Electronic Excitations and UV–Visible Spectra

An electronic absorption spectrum is assembled from transitions between many-electron states, not from the difference between two frontier-orbital energies. Each transition has an excitation energy and a transition probability; an experimental band then reflects those discrete transitions together with nuclear motion, solvent, temperature, conformational sampling, finite lifetime, and instrumental response.

Tako calculates approximate vertical electronic excitations with an sTDA-xTB model and constructs a Gaussian-folded curve from their oscillator strengths. The result is intended for rapid molecular screening and qualitative assignment. It is not a full linear-response TD-DFT calculation, it does not propagate excited-state dynamics, and the smooth curve is not a prediction of an experimental line shape.

Absorption as a transition between electronic states

Within the Born–Oppenheimer picture, an electronic state is solved at fixed nuclear coordinates $\mathbf R$. Absorption of a photon with angular frequency $\omega$ is resonant when

where $E_0$ and $E_I$ are the ground- and excited-state electronic energies. Because electronic motion is much faster than nuclear rearrangement, a conventional absorption calculation evaluates both states at the same geometry. The transition is therefore vertical on a potential-energy diagram. It is not an adiabatic gap between separately optimized minima.

The electric-dipole transition moment is

In atomic units, the dimensionless electric-dipole oscillator strength is commonly written

The excitation energy determines where a stick appears; $f_{0I}$ measures its electric-dipole intensity. A low-energy state can be dark, while a higher state can dominate the absorption envelope. Symmetry, spin character, orbital overlap, and state mixing all affect the transition moment. “HOMO to LUMO” is consequently an orbital description, not an intensity rule.

Energy and wavelength are related by

The transformation is nonlinear. Equal spacing in energy does not give equal spacing in wavelength, and broadening a spectrum in eV is not equivalent to broadening it by a constant number of nanometres.

From independent-particle transitions to TDA

A ground-state one-electron model supplies occupied orbitals $i,j$ and virtual orbitals $a,b$ with orbital energies $\varepsilon_p$. An uncoupled orbital promotion has the gap

This gap is only a starting point. Creation of a hole and an excited electron changes the electronic response, and different promotions couple. In the Tamm–Dancoff approximation (TDA), excited states are expanded in singly excited configurations,

and obtained from the Hermitian eigenvalue problem

The diagonal contains orbital-energy differences and the off-diagonal terms couple promotions through an approximate interaction kernel. Each eigenvalue $\Omega_I$ is an excitation energy. Each eigenvector contains the configuration amplitudes that define the state character and transition moment.

Full linear-response TD methods contain resonant and anti-resonant blocks,

TDA neglects the $\mathbf B$ coupling and solves only the $\mathbf A$ problem. This reduces cost and gives a symmetric eigenproblem, but it remains a response calculation rather than a list of raw orbital gaps.

What “simplified” means

The configuration space grows as $N_\mathrm{occ}N_\mathrm{virt}$. Constructing and diagonalizing an exact response matrix becomes costly for large molecules. Simplified TDA reduces this cost in two related ways: it selects a restricted transition space around the energy range of interest, and it replaces expensive four-centre interaction terms with atom-resolved transition-charge interactions and empirical damping/parameterization.

Schematically, an sTDA matrix can be read as

where the Coulomb- and exchange-like contributions are evaluated from simplified transition charges rather than a full integral transformation. The precise parameterization is part of the method and does not reduce to “orbital gaps with Gaussian peaks”: coupling can move roots and mix several promotions into one state.

Tako exposes two ground-state routes into the sTDA response. The default GFN2 route builds the dedicated parameterized xtb4stda-like ground state from the structure. The g-xTB route instead imports the accepted g-xTB orbitals, orbital energies, and occupations without reconstruction, preserves the cached density unchanged, and evaluates response dipole integrals in the calculator-owned q-vSZP basis. Both then form closed- or open-shell transition spaces, construct the same simplified Coulomb/exchange contractions, solve for the requested low roots, and evaluate transition dipoles and oscillator strengths. Choosing GFN2+D4 does not add a dispersion term to the electronic response; D4 can matter indirectly when it produced a different input geometry.

The frozen 20-compound benchmark does not support declaring one route universally superior. GFN2 gave the lower assigned-band error (0.221 eV MAE versus 0.380 eV) and won 13 of 20 cases, so it remains the accuracy-first default. g-xTB was faster for every case, with a 4.7-fold paired geometric-mean speedup, and had better aggregate color error on the eight full spectra, although those spectrum-level wins split four to four by molecule. Tako therefore keeps both choices explicit and never switches per molecule.

The method accepts total charge and an unpaired-electron count through the structure. Integral charge, a consistent electron/spin occupation, and complete method-parameter coverage are required. Open-shell calculations use separate alpha and beta orbital channels and can report a spin label on dominant promotions. These labels are orbital-channel metadata, not a computed spin-purity analysis or an excited-state multiplicity assignment.

Transition vectors and state assignments

For a normalized TDA eigenvector, squared configuration amplitudes can provide a useful measure of promotion character. Tako instead returns the four components with the largest absolute amplitudes, retaining each component’s sign, when transition-vector output is enabled. The public field is named weight, but it is a signed eigenvector coefficient: it is neither a squared population nor an oscillator-strength fraction.

A state such as

is not two separate spectral peaks. It is one eigenstate with mixed configuration character. Signs can matter for interference and transition properties. Four retained coefficients are suitable for a compact composition summary, not reconstruction of the response eigenvector.

An orbital assignment does not by itself identify a chemical excitation. Local excitation, charge transfer, Rydberg character, and metal-to-ligand or ligand-to-metal transfer require inspection of the relevant orbitals or transition densities. Tako currently returns neither natural transition orbitals nor attachment/detachment densities in the public sTDA result. Avoid presenting a frontier label as a spatial assignment that was never calculated.

How the viewer renders these coefficients as percentages and infers HOMO/LUMO labels is display heuristics, documented in UV–Vis viewer semantics.

Oscillator strengths and selection rules

Oscillator strength is dimensionless and nonnegative in Tako’s public result. Missing oscillator strengths are replaced by zero; negative numerical values are clamped to zero. The viewer uses descriptive thresholds of $f<0.001$ for “dark,” $0.001\le f<0.05$ for “weak,” and $f\ge0.05$ for “bright” in its normalized state model. These are display categories, not universal spectroscopic laws.

An electric-dipole-forbidden transition can acquire intensity through symmetry breaking, state mixing, vibronic coupling, spin–orbit coupling, or finite-temperature sampling. Conversely, a large orbital overlap does not guarantee a strong band. Tako calculates the electronic vertical electric-dipole response of the supplied geometry. It does not calculate vibronic intensity borrowing, magnetic-dipole/rotatory strengths, spin–orbit-coupled spectra, fluorescence, or excited-state lifetimes.

The returned transition dipole is in atomic units when available. It is the vector used to characterize the electronic transition; it is not the permanent dipole of the ground state. The public sTDA operation does not expose state gradients, relaxed excited-state densities, or nonadiabatic couplings.

From sticks to the plotted curve

Tako folds each state with a normalized Gaussian in energy:

Here $\sigma$ is the user-selected standard deviation in eV, not the full width at half maximum. For a Gaussian,

The default $\sigma=0.15$ eV therefore corresponds to an energy-domain FWHM of approximately 0.353 eV. Because each Gaussian is normalized, its integrated area equals the oscillator strength. Changing $\sigma$ redistributes height and width but does not create new electronic states.

The grid is

The maximum is included only when the interval is an integer multiple of the spacing. A finer spacing samples the same analytic sum more densely; it does not improve the electronic-structure model. A window can exclude a state centre while retaining a Gaussian tail, or exclude both a state and its visible contribution. Request enough states and a wide enough energy interval independently.

Detecting local maxima in the sampled array, associating nearby states, and normalizing their oscillator strengths into displayed “Peaks” percentages is viewer-derived post-processing, not sTDA output; see UV–Vis viewer semantics.

Number of states, energy window, and convergence of interpretation

States controls how many low-energy response roots are requested, from 1 to 256 at the Tako boundary. It does not mean “include every transition below E max.” If 16 roots end below the desired spectral window, increasing E max alone produces only an empty high-energy tail. Increase the root count until all states relevant to the window and assignment are present.

The reverse is also possible: states above E max remain in states but their Gaussian centres lie outside the plotted interval. Treat root selection and spectrum sampling as separate convergence dimensions:

1. Increase the number of roots until peak positions and oscillator-strength distribution in the target interval stop changing materially.
2. Widen the energy interval enough to contain the relevant centres and Gaussian tails.
3. Reduce the spacing until plotted maxima are stable relative to the precision being reported.
4. Choose $\sigma$ for a declared comparison purpose.

Large state requests can exhaust the available selected single-excitation space and fail rather than returning fewer roots silently. Cost and memory grow with transition-space construction and diagonalization, not with atom count alone. Diffuse systems, many frontier orbitals, open shells, and high requested roots can be substantially more demanding than a small default calculation.

Geometry, charge, spin, and environment

Vertical excitations are geometry sensitive. Bond alternation, planarity, torsion between chromophores, coordination geometry, and intermolecular arrangement can shift energies and oscillator strengths. An optimized minimum is often a sensible starting point, but flexible molecules may require conformer sampling and population-weighted spectra.

Charge and unpaired electrons define the electronic occupation. A calculation that converges for the wrong charge/spin answers the wrong physical question. Record both with every spectrum. For radicals and transition-metal systems, inspect whether a single-reference simplified model is credible; the presence of alpha/beta labels does not prove it.

Tako’s public sTDA setup does not expose a solvent model. Experimental solution spectra can shift and reshape through dielectric stabilization, specific hydrogen bonding, protonation equilibria, ion pairing, aggregation, and conformer populations. Do not interpret a gas-phase model–experiment offset as a universal correction. If a structure was optimized with a different model, state that provenance explicitly; the excitation calculation still uses its own dedicated response model.

Reading the result without confusing layers

The calculation result contains four conceptually distinct layers:

The workspace stores excitations.json and uv_vis.json as separate artifacts in addition to the complete result. The viewer combines them. Wavelengths, brightness labels, compact orbital strings, detected peaks, relative peak heights, spectral-region names, and peak contribution chips are derived in the frontend. Preserve the raw artifacts whenever quantitative reuse matters.

states and excitations.states duplicate the same array in the current result. Duplication is a compatibility boundary, not two independent calculations. The uv_vis intensity has the units implied by a normalized Gaussian per eV and is normally displayed after vertical normalization; it is not molar absorptivity, absorbance, or detector counts.

Method domain and limitations

sTDA-xTB is valuable because it makes approximate spectra accessible for systems where a higher-level response calculation is too expensive. Its speed does not remove the usual excited-state hazards. Particular caution is required for strong charge-transfer states, Rydberg states needing diffuse basis character, near-degenerate and multireference states, transition-metal manifolds, conical-intersection regions, double-excitation character, strong spin–orbit coupling, and quantitative band-shape prediction.

The root energy is model dependent. No single shift is transferable across unrelated chromophores and state characters. Benchmark representative compounds against appropriate experiment or higher-level calculations, using the same geometry/environment policy. Report trends at a precision supported by that benchmark; the viewer’s displayed decimals exceed the method’s accuracy.

The smooth spectrum omits Franck–Condon structure, Herzberg–Teller intensity, finite-temperature nuclear distributions, homogeneous and inhomogeneous broadening, solvent dynamics, excited-state relaxation, internal conversion, intersystem crossing, and instrument response. A Gaussian chosen for readability is a visualization convention. A Gaussian calibrated against a declared experimental resolution is part of a comparison model. Those are different claims.

A defensible analysis sequence

Begin with the discrete state table, not the envelope. Establish that charge, spin, geometry, root count, and method are correct. Identify bright and dark roots and inspect whether strong bands are single-state or mixed-state features. Use orbital weights as hypotheses, then confirm spatial character with orbital or density analysis outside the current public sTDA result when the assignment matters.

Next inspect the folded curve. Confirm energy window, actual last grid point, spacing, and $\sigma$. Separate changes caused by the electronic roots from changes caused only by broadening. When comparing structures, keep the folding parameters fixed. When comparing with experiment, declare solvent, temperature, conformer policy, and any empirical shift or normalization.

Finally, test robustness: vary geometry or conformer, increase roots, compare a chemically relevant benchmark, and challenge assignments that depend on one frontier-orbital label. A useful spectrum is an evidence chain from structure to response roots to plotted representation, with every approximation visible.

The visible control sequence is documented in Simulate a UV–Visible Spectrum. Programmatic operation and result handling are documented separately in UV–Visible Spectra with Tako Script. Exhaustive field names and normalized defaults are generated in tako.stda Settings.

References and credit

• S. Grimme, “A simplified Tamm–Dancoff density functional approach for the electronic excitation spectra of very large molecules,” J. Chem. Phys. 138, 244104 (2013), DOI: 10.1063/1.4811331.
• C. Bannwarth and S. Grimme, “A simplified time-dependent density functional theory approach for electronic ultraviolet and circular dichroism spectra of very large molecules,” Comput. Theor. Chem. 1040–1041, 45–53 (2014), DOI: 10.1016/j.comptc.2014.02.023.
• S. Grimme and C. Bannwarth, “Ultra-fast computation of electronic spectra for large systems by tight-binding based simplified Tamm–Dancoff approximation (sTDA-xTB),” J. Chem. Phys. 145, 054103 (2016), DOI: 10.1063/1.4959605.