chapter I

The forward problem

There is no unique, model-free inversion from a spectrum to a structure. A common workflow proposes a structure, simulates its spectrum, and lets the mismatch drive the next proposal. Every simulation on the ladder approximates one compact statement:

|i⟩ · the core state → element specificityε̂·r · the dipole filter → selection rules|f⟩ · empty states with a core hole → the spectrum

Simulation closes the loop

Part I ended with fits whose ingredients, and , were taken on faith. They come from here: a forward calculation on a candidate structure, compared against the measurement, adjusted, and run again.

Fig. 1 | The forward loop. Propose a structure, simulate its spectrum, compare against the measurement, and revise; the equation names each arrow.

The photoelectron scatters along paths

Every trip the photoelectron can take from the absorber and back contributes one oscillating term. Count the trips by their number of legs and watch the trade that defines both regimes of part I.

absorber, two shells, one example path

model-cluster path count

12 in model cluster

normalized relative amplitude

single scattering

Out to one neighbor and straight back. These paths carry the EXAFS equation of part I: one shell, one distance, one frequency.

Fig. 2 | Scattering paths. Select a path family and its term lights up in the path sum; longer and multiple-scattering paths carry higher effective distances.

The path expansion. Part I's single-shell equation is this sum's first and largest term, with and computed per path. The factor is the next section's subject.

The mean free path sets the horizon

A photoelectron contributes coherently while it avoids inelastic loss and the core-excited state retains coherence. The effective controls a continuous attenuation, not a hard cutoff. Drag along the universal curve and watch the modeled attenuation length around the absorber change.

universal curve · λ(k)

wavenumber k: 8.0 Å⁻¹

the horizon around the absorber

λ = 8.5 Å · first-shell amplitude 62%

The factor fades paths continuously: at only of the amplitude remains. The dashed circle marks only as a visual reference, not a visibility boundary. Together with geometric spreading and lifetime damping, this helps make XAS a local probe.

Fig. 3 | The universal curve. The inelastic mean free path sets how far the photoelectron travels, and therefore which shells can contribute at each k.

chapter II

The theory ladder

Five families of methods, one recurring trade: fidelity near the edge against cost per calculation. Each family has working codes you can download today.

Pick your rung

The right method depends on the question. Fitting EXAFS paths, screening many candidate structures, and nailing one pre-edge exciton are three different jobs.

method families · select a rungruntime depends on hardware, system size, and convergence settings

Multiple scattering

seconds–minutes per absorber

Use real-space multiple scattering with effective atomic potentials. For EXAFS, calculate path amplitudes and phases once, then reuse them while fit parameters vary; the same family also supports fast XANES surveys.

FEFF

best for

EXAFS paths for fitting
quick K-edge XANES surveys
broad elemental and structural coverage

watch out

muffin-tin potentials blur near-edge detail
self-consistency optional and often needed

Fig. 4 | The theory ladder. Each rung trades cost for physics, from fast path expansions to full many-body treatments; the chips name the codes that live there.

before trusting a spectrum

Converge the numerical representation appropriate to the method: cluster or supercell size, basis and cutoffs, k-point sampling, empty states, broadening, and energy grid. Change one control at a time, compare the spectrum over a declared energy window, and keep the inputs with the result. A plausible line shape is not a convergence test.

code	family	reach for it when
FEFF	multiple scattering	you need f(k) and φ(k) for an EXAFS fit, or a fast XANES survey
FDMNES	multiple scattering / finite difference	you want to test full-potential effects on distorted or low-symmetry sites
XSpectra	core-hole DFT (Quantum ESPRESSO)	periodic solids, K-edges, and high-throughput campaigns
OCEAN	BSE	excitonic pre-edges and quantitative near-edge intensities
exciting	all-electron BSE	all-electron benchmarks that avoid pseudopotential approximations
Quanty / CTM4XAS	ligand-field multiplets	transition-metal L-edges, spin states, correlated systems
Larch / Larix / Athena / Artemis	processing & fitting	Larch, Larix, or Athena for processing; Larch or Artemis for EXAFS path fitting

The core hole reshapes the final states

The golden rule above is explicit: the final states contain a core hole. Leave it out of the model and the computed XANES misplaces near-edge weight. The appropriate treatment depends on screening, edge, and material.

computed XANES vs. measurement

measuredcomputed

mismatch to measurement

25%

In the independent-particle model, omitting the attractive core-hole potential removes the bound pre-edge feature and shifts near-edge weight.

Fig. 5 | The core-hole approximation. Toggle how the missing core electron is treated and watch the white line and near-edge shape respond.

chapter III

Computed vs measured

A theory–experiment comparison needs a declared energy alignment and physically motivated broadening. Agreement after those operations is evidence, not proof that the model is complete.

Broaden, shift, then compare

Calculations produce sharp features on an arbitrary energy zero; measurements are lifetime- and instrument-broadened on an absolute one. The comparison below uses one effective Lorentzian width and a rigid . Real comparisons often combine Lorentzian lifetime and Gaussian instrument terms as a Voigt profile. Try to reach a mismatch below 2%.

match the computed spectrum to the measured one

measuredcomputed

broadening : 0.6 eVshift : +0.0 eV

rms mismatch

26.0%

keep adjusting

The calculation is too sharp and uses a different energy zero. Adjust the declared comparison model; agreement should not be manufactured with unconstrained feature-by-feature shifts.

Fig. 6 | Broaden and align. Convolve and shift the raw calculation until it is comparable with the measurement; judging before this step is meaningless.

Two candidates, one measurement

Both candidates share the same forward model and broadening, so the smaller residual identifies which candidate generated the simulated measurement. With experimental data, residuals can also reflect model inadequacy, calibration, and unmodeled sample complexity.

which structure produced this measurement?

measuredsimulated

rms · candidate A

6.8%

rms · candidate B

1.0%

the residual says otherwise

isolated octahedra, one broad white line. Both candidates were simulated on the same forward model and broadened identically, so the residual difference identifies the generating structure. With measured data, forward-model error, calibration, mixtures, and omitted candidates must also be tested.

Fig. 7 | Candidate discrimination. Two proposed structures, one measurement; the residual decides which forward model earns belief.

A computed spectrum is a hypothesis stated precisely enough to be wrong. Broadening and alignment are part of the hypothesis, which is why they belong in the caption of every theory-experiment figure you publish.