max_by(|a,b| a.oscillator_strength.partial_cmp(&b.oscillator_strength).unwrap())
        .map(|e| e.wavelength_nm).unwrap_or(0.0),
    spectrum.excitations.iter()
        .map(|e| e.oscillator_strength).fold(0.0_f64, f64::max),
);

Python

from sci_form import stda_uvvis, embed

conf = embed("c1ccc(cc1)C=O", seed=42)
spec = stda_uvvis(conf.elements, conf.coords, sigma=0.3, e_min=1.0, e_max=8.0)
print(f"Gap: {spec.gap:.2f} eV, {spec.n_transitions} transitions")
print(f"λ_max ≈ {spec.wavelengths_nm[spec.intensities.index(max(spec.intensities))]:.1f} nm")

TypeScript/WASM

import init, { embed, compute_stda_uvvis } from 'sci-form-wasm';
await init();

const r = JSON.parse(embed('c1ccc(cc1)C=O', 42));
const elements = JSON.stringify(r.elements);
const coords   = JSON.stringify(r.coords);

const spec = JSON.parse(compute_stda_uvvis(elements, coords, 0.3, 1.0, 8.0, 500, 'gaussian'));
console.log(`Gap: ${spec.gap.toFixed(2)} eV`);
const maxIdx = spec.intensities.indexOf(Math.max(...spec.intensities));
console.log(`λ_max ≈ ${spec.wavelengths_nm[maxIdx].toFixed(1)} nm`);

Limitations

• Excitation energies from EHT are systematically blue-shifted compared to TDDFT or experimental values; use for trends and fingerprinting, not absolute accuracy.
• Transition dipole moments are computed in the minimal STO basis; heavy elements may have reduced accuracy.
• No solvent effects are included.
• For transition-metal chromophores, GFN-xTB MOs are used automatically when EHT unsupported elements are detected.