12 Case study: short-range order in Li₂FeSO

This chapter presents a case study illustrating the computational workflow developed in the preceding chapters. The example — cation ordering in the antiperovskite cathode material Li₂FeSO — demonstrates how cluster expansion, Monte Carlo sampling, and pair distribution function analysis combine to characterise short-range order that Bragg diffraction cannot detect. Full details are available in Coles et al., Journal of Materials Chemistry A (2023).

12.1 The puzzle

Li₂FeSO adopts the antiperovskite structure. Sulfur occupies the 12-coordinate site, oxygen occupies the 6-coordinate octahedral site, and lithium and iron share the remaining sites in a 2:1 ratio (Fig. X). Previous diffraction studies found no superlattice reflections — no evidence of long-range cation order — leading to the assumption that Li and Fe are randomly distributed over their shared sites.

[Figure: Li₂FeSO antiperovskite structure showing the shared Li/Fe sites]

But this assumption is puzzling on physical grounds. Fe²⁺ and Li⁺ have different formal charges. Simple electrostatic arguments suggest that configurations which separate the higher-charged Fe²⁺ ions should be lower in energy than configurations that cluster them together. If an energetic preference exists, why would the cation distribution be random?

The absence of long-range order does not necessarily imply the absence of short-range order. Diffraction probes the average, periodic structure; local correlations that do not produce a periodic pattern are invisible to Bragg peaks. The question is whether Li₂FeSO exhibits preferential short-range cation ordering, and if so, what form it takes.

12.2 The computational approach

To address this question, we calculated DFT energies for approximately 100 different Li/Fe configurations in supercells of Li₂FeSO. These energies were used to fit a cluster expansion — a parameterised model that expresses the configurational energy as a function of site occupancies, as described in Chapter 102.

With the cluster expansion in hand, Monte Carlo sampling becomes computationally tractable. We performed MC simulations in an 8×8×8 supercell at 1025 K, corresponding to the experimental synthesis temperature. The simulation generates an ensemble of configurations representative of thermal equilibrium at that temperature.

From the MC trajectory, we can extract the probability distribution of local coordination environments. Each oxygen in the structure is surrounded by six cation sites. In a material with 2:1 Li:Fe stoichiometry, the possible oxygen coordinations range from OLi₆ (all lithium) to OFe₆ (all iron), with intermediate compositions OLi₅Fe, OLi₄Fe₂, OLi₃Fe₃, OLi₂Fe₄, and OLiFe₅.

For OLi₄Fe₂ coordination — four lithium and two iron around the central oxygen — there are two distinct geometric arrangements. In cis-OLi₄Fe₂, the two iron atoms occupy adjacent vertices of the octahedron. In trans-OLi₄Fe₂, they occupy opposite vertices.

[Figure: Schematic of cis versus trans OLi₄Fe₂ coordination environments]

12.3 The result: preferential short-range order

The Monte Carlo simulations predict that the cation distribution is not random.

For a random arrangement of Li and Fe over the available sites, the probability of each oxygen coordination environment follows a binomial distribution. The most probable coordination would be OLi₄Fe₂, occurring in approximately 33% of oxygen environments.

The DFT-derived cluster expansion predicts a markedly different distribution. OLi₄Fe₂ coordination accounts for 65% of oxygen environments — roughly twice the random expectation. This enhanced preference for OLi₄Fe₂ is consistent with Pauling’s second rule: coordination environments that achieve local electroneutrality are energetically favoured.

[Figure: Oxygen coordination probability distributions comparing DFT-CE model, random distribution, and point-charge model]

More unexpectedly, within the OLi₄Fe₂ environments, 81% adopt the cis configuration and only 19% adopt the trans configuration.

This result contradicts simple electrostatic reasoning. If Li⁺ and Fe²⁺ are treated as point charges at their formal crystallographic positions, the electrostatic energy of an OLi₄Fe₂ octahedron is minimised when the two Fe²⁺ ions occupy opposite vertices — the trans configuration — maximising their separation. A point-charge model predicts preferential trans-OLi₄Fe₂ coordination and, consequently, long-range cation order. The DFT-derived model predicts preferential cis-OLi₄Fe₂ coordination, which produces long-range disorder.

12.4 Validation against experiment

The PDF provides a direct test of these competing structural models. From the Monte Carlo configurations, we can calculate the pair distribution function and compare it to experimental total scattering data. This particular study used X-ray total scattering, which is most sensitive to the heavier Fe and S atoms; neutron PDF on the same system would provide complementary sensitivity to the lithium correlations. The principles of the comparison — generating configurations computationally and comparing calculated g(r) to experiment — are identical regardless of the probe.

Three models were tested. The DFT-derived cluster expansion model, with its preference for cis-OLi₄Fe₂ coordination, gives the best agreement with experiment (\(R_\mathrm{w} = 14.0%\)). A random Li/Fe distribution gives poorer agreement (\(R_\mathrm{w} = 16.5%\)) — the difference is modest but systematic, reflecting the failure to capture the enhanced OLi₄Fe₂ coordination that the DFT model predicts. Most tellingly, a point-charge ground state with 100% trans-OLi₄Fe₂ coordination gives substantially worse agreement (\(R_\mathrm{w} = 35.2%\)). The structure that simple electrostatics predicts is incompatible with the experimental PDF at short range.

[Figure: Comparison of experimental and simulated PDFs for the three models]

The PDF comparison validates the DFT-derived model. The short-range order in Li₂FeSO is real, and it takes the form of preferential cis-OLi₄Fe₂ coordination — not the trans preference that point-charge electrostatics would suggest.

12.5 The physical origin: anion polarisation

Why does DFT favour cis over trans, when simple electrostatics predicts the opposite?

The key lies in the symmetry of the anion coordination environment. In trans-OLi₄Fe₂ coordination, the oxygen sits at a centre of symmetry — the distribution of cations around it is centrosymmetric. In cis-OLi₄Fe₂ coordination, the oxygen sits in a non-centrosymmetric environment.

Anions in non-centrosymmetric environments experience an asymmetric electric field from the surrounding cations. This field induces electronic polarisation — the anion’s electron density shifts in response. The induced dipole lowers the total electrostatic energy of the system.

Point charges cannot polarise. A model that treats ions as fixed charges at crystallographic positions misses this physics entirely. It correctly captures the preference for OLi₄Fe₂ coordination (local electroneutrality), but incorrectly predicts trans over cis because it cannot account for the stabilisation of polar coordination environments through anion polarisation.

Analysis of the DFT calculations confirms this picture. In structures with cis-OLi₄Fe₂ coordination, the oxygen anions exhibit significant electronic dipole moments. In structures with trans-OLi₄Fe₂ coordination, the oxygen dipole moments are zero by symmetry. The polarisation energy tips the balance from trans to cis.

12.6 What this example illustrates

This case study demonstrates the interplay between experiment and computation that runs through the preceding sections. Bragg diffraction correctly reported no long-range order, but could not distinguish between a truly random cation distribution and one with short-range order that lacks long-range periodicity. The combination of DFT, cluster expansion, and Monte Carlo sampling predicted a specific form of short-range order — preferential cis-OLi₄Fe₂ coordination — and the PDF comparison validated this prediction against experiment. Further analysis then revealed the physical origin of this preference: anion polarisation in non-centrosymmetric coordination environments, physics that the experiment could not directly access but that computation reveals.

The relationship was bidirectional throughout. The experimental observation of “no long-range order” prompted the computational investigation. The computational result was validated against experimental PDF data. And the physical insight emerged from analysing why the DFT results differed from simple electrostatic expectations.

For this material, “disordered” does not mean “random.” The distinction matters: short-range cation order in cathode materials affects lithium transport, redox behaviour, and electrochemical performance. Characterising that order required the combination of total scattering and computation that this case study illustrates.