# Anion-polarisation–directed short-range-order in antiperovskite Li2FeSO

Samuel W. Coles<EMAIL_ADDRESS>Department of Chemistry, University of Bath,
Claverton Down, BA2 7AY, United Kingdom  The Faraday Institution, Quad One,
Harwell Science and Innovation Campus, Didcot, OX11 0RA, United Kingdom
Viktoria Falkowski  Department of Chemistry, University of Oxford, Inorganic
Chemistry Laboratory, Oxford OX1 3QR, United Kingdom  The Faraday Institution,
Quad One, Harwell Science and Innovation Campus, Didcot, OX11 0RA, United
Kingdom  Harry S. Geddes  Department of Chemistry, University of Oxford,
Inorganic Chemistry Laboratory, Oxford OX1 3QR, United Kingdom  The Faraday
Institution, Quad One, Harwell Science and Innovation Campus, Didcot, OX11
0RA, United Kingdom  Gabriel E. Pérez  ISIS Neutron and Muon Source, STFC
Rutherford Appleton Laboratory, Didcot OX11 0QX, United Kingdom  The Faraday
Institution, Quad One, Harwell Science and Innovation Campus, Didcot, OX11
0RA, United Kingdom  Samuel G. Booth  Department of Materials Science and
Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom  The
Faraday Institution, Quad One, Harwell Science and Innovation Campus, Didcot,
OX11 0RA, United Kingdom  Alexander G. Squires  Department of Chemistry,
University of Bath, Claverton Down, BA2 7AY, United Kingdom  Department of
Chemistry, University College London, London WC1H 0AJ, United Kingdom  The
Faraday Institution, Quad One, Harwell Science and Innovation Campus, Didcot,
OX11 0RA, United Kingdom  Conn O’Rourke  Department of Chemistry, University
of Bath, Claverton Down, BA2 7AY, United Kingdom  The Faraday Institution,
Quad One, Harwell Science and Innovation Campus, Didcot, OX11 0RA, United
Kingdom  Kit McColl  Department of Chemistry, University of Bath, Claverton
Down, BA2 7AY, United Kingdom  The Faraday Institution, Quad One, Harwell
Science and Innovation Campus, Didcot, OX11 0RA, United Kingdom  Andrew L.
Goodwin  Department of Chemistry, University of Oxford, Inorganic Chemistry
Laboratory, Oxford OX1 3QR, United Kingdom  The Faraday Institution, Quad One,
Harwell Science and Innovation Campus, Didcot, OX11 0RA, United Kingdom
Serena A. Cussen  Department of Materials Science and Engineering, University
of Sheffield, Sheffield S1 3JD, United Kingdom  The Faraday Institution, Quad
One, Harwell Science and Innovation Campus, Didcot, OX11 0RA, United Kingdom
Simon J. Clarke  Department of Chemistry, University of Oxford, Inorganic
Chemistry Laboratory, Oxford OX1 3QR, United Kingdom  The Faraday Institution,
Quad One, Harwell Science and Innovation Campus, Didcot, OX11 0RA, United
Kingdom  M. Saiful Islam  Department of Chemistry, University of Bath,
Claverton Down, BA2 7AY, United Kingdom  Department of Materials, University
of Oxford, Oxford, OX1 3PH, United Kingdom  The Faraday Institution, Quad One,
Harwell Science and Innovation Campus, Didcot, OX11 0RA, United Kingdom
Benjamin J. Morgan<EMAIL_ADDRESS>Department of Chemistry, University
of Bath, Claverton Down, BA2 7AY, United Kingdom  The Faraday Institution,
Quad One, Harwell Science and Innovation Campus, Didcot, OX11 0RA, United
Kingdom

###### Abstract

Short-range ordering in cation-disordered cathodes can have a significant
effect on their electrochemical properties. Here, we characterise the cation
short-range order in the antiperovskite cathode material Li2FeSO, using
density functional theory, Monte Carlo simulations, and synchrotron X-ray
pair-distribution-function data. We predict partial short-range cation-
ordering, characterised by favourable OLi4Fe2 oxygen coordination with a
preference for polar _cis_ -OLi4Fe2 over non-polar _trans_ -OLi4Fe2
configurations. This preference for polar cation configurations produces long-
range disorder, in agreement with experimental data. The predicted short-
range-order preference contrasts with that for a simple point-charge model,
which instead predicts preferential _trans_ -OLi4Fe2 oxygen coordination and
corresponding long-range crystallographic order. The absence of long-range
order in Li2FeSO can therefore be attributed to the relative stability of
_cis_ -OLi4Fe2 and other non-OLi4Fe2 oxygen-coordination motifs. We show that
this effect is associated with the polarisation of oxide and sulfide anions in
polar coordination environments, which stabilises these polar short-range
cation orderings. We propose similar anion-polarisation–directed short-range-
ordering may be present in other heterocationic materials that contain cations
with different formal charges. Our analysis also illustrates the limitations
of using simple point-charge models to predict the structure of cation-
disordered materials, where other factors, such as anion polarisation, may
play a critical role in directing both short- and long-range structural
correlations.

##  I Introduction

Crystallographically disordered materials, in which two or more heteroatomic
species are distributed over otherwise equivalent sites, find use in several
applications [1, 2], including solar cells [3, 4, 5, 6, 7], hydrogen storage
[8, 9], and lithium-ion batteries [10, 11, 12, 13, 14]. The material
properties of crystallographically disordered systems depend not only on their
stoichiometry and crystal structure, but also on the short-range configuration
of their disordered heteroatoms [1, 15, 2, 16, 17]. One such example is
provided by cation-disordered lithium-ion cathode materials, in which the
short-range cation configuration controls lithium-transport rates, charge and
discharge behaviour, and redox properties [10, 18, 16, 19, 14].

A full understanding of heteroatomic materials requires both an accurate
description of their short-range structures and an understanding of the
physical principles that promote or inhibit specific short-range orderings.
Such mechanistic understanding is particularly valuable for technologically
relevant materials, where targeted synthesis protocols that promote or inhibit
particular local structure motifs may allow the optimisation of key material
properties.

While many anion-disordered heteroanionic materials have been structurally
well-characterised [17, 20, 2, 1, 21, 22, 23], cation-disordered
heterocationic materials have been generally less studied. For heteroanionic
materials, various general design rules have been proposed to explain
particular examples of partial or full anion-ordering, based on electronic,
strain, or electrostatic effects [1, 15, 2, 21]. For heterocationic materials,
however, the factors that direct short-range order preferences are less well
understood [24].

The antiperovskite-structured lithium oxychalcogenides (Li2$M$) _Ch_ O ( _M_
= a transition metal; _Ch_ = S, Se) are one family of cation-disordered
materials that have been proposed as high-capacity cathodes for lithium-ion
batteries [25, 26, 27, 28, 29]. The most promising of these is the oxysulfide
Li2FeSO [27, 25, 26], which has a first-cycle capacity of
$275\text{\,}\mathrm{mA}\text{\,}\mathrm{h}\text{\,}{\mathrm{g}}^{-1}$ at
C/10. The practical use of Li2FeSO as a cathode material is limited by
capacity-fade on cycling, associated with progressive amorphisation [29]. The
underlying mechanism of this cycling-induced amorphisation remains unclear; in
part, because the atomic structure of pristine Li2FeSO is not yet fully
characterised.

Figure 1: The reported antiperovskite structure (space group $Pm\bar{3}m$) of
Li2FeSO [25]. Oxygen occupies the 6-coordinate octahedral Wyckoff $1a$ site.
Sulfur occupies the 12-coordinate Wyckoff $1a$ site. Lithium and iron are
distributed over the Wyckoff $3d$ sites in a 2:1 ratio.

Cubic antiperovskites, such as Li2FeSO, are charge-inverted structural
analogues of the well-known conventional cubic perovskites. In Li2FeSO (space
group $Pm\bar{3}m$; $a=$3.914\text{\,}\mathrm{\text{\AA}}$$ [25]), sulfur
occupies the 12-coordinate Wyckoff $1b$ site, oxygen occupies the 6-coordinate
Wyckoff $1a$ site, and lithium and iron occupy the Wyckoff $3d$ sites in a 2:1
Li:Fe ratio (Fig. 1). Previous diffraction studies of Li2FeSO have shown an
absence of cation ordering at long ranges [25, 26], leading to the assignment
of Li and Fe as randomly distributed over the available Wyckoff $3d$ sites
[25]. While a “fully random” cation-distribution is consistent with the
previous experimental diffraction data [25, 26], alternative structural models
with some preferential short-range order but no long-range order would also be
compatible. For mixed Li/Fe systems, such as Li2FeSO, some degree of short-
range order might in fact be expected: Fe2+ ions have a higher formal charge
than Li+ ions, and simple electrostatic arguments predict that cation
configurations that maximise Fe–Fe separations should be energetically
favoured, resulting in some form of short-range, and possibly even long-range,
cation ordering.

While previous experimental data have been interpreted as evidence of random
Li/Fe distribution in Li2FeSO [25], computational studies of Li2FeSO have
predicted that different Li/Fe configurations give different structural
energies [27, 30], indicating a preference for some specific cation
configurations. This apparent inconsistency between proposed structural models
raises the question of whether the Li/Fe cation distribution in Li2FeSO does
in fact demonstrate preferential short-range order, and, if so, what form this
takes. Secondly, if Li2FeSO does indeed exhibit short-range cation order, what
is the physical origin of this cationic ordering, and can this be explained
by, for example, simple models of point-charge electrostatics [27]?

To characterise the short-range order in Li2FeSO, we have performed density
functional theory (DFT) calculations, cluster-expansion–based Monte Carlo
sampling, and X-ray total scattering experiments and pair-distribution
function (PDF) analysis. Our computational model predicts partial short-range
ordering, characterised by a strong preference for OLi4Fe2 oxygen-
coordination, with a weaker preference for _cis_ -OLi4Fe2 over _trans_
-OLi4Fe2 oxygen-coordination. To validate this structural model, we have
compared simulated pair-distribution functions (PDFs) against our experimental
PDF data. Our DFT-derived computational model gives better agreement with the
experimental data than models generated assuming either a random Li/Fe
distribution or using the ground-state structure from a simple point-charge
electrostatic model.

We find that Li/Fe configurations that give polar anion-coordination are
stabilised relative to configurations that give non-polar-anion coordination,
when compared to the energy ranking predicted by a simple point-charge
electrostatic model. The stabilisation of polar Li/Fe configurations can be
understood as a consequence of anions with polar coordination being
electronically polarised, which lowers the net electrostatic energy for these
configurations. We attribute this anion-polarisation–induced cationic short-
range order in Li2FeSO as a consequence of high anion polarisabilities
combined with a capacity for highly polar local cation configurations, and we
expect this effect to be generally applicable in cation-disordered materials
where the cations have different formal charges.

Finally, we discuss the role of preferential short-range order on the presence
or absence of long-range order, and characterise this in terms of the
configurational density of states and the temperature dependence of short- and
long-range order parameters. Preferential _cis_ -OLi4Fe2 oxygen coordination,
as predicted by our DFT calculations, means that neighbouring pairs of OLi4Fe2
octahedra are configurationally underconstrained, and can adopt various
different relative orientations. This produces long-range disorder, in
agreement with experimental diffraction data, and is associated with a
continuous density of states, and short- and long-range order parameters vary
continuously at any non-zero temperature. In contrast, preferential _trans_
-OLi4Fe2 oxygen coordination fully constrains the relative orientations of
neighbouring OLi4Fe2 pairs in two dimensions, producing long-range order. In
this case, short- and long-range order parameters show strong ordering to
relatively high temperatures, before undergoing a more sudden change to
partial disorder, characteristic of a formal order–disorder transition.

Beyond the specific case of Li2FeSO, our results demonstrate how going beyond
simple point-charge models can be necessary to understand local structure in
such cation-disordered materials, and highlight the role of anion polarisation
in directing short-range order, and consequently the presence or absence of
long-range order, in these materials.

Figure 2: (a–c) Oxygen coordination environment probability distributions for
Li2FeSO for (a) a density-functional theory derived cluster expansion model
(DFT-CE), from Monte Carlo simulations at $1025\text{\,}\mathrm{K}$, (b)
random Li/Fe positions (RANDOM), and (c) an on-site point-charge cluster
expansion model (PC-CE), from Monte Carlo simulations at
$1025\text{\,}\mathrm{K}$. (d) Schematic of _cis_ versus _trans_ (top) and
_fac_ versus _mer_ (bottom) pseudo-octahedral coordination environments.

##  II Results and Discussion

###  II.1 Quantifying short-range order in Li2FeSO

The degree to which Li2FeSO exhibits cationic short-range order can be
quantified by considering the relative probabilities of different competing
cation configurations. Here, we consider the set of [LixFe6-x] oxygen
coordination environments as a descriptor of possible short-range ordering. To
sample the different local configurations expected in as-synthesised Li2FeSO,
we first parametrise a cluster-expansion (CE) model from a set of DFT
calculations with different {Li,Fe} configurations (see the Computational
Methods section for details), and then use the resulting DFT-CE model to
perform a Monte Carlo simulation in a $8\times 8\times 8$ supercell. The
relative probabilities of competing OLixFe6-x coordination environments are
obtained directly from analysis of the Monte Carlo simulation trajectory.

Our DFT-CE model (Fig. 2a) predicts that the most likely oxygen coordination
is OLi4Fe2, which accounts for $65\text{\,}\mathrm{\char 37\relax}$ of the
oxygen environments. Within this preferential OLi4Fe2 coordination,
$81\text{\,}\mathrm{\char 37\relax}$ of oxygen coordination environments are
_cis_ -OLi4Fe2 and $19\text{\,}\mathrm{\char 37\relax}$ are _trans_ -OLi4Fe2.
We also predict moderate amounts of OLi3Fe3 and OLi5Fe2 oxygen-coordination,
with the OLi3Fe3 environments divided 71:29 into _fac_ and _mer_
configurations.

For a fully random arrangement of Li and Fe, the corresponding oxygen-
coordination probability distribution is a binomial distribution, with
$p=\frac{2}{3}$ and $n=6$ (Fig. 2b). This distribution is visually distinct
from that obtained from our Monte Carlo simulations, demonstrating the
existence of short-range order in the DFT-parametrised computational model.

The preferential OLi4Fe2 coordination predicted by Monte Carlo simulation is
consistent with the predictions of simple point-charge electrostatics.
Coordination environments that give “local electroneutrality” are generally
expected to be favoured, as expressed by Pauling’s second rule [31]. Assuming
formal oxidation states, for corner-sharing
O2-[Li${}^{+}_{x}$Fe${}^{2+}_{6-x}$] octahedra local electroneutrality is
achieved when $x=4$, corresponding to the preferential OLi4Fe2 coordination
predicted by our DFT-CE model.

While a simple point-charge electrostatic model is consistent with
preferential OLi4Fe2 coordination, this model also predicts _trans_ -OLi4Fe2
being favoured over _cis_ -OLi4Fe2. Considering Fe and Li as point-charges
with formal 2+ and 1+ charges, respectively, that occupy ideal Wyckoff $3d$
crystallographic sites, the electrostatic energy of a OLi4Fe2 octahedron is
minimised when the Fe ions maximise their separation, by occupying opposing
octahedral vertices in a _trans_ configuration. Indeed, the ground state for
an on-site point-charge model is comprised of $100\text{\,}\mathrm{\char
37\relax}$ _trans_ -OLi4Fe2 coordination. At non-zero temperatures, entropic
contributions mean some proportion of non-_trans_ -OLi4Fe2 oxygen coordination
is expected. Yet even at relatively high temperatures, a simple point-charge
model predicts a strong preference for _trans_ -OLi4Fe2 over _cis_ -OLi4Fe2
oxygen coordination—for a formal-charge on-site point-charge model for Li2FeSO
with relative permittivity $\varepsilon_{\mathrm{r}}=4$ at
$T=$1025\text{\,}\mathrm{K}$$ (Fig. 2c), the predicted _trans_ :_cis_ OLi4Fe2
ratio is 69:31. Even in the high-temperature limit, where the point-charge
model recovers the fully random model distribution, the ratio of OLi4Fe2
_trans_ versus _cis_ environments is 25:75, and therefore never reaches the
ratio of 19:81 _trans_ \- to _cis_ -OLi4Fe2 predicted by the DFT-CE model at
$1025\text{\,}\mathrm{K}$.

To validate our DFT-CE model, we compared the pair distribution function (PDF)
obtained from X-ray total scattering of Li2FeSO to simulated PDFs for (a) the
DFT-CE model, (b) the RANDOM model, and (c) the $100\text{\,}\mathrm{\char
37\relax}$ _trans_ -OLi4Fe2 structure that would be obtained from a simple
point-charge electrostatic ranking of all possible Li/Fe configurations (Fig.
3). The best quality-of-fit is obtained for the DFT-CE model
($R_{\mathrm{w}}=$14.03\text{\,}\mathrm{\char 37\relax}$$), versus
$R_{\mathrm{w}}=$16.47\text{\,}\mathrm{\char 37\relax}$$ for the RANDOM model,
and $R_{\mathrm{w}}=$35.22\text{\,}\mathrm{\char 37\relax}$$ for the ground-
state point-charge model. The particularly poor quality-of-fit for the
$100\text{\,}\mathrm{\char 37\relax}$ _trans_ -OLi4Fe2 point-charge ground-
state illustrates how a simple ranking of structures based on point-charge
electrostatic energies would predict a structure that is incompatible with the
experimental PDF data at short range ($<$20\text{\,}\mathrm{\text{\AA}}$$).

Figure 3: Comparison of pair-distribution functions (PDFs) obtained from X-ray
total scattering and simulated PDF data for Li2FeSO for structural models
generated for (a) the DFT-fitted cluster-expansion model (DFT-CE), (b) the
RANDOM model, and (c) the PC-CE point-charge model.

The oxygen-coordination populations obtained from the DFT-derived cluster
expansion model indicate that, on average, structures with _cis_ -OLi4Fe2 are
lower in energy than those with _trans_ -OLi4Fe2 coordination. To better
characterise the relative energies of competing OLi4Fe2 cation configurations
as a function of the proportion of _cis_ \- versus _trans_ -OLi4Fe2 oxygen
coordination, we calculated the energies of all symmetry-inequivalent $2\times
2\times 2$ supercells containing only Li4Fe2 oxygen-coordination—i.e., every
oxygen is coordinated by Li4Fe2 in either a _cis_ or _trans_ configuration.

Fig. 4(a) shows the distribution of these energies, relative to the ground-
state, grouped by the proportion of _cis_ oxygen coordination environments in
each structure. The lowest energy structure has $100\text{\,}\mathrm{\char
37\relax}$ _cis_ -OLi4Fe2 coordination, as expected from the oxygen-
coordination probabilities obtained from Monte Carlo simulation.
Interestingly, the _highest_ energy structure containing only OLi4Fe2
coordination also has $100\text{\,}\mathrm{\char 37\relax}$ _cis_ -oxygen
coordination. This suggests the existence of a more complex relationship
between cation configuration and energy than a simple preference for _cis_ \-
versus _trans_ -oxygen-coordination.

Fig. 4(b) shows the equivalent distribution of energies for all
$100\text{\,}\mathrm{\char 37\relax}$ OLi4Fe2 $2\times 2\times 2$ supercells
for the point-charge model ($\varepsilon=4.0$). Here, the lowest energy
structures are $100\text{\,}\mathrm{\char 37\relax}$ _trans_ -OLi4Fe2, and the
energy generally increases with increasing proportion of _cis_ -OLi4Fe2
coordination, as expected.

Figure 4: Distributions of structural energies for all 2$\times$2$\times$2
structures consisting solely of OLi_4Fe_2 oxygen coordination environments,
calculated using (a) the DFT-CE DFT-fitted cluster expansion model, and (b)
the PC-CE point-charge model. For each model, the distributions are subdivided
according to the fraction of _cis_ \- versus _trans_ -OLi_4Fe_2 octahedra in
each structure.

###  II.2 Dipolar stabilisation of polar anion coordination

To better understand the physical origin of the energy variation between
structures with different OLi4Fe2 configurations, we performed further DFT
calculations on three exemplar OLi4Fe2 $2\times 2\times 2$ configurations: the
lowest energy all-_trans_ structure, the highest energy all-_cis_ structure,
and the lowest energy all-_cis_ structure. These three structures are shown
schematically in Fig. 5, with their corresponding cation configurations around
oxygen and sulfur.

In our analysis, we consider two possible contributions to the total energy
that are not accounted for in a simple point-charge electrostatic model.
First, anions with polar coordination can move off their formal positions to
give some shorter anion–cation distances, potentially resulting in stronger
anion–cation interactions. Second, anions with polar coordination can be
electronically polarised, and introducing anion dipoles may lower the net
electrostatic energy of the system relative to a simple sum over point-charge
Coulomb terms. This effect is potentially more significant in heterocationic
materials, such as Li2FeSO, where the central atom is a relatively more
polarisable anion, than in conventional heteroanionic materials, such as
NbO2F[32] and SrNbO2N[33], where the central atom is typically a relatively
unpolarisable high-formal-charge cation.

Figure 5: Illustration of the three structures of interest (top) with their
oxygen (middle) and sulfur (bottom) environments shown with them. For the
environment examples the prevailing dipole is marked with an arrow or a non-
polar environment with a  glyph.

For each DFT-optimised structure, we consider the Fe–O and Fe–S distances and
the magnitudes of the O and S electronic dipoles (Table 1), which we calculate
from Wannier analysis of the converged DFT calculations. In the _trans_
structure, oxygen and sulfur both have non-polar coordination, and
corresponding dipole moments of
$0.00\text{\,}\mathrm{\text{e}}\text{\,}\mathrm{\text{\AA}}$. This structure
can therefore be considered a reference system in which ionic and electronic
polarisation effects are absent. We also obtain reference O–Fe and S–Fe
nearest-neighbour distances of $1.97\text{\,}\mathrm{\text{\AA}}$ and
$2.74\text{\,}\mathrm{\text{\AA}}$ respectively.

In the _cis_ high-energy structure, the oxygen ions have polar coordination
and exhibit a dipole moment of
$\mu_{\mathrm{O}}=$0.41\text{\,}\mathrm{\text{e}}\text{\,}\mathrm{\text{\AA}}$$.
The sulfur ions have non-polar coordination, and a dipole moment of
$\mu_{\mathrm{S}}=$0.00\text{\,}\mathrm{\text{e}}\text{\,}\mathrm{\text{\AA}}$$.
The O–Fe nearest-neighbour distance is effectively unchanged from the value
for the reference _trans_ structure, which suggests that the relative
stability of this structure relative to the $100\text{\,}\mathrm{\char
37\relax}$ _trans_ structure cannot be attributed to a change in O–Fe
distance, and instead is due to the polarisation of oxygen anions in this
_cis_ -OLi4Fe2-coordination structure.

In the _cis_ low-energy structure, all the oxygen ions and half the sulfur
ions have polar coordination, and the calculated dipole moments for these
polar-coordinated anions are
$\mu_{\mathrm{O}}=$0.33\text{\,}\mathrm{\text{e}}\text{\,}\mathrm{\text{\AA}}$$
and
$\mu_{\mathrm{S}}=$0.36\text{\,}\mathrm{\text{e}}\text{\,}\mathrm{\text{\AA}}$$,
respectively. The polarisation of both oxygen and sulfur correlates with the
lower energy of this $100\text{\,}\mathrm{\char 37\relax}$ _cis_ -OLi4Fe2
structure relative to the _cis_ high-energy structure. In the _cis_ high-
energy structure, only the oxygen anions are polarised. In the _cis_ ground-
state structure, half the sulfur anions are also polarised, which provides
additional dipole stabilisation, bringing the energy of this structure _lower_
than that of the _trans_ reference. For the _cis_ ground-state structure, the
O–Fe nearest-neighbour distance again is effectively unchanged from the
reference _trans_ value, while the S–Fe nearest-neighbour distance decreases
to $2.68\text{\,}\mathrm{\text{\AA}}$. This off-site sulfur displacement
potentially further stabilises this structure relative to the reference on-
site point-charge model.

In heteroanionic oxyfluorides and oxynitrides, transition metals with polar
six-coordinate _cis_ or _fac_ coordination move off-site to give shorter
transition metal–oxygen distances, which stabilises these coordination
geometries due to increased transition metal–oxygen covalent bonding [34, 33,
35, 2, 23, 17]. To characterise the degree of covalency in Li2FeSO, and
whether this varies with different short-range anion–cation coordination
configurations, we calculated integrated crystal orbital bond index
(ICOBI)[36] values for all adjacent Fe–O pairs in each of our exemplar
structures. We obtain values of
$0.22\text{\,}\mathrm{e}\mathrm{V}0.25\text{\,}\mathrm{e}\mathrm{V}$ for the
three structures, which are indicative of Li2FeSO being highly ionic 111The
ICOBI values reported here are comparable to those of ionic salts, such as
LiCl [36]. We also find no correlation between these ICOBI values and the
specific _cis_ versus _trans_ oxygen coordination in each structure (see the
Supporting Information for details).

Structure | $X$ |  $r$(Fe–$X$) / $\mathrm{\text{\AA}}$ |  $\left|\mu_{X}\right|$ / $\mathrm{\text{e}}\text{\,}\mathrm{\text{\AA}}$
---|---|---|---
_trans_ low-energy | Onon-polar | 1.97 | 0.00
Snon-polar | 2.74 | 0.00
_cis_ high-energy | Opolar | 1.96 | 0.41
Snon-polar | 2.75 | 0.00
_cis_ ground-state | Opolar | 1.99 | 0.33
Spolar | 2.68 | 0.36
Snon-polar | 2.73 | 0.00
Table 1: Iron anion bond lengths and anion dipoles for the different types of
anions in the different structures of interest.

The observation that anions with polar coordination are electronically
polarised, and that the lowest energy Li2FeSO structures are those with both
oxygen and sulfur polarisation, provides a general explanation for the
relative increased stability of structures with _cis_ -OLi4Fe2 relative to the
predictions of the simple point-charge model. _cis_ -OLi4Fe2 coordination
necessarily means polar oxygen coordination, and these oxygen ions will be
electronically polarised. For some _cis_ -OLi4Fe2 configurations, some
proportion of sulfur ions additionally will have polar coordination. These
sulfur ions therefore will also be polarised, further stabilising these
particular structures. We note that this effect of polarisation of anions with
polar coordination is not limited to _cis_ -OLi4Fe2 versus _trans_ -OLi4Fe2
coordination, and we expect all structures with polar O or S coordination to
exhibit some degree of stabilisation relative to a simple point-charge
description.

###  II.3 Short-range ordering dictates long-range ordering

The results presented above predict that Li2FeSO exhibits short-range order
characterised by a dual preference for OLi4Fe2 oxygen coordination and for
polar cation coordination around the O and S anions, e.g., _cis_ -OLi4Fe2 over
_trans_ -OLi4Fe2. The preference for polar cation-coordination around the
anions is attributed to the capacity for these asymmetrically-coordinated
anions to then polarise, with the resulting dipoles lowering the net
electrostatic energy for these cation configurations.

These short-range order preferences have a direct effect on the form and
degree of long-range order in Li2FeSO. The role of preferential short-range
order in directing long-range order is illustrated schematically in Fig. 6.
This figure shows examples for idealised $100\text{\,}\mathrm{\char 37\relax}$
_cis_ versus $100\text{\,}\mathrm{\char 37\relax}$ _trans_ -OLi4Fe2
coordination, i.e., the preferred local ordering as
$T\to$0\text{\,}\mathrm{K}$$ for OLi4Fe2 as modelled using DFT and as
predicted from a simple point-charge model, respectively.

Figure 6: Schematic illustrating how _cis_ versus _trans_ -OLi4Fe2 short-range
order produces long-range disorder—in the case of an all-_cis_ ground-state—or
long-range order—in the case of an all-_trans_ ground state—in the limit of
$T=$0\text{\,}\mathrm{K}$$.

For a Li/Fe configuration with $100\text{\,}\mathrm{\char 37\relax}$ _cis_
-OLi4Fe2 oxygen coordination, the relative cation configurations around pairs
of corner-sharing OLi4Fe2 octahedra are only partially correlated. For any
individual _cis_ -OLi4Fe2 octahedron, the neighbouring octahedra can adopt any
of four possible configurations that still preserve the local _cis_
coordination. This configurational underconstraint [38] means that even in a
$100\text{\,}\mathrm{\char 37\relax}$ _cis_ -OLi4Fe2 system, cation site
occupations will progressively decorrelate with increasing cation–cation
separation. At long range the Fe and Li site occupations are fully
uncorrelated (Fig. 6(a)), mirroring the lack of long-range cation order
observed in diffraction experiments [25, 26].

The configurational underconstraint exhibited in a $100\text{\,}\mathrm{\char
37\relax}$ _cis_ -OLi4Fe2 system can be contrasted with the character of a
$100\text{\,}\mathrm{\char 37\relax}$ _trans_ -OLi4Fe2 system. For this
different short-range order preference, the relative orientation of each
_trans_ -OLi4Fe2 coordinated octahedron fully constrains the orientations of
four “in-plane” neighbouring octahedra (Fig. 6b) This short-range
configurational constraint enforces long-range cation order in two dimensions;
adjacent 2D planes can have mutually parallel or perpendicular relative
orientations 222The lowest energy $100\text{\,}\mathrm{\char 37\relax}$
_trans_ -OLi4Fe2 configuration for both the DFT-CE and PC-CE models
corresponds to perpendicular Fe–O–Fe chains in each alternating 2D plane,
which maximises the distance between Fe––Fe pairs in these adjacent layers..
Hence, a structure comprised of corner-sharing A2B4-octahedra with a short-
range preference for _trans_ -coordination is expected to be long-range
ordered, while an otherwise equivalent structure with a short-range preference
for _cis_ -coordination is expected to be long-range disordered 333This
effect, where the geometry of structural building blocks dictates the degree
and form of long-range order in different materials, has previously been
discussed by Overy _et al_ in the context of generalised ice rules [64]. The
all-_cis_ and all-_trans_ structures discussed here (Fig. 6) correspond to the
C4C and C4T procrystalline systems, respectively..

This analysis of idealised $100\text{\,}\mathrm{\char 37\relax}$ _cis_ \- or
_trans_ -OLi4Fe2 systems illustrates that in the limit of
$T=$0\text{\,}\mathrm{K}$$ different short-range order preferences are
predicted to lead to qualitative differences in the degree of long-range
order. In as-synthesised Li2FeSO we expect a range of local coordination
environments, as sampled in our Monte Carlo simulations. The relationship
between short-range order and long-range order as a function of
configurational temperature can be quantified by considering appropriate
short-range and long-range order parameters. Fig. 7(a) and (b) show calculated
short-range and long-range order parameters, $\Phi_{\mathrm{SR}}$ and
$\Phi_{\mathrm{LR}}$, for the DFT-derived DFT-CE cluster expansion model and
the on-site point-charge PC-CE model, respectively, calculated as a thermal
average over all possible $2\times 2\times 2$ Li2FeSO supercells. For both
models the long-range order parameter is defined as the proportion of
collinear Fe–O–Fe units; in a $2\times 2\times 2$ supercell this is equivalent
to long-range Fe–O–Fe–O–Fe ordering. The short-range order parameter in each
case is defined as the proportion of _cis_ or _trans_ OLi4Fe2 units in the
DFT-CE and PC-CE models, respectively, i.e., it is the proportion of oxygen
coordination environments that adopt the preferential
$T=$0\text{\,}\mathrm{K}$$ short-range ordering.

For the DFT-CE model, at $T=$0\text{\,}\mathrm{K}$$, $\Phi_{\mathrm{SR}}=1$
and $\Phi_{\mathrm{LR}}=0$, as expected from the discussion above (Fig. 6). As
the temperature is increased, the short-range order parameter starts to
decrease, even at relatively low temperatures, and the long-range order
parameter shows a small increase. These short-range and long-range order
parameters continuously decrease and increase, respectively, as the
temperature is increased. Even at relatively high temperature, however, the
long-range order parameter $\Phi_{\mathrm{LR}}$ is still low, and there is no
temperature regime where significant long-range ordering is predicted. In
contrast, for the PC-CE ($\epsilon_{\mathrm{r}}=4$) model, there is strong
ordering at both short- and long-ranges until nearly
$T=$1000\text{\,}\mathrm{K}$$. Above $T\approx$1000\text{\,}\mathrm{K}$$ both
order parameters start to decrease. The correlation between
$\Phi_{\mathrm{SR}}$ and $\Phi_{\mathrm{LR}}$ is expected because of the
similarity in how these are defined for this system, and because of the long-
range ordering promoted by preferential _trans_ -OLi4Fe2 coordination.

The DFT-CE and PC-CE models therefore show somewhat qualitatively different
changes in their short-range order, and hence in their long-range order, as a
function of temperature. For the DFT-CE model, any increase in temperature
above $T=$0\text{\,}\mathrm{K}$$ progressively disrupts the preferential _cis_
-OLi4Fe2 coordination. For the PC-CE model, however, the short- and long-range
order are resistant to thermal disordering up to
$T=$1000\text{\,}\mathrm{K}$$. This difference in behaviour can be understood
by comparing the configurational densities of states (cDOS) for these two
models (Fig. 7(c) and (d)). The DFT-CE model gives a relatively narrow cDOS.
Not all cation configurations are thermally accessible at
$1025\text{\,}\mathrm{K}$, but the distribution of thermally accessible states
forms a continuous distribution that includes a relatively large number of
inequivalent cation configurations. For the point-charge PC-CE model, in
contrast, the thermally accessible states at $1025\text{\,}\mathrm{K}$ are
largely restricted to a very narrow distribution at low energy that is split
off from the other states by a large energy-gap. This low energy peak consists
entirely of $100\text{\,}\mathrm{\char 37\relax}$ _trans_ -OLi4Fe2
configurations, and we therefore observe strong long-range ordering even at
$1025\text{\,}\mathrm{K}$, with the sudden onset of disordering above this
temperature indicative of a formal order–disorder transition.

Figure 7: (a) and (b) Short- and long-range order parameters,
$\Phi_{\mathrm{SR}}$ and $\Phi_{\mathrm{LR}}$ computed as a thermally weighted
statistical average over all $2\times 2\times 2$ Li2FeSO supercells for the
DFT-CE and PC-CE ($\varepsilon_{\mathrm{r}}=4$) models. (c) and (d)
Configurational densities of states (cDOS) for all $2\times 2\times 2$ Li2FeSO
supercells for the DFT-CE and PC-CE ($\varepsilon_{\mathrm{r}}=4$) models. Red
shading indicates the distribution of thermally accessible states at
$T=$1025\text{\,}\mathrm{K}$$, computed by Boltzmann-weighting the full cDOS.

##  III Summary and Discussion

Here, we have reported a combined computational and experimental study of
cation order in the cation-disordered heterocationic lithium-ion cathode
material Li2FeSO. Our computational model predicts that lithium and iron
cations in as-synthesised Li2FeSO are not randomly distributed, as previously
proposed based on experimental diffraction data [25], but instead exhibit
partial short-ranged order, characterised by a strong preference for OLi4Fe2
oxygen coordination environments, with a weaker preference for these to be
_cis_ -OLi4Fe2 over _trans_ -OLi4Fe2. This result is supported by experimental
X-ray pair-distribution function (PDF) analysis: we show that our
computational model gives better agreement with these experimental PDF data
than either a fully-random cation distribution, or an ordered structure
constructed on the basis of point-charge electrostatics.

To better understand the physical basis for these short-range order
preferences, we performed an analysis of the energies of all possible $2\times
2\times 2$ Li2FeSO supercells, using our DFT-derived cluster-expansion model,
and compared these results to the corresponding energies predicted by a simple
electrostatic model, wherein the Li and Fe cations are treated as point
charges fixed at their formal crystallographic positions. This analysis showed
that, relative to the simple point-charge model, cation configurations with
polar anion coordination, e.g., _cis_ -OLi4Fe2, are stabilised relative to
configurations with non-polar anion coordination, e.g., _trans_ -OLi4Fe2.

The stabilisation of polar anion coordination environments in heterocationic
Li2FeSO is functionally similar to the stabilisation of polar cation
coordination environments in heteroanionic transition-metal oxyfluorides and
oxynitrides [2], such as NbO2F[32] and SrNbO2N[33] . In Li2FeSO, however, this
stabilisation of lower-symmetry polar coordination environments appears to
have a different physical origin. In transition-metal oxyfluorides and
oxynitrides, polar cation coordination environments are stabilised through
off-centre displacements of the central transition-metal cation, to give
enhanced covalent bonding between the central cation and the coordinating
heteroanions [34, 23, 33, 35, 17]. Here, we identify a different mechanism,
whereby _anions_ with polar heterocationic coordination are electronically
polarised, with the resulting dipoles lowering the net electrostatic energy of
the system. We expect this mechanism to be quite general for heterovalent
(cations with different formal oxidation states) heterocationic materials.
This mechanism of anion-polarisation–mediated stabilisation of polar
heterocationic coordination environments requires only that the central ion is
polarisable, and that polar coordination configurations of the heterocations
produce an electric field. This provides a possible explanation for the
difference in physical mechanism responsible for short-range order in Li2FeSO
versus previously studied heteroanionic transition-metal systems: the central
ions in heterocationic materials are softer more-polarisable anions, while in
heteroanionic materials the central ions are typically harder less-polarisable
high-valence cations.

We have also examined how the precise form of preferential short-range order
dictates the presence or absence of long-range order. In Li2FeSO, where the
preferential short-range ordering gives configurationally underconstrained
preferential _cis_ -OLi4Fe2 coordination, there is no long-range cation order,
even at $T=$0\text{\,}\mathrm{K}$$. We have contrasted this with the behaviour
in a system where short-range ordering is directed purely by point-charge
electrostatics. In this case, the preferential short-range ordering gives
configurationally constrained _trans_ -OLi4Fe2 coordination, and at
$T=$0\text{\,}\mathrm{K}$$ the system is fully long-range ordered. By
comparing different models for Li2FeSO we show that in the configurationally
underconstrained DFT-predicted system, short- and long-range order parameters
vary continuously at any non-zero temperature. In contrast, in the
configurationally constrained point-charge model, short- and long-range order
parameters show strong ordering to relatively high temperatures, before the
onset of partial disordering. This temperature dependent onset of disorder is
characteristic of a formal order–disorder transition, and can be understood as
a consequence of a significant energy gap between fully ordered and disordered
structures.

More generally, the results and analysis presented here illustrate how it can
be necessary to go beyond simple point-charge models to predict or understand
local structure in cation-disordered materials, and highlight the role of
anion polarisation in directing short-range order and consequently explaining
the presence or absence of long-range order.

##  IV Methods

###  IV.1 Computational Methods

All density functional theory (DFT) calculations were performed using the
plane-wave DFT code VASP [41, 42]. Interactions between core and valence
electrons were described using the projector-augmented-wave (PAW) method, [43]
with cores of [Mg] for Fe, [He] for O, [Ar] for S, and all electrons treated
as valence for Li.. We used the GGA functional PBEsol with a Dudarev $+U$
correction applied to the Fe $d$ states (GGA$+U$), with
$U_{\mathrm{Fe},d}=~{}$$5.3\text{\,}\mathrm{e}\mathrm{V}$ [44]. All
calculations used a plane-wave basis-set cut-off of
$720\text{\,}\mathrm{e}\mathrm{V}$. Reciprocal space was sampled using a
minimum $k$-point spacing of $0.25\text{\,}{\mathrm{\text{\AA}}}^{-1}$. For
each structure, the ionic positions and the cell parameters were relaxed until
all atomic forces were less than
$1\text{\times}{10}^{-2}\text{\,}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{\AA}}$.
To quantify anion polarisation in select structures we performed additional
post-processing to compute the set of maximally-localised Wannier functions
[45] using the Wannier90 code [46]. Dipoles on the ions of the cathode
material are obtained by associating Wannier centres with ions and calculation
of the dipole from the vectors between positively charged ionic cores and the
negatively charged ionic centres. Full details are provided in the supporting
information [reference to be added at publication].

To allow the computationally efficient evaluation of relative energies of
Li2FeSO structures with different Li/Fe configurations, we parametrised a
cluster-expansion effective Hamiltonian [47, 48] by fitting to the DFT-
calculated energies for 111 Li2FeSO configurations, using the icet and
trainstation packages [49], with a limit of a maximum of 40 non-zero features.
To reduce the risk of over-fitting, we use the Least Absolute Shrinkage and
Selection Operator (LASSO), in combination with recursive feature elimination.
The resulting cluster-expansion model gives a cross-validation score of
$8\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$.
To construct our simple point-charge model, we fit a second cluster expansion
Hamiltonian (PC-CE) to the energies obtained from an Ewald sum, for ions with
formal charges positioned at their corresponding crystallographic sites. This
model is fitted against all symmetry inequivalent arrangements of ions within
$2\times 1\times 1$ and $2\times 2\times 1$ supercells [50].

To model the probable distribution of different Li/Fe configurations, we
performed lattice Monte Carlo simulations using our parametrised cluster-
expansion Hamiltonian, using the mchammer software package [49]. These Monte
Carlo simulations were performed in the canonical ensemble for $8\times
8\times 8$ supercells using both the DFT and electrostatic fitted
Hamiltonians. Initial configurations were generated at random, and then
annealed from $20\,000\text{\,}\mathrm{K}$ to $1025\text{\,}\mathrm{K}$, which
corresponds to the experimental synthesis temperature, with $500\,000$
attempted steps (approximately $32.5$ MC Cycles), followed by a production run
at $1025\text{\,}\mathrm{K}$ of $1\,000\,000$ attempted steps (approximately
$641$ MC Cycles). Our reference distributions for fully random Li/Fe
configurations were generated from a sample of $1000$ random arrangements of
cations. For the modelling of PDF data we generated $4\times 4\times 4$
supercells using this same procedure, that were then relaxed using DFT using
the same protocol as for the initial training set. For PDF modelling of a
“random” Li/Fe distribution we used a special quasi-random structure [51]
within the same $4\times 4\times 4$ supercell.

The three structures used for comparison with the experimental PDF were
generated as follows: the $4\times 4\times 4$ DFT-CE structure was generated
from lattice Monte Carlo simulations using the icet library, following the
same MC protocal as described above; the $4\times 4\times 4$ RANDOM structure
was generated as a special quasi-random structure using icet; the ordered
_trans_ -structure is the electrostatic ground state, as obtained from a full
enumeration of all $2\times 2\times 2$ cells, with energies calculated by
Ewald summation using pymatgen.

###  IV.2 Synthesis and X-ray total scattering methods

For experimental characterisation, samples of Li2FeSO were prepared from
stoichiometric amounts of Li2O (Alfa Aesar, $99.5\text{\,}\mathrm{\char
37\relax}$), Fe (Alfa Aesar, $99.9\text{\,}\mathrm{\char 37\relax}$), and S
(Alfa Aesar, $99.5\text{\,}\mathrm{\char 37\relax}$) using a slightly modified
version of the previously reported method [25]. The homogenised reactants were
pressed into pellets, transferred in alumina crucibles and sealed within fused
silica tubes under vacuum. Deviating from the previously reported procedure,
the samples were placed directly in preheated furnaces at
$750\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The pellets were annealed for
$4\text{\,}\mathrm{h}$ at this temperature, with intermittent grinding, and
subsequently quenched in ice water. The resulting product was manually ground
to obtain a fine powder. All handling of the starting materials and products
was performed under dry inert gas atmosphere in an Ar-filled glove box.

X-ray total scattering data were collected at beamline I15-1 at the Diamond
Light Source with an X-ray beam of energy $76.69\text{\,}\mathrm{keV}$
($\lambda=$0.1617\text{\,}\mathrm{\text{\AA}}$$) and a PerkinElmer XRD 1611
CP3 area detector. Data reduction and normalisation were performed using DAWN
[52] and GudrunX [53, 54] respectively, with
$Q_{\textrm{min}}=$0.5\text{\,}{\mathrm{\text{\AA}}}^{-1}$$ and
$Q_{\textrm{max}}=$28.0\text{\,}{\mathrm{\text{\AA}}}^{-1}$$. Pair
distribution function (PDF) refinements were performed using the PDFgui
software [55]. PDF fits were performed in the range $1.5\,\leq r\leq 20\,$Å
the following parameters were refined in each case: scale factor, lattice
parameters $a$, $b$, and $c$, atomic correlation factor, and isotropic
displacement parameters for each element. A LiFeO2 disordered rocksalt side
phase was identified from conventional Rietveld analysis and was included in
the real-space refinements.

##  V Data and code availability

A dataset containing the inputs and outputs for all the DFT calculations
described in this paper is available from the University of Bath Research Data
Archive [56]. All code used for analysis of our raw DFT data is available on
GitHub [57]. This analysis uses the bsym [50], polyhedral-analysis [58], scipy
[59], matplotlib [60], numpy [61], pymatgen [62], and ase [63] packages.

##  VI Acknowledgements

The authors thank the Faraday Institution CATMAT and FutureCat projects
(EP/S003053/1, FIRG016, FIRG017) for financial support. This work used the
Michael computing cluster (FIRG019), the ARCHER2 UK National Supercomputer
Service, with access provided by our membership of the UK’s HPC Materials
Modelling Consortium (EP/R029431), and the Isambard 2 UK National Tier-2 HPC
Service operated by GW4 and the UK Met Office, and funded by EPSRC
(EP/T022078/1). The authors acknowledge the Diamond Light Source for
allocation of beamtime under proposal number CY27702. BJM thanks the Royal
Society for financial support (URF/R/191006). AGS thanks the STFC Batteries
Network for an Early Career Researcher Award (ST/R006873/1). HSG and ALG
acknowledge funding from the ERC (grant 788144).

## References

  * Kageyama  _et al._ [2018] H. Kageyama, K. Hayashi, K. Maeda, J. P. Attfield, Z. Hiroi, J. M. Rondinelli, and K. R. Poeppelmeier, Expanding frontiers in materials chemistry and physics with multiple anions, Nat. Commun. 9, 772 (2018).
  * Harada  _et al._ [2019] J. K. Harada, N. Charles, K. R. Poeppelmeier, and J. M. Rondinelli, Heteroanionic materials by design: Progress toward targeted properties, Adv. Mater. 31, 1805295 (2019).
  * Scanlon and Walsh [2012] D. O. Scanlon and A. Walsh, Bandgap engineering of ZnSnP2 for high-efficiency solar cells, Appl. Phys. Lett. 100, 251911 (2012).
  * Veal  _et al._ [2015] T. D. Veal, N. Feldberg, N. F. Quackenbush, W. M. Linhart, D. O. Scanlon, L. F. J. Piper, and S. M. Durbin, Band gap dependence on cation disorder in ZnSnN2 solar absorber, Adv. Energy Mater. 5, 1501462 (2015).
  * Chen  _et al._ [2021] W. Chen, D. Dahliah, G.-M. Rignanese, and G. Hautier, Origin of the low conversion efficiency in Cu2ZnSnS4 kesterite solar cells: the actual role of cation disorder, Energy Environ. Sci. 14, 3567 (2021).
  * Yong  _et al._ [2018] Z.-J. Yong, S.-Q. Guo, J.-P. Ma, J.-Y. Zhang, Z.-Y. Li, Y.-M. Chen, B.-B. Zhang, Y. Zhou, J. Shu, J.-L. Gu, L.-R. Zheng, O. M. Bakr, and H.-T. Sun, Doping-enhanced short-range order of perovskite nanocrystals for near-unity violet luminescence quantum yield, J. Am. Chem. Soc. 140, 9942 (2018).
  * Mangelis  _et al._ [2019] P. Mangelis, A. Aziz, I. da Silva, R. Grau-Crespo, P. Vaqueiro, and A. V. Powell, Understanding the origin of disorder in kesterite-type chalcogenides A2ZnBQ4 (A = Cu, Ag; B = Sn, Ge; Q = S, Se): the influence of inter-layer interactions, Phys. Chem. Chem. Phys. 21, 19311 (2019).
  * Hirscher  _et al._ [2020] M. Hirscher, V. A. Yartys, M. Baricco, J. B. von Colbe, D. Blanchard, R. C. Bowman, D. P. Broom, C. E. Buckley, F. Chang, P. Chen, Y. W. Cho, J.-C. Crivello, F. Cuevas, W. I. David, P. E. de Jongh, R. V. Denys, M. Dornheim, M. Felderhoff, Y. Filinchuk, G. E. Froudakis, D. M. Grant, E. M. Gray, B. C. Hauback, T. He, T. D. Humphries, T. R. Jensen, S. Kim, Y. Kojima, M. Latroche, H.-W. Li, M. V. Lototskyy, J. W. Makepeace, K. T. Møller, L. Naheed, P. Ngene, D. Noréus, M. M. Nygård, S. ichi Orimo, M. Paskevicius, L. Pasquini, D. B. Ravnsbæk, M. V. Sofianos, T. J. Udovic, T. Vegge, G. S. Walker, C. J. Webb, C. Weidenthaler, and C. Zlotea, Materials for hydrogen-based energy storage — past, recent progress and future outlook, J. Alloys Comp. 827, 153548 (2020).
  * Marques  _et al._ [2021] F. Marques, M. Balcerzak, F. Winkelmann, G. Zepon, and M. Felderhoff, Review and outlook on high-entropy alloys for hydrogen storage, Energy Environ. Sci. 14, 5191 (2021).
  * Lee  _et al._ [2014] J. Lee, A. Urban, X. Li, D. Su, G. Hautier, and G. Ceder, Unlocking the potential of cation-disordered oxides for rechargeable lithium batteries, Science 343, 519 (2014).
  * Clément  _et al._ [2020] R. J. Clément, Z. Lun, and G. Ceder, Cation-disordered rocksalt transition metal oxides and oxyfluorides for high energy lithium-ion cathodes, Energy Environ. Sci. 13, 345 (2020).
  * Minafra  _et al._ [2020] N. Minafra, M. A. Kraft, T. Bernges, C. Li, R. Schlem, B. J. Morgan, and W. G. Zeier, Local charge inhomogeneity and lithium distribution in the superionic argyrodites Li6PS5${X}$ (${X}$ = Cl, Br, I), Inorg. Chem. 59, 11009 (2020).
  * Morgan [2021] B. J. Morgan, Mechanistic origin of superionic lithium diffusion in anion-disordered Li6PS5${X}$ argyrodites, Chem. Mater. 33, 2004 (2021).
  * McColl  _et al._ [2022] K. McColl, R. A. House, G. J. Rees, A. G. Squires, S. W. Coles, P. G. Bruce, B. J. Morgan, and M. S. Islam, Transition metal migration and O2 formation underpin voltage hysteresis in oxygen-redox disordered rocksalt cathodes, Nat. Commun. 13, 5275 (2022).
  * Charles  _et al._ [2018] N. Charles, R. J. Saballos, and J. M. Rondinelli, Structural diversity from anion order in heteroanionic materials, Chem. Mater. 30, 3528 (2018).
  * Ji  _et al._ [2019] H. Ji, A. Urban, D. A. Kitchaev, D.-H. Kwon, N. Artrith, C. Ophus, W. Huang, Z. Cai, T. Shi, J. C. Kim, H. Kim, and G. Ceder, Hidden structural and chemical order controls lithium transport in cation-disordered oxides for rechargeable batteries, Nature Comm. 10, 10.1038/s41467-019-08490-w (2019).
  * Legein  _et al._ [2022] C. Legein, B. J. Morgan, M. Body, W. Li, M. Burbano, M. Salanne, T. Charpentier, O. J. Borkiewicz, and D. Dambournet, Correlated anion-disorder in heteroanionic cubic TiOF2 (2022), in preparation.
  * Seo  _et al._ [2016] D.-H. Seo, J. Lee, A. Urban, R. Malik, S. Kang, and G. Ceder, The structural and chemical origin of the oxygen redox activity in layered and cation-disordered Li-excess cathode materials, Nature Chem. 8, 692 (2016).
  * Cai  _et al._ [2021] Z. Cai, H. Ji, Y. Ha, J. Liu, D.-H. Kwon, Y. Zhang, A. Urban, E. E. Foley, R. Giovine, H. Kim, Z. Lun, T.-Y. Huang, G. Zeng, Y. Chen, J. Wang, B. D. McCloskey, M. Balasubramanian, R. J. Clément, W. Yang, and G. Ceder, Realizing continuous cation order-to-disorder tuning in a class of high-energy spinel-type Li-ion cathodes, Matter 4, 3897 (2021).
  * Morelock  _et al._ [2013] C. R. Morelock, B. K. Greve, M. Cetinkol, K. W. Chapman, P. J. Chupas, and A. P. Wilkinson, Role of anion site disorder in the near zero thermal expansion of tantalum oxyfluoride, Chem. Mater. 25, 1900 (2013).
  * Pilania  _et al._ [2020] G. Pilania, A. Ghosh, S. T. Hartman, R. Mishra, C. R. Stanek, and B. P. Uberuaga, Anion order in oxysulfide perovskites: origins and implications, npj Comput. Mater. 6, 10.1038/s41524-020-0338-1 (2020).
  * Johnston  _et al._ [2018] H. Johnston, A. P. Black, P. Kayser, J. Oró-Solé, D. A. Keen, A. Fuertes, and J. P. Attfield, Dimensional crossover of correlated anion disorder in oxynitride perovskites, Chem. Commun. 54, 5245 (2018).
  * Wolff and Dronskowski [2008] H. Wolff and R. Dronskowski, First-principles and molecular-dynamics study of structure and bonding in perovskite-type oxynitrides ${AB}$O2N (${A}$ = Ca, Sr, Ba; ${B}$ = Ta, Nb), J. Comput. Chem. 29, 2260 (2008).
  * Urban  _et al._ [2017] A. Urban, A. Abdellahi, S. Dacek, N. Artrith, and G. Ceder, Electronic-structure origin of cation disorder in transition-metal oxides, Phys. Rev. Lett. 119, 176402 (2017).
  * Lai  _et al._ [2017] K. T. Lai, I. Antonyshyn, Y. Prots, and M. Valldor, Anti-perovskite Li-battery cathode materials, J. Am. Chem. Soc. 139, 9645 (2017).
  * Mikhailova  _et al._ [2018] D. Mikhailova, L. Giebeler, S. Maletti, S. Oswald, A. Sarapulova, S. Indris, Z. Hu, J. Bednarcik, and M. Valldor, Operando studies of antiperovskite lithium battery cathode material Li2FeSO, ACS Appl. Energy Mater. 1, 6593 (2018).
  * Lu and Ciucci [2018] Z. Lu and F. Ciucci, Anti-perovskite cathodes for lithium batteries, J. Mater. Chem. A 6, 5185 (2018).
  * Lai  _et al._ [2018] K. T. Lai, I. Antonyshyn, Y. Prots, and M. Valldor, Extended chemical flexibility of cubic anti-perovskite lithium battery cathode materials, Inorg. Chem. 57, 13296 (2018).
  * Gorbunov  _et al._ [2020] M. V. Gorbunov, S. Carrocci, S. Maletti, M. Valldor, T. Doert, S. Hampel, I. G. G. Martinez, D. Mikhailova, and N. Gräßler, Synthesis of (Li2Fe1-yMny)SO antiperovskites with comprehensive investigations of (Li2Fe0.5Mn0.5)SO as cathode in Li-ion batteries, Inorg. Chem. 59, 15626 (2020).
  * Zhu and Scanlon [2022] B. Zhu and D. O. Scanlon, Predicting lithium iron oxysulfides for battery cathodes, ACS Appl. Energy Mater. 5, 575 (2022).
  * Pauling [1929] L. Pauling, The principles determining the structure of complex ionic crystals, J. Am. Chem. Soc. 51, 1010 (1929).
  * Brink  _et al._ [2002] F. J. Brink, R. L. Withers, and L. Norén, An electron diffraction and crystal chemical investigation of oxygen/fluorine ordering in niobium oxyfluoride, NbO2F, J. Sol. Stat. Chem 166, 73 (2002).
  * Yang  _et al._ [2010] M. Yang, J. Oró-Solé, J. A. Rodgers, A. B. Jorge, A. Fuertes, and J. P. Attfield, Anion order in perovskite oxynitrides, Nature Chem. 3, 47 (2010).
  * Withers  _et al._ [2007] R. L. Withers, F. Brink, Y. Liu, and L. Noren, Cluster chemistry in the solid state: Structured diffuse scattering, oxide/fluoride ordering and polar behaviour in transition metal oxyfluorides, Polyhedron 26, 290 (2007).
  * Attfield [2013] J. P. Attfield, Principles and applications of anion order in solid oxynitrides, Crys. Grow. Des. 13, 4623 (2013).
  * Müller  _et al._ [2021] P. C. Müller, C. Ertural, J. Hempelmann, and R. Dronskowski, Crystal orbital bond index: Covalent bond orders in solids, J. Phys. Chem. C 125, 7959 (2021).
  * Note [1] The ICOBI values reported here are comparable to those of ionic salts, such as LiCl [36].
  * Keen and Goodwin [2015] D. A. Keen and A. L. Goodwin, The crystallography of correlated disorder, Nature 521, 303 (2015).
  * Note [2] The lowest energy $100\text{\,}\mathrm{\char 37\relax}$ _trans_ -OLi4Fe2 configuration for both the DFT-CE and PC-CE models corresponds to perpendicular Fe–O–Fe chains in each alternating 2D plane, which maximises the distance between Fe––Fe pairs in these adjacent layers.
  * Note [3] This effect, where the geometry of structural building blocks dictates the degree and form of long-range order in different materials, has previously been discussed by Overy _et al_ in the context of generalised ice rules [64]. The all-_cis_ and all-_trans_ structures discussed here (Fig. 6) correspond to the C4C and C4T procrystalline systems, respectively.
  * Kresse and Joubert [1999] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  * Kresse and Hafner [1994] G. Kresse and J. Hafner,  _Ab initio_ molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium, Phys. Rev. B 49, 14251 (1994).
  * Blöchl [1994] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  * Wang  _et al._ [2006] L. Wang, T. Maxisch, and G. Ceder, Oxidation energies of transition metal oxides within the GGA$+{U}$ framework, Phys. Rev. B 73, 195107 (2006).
  * Marzari  _et al._ [2012] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012).
  * Pizzi  _et al._ [2020] G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Poncé, T. Ponweiser, J. Qiao, F. Thöle, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A. Mostofi, and J. R. Yates, Wannier90 as a community code: new features and applications, J. Phys.: Condens. Matter 32, 165902 (2020).
  * Sanchez  _et al._ [1984] J. Sanchez, F. Ducastelle, and D. Gratias, Generalized cluster description of multicomponent systems, Phys. A: Stat. Mech. Appl. 128, 334 (1984).
  * Sanchez [2010] J. M. Sanchez, Cluster expansion and the configurational theory of alloys, Phys. Rev. B 81, 224202 (2010).
  * Ångqvist  _et al._ [2019] M. Ångqvist, W. A. Muñoz, J. M. Rahm, E. Fransson, C. Durniak, P. Rozyczko, T. H. Rod, and P. Erhart, ICET – a Python library for constructing and sampling alloy cluster expansions, Adv. Theory Simul. 2, 1900015 (2019).
  * Morgan [2017] B. J. Morgan, bsym: A basic symmetry module, J. Open. Sour. Soft. 2, 370 (2017).
  * van de Walle  _et al._ [2013] A. van de Walle, P. Tiwary, M. de Jong, D. Olmsted, M. Asta, A. Dick, D. Shin, Y. Wang, L.-Q. Chen, and Z.-K. Liu, Efficient stochastic generation of special quasirandom structures, Calphad 42, 13 (2013).
  * Basham  _et al._ [2015] M. Basham, J. Filik, M. T. Wharmby, P. C. Y. Chang, B. E. Kassaby, M. Gerring, J. Aishima, K. Levik, B. C. A. Pulford, I. Sikharulidze, D. Sneddon, M. Webber, S. S. Dhesi, F. Maccherozzi, O. Svensson, S. B. G. Náray, and A. W. Ashtona, Data analysis workbench (DAWN), J. Synchro. Rad. 22, 853 (2015).
  * Soper and Barney [2011] A. K. Soper and E. R. Barney, Extracting the pair distribution function from white-beam X-ray total scattering data, J. Appl. Cryst. 44, 714–726 (2011).
  * Soper [2012] A. K. Soper, GudrunN and gudrunX: Programs for correcting raw neutron and X-ray total scattering data to differential cross section (2012).
  * Farrow  _et al._ [2007] C. L. Farrow, P. Juhas, J. Liu, D. Bryndin, E. S. Božin, J. Bloch, T. Proffen, and S. J. L. Billinge, PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals, J. Phys.: Condens. Matter 19, 335219 (2007).
  * S. W. Coles [a] S. W. Coles, Dft dataset for “anion-polarisation—directed short-range-order in antiperovskite Li2FeSO” (a), dOI TO BE ADDED.
  * S. W. Coles [b] S. W. Coles, Data analysis for “anion-polarisation—directed short-range-order in antiperovskite Li2FeSO” (b), github.com/user200000/data_Li2FeSO_structure.
  * Morgan [2020] B. Morgan, bjmorgan/polyhedral-analysis (2020), original-date: 2018-03-20T10:01:32Z.
  * Virtanen  _et al._ [2020] P. Virtanen, , R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, and P. van Mulbregt, SciPy 1.0: fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  * Hunter [2007] J. D. Hunter, Matplotlib: A 2D graphics environment, Comput. Sci. Eng. 9, 90 (2007).
  * Harris  _et al._ [2020] C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del Río, M. Wiebe, P. Peterson, P. Gérard-Marchant, K. Sheppard, T. Reddy, W. Weckesser, H. Abbasi, C. Gohlke, and T. E. Oliphant, Array programming with NumPy, Nature 585, 357 (2020).
  * Ong  _et al._ [2013] S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V. L. Chevrier, K. A. Persson, and G. Ceder, Python Materials Genomics (pymatgen): A robust, open-source Python library for materials analysis, Comp. Mat. Sci. 68, 314 (2013).
  * Hjorth Larsen  _et al._ [2017] A. Hjorth Larsen, J. J. Mortensen, J. Blomqvist, I. E. Castelli, R. Christensen, M. Dułak, J. Friis, M. N. Groves, B. Hammer, C. Hargus, E. D. Hermes, P. C. Jennings, P. B. Jensen, J. Kermode, J. R. Kitchin, E. L. Kolsbjerg, J. Kubal, K. Kaasbjerg, S. Lysgaard, J. B. Maronsson, T. Maxson, T. Olsen, L. Pastewka, A. Peterson, C. Rostgaard, J. Schiøtz, O. Schütt, M. Strange, K. S. Thygesen, T. Vegge, L. Vilhelmsen, M. Walter, Z. Zeng, and K. W. Jacobsen, The atomic simulation environment—a Python library for working with atoms, J. Phys.: Condens. Matter 29, 273002 (2017).
  * Overy  _et al._ [2016] A. R. Overy, A. B. Cairns, M. J. Cliffe, A. Simonov, M. G. Tucker, and A. L. Goodwin, Design of crystal-like aperiodic solids with selective disorder–phonon coupling, Nature Commun. 7, 10.1038/ncomms10445 (2016).