# A Hybrid Density Functional Theory Benchmark Study on Lithium Manganese
Oxides

Marco Eckhoff<EMAIL_ADDRESS>Universität Göttingen,
Institut für Physikalische Chemie, Theoretische Chemie, Tammannstraße 6, 37077
Göttingen, Germany.  Peter E. Blöchl<EMAIL_ADDRESS>Technische
Universität Clausthal, Institut für Theoretische Physik, Leibnizstraße 10,
38678 Clausthal-Zellerfeld, Germany.  Universität Göttingen, Institut für
Theoretische Physik, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany.  Jörg
Behler<EMAIL_ADDRESS>Universität Göttingen, Institut für
Physikalische Chemie, Theoretische Chemie, Tammannstraße 6, 37077 Göttingen,
Germany.  Universität Göttingen, International Center for Advanced Studies of
Energy Conversion (ICASEC), Tammannstraße 6, 37077 Göttingen, Germany

###### Abstract

The lithium manganese oxide spinel LixMn2O4, with $0\leq x\leq 2$, is an
important example for cathode materials in lithium ion batteries. However, an
accurate description of LixMn2O4 by first-principles methods like density
functional theory is far from trivial due to its complex electronic structure,
with a variety of energetically close electronic and magnetic states. It was
found that the local density approximation as well as the generalized gradient
approximation (GGA) are unable to describe LixMn2O4 correctly. Here, we report
an extensive benchmark for different LixMnyOz systems using the hybrid
functionals PBE0 and HSE06, as well as the recently introduced local hybrid
functional PBE0r. We find that all of these functionals yield energetic,
structural, electronic, and magnetic properties in good agreement with
experimental data. The notable benefit of the PBE0r functional, which relies
on on-site Hartree-Fock exchange only, is a much reduced computational effort
that is comparable to GGA functionals. Furthermore, the Hartree-Fock mixing
factors in PBE0r are smaller than in PBE0, which improves the results for
(lithium) manganese oxides. The investigation of LixMn2O4 shows that two Mn
oxidation states, +III and +IV, coexist. The Mn${}^{\text{III}}$ ions are in
the high-spin state and the corresponding MnO6 octahedra are Jahn-Teller
distorted. The ratio between Mn${}^{\text{III}}$ and Mn${}^{\text{IV}}$ and
thus the electronic structure changes with the Li content while no major
structural changes occur in the range from $x=0$ to $1$. This work
demonstrates that the PBE0r functional provides an equally accurate and
efficient description of the investigated LixMnyOz systems.

Lithium Manganese Oxide, PBE0r, HSE06, PBE0, Benchmark

##  I Introduction

Today, a world without lithium ion batteries is hard to imagine because they
are essential for the energy supply of almost all portable electronic devices
from mobile phones to laptop computers. The lithium manganese oxide spinel
LixMn2O4, an intercalation compound with Li contents $0\leq x\leq 2$, is a
prominent example for cathode materials in lithium ion batteries,[1] which
offers advantages such as low costs and non-toxicity.[2]

Figure 1: Spinel structure of LiMn2O4.[3] Li is colored green, O red, and Mn
purple. The unit cell is marked by black lines. Panel (a) shows the
coordination polyhedra of Li (tetrahedra) and Mn (octahedra) while (b)
includes only the atoms of the unit cell containing eight formula units. This
and all other figures in this work were created with VESTA version 3.4.4.[4]

The Li content can be varied over a wide range of $0\leq x\leq 2$ using, for
example, electrochemical (de)intercalation.[5] At temperatures above $\sim$
290 K, for compositions $0\leq x\leq 1$ the crystal exhibits a cubic spinel
structure with the space group Fd$\overline{3}$m in which Li occupies the
tetrahedral 8a sites and Mn the octahedral 16d sites (FIG. 1).[3] The MnO6
octahedra share one half of their edges with other MnO6 octahedra. Each of
their corners is shared with one LiO4 tetrahedron in the case of LiMn2O4. The
MnO6 octahedra build a superstructure of corner-sharing
$\left(\text{MnO}_{6}\right)_{4}$ tetrahedra. Complete delithiation results in
the formation of $\uplambda$-Mn2O4 exhibiting the same manganese oxide host
lattice, while all tetrahedral sites are unoccupied. Thus, no major internal
structural changes occur when reducing the Li contents from $x=1$ to $x=0$,
while the lattice constant is approximately linearly decreasing. The varying
content of Li ions is balanced by changes in the oxidation states of the Mn
ions to maintain overall charge neutrality. Consequently, in LiMn2O4, one half
of the Mn ions is in the high-spin (hs) Mn${}^{\text{III}}$ state
(t${}_{\mathrm{2g}}^{3}$e${}_{\mathrm{g}}^{1}$) while the other half is in the
oxidation state Mn${}^{\text{IV}}$
(t${}_{\mathrm{2g}}^{3}$e${}_{\mathrm{g}}^{0}$).[2] The lithium-free
$\uplambda$-Mn2O4 contains exclusively Mn${}^{\text{IV}}$ ions.
The hs-Mn${}^{\text{III}}$ ions lead to Jahn-Teller (JT) distorted MnO6
octahedra.[6] Therefore, the cubic crystal structure can be understood as a
disordered arrangement of Mn${}^{\text{III}}$ and Mn${}^{\text{IV}}$,[7] in
which the JT distortions are thermally averaged in all spatial directions.
Below $\sim$ 290 K, an increasing ordering of the Mn ions and the alignment of
the distorted octahedra results in a transformation of the spinel structure to
an orthorhombic phase with space group Fddd.[8] At very low temperatures, an
antiferromagnetic long-range order as well as spin-glass behavior were
observed.[9, 10]
For $x>1$, a phase transition takes place to a tetragonal spinel structure
with space group I41/amd.[2] This transition is caused by the increasing
number of Mn${}^{\text{III}}$ ions and the associated JT distorted MnO6
octahedra upon lithiation. A miscibility gap exists between the cubic ($x=1$)
and tetragonal ($x=2$) spinel structure.[11] If the average Li content is
higher than $x=1$, the tetragonal phase with $x=2$ will form and both phases
coexist in the range $1<x<2$.
Detailed insights into the structure of LixMn2O4 have been gained in various
experiments, e.g. X-ray diffraction, transmission electron microscopy, and
atom probe tomography.[12, 11] Theoretical studies are able to provide
complementary information, for example, about the underlying atomistic
processes of the phase transitions or about the Li diffusion pathways.
However, the underlying electronic structure of LixMn2O4 is very complex due
to the large number of energetically close electronic and magnetic states.
Therefore, a theoretical treatment using density functional theory (DFT) is
far from trivial. Previous studies could show that the local density
approximation (LDA) as well as the generalized gradient approximation (GGA) do
not yield a qualitatively correct electronic structure,[7, 13, 14, 15] since
all Mn ions in LiMn2O4 are found to be in an averaged oxidation state of 3.5.
Further, in contrast to experiment no band gap exists. Consequently, GGA$+U$
or hybrid functionals are required to correctly obtain distinct
Mn${}^{\text{III}}$ and Mn${}^{\text{IV}}$ ions as well as a qualitatively
correct band gap,[7, 16] which is about 1.2 eV in experiment.[17]
Manganese is known for its wide range of possible oxidation states and complex
magnetic structures of its compounds. Experimental studies could show that
even elemental $\upalpha$-Mn exhibits a non-collinear antiferromagnetic
structure.[18] The Mn${}^{\text{II}}$ ions in MnO are arranged in a way that
the antiferromagnetic order is present in all three cubic directions.[19] The
magnetic structure of the Mn${}^{\text{III}}$ ions in $\upalpha$-Mn2O3 was
investigated by a combined GGA$+U$ and neutron diffraction study.[20] The
outcome is a complex non-collinear antiferromagnetic order. In $\upbeta$-MnO2
the Mn${}^{\text{IV}}$ ions crystallise in an antiferromagnetic structure with
helically ordered magnetic moments.[21]

Figure 2: Structures of (a): $\uplambda$-Mn2O4,[22] (b): Li0.5Mn2O4,[23] (c):
LiMn2O4,[3] (d): Li2Mn2O4,[24] (e): Li,[25] (f): Li2O2,[26] (g): Li2O,[27]
(h): $\upalpha$-Mn,[25] (i): MnO,[28] (j): Mn3O4,[29] (k):
$\upalpha$-Mn2O3,[30] (l): $\upbeta$-MnO2,[31] and (m): LiMnO2.[32] Li is
colored green, O red, and Mn purple. The black lines represent the unit
cell.[4]

In this study, we investigate the accuracy of the well-established PBE0[33,
34] and HSE06[35, 36, 37] functionals as well as the recently developed local
hybrid functional PBE0r[38] for these structures and a variety of other
systems containing lithium, manganese, and oxygen. The benchmarked properties
include formation energies, structural properties, and the density of states.
Furthermore, the magnetic order and the intercalation potential of LixMn2O4
are determined. With this extensive and rigorous benchmark we examine the
quality of hybrid DFT calculations for this class of materials and test the
approximations in HSE06 and PBE0r compared to PBE0 to increase the efficiency.
Especially, we optimize the admixture of Hartree-Fock (HF) exchange in the
PBE0r hybrid functional for the calculation of LixMnyOz systems. We validate
its approach of including only on-site HF exchange terms and keep the off-site
terms on the GGA level aiming a very efficient functional which provides a
high accuracy for these systems. Additionally, in the Supporting Information
(SI) the D3 method[39] is evaluated for these systems as a possible
approximate correction to overcome limitations in the description of van der
Waals interactions in current hybrid functionals.
The main focus of our work is on the LixMn2O4 spinel structure. Therefore, our
benchmark set includes spinels with varying Li contents, specifically the
X-ray diffraction structures of $\uplambda$-Mn2O4 (at room temperature),[22]
Li0.5Mn2O4 (at 293 K),[23] LiMn2O4 (at 330 K),[3] and Li2Mn2O4 (at room
temperature)[24] (FIG. 2 (a)-(d)). Moreover, several related systems, which
were thoroughly investigated theoretically and experimentally before, were
chosen to benchmark the exchange-correlation functionals. They comprise the
X-ray diffraction structures of Li (at 78 K),[25] Li2O2,[26], Li2O,[27]
$\upalpha$-Mn,[25] MnO,[28] Mn3O4,[29] $\upalpha$-Mn2O3,[30] $\upbeta$-MnO2
(neutron diffraction),[31] and orthorhombic LiMnO2[32] (FIG. 2 (e)-(m)). These
structures, which refer to room temperature unless stated differently, cover
both common Li oxidation states of 0 and +I and the Mn oxidation states 0,
+II, +III, and +IV. This benchmark set enables us to find optimal settings for
the PBE0r functional to describe most of the LixMnyOz systems accurately.
Furthermore, the molecules H2, O2, and H2O are part of the benchmark set
because O2 is required as reference for the calculation of the formation
energies and its total energy is checked using the formation energy of water.
Moreover, we use a water molecule as an example for a covalently bonded system
in the discussion of the results.

##  II Methods

While Kohn-Sham DFT is in principle exact, approximate exchange-correlation
functionals $E_{\mathrm{xc}}$ need to be employed. In recent decades, a
hierarchy of functionals was proposed, which can have a notable impact on the
quality of the obtained results.[40, 41, 42] In this work, we will address the
performance of modern hybrid functionals, which currently represent the state-
of-the-art, using the examples PBE0, HSE06, and PBE0r.
PBE0[33, 34] is based on the PBE[43] GGA functional, but 25% of the PBE
exchange $E_{\mathrm{x}}^{\mathrm{PBE}}$ are replaced by exact HF exchange
$E_{\mathrm{x}}^{\mathrm{HF}}$ resulting in

$\displaystyle
E_{\mathrm{xc}}^{\mathrm{PBE0}}=E_{\mathrm{xc}}^{\mathrm{PBE}}+\dfrac{1}{4}\left(E_{\mathrm{x}}^{\mathrm{HF}}-E_{\mathrm{x}}^{\mathrm{PBE}}\right)\
.$ (1)

The calculation of the HF exchange increases the computational costs
drastically due to the long-range nature of the Coulomb interaction. Replacing
the Coulomb interaction in the exchange by a screened interaction reduces the
number of integrals to be evaluated, and it recovers the subtle balance of
exchange and correlation for electrons at large distances. In the HSE06
functional [35, 36, 37] this is realized by a screened Coulomb potential,

$\displaystyle\dfrac{1}{r}=\dfrac{1-\mathrm{erf}(\omega
r)}{r}+\dfrac{\mathrm{erf}(\omega r)}{r}\ ,$ (2)

i.e. the Coulomb potential is separated into a short-range and a long-range
part. The separation range is determined by the screening parameter,
$\omega=0.11\,a_{0}$, which was empirically obtained by calibration to
experimental properties.[37] $a_{0}$ is the Bohr radius. Consequently, the HF
exchange is only calculated for the short-range part (superscript s) but not
for the long-range part (superscript l), which reduces the computational cost
substantially for extended systems,

$\displaystyle
E_{\mathrm{xc}}^{\mathrm{HSE06}}=\dfrac{1}{4}E_{\mathrm{x}}^{\mathrm{HF,\,s}}+\dfrac{3}{4}E_{\mathrm{x}}^{\mathrm{PBE,\,s}}+E_{\mathrm{x}}^{\mathrm{PBE,\,l}}+E_{\mathrm{c}}^{\mathrm{PBE}}\
.$ (3)

The correlation part $E_{\mathrm{c}}^{\mathrm{PBE}}$ is not affected by the
screened Coulomb potential. For a screening parameter $\omega=0$ the
functional is equal to PBE0. For $\omega=\infty$ the functional is identical
to PBE.
The recently published PBE0r functional[38] is a local hybrid exchange-
correlation functional which is also derived from PBE0. The Kohn-Sham orbitals
are mapped onto a minimal basis of localized atom-centered tight-binding
orbitals. The tight-binding orbitals are used to calculate the on-site HF
exchange terms including the exchange interaction between core and valence
electrons. All other exchange contributions, i.e. those with tight-binding
orbitals centered on different atoms, are neglected. Hence, PBE0r can be
regarded as range-separated hybrid functional where the cutoff of the exchange
interaction is defined by the localized tight-binding orbitals. The PBE0r
exchange-correlation functional $E_{\mathrm{xc}}^{\mathrm{PBE0r}}$ is given by

$\displaystyle
E_{\mathrm{xc}}^{\mathrm{PBE0r}}=E_{\mathrm{xc}}^{\mathrm{PBE}}+\sum_{n=1}^{N}a_{n}\left(E_{\mathrm{x},\,n}^{\mathrm{HF,\,r}}-E_{\mathrm{x},\,n}^{\mathrm{PBE,\,r}}\right)\
.$ (4)

Since the inclusion of HF exchange is restricted to the on-site terms,
$E_{\mathrm{x}}^{\mathrm{HF,\,r}}$, only the corresponding PBE exchange terms,
$E_{\mathrm{x}}^{\mathrm{PBE,\,r}}$, are subtracted in order to avoid double
counting. The HF mixing factor $a_{n}$ of the $N$ atoms can vary for the
chemical elements in a given system. For the determination of the HF mixing
factors different routes were applied in previous studies: fitting according
to ground-state properties,[44] using the inverse of the dielectric
constant,[45, 46] employing a dielectric model dependence[47, 48, 49],
applying self-consistent schemes,[50, 51] or derivation from perturbation
theory arguments as it was the case for PBE0.[33] We choose to perform a
systematic search on a grid of HF mixing factors which can be different for
each element to get a set of mixing factors which yields good agreement with
known experimental and theoretical reference data. Using the grid approach we
can identify trends which lead us to a good compromise for all reference data.
The procedure of the search and the trends are described in the SI.
Our empirically determined optimal mixing factors for the LixMnyOz systems are
between 0.05 and 0.09. Lower mixing factors than the PBE0 value of 0.25 are
also applied in previous studies on transition-metal perovskites including Mn,
which employ mixing factors of 0.15 in HSE06[44] and between 0.07 and 0.15 in
PBE0r.[38] Moreover, a work on transition-metal complexes concludes that the
optimal admixture of HF exchange is between 0.08 and 0.16 in B3LYP[52] for the
calculation of FeII-S complexes.[53]
Replacing the local exchange used in LDA and GGA functionals by exact exchange
terms of HF has a strong affect on the Kohn-Sham band structure. Note, that
the GW method[54], a many-body Green’s functional method, has a similar
structure as the HF method, albeit with a screened interaction in the exchange
term instead of the long-range Coulomb interaction. Therefore, it is not
surprising that the Kohn-Sham band structure of hybrid functionals tend to
agree better with quasiparticle spectra than that of local density functionals
such as LDA and GGA. In order to rationalize different types of hybrid
functionals, we find it useful to distinguish three types of HF exchange
terms:
1\. On-site exchange acts between orbitals centered on the same site. The main
effect can be attributed to the self-interaction correction: Otherwise
degenerate states split into a multiplet of filled and another one of empty
orbitals. The two bands can be attributed to the Mott-Hubbard bands and their
separation is roughly proportional to the $U$-parameter. On-site exchange is
important in narrow, partially filled d and f shells as they are present in
many transition metal oxides.
2\. Bond exchange consists of an exchange term, for which a density on one
site interacts with that of another site. This exchange term is sensitive to
the phase relation of the orbitals on the two sites and it distinguishes
bonding and antibonding states. Bond exchange opens the band gap of covalent
materials such as silicon.
3\. Long-range exchange is analogous to bond exchange. It acts over longer
distances than bond distances. This term affects metallic solids and results
for a free electron gas in a vanishing density of states at the Fermi level.
This behavior is caused by the long-range tail of the Coulomb interaction in
the exchange terms, which is effectively removed by screening.
GGAs such as PBE describe many transition metal oxides poorly, because they
lack exact on-site exchange which splits the d shell into filled and empty
orbitals. The on-site exchange terms are, however, well captured by the PBE0r
functional. On the other extreme of local functionals, the PBE0 functional
overestimates long-range exchange, so that solids, in particular metals, are
not described adequately.[55] This problem is remedied by range separated
hybrid functionals such as HSE06. While one of the motivations for HSE06 was
to limit the computational effort, it also captures an important physical
effect, namely screening. In the PBE0r functional the range separation is
carried further: it removes not only long-range exchange but also bond
exchange. Except for the replaced on-site exchange terms PBE0r is the same as
PBE, i.e. PBE0r does not miss any terms. Due to the drastically reduced number
of included HF terms compared to PBE0, whose calculation would account for
most of the computational costs, the computational effort of PBE0r is
comparable to GGA functionals. While PBE0r is inadequate to describe covalent
materials such as silicon on a higher level than PBE, it is suitable for an
accurate description of the LixMnyOz systems discussed in the present study.

##  III Computational Details

The PBE0r calculations were performed using the Car-Parrinello Projector
Augmented-Wave (CP-PAW) code (version from 28${}^{\text{th}}$ September 2016)
which applies the projector augmented-wave (PAW) method[56] for electronic
structure calculations. The augmentation of the PAW method included the 1s
orbital of H, the 2s and 2p orbitals of Li, the 2s, 2p, and 3d orbitals of O,
and the 3s, 3p, 3d, 4s, and 4p orbitals of Mn. In the case of Mn, besides the
4s and 3d orbitals also the 3s and 3p orbitals were treated as valence
electrons, because these semi-core states are required to describe the
electronic structure of Mn in the systems of the benchmark set properly. The
matching radii for the construction of the auxiliary partial waves in units of
the covalent radii were set to 0.7 for all orbitals. The covalent radii were
set to 0.32 Å for H, 1.23 Å for Li, 0.73 Å for O, and 1.17 Å for Mn. The
auxiliary wave functions were constructed as node-less partial waves.[57] The
tight-binding orbitals include the 1s orbital of H, the 2s orbital of Li, the
2s and 2p orbitals of O, and the 3s, 3p, 3d, and 4s orbitals of Mn. The mixing
factors $a_{m}$ for the HF exchange were adjusted to minimize the errors of
the formation energies and band gaps of the benchmark set using experimental
reference data, which are given and referenced in the following chapter. The
obtained $a_{m}$ values are 0.07 for H, 0.07 for Li, 0.05 for O, and 0.09 for
Mn. The determination of the given mixing factors is described in detail in
the SI. Moreover, the complete settings for each element are given in the SI.
As the non-collinear treatment of the spins would increase the computational
effort for the benchmark systems significantly, the approximation of collinear
spin-polarization was applied. The plane wave cutoff was 25 $E_{\mathrm{H}}$
(Hartree) for the auxiliary wave functions and 100 $E_{\mathrm{H}}$ for the
auxiliary densities. With these settings, the obtained formation energies
deviate less than 0.01 eV per atom from the complete basis set limit. The
$\boldsymbol{\Gamma}$-centered $\mathbf{k}$-point grid was set to $2\times
2\times 2$ for the LiMn2O4 unit cell, and for the other systems
$\mathbf{k}$-point grids of a comparable $\mathbf{k}$-point density were
chosen. For metallic systems, the improved tetrahedron method was used.[58]
This ensures a convergence level of about 0.001 eV per atom for energies
differences. Molecular systems were placed in a large periodic cell with
lattice vectors (0 11.5 11.5)T Å, (12 0 12)T Å, and (12.5 12.5 0)T Å using
only the $\boldsymbol{\Gamma}$-point. The long-ranged electrostatic
interactions were decoupled from the periodic images for the molecules.[59]
The cell size was converged so that no artificial interactions between
periodic images are taken into account for molecular systems. Wave function
and geometry optimizations were performed using the Car-Parrinello ab-initio
molecular dynamics method[60] with a friction term which quenches the system
to the ground state. This enabled efficient optimizations of the atomic
positions in the unit cell. The computational costs of geometry optimizations
increased only by roughly a factor of 2 compared to single-point calculations
depending on the initial structure. For metallic systems, the Mermin
functional[61] was applied to treat variable occupations of the one-electron
energy eigenstates. The total energy was minimized up to a numerical
convergence of $10^{-5}\,E_{\mathrm{H}}$ for the given settings. PBEPAW
calculations were performed with the CP-PAW code as well using the same
settings with the exception that all HF mixing factors were set to zero.
The PBE, HSE06, and PBE0 calculations were performed using the Fritz-Haber-
Institute ab initio molecular simulations (FHI-aims) package (version
160328_3)[62] which is an all-electron electronic structure code with numeric
atom-centered basis functions. Again, a collinear treatment of the spin-
polarization was applied. The default light basis set of FHI-aims was used
which achieves a finite basis set error of less than 0.04 eV per atom for
energy differences like formation energies. A $\boldsymbol{\Gamma}$-centered
$\mathbf{k}$-point grid was used for periodic systems. The density of the
$\mathbf{k}$-point grid was the same as in the CP-PAW calculations. The error
of the finite $\mathbf{k}$-point grid is less than 0.001 eV per atom for
energy differences unless metallic systems are present, then the error can be
up to 0.04 eV per atom. Molecular systems were calculated in a non-periodic
environment which saved computational resources while full numerical
consistency of the settings with periodic calculations was maintained.
Geometry optimizations were performed using the Broyden-Fletcher-Goldfarb-
Shanno algorithm[63, 64, 65, 66] up to a numerical convergence of 0.001 eV of
the system’s total energy whereby the used forces provide an accuracy of
$2\cdot 10^{-4}$ eV/Å. An exception were the $\upalpha$-Mn calculations where
the total energy and forces were only converged up to 0.01 eV and $2\cdot
10^{-3}$ eV/Å, respectively. The total energies themselves were converged in
every iteration of the geometry optimizations in a self-consistent field
procedure to a numerical accuracy of $10^{-5}\,\mathrm{eV}$. For metallic
systems, the zero-broadening corrected energies were used. Further details of
the FHI-aims calculations are given in the SI.

##  IV Results and Discussion

###  IV.1 Magnetic Order

The experimentally determined atomic structures referenced in Sec. I were
taken as starting geometry of the electronic structure calculations. The
structures were optimized by the individual DFT functionals under the
constraint of fixed lattice vectors which were taken from experimental data.
For most of the systems the initial spin configurations were taken from the
references mentioned in Sec. I or they were derived by projecting the given
non-collinear spins onto a collinear arrangement. Otherwise, a search for the
minimum energy spin configuration was performed. The procedure for identifying
the minimum energy spin configuration as well as the classification of the hs-
Mn${}^{\text{III}}$ and Mn${}^{\text{IV}}$ ions is described in the SI. The
initial spins were fully optimized including a possible reordering – but no
spin-flips – in the subsequent electronic structure calculations.
The spin configuration of cubic LiMn2O4 was investigated in several previous
theoretical and experimental studies[7, 10, 67, 9, 68] which showed that the
corner-sharing $\left(\text{MnO}_{6}\right)_{4}$ tetrahedra, formed for
example by the four upper left Mn ions shown in FIG. 3, generally contain two
Mn${}^{\text{III}}$ and two Mn${}^{\text{IV}}$ ions. Furthermore, an
antiferromagnetic long-range order at low temperatures was proposed. Our
calculations confirm this result in that a ferromagnetic structure is
energetically less stable then a configuration with an overall zero magnetic
moment.

Figure 3: The lowest-energy oxidation and spin states of the Mn atoms in
LiMn2O4 obtained by PBE0r (a) as well as by PBE0 and HSE06 (b). The size and
color represent the position of Mn in the $z$ direction of the unit cell: from
large/bright (top layer) to small/dark (bottom layer). The Roman numerals
correspond to the oxidation states. Filled and empty circles represent the two
spin channels. The lines define the $\left(\text{MnO}_{6}\right)_{4}$
tetrahedra.

FIG. 3 illustrates the lowest-energy oxidation and spin states of the Mn ions
calculated by the hybrid functionals. First of all, we note that there are
many other configurations with very similar energies differing only by a few
meV per atom. While the PBE0 and HSE06 calculations yield the same
configuration, the result obtained in the PBE0r calculations differs slightly.
Both configurations have in common that the spins of the Mn ions are ordered
in antiferromagnetically coupled (100) planes (planes of filled and empty
circles in FIG. 3). Moreover, in each plane there is an equal number of
Mn${}^{\text{III}}$ and Mn${}^{\text{IV}}$ ions. Additionally, all
$\left(\text{MnO}_{6}\right)_{4}$ tetrahedra consist of two
Mn${}^{\text{III}}$ and two Mn${}^{\text{IV}}$ ions. This is in agreement with
the previous studies mentioned above. However, the distribution of Mn atoms
with different oxidation states within the network of tetrahedra is not the
same in PBE0 and HSE06 on the one hand and PBE0r on the other hand. Still, the
second lowest minimum found by HSE06 is the minimum of PBE0r. The total energy
difference between these two spin configurations is only 0.002 eV per atom,
which is within the remaining uncertainty of the hybrid functionals. The PBE0r
minimum configuration is also among the energetically lowest configurations of
PBE0. The PBE0 energy difference between the PBE0r minimum configuration and
the PBE0 minimum is 0.003 eV per atom and thus very small. In conclusion, the
energy differences are one order of magnitude smaller than the error of the
finite basis set (0.01 eV per atom) in all of these calculations. The latter
is in the range of the experimental uncertainty for formation enthalpies.[69,
70] Therefore, the two magnetic orders in FIG. 3 can be considered as
degenerate within the given accuracy.
Because both spin configurations show antiferromagnetically coupled (100)
planes with an equal number of Mn${}^{\text{III}}$ and Mn${}^{\text{IV}}$ ions
and all $\left(\text{MnO}_{6}\right)_{4}$ tetrahedra also contain two
Mn${}^{\text{III}}$ and two Mn${}^{\text{IV}}$ ions, there is no fundamental
difference in the description of the magnetic order by PBE0, HSE06, and PBE0r.
Therefore, and because the energies of all configurations are very similar, in
the remaining part of this work the PBE0r minimum configuration is used for
all calculations.

Figure 4: The lowest-energy oxidation and spin states of the Mn atoms
calculated consistently by all hybrid functionals for (a): $\uplambda$-Mn2O4
and (b): Li0.5Mn2O4. The size and color represent the position of Mn in the
$z$ direction of the unit cell: from large/bright (top layer) to small/dark
(bottom layer). The Roman numerals correspond to the oxidation states. Filled
and empty circles represent the two spin channels. The lines define the
$\left(\text{MnO}_{6}\right)_{4}$ tetrahedra.

Next, the oxidation states and spin directions in the minimum energy
configurations of Li0.5Mn2O4 and $\uplambda$-Mn2O4 are investigated. For both
systems, the HSE06, PBE0, and PBE0r fully agree with each other (FIG. 4). The
spin directions of the Mn ions in Li0.5Mn2O4 are oriented in the same way as
in LiMn2O4. As a consequence of the reduced Li content the number of
Mn${}^{\text{III}}$ ions in Li0.5Mn2O4 is reduced to four per unit cell
resulting in the ratio Mn${}^{\text{III}}$:Mn${}^{\text{IV}}$ of 1:3 in each
spin plane. The energetically lowest order of the Mn spins is different in the
case of $\uplambda$-Mn2O4 compared to those of the lithiated compounds. The
planes of equal spin are replaced by an alternating sequence in which the spin
direction changes every second Mn ion. As in the case of LiMn2O4 also for
Li0.5Mn2O4 and $\uplambda$-Mn2O4 several magnetic orders with energies
differing only in the order of a few meV per atom exist. The resulting minimum
energy spin configurations of the PBE0r geometry optimized structures are
given in the SI for all benchmark systems considered in this study.

###  IV.2 Formation Energies

Arguably, the most important benchmark property is the relative stability of
different structures. In particular, formation energies $\Delta
E_{\mathrm{f}}$ can be compared to other levels of theory and to experimental
formation enthalpies $\Delta H_{\mathrm{f}}$ to judge the quality of a given
exchange-correlation functional. The formation energy is defined as the
difference between the total energy of a given structure and the sum of the
total energies of the elements in their reference modification, i.e.
H${}_{2}\mathrm{(g)}$, O${}_{2}\mathrm{(g)}$, bulk bcc Li, and bulk
$\upalpha$-Mn in the present case. In contrast to formation energies,
experimental formation enthalpies also include the zero point energy.
Additionally, formation enthalpies are determined at standard conditions, i.e.
at a temperature of 298.15 K and a pressure of 1 bar. However, the additional
contributions are typically small compared to potential energy differences of
different chemical compounds or structures. Therefore, comparing formation
energies $\Delta E_{\mathrm{f}}$ and standard formation enthalpies $\Delta
H_{\mathrm{f}}^{\circ}$ is a reasonable and frequently used approximation,
which we will also employ here.

Table 1: Calculated formation energies $\Delta E_{\mathrm{f}}$ and experimental standard formation enthalpies $\Delta H_{\mathrm{f}}^{\circ}$ in eV per formula unit for the investigated benchmark systems. The structures were optimized under the constraint of fixed lattice vectors which were taken from experimental data. The mean absolute error ($\mathrm{MAE}=\tfrac{1}{n_{\mathrm{exp}}}\sum_{i=1}^{n_{\mathrm{exp}}}\left|{\Delta E_{\mathrm{f}}}_{i}-{\Delta H_{\mathrm{f}}^{\circ}}_{i}\right|$) is calculated using the $n_{\mathrm{exp}}$ systems for which experimental data are given excluding H2O(g). The mean absolute error per atom ($\mathrm{MAE\,/\,atom}=\tfrac{1}{n_{\mathrm{exp}}}\sum_{i=1}^{n_{\mathrm{exp}}}\tfrac{1}{{n_{\mathrm{atoms}}}_{i}}\left|{\Delta E_{\mathrm{f}}}_{i}-{\Delta H_{\mathrm{f}}^{\circ}}_{i}\right|$) uses for each system the error per atom instead of the error per formula unit which includes $n_{\mathrm{atoms}}$ atoms. | $\Delta E_{\mathrm{f}}$ | $\Delta H_{\mathrm{f}}^{\circ}$
---|---|---
System | PBE | PBEPAW | PBE0 | HSE06 | PBE0r | Exp.
H2O(g) |  $-$2.49 |  $-$2.52 |  $-$2.64 |  $-$2.63 |  $-$2.38 |  $-$2.51[71]
Li2O2 |  $-$5.79 |  $-$6.18 |  $-$5.99 |  $-$5.98 |  $-$5.91 |  $-$6.56[72]
Li2O |  $-$5.60 |  $-$6.00 |  $-$5.88 |  $-$5.87 |  $-$5.78 |  $-$6.21[71]
MnO |  $-$2.54 |  $-$2.58 |  $-$4.49 |  $-$4.46 |  $-$3.75 |  $-$3.99[71]
Mn3O4 |  $-$11.56 |  $-$11.61 |  $-$16.00 |  $-$16.05 |  $-$14.30 |  $-$14.38[71]
$\upalpha$-Mn2O3 |  $-$8.35 |  $-$8.35 |  $-$10.91 |  $-$10.98 |  $-$9.95 |  $-$9.94[71]
$\upbeta$-MnO2 |  $-$5.07 |  $-$5.07 |  $-$5.51 |  $-$5.61 |  $-$5.51 |  $-$5.39[71]
LiMnO2 |  $-$7.48 |  $-$7.56 |  $-$8.98 |  $-$8.99 |  $-$8.31 |  $-$8.70[69]
$\uplambda$-Mn2O4 |  $-$9.81 |  $-$9.81 |  $-$10.60 |  $-$10.71 |  $-$10.76 | —
Li0.5Mn2O4 |  $-$11.58 |  $-$11.65 |  $-$12.81 |  $-$12.91 |  $-$12.55 | —
LiMn2O4 |  $-$13.15 |  $-$13.26 |  $-$14.91 |  $-$14.98 |  $-$14.30 |  $-$14.32[70]
Li2Mn2O4 |  $-$15.18 |  $-$15.36 |  $-$18.07 |  $-$18.09 |  $-$16.78 |  $-$17.34[69]
MAE | 1.35 | 1.21 | 0.63 | 0.67 | 0.28 |
MAE / atom | 0.31 | 0.27 | 0.15 | 0.16 | 0.08 |
Figure 5: Differences between calculated formation energies and experimental
standard formation enthalpies $\Delta E_{\mathrm{f}}-\Delta
H_{\mathrm{f}}^{\circ}$ in eV per formula unit for the benchmark systems
obtained by the investigated functionals. The energy axis is inverted to show
overestimated formation energies above the zero line and underestimated
formation energies below. All calculated formation energy values and standard
formation enthalpies are given in TABLE 1.

The calculated formation energies and the corresponding experimental formation
enthalpies of the benchmark systems are given in TABLE 1. The internal
structures of the benchmark systems were optimized by the respective
functionals under the constraint of fixed lattice vectors. The unit cell
parameters were taken from the experimental data cited in Sec. I. The
deviations between the theoretical and experimental results are plotted in
FIG. 5. A comparison to theoretical results from previous studies is given in
the SI.
First, the agreement between FHI-aims and CP-PAW calculations is investigated
by performing PBE calculations with both codes. As shown in the columns PBE
(FHI-aims) and PBEPAW (CP-PAW) of TABLE 1, the agreement for the formation
energies of the manganese oxides and water is very good. However, the
formation energies of the lithium oxides show larger deviations of up to 0.13
eV per atom. This is mainly related to the description of the Li atoms, which
is also visible in the deviations between the PBE formation energies for the
lithium manganese oxides calculated with both codes. Also in this case the
discrepancies increase with larger Li contents. Tests have shown that if the
1s electrons of Li would also have been treated as valence electrons in the
CP-PAW calculations, the deviation, for example, for the Li2O formation energy
could have be reduced by 0.05 eV per atom. The PBE value from CP-PAW for the
formation energy of Li2O2, which deviates by 0.10 eV per atom from the value
obtained by FHI-aims, agrees within 0.02 eV per atom with the Vienna Ab initio
Simulation Package (VASP).[73, 74] For all benchmark system except the lithium
oxides the maximum deviation of the formation energies is 0.02 eV per atom,
which is within the basis set error.
Irrespective of the DFT code, the PBE formation energies of all (lithium)
manganese oxides are much smaller, i.e. less negative, than the experimental
enthalpies. These systematic deviations have previously been explained by the
self-interaction error of GGA functionals.[75] (Lithium) manganese oxides are
highly correlated systems as the Mn d electrons are strongly localized. The
size of the Mn 3d valence orbitals is similar to the size of the 3s and 3p
core orbitals which leads to a strong Coulomb interaction. Those systems are
typically most severely affected by the self-interaction error. Specifically,
the energy of spatially localized states is unphysically increased by the
interaction of an electron with its own charge density. Therefore, delocalized
charge distributions, i.e. metallic behavior, are favored. As a consequence,
metallic Mn is more favored than oxidic Mn and consequently the formation
energies of the (lithium) manganese oxides are smaller. The introduction of
exact exchange contributions aims to reduce the self-exchange error, which
results in a more accurate description of localized states. In conclusion, the
formation energies of the (lithium) manganese oxides should be larger in case
of hybrid functionals compared to the PBE GGA functional, which is consistent
with our improved results for the oxide benchmark systems reported in TABLE 1.
The results of the PBE0 and HSE06 functionals are very similar. Therefore, the
exclusion of the long-range exact exchange contributions seems to be a good
approximation leading only to small errors. However, both functionals
overestimate the formation energies of all (lithium) manganese oxides compared
to the experimental values, while PBE underestimates the formation energies.
Therefore, a HF mixing factor between 0 and 25% should improve the agreement
with experiment. This is the case for PBE0r, which uses only 9% HF on-site
exchange for Mn while off-site exchange terms are described by GGA only, and
indeed we find a better agreement with the experimental data of the (lithium)
manganese oxides as shown in FIG. 5. The freedom of choosing the amount of on-
site exact exchange allows – to a certain extent – to compensate for errors
introduced by the local approximation in PBE0r. The formation energies of the
water molecule and the lithium oxides are not as much affected by the
inclusion of HF exchange. The PBE0r formation energies for the oxides are
systematically higher than the PBE results but smaller than the PBE0 data.
This is in accordance with the intermediate HF mixing factor. The formation
energy of water is an exception of this trend. The water molecule is
covalently bonded. Here, the restriction to on-site exchange terms seems to
miss relevant contributions.
Compared to experiment, the PBE0r formation energy of the H2O monomer is
underestimated by 0.13 eV, while it is overestimated in case of PBE0 and HSE06
by about 0.13 and 0.12 eV, respectively. The experimental enthalpy is 0.03 eV
smaller at 0 K compared to the standard value at 298 K[76] given in TABLE 1.
However, the experimental formation enthalpy includes a reduction by the zero
point energy[77] while the calculated formation energies in TABLE 1 have not
been corrected for the zero point energy and are thus expected to be too high.
Since the contribution of the experimental zero point energy is about 0.24
eV[78, 79, 80], the experimental formation energy at 0 K without zero point
energy correction is $-$2.72 eV. Thus, the PBE0 and HSE06 results are more
accurate for the formation energy of the H2O monomer. The less accurate result
of PBE0r could be caused by the lack of off-site exact exchange, which is
important for the covalent bonds in H2, O2, and H2O. The zero point energies
are typically much smaller if no H atoms are present in the system.

Table 2: Formation energies $\Delta E_{\mathrm{ox}}$ and experimental standard formation enthalpies $\Delta H_{\mathrm{ox}}^{\circ}$ for the investigated benchmark systems calculated with respect to the oxides Li2O, MnO, and $\upbeta$-MnO2 in eV per formula unit. The structures were optimized under the constraint of fixed lattice vectors, which were taken from experimental data. The calculation of the MAE is performed as described in FIG. 5. | $\Delta E_{\mathrm{ox}}$ | $\Delta H_{\mathrm{ox}}^{\circ}$
---|---|---
System | PBE0 | HSE06 | PBE0r | Exp.
Mn3O4 |  $-$1.51 |  $-$1.51 |  $-$1.29 |  $-$1.01[71]
$\upalpha$-Mn2O3 |  $-$0.91 |  $-$0.91 |  $-$0.69 |  $-$0.56[71]
LiMnO2 |  $-$1.04 |  $-$1.02 |  $-$0.79 |  $-$0.91[71, 69]
LiMn2O4 |  $-$1.45 |  $-$1.41 |  $-$1.27 |  $-$1.14[71, 70]
Li2Mn2O4 |  $-$2.19 |  $-$2.16 |  $-$1.74 |  $-$1.75[71, 69]
MAE | 0.35 | 0.33 | 0.13 |
MAE / atom | 0.05 | 0.05 | 0.02 |

The PBE0r approximation of on-site exchange should work best for oxides. To
check this hypothesis, formation energies $\Delta E_{\mathrm{ox}}$ and
experimental standard formation enthalpies $\Delta H_{\mathrm{ox}}^{\circ}$
have also been calculated with respect to a reference of the oxides Li2O, MnO,
and $\upbeta$-MnO2 instead of the elemental reference states for the
investigated benchmark systems (TABLE 2). Therefore, these energy difference
do not include information from the covalently bonded H2, O2, and H2O and
metallic Li and $\upalpha$-Mn, which are most critical in case of the PBE0r
functional. Again, the results of PBE0 and HSE06 are overall very similar with
a slightly smaller mean absolute error (MAE) of the HSE06 results compared to
experimental data. As expected, the PBE0r results now show a much reduced MAE
of 0.13 eV which is about 2.5 times smaller than the MAE for the other two
hybrid functionals. In general, the MAEs for the formation energies with
respect to the oxides are almost half of the MAEs for the formation energies
from the elements (TABLE 1) in the case of all hybrid functionals.
In summary, the MAE of the formation energies from the elements for all
benchmark systems can be reduced from 1.35 eV (PBE) to 0.63 eV (PBE0) by the
inclusion of 25% HF exchange confirming that the inclusion of exact exchange
is very important for these systems. However, this increases the average
computation time of the benchmark systems by a factor of about 25. The MAE is
only very little affected by the neglect of the long-range exact exchange
terms (PBE0 vs. HSE06). This decreases the computational effort of HSE06
compared to PBE0 by a factor of 0.8. The MAE can further be decreased to 0.28
eV (PBE0r) for these benchmark systems if smaller HF mixing factors are used.
For the formation energy with respect to reference oxides the MAE of the PBE0r
functional is only 0.13 eV providing the best agreement with the experimental
data and demonstrarting a very good description of energy differences.
However, the main advantage of the PBE0r functional is that the average
computation time per iteration of the benchmark systems compared to PBEPAW is
only increased by a factor of 1.3 while the error is reduced by a factor of 4.

###  IV.3 Intercalation Potential

An important property of the LixMn2O4 spinel is the possibility to remove or
incorporate Li atoms by (de)intercalation. This enables the application as
cathode material for lithium ion batteries. Experiments show that the standard
electrochemical potential of a Li/LixMn2O4 cell is about 4 V vs. Li/Li+ in the
range of $0\leq x\leq 1$ and 3 V in the range of $1<x\leq 2$.[81]
The intercalation potential $\Delta G_{\mathrm{p}}$ can be calculated by

$\displaystyle\Delta
G_{\mathrm{p}}=\dfrac{(x_{2}-x_{1})G_{\mathrm{Li}}+G_{\mathrm{Li}_{x_{1}}\mathrm{Mn}_{2}\mathrm{O}_{4}}-G_{\mathrm{Li}_{x_{2}}\mathrm{Mn}_{2}\mathrm{O}_{4}}}{(x_{2}-x_{1})}\
,$ (5)

where $x_{1}$ and $x_{2}$ specify the Li content of the LixMn2O4
structure.[82] It corresponds to the Gibbs free energy difference of the
reaction

$\displaystyle\mathrm{Li}_{x_{2}}\mathrm{Mn}_{2}\mathrm{O}_{4}\rightleftharpoons\mathrm{Li}_{x_{1}}\mathrm{Mn}_{2}\mathrm{O}_{4}+(x_{2}-x_{1})\mathrm{Li}$
(6)

divided by $(x_{2}-x_{1})$. The Gibbs free energy $G$ can be approximated by
the total energy $E$ when neglecting the influence of zero point energies and
entropic contributions at finite temperatures. The resulting error of this
approximate treatment is typically small for this reaction since the change in
the neglected terms is much smaller than the change in the total energy $E$
during this reaction.[82] If the intercalation potential of Li2Mn2O4/LiMn2O4
is calculated from the experimental formation enthalpies given in TABLE 1, the
result is an intercalation potential of 3.02 eV which matches the
electrochemically determined data given in TABLE 3 very well.

Table 3: Calculated intercalation potentials $\Delta E_{\mathrm{p}}$ and experimentally measured standard electrochemical potentials vs. Li/Li+ $E_{\mathrm{p}}^{0}$ in eV. Potential | PBE | PBE0 | HSE06 | PBE0r | Exp.
---|---|---|---|---|---
Li0.5Mn2O4/$\uplambda$-Mn2O4 | 3.54 | 4.43 | 4.39 | 3.59 | 4.1[83]
LiMn2O4/Li0.5Mn2O4 | 3.14 | 4.19 | 4.15 | 3.49 | 4.0[83]
Li2Mn2O4/LiMn2O4 | 2.03 | 3.16 | 3.11 | 2.48 | 3.0[84]
MAE | 0.80 | 0.23 | 0.18 | 0.51 |
MAE / atom | 0.11 | 0.03 | 0.03 | 0.07 |

The experimentally measured standard electrochemical potentials vs. Li/Li+
$E_{\mathrm{p}}^{0}$ are in between the calculated PBE and PBE0 intercalation
potentials $\Delta E_{\mathrm{p}}$ (TABLE 3). The MAE of the three PBE0
intercalation potentials with respect to to the experiment is 0.23 eV and thus
much smaller than the MAE of PBE, which is 0.80 eV. The HSE06 results
exhibiting an MAE of 0.18 eV are a little closer to the experimental data than
PBE0. As expected, the PBE0r values (MAE 0.51 eV) are in between the PBE and
PBE0 data, but only slightly better than PBE and clearly less accurate than
PBE0 and HSE06. Still, the trend of the intercalation potential as a function
of the Li content $x$ is described reliably by PBE0r in contrast to PBE which
predicts the relative differences among the potentials at different lithium
contents inaccurately. PBE0 and HSE06 describe the relative differences well
but not as good as PBE0r whose results show a relatively constant
underestimation of approximately 0.5 eV for all potentials.

###  IV.4 Structural Properties

Up to this point, all properties have been obtained from optimized atomic
positions using the experimental lattice parameters. A natural next question
is how well the tested functionals are able to describe the structural and
structure-related properties such as the equilibrium lattice constants and
bulk moduli. To answer this question, we used the Birch-Murnaghan equation of
state [85, 86]

$\displaystyle\begin{split}U={}&U_{0}+\tfrac{9}{16}B_{0}V_{0}\left[\left(\tfrac{V_{0}}{V}\right)^{\tfrac{2}{3}}-1\right]^{2}\\\
&\left\\{B_{0}^{\prime}\left[\left(\tfrac{V_{0}}{V}\right)^{\tfrac{2}{3}}-1\right]-4\left(\tfrac{V_{0}}{V}\right)^{\tfrac{2}{3}}+6\right\\},\end{split}$
(7)

to derive the equilibrium energy $U_{0}$, equilibrium volume $V_{0}$, the bulk
modulus at zero pressure $B_{0}$, and its pressure derivative
$B_{0}^{\prime}$. To reduce the search space, in case of non-cubic cells we
have kept the ratios of the lattice parameters and scaled the experimental
lattice constants isotropically by factors in the range from 0.95 to 1.05 in
steps of 0.01. Again, for each calculation the atomic positions in the cells
were optimized under the constraint of the respective lattice constants.
Approximating the internal energy $U$ by the total energy $E$, the Birch-
Murnaghan equation of state describes the $E(V)$ relation for all periodic
benchmark systems very well. The formation energies using the oxides Li2O,
MnO, and $\upbeta$-MnO2 as reference were calculated using the resulting
equilibrium energies $E_{0}$ from the Birch-Murnaghan equation of state for
each hybrid functional. The values are given in the SI. These energies deviate
only very little from the DFT data using the experimental lattice constants,
which are given in TABLE 2. The MAE of the energies per formula unit changes
only by 0.009 eV for PBE0, $-$0.001 eV for HSE06, and $-$0.013 eV for PBE0r.

Table 4: Relative deviations between the experimental equilibrium lattice constants and the calculated values using the examined functionals in %. System | PBE | PBE0 | HSE06 | PBE0r
---|---|---|---|---
Li |  $-$0.9 | 0.0 |  $-$0.3 |  $-$2.1
Li2O2 | 0.9 |  $-$0.4 |  $-$0.4 |  $-$0.1
Li2O | 0.8 |  $-$0.5 |  $-$0.4 |  $-$0.5
MnO | 0.1 |  $-$0.1 | 0.1 | 1.0
Mn3O4 | 0.3 | 0.0 | 0.1 | 1.3
$\upalpha$-Mn2O3 | 0.5 | 0.0 | 0.1 | 1.5
$\upbeta$-MnO2 | 0.3 |  $-$0.7 |  $-$0.6 | 0.7
LiMnO2 | 1.1 | 0.1 | 0.2 | 1.1
$\uplambda$-Mn2O4 | 0.7 |  $-$0.4 |  $-$0.2 | 1.2
Li0.5Mn2O4 | 0.2 |  $-$0.1 | 0.0 | 1.1
LiMn2O4 |  $-$0.2 |  $-$0.1 | 0.1 | 0.8
Li2Mn2O4 | 0.7 | 0.0 | 0.1 | 0.9
Mean Error | 0.6 | 0.2 | 0.2 | 1.0

TABLE 4 shows the relative deviations between the calculated equilibrium
lattice constants and the experimental data. In contrast to the formation
energies, the PBE equilibrium lattice constants do not show large deviations
from the experimental data with an overall relative error of only 0.6%, which
is in agreement with the general finding that structural properties are well
described already at the GGA level. PBE0 and HSE06, which have a tendency to
only very slightly underestimate the lattice parameters, yield an even smaller
error as low as 0.2% with respect to experiment. This is reasonable as the
experimental results were determined at finite temperature, and should be
somewhat higher than the theoretical results obtained at 0 K due to thermal
expansion.
PBE0r underestimates in particular the lattice constants of Li by 2.1%. The
lattice constants of the lithium oxides are described with an accuracy
comparable to the other hybrid functionals. The equilibrium lattice constants
of all given (lithium) manganese oxides are overestimated between 0.7 and 1.5%
thus covering a range of 0.8%. PBE0 and HSE06 span exactly the same range of
0.8% (from $-$0.7 to 0.1 and from $-$0.6 to 0.2, respectively). Thus, the
relative differences are described similarly by all hybrid functionals. Still,
there is a constant shift of about 1% to larger lattice parameters with PBE0r
compared to PBE0 and HSE06. Consequently, this leads to a mean relative error
of approximately 1.0% for PBE0r which is still a very good agreement. We
attribute the overestimation of the lattice constants to the neglect of off-
site HF exchange terms which we expect to strengthen bonds. The equilibrium
bond lengths for the molecular systems are reported in the SI.
The results for $\upalpha$-Mn are not listed in TABLE 4 and 5 because no
minimum is observed in the isotropic compression/expansion range from 0.95 to
1.05 using any of the hybrid functionals. If the equilibrium lattice constant
is calculated by PBE with collinear spin, the outcome is an underestimation of
2.7%, which is very similar to a previous PBE study employing non-collinear
spin.[87] The error of PBE0, HSE06, and PBE0r could originate from the
restriction to a collinear spin arrangement in our calculations, as
$\upalpha$-Mn has a complex non-collinear magnetic electronic structure, whose
characterization is beyond the scope of the present work due to the large
computational effort.

Table 5: Calculated and experimental bulk moduli $B_{0}$ in GPa for the benchmark systems. The bulk modulus of $\upbeta$-MnO2 has not been included in the calculation of the MAE with respect to experiment because of the large uncertainty in the experimental value. System | PBE | PBE0 | HSE06 | PBE0r | Exp.
---|---|---|---|---|---
Li | 13 | 13 | 13 | 14 | 12[88]
Li2O2 | 72 | 79 | 79 | 76 | —
Li2O | 74 | 82 | 81 | 80 | 82[89]
MnO | 124 | 162 | 161 | 148 | 154[90]
Mn3O4 | 127 | 149 | 149 | 131 | 133[91]
$\upalpha$-Mn2O3 | 140 | 173 | 173 | 151 | 169[92]
$\upbeta$-MnO2 | 224 | 263 | 262 | 229 |  $260-280$[93]
LiMnO2 | 111 | 127 | 127 | 112 | —
$\uplambda$-Mn2O4 | 100 | 119 | 118 | 100 | —
Li0.5Mn2O4 | 108 | 126 | 126 | 108 | —
LiMn2O4 | 118 | 129 | 133 | 126 | 119[94]
Li2Mn2O4 | 114 | 129 | 129 | 115 | —
MAE | 13 | 7 | 7 | 6 |

The calculated and experimental bulk moduli $B_{0}$ are given in TABLE 5. PBE
generally underestimates the bulk moduli except for the bulk modulus of Li,
with a overall MAE of about 13 GPa. PBE0 matches the experimental results with
a MAE of only 7 GPa very well, and except for $\upbeta$-MnO2 the bulk modulus
is typically overestimated. However, this comparison between theory and
experiment has to be made with care, as in contrast to formation energies and
lattice constants that can be measured quite accurately, the experimental bulk
moduli have larger uncertainties. Especially, in the case of $\upbeta$-MnO2
the experimental values of the bulk modulus in the literature show large
differences depending on the method which was used.[93] From our calculations
the reported value of 260 GPa fits the trend that the experimental values are
between the PBE and PBE0 results better. The HSE06 results are again very
similar to the PBE0 results, promoting the general use of HSE06 instead of
PBE0 for these types of systems because the same accuracy is obtained with
less computational effort. For the PBE0r results we would expect that the
values of the oxide benchmark systems are in between the results of PBE and
PBE0, which is indeed confirmed. The PBE0r MAE of 6 GPa is similar to the ones
of PBE0 and HSE06 and thus in very good accordance with the experimental
measurements.
All four functionals predict that the bulk modulus increases upon lithiation
of LixMn2O4 in the range $0\leq x\leq 1$. The bulk modulus of tetragonal
Li2Mn2O4 is however predicted to be smaller than the one of LiMn2O4 with the
exception of the PBE0 bulk modulus of Li2Mn2O4 which is equal to the one of
LiMn2O4.

###  IV.5 Band Gaps

Important properties in electronic applications and in optical absorption are
the indirect and direct band gaps, respectively. The indirect band gap is the
smallest overall energetic difference between occupied and unoccupied states.
For the direct band gap, the differences between highest occupied and lowest
unoccupied state are individually calculated for each $\mathbf{k}$-point, and
then the minimum of those is determined. The calculated direct band gaps are
given in TABLE 6 and compared to spectroscopic data. In case of molecular
water, the first spin-allowed electronic transition is considered.

Table 6: Calculated direct band gaps or first spin-allowed electronical transitions of the non-metallic benchmark systems in eV compared to experimental data. In the calculation of the MAE the differences to the given experimental data excluding H2O(g) are used including predicted zero band gaps. System | PBE | PBE0 | HSE06 | PBE0r | Exp.
---|---|---|---|---|---
H2O(g) | 7.1 | 10.1 | 9.3 | 6.3 | 7.4[95]
Li2O2 | 2.0 | 5.3 | 4.5 | 2.1 | —
Li2O | 5.0 | 7.3 | 6.6 | 5.0 | 6.0[96]
MnO | 0.1 | 4.9 | 4.1 | 2.3 | 4.1[97]
Mn3O4 | 0.9 | 3.9 | 3.1 | 1.6 | 1.9[98]
$\upalpha$-Mn2O3 | 0.1 | 3.0 | 2.2 | 0.8 | 1.2[99]
$\upbeta$-MnO2 | 0.3 | 2.3 | 1.6 | 0.4 | 1.0[100]
LiMnO2 | 0.5 | 4.0 | 3.2 | 1.5 | —
$\uplambda$-Mn2O4 | 1.4 | 4.3 | 3.6 | 2.2 | —
Li0.5Mn2O4 | 0.0 | 3.2 | 2.4 | 0.9 | —
LiMn2O4 | 0.0 | 2.7 | 1.9 | 1.1 | 1.2[17]
Li2Mn2O4 | 1.4 | 4.5 | 3.8 | 2.2 | —
MAE | 1.5 | 1.5 | 0.7 | 0.7 |

Based on the results of Sec. IV.2 we have already concluded that the self-
interaction error is much larger for PBE than for the hybrid functionals.
Strong self-interaction leads to a prediction of a too small or even non-
existent band gap,[75, 101] since localized electrons become less favored,
resulting in an increased metallicity of the system. The data in TABLE 6
confirms this trend. The band gaps calculated by PBE are generally smaller
than those calculated by PBE0. The PBE band gaps of all given oxides are
underestimated with respect to experiment. Some of the (lithium) manganese
oxides are even predicted to be metallic, while the PBE0 functional correctly
predicts their non-metallic character. However, the band gaps of all benchmark
systems are always overestimated by PBE0 compared to the experimental data.
The MAE is 1.5 eV for the given oxide data. Despite the similar description of
energetic and structural properties by PBE0 and HSE06, HSE06 predicts for all
of these systems smaller band gaps which are in better agreement with
experiment exhibiting a MAE for the oxides of 0.7 eV only.
As PBE underestimates the band gaps while PBE0 yields too large band gaps,
there should be an intermediate HF mixing factor which leads to a very good
agreement with experiment as in the case of the formation energies (Sec.
IV.2). We tested different values for the HF mixing factor of Mn in the PBE0r
functional and we experienced that increasing the HF mixing factor of Mn opens
the band gap of the (lithium) manganese oxides. For example, in the range from
5 to 10% of the HF mixing factor the band gap of LiMn2O4 widens from 0.7 to
1.2 eV. The chosen intermediate value of 9% does not only lead to a good
prediction of the formation energies, but it also yields very accurate band
gaps. The MAE for the band gaps of the oxides is only 0.7 eV, and the
individual values are, as expected, in between the PBE and PBE0 gaps. For the
covalently bonded H2O this trend is not present. This again confirms that the
approximation of the PBE0r functional is not well-suited for covalent bonds,
but performs very well for the (lithium) manganese oxides.

###  IV.6 Density of States

The Kohn-Sham density of states (DoS) $D(\epsilon)$ provides detailed insights
into the electronic structure of a system, and, for example, the influence of
Li insertion can be studied. Furthermore, some features like band gaps or
orbital occupations can be directly compared to experimental data.
First, we will compare the DoS of LiMn2O4 calculated by PBE, PBE0, HSE06, and
PBE0r (FIG. 6 and 7) in order to understand why the PBE results are less
accurate than the results of the hybrid functionals (more information on the
calculation procedure of the DoS in both DFT codes are given in the SI).
Afterwards, we will study the effect of Li insertion on the electronic
structure using the PBE0r DoS of LixMn2O4 with $x=0,0.5,1,$ and $2$ (FIG. 7
(b) and 8). In FIG. 6, 7, and 8 the peaks of the DoS are broadened for
visualization which is done differently by FHI-aims and CP-PAW and must be
considered in the comparison. All figures show only the DoS of one spin
channel since the DoS of the other spin channel is identical for these
optimized structures. The partial DoS of Li, O, Mn s and p, and Mn d orbitals
are shown as a stacked plot in front of the total DoS plot. For Mn d states
the individual contribution of each atom in the unit cell is plotted
separately. In the DoS plots obtained from PBEPAW and PBE0r calculations, the
Mn d partial DoS is further splitted into the t2g and eg contributions.
The PBE DoS calculated by FHI-aims (FIG. 6 (a)) and CP-PAW (FIG. 7 (a)) are
very similar. They reveal the missing band gap in those calculations.
Moreover, they show that all Mn atoms have the same electron density
corresponding to an oxidation state of 3.5. Accordingly, the d electrons are
not localized at the Mn ions, because then the oxidation state would be an
integer number. Instead, they are strongly delocalized electrons. CP-PAW
enables resolving the d electron density into the t2g and eg contributions,
which are present in an octahedral ligand field. This reveals that all Mn d
electrons have mainly t2g character. The stability underestimation of hs-
states is again typical for self-interaction errors.[75]

Figure 6: The density of states $D(\epsilon)$ of one spin channel plotted for
the atoms in the LiMn2O4 unit cell calculated by (a): PBE, (b): PBE0, and (c):
HSE06. Unoccupied orbitals are in lighter colors. Figure 7: The density of
states $D(\epsilon)$ of one spin channel plotted for the atoms in the LiMn2O4
unit cell calculated by (a): PBEPAW and (b): PBE0r. Unoccupied orbitals are in
lighter colors.

In conclusion, PBE predicts metallic LiMn2O4 with all Mn being in the low-spin
Mn3.5 state. The MnO6 octahedra are slightly distorted. The PBEPAW result
shows that two Mn-O bonds are 1.94 Å and four are 1.97 Å. However, if this
bond length difference is compared to the one of the JT distorted
Mn${}^{\text{III}}$ in case of the hybrid functionals mentioned later, it is
obvious that this is not an accurate description of the JT distortion. The PBE
outcome is contradicting experimental results which show that LiMn2O4 has a
band gap and one half of the Mn is in the JT distorted hs-Mn${}^{\text{III}}$
state while the other half is in the Mn${}^{\text{IV}}$ state.
The DoS of the hybrid functionals (FIG. 6 (b) and (c) and FIG. 7 (b)) are very
similar but clearly differ from the PBE DoS. The main effect of exact exchange
is the self-interaction correction, which shifts filled electron levels
downward. For manganese oxides this shifts the majority spin t2g states into
the O 2p valence band. Similarly, it shifts the majority spin eg orbitals
below the minority spin t2g orbitals and thus favors high-spin Mn.
In particular PBE0 and HSE06 yield an almost equal DoS except for the size of
the band gap. This is in accordance with the similar energetic and structural
results. Hence, the reliability of the range-separated hybrid functional is
again confirmed. The Mn d orbitals are split into two states of different
energy due to the ligand field. In the PBE0r DoS this separation into the t2g
and eg states is highlighted. The lower band is similar for each Mn since both
Mn${}^{\text{III}}$ and Mn${}^{\text{IV}}$ have three t2g electrons. However,
FIG. 7 (b) shows that only half of the Mn in one spin channel in the LiMn2O4
unit cell (four of eight) have an eg electron which corresponds to the
energetically highest occupied band. Thus, there are two different types of Mn
in the LiMn2O4 unit cell: eight Mn${}^{\text{III}}$ and eight
Mn${}^{\text{IV}}$ (in both cases four with spin up and four with spin down).
Separately plotted DoS for both Mn states are given in the SI. They show that
the t2g and eg electrons of a Mn have always the same spin direction, i.e. the
Mn is in the hs-state. The hs-Mn${}^{\text{III}}$ leads to a JT distortion of
the corresponding MnO6 octahedron.
A geometrical study of the structure which was optimized by PBE0 or HSE06
using the experimental lattice parameters, i.e. a cubic unit cell, shows that
two Mn-O bonds are 1.92 Å, two are 2.05 Å, and two are 2.07 Å for the MnO6
octahedra of Mn${}^{\text{III}}$. PBE0r predicts the averaged bond distances
1.93 Å, 2.05 Å, and 2.10 Å for these bonds. These confirm the presence of a JT
distortion. The MnO6 octahedra of the Mn${}^{\text{IV}}$ are not JT distorted.
PBE0 and HSE06 predict two Mn-O bond lengths of 1.89 Å and four of 1.90 Å,
i.e. the MnO6 octahedron is only very slightly distorted for
Mn${}^{\text{IV}}$. PBE0r yields for these bond lengths the averaged values
1.89 Å and 1.94 Å. The Mn-O bond lengths also show that the MnO6 octahedra of
Mn${}^{\text{III}}$ are larger than those of Mn${}^{\text{IV}}$ because of the
additional eg electron of Mn${}^{\text{III}}$ in the antibonding Mn-O orbital.
All these features of the DoS of the hybrid functionals are in accordance with
experimental results.
The LixMn2O4 structure does not undergo any structural transformations if the
Li content is varied in the range $0\leq x\leq 1$. Only the lattice constant
and the number of JT distorted MnO6 octahedra decrease with decreasing Li
content. Thus, one would expect that the DoS changes only slightly except for
the number of eg electrons. This is confirmed by the data shown in FIG. 7 (b)
and 8 (a) and (b). The number of eg electrons is in all cases equal to the
number of Li+ ions. There is no Mn${}^{\text{III}}$ present in
$\uplambda$-Mn2O4. As a consequence, the band gap of $\uplambda$-Mn2O4 is much
larger than the band gaps of LixMn2O4 with $0<x\leq 1$: the band gap in
$\uplambda$-Mn2O4 is between the mixed O 2p/Mn t2g valence band and the
unoccupied d states, while in LixMn2O4 the highest occupied orbitals are the
mixed O 2p/Mn eg states which are higher in energy as the O 2p/Mn t2g states.
The band gap between the O 2p/Mn eg states and the unoccupied d states is
determined by the much smaller JT splitting of the eg states.

Figure 8: The PBE0r density of states $D(\epsilon)$ of one spin channel
plotted for the atoms (a): in the $\uplambda$-Mn2O4 unit cell (48 atoms), (b):
in the Li0.5Mn2O4 unit cell (52 atoms), and (c): in the Li2Mn2O4 unit cell (32
atoms). Unoccupied orbitals are shown in lighter colors.

The DoS of Li2Mn2O4 (FIG. 8 (c)) differs more from the DoS of LixMn2O4 with
$0\leq x\leq 1$ because the unit cell is tetragonally distorted. However, the
trend of the increasing number of eg electrons is retained. All Mn in the unit
cell have one eg electron which means that they are all in the
Mn${}^{\text{III}}$ state. The JT distortion opens a gap between both eg
states of the same spin channel. These two states are the highest occupied and
lowest unoccupied state for Mn${}^{\text{III}}$. This is shown in the DoS of
one Mn${}^{\text{III}}$ which is given in the SI. In Li2Mn2O4 only
Mn${}^{\text{III}}$ ions are present and the entire structure is tetragonally
distorted. Therefore, the band gap is larger than for LixMn2O4 with $0<x\leq
1$ because only strongly JT distorted Mn${}^{\text{III}}$ are present.
Moreover, the JT distortion decreases the difference between the t2g and eg
state. This trend is also observed if we compare FIG. 7 (b) with FIG. 8 (c).
The DoS of the various (lithium) manganese oxides can be analyzed in a similar
way. The hs-Mn state is preferred for all given systems by the hybrid
functionals. The PBE0r spin values of the Mn, from which one can derive their
oxidation states, are given in the SI.

##  V Conclusion

The PBE, PBE0, HSE06, and PBE0r exchange-correlation functionals have been
benchmarked for various LixMnyOz systems. The deviations of predicted
energetic, structural, and electronic properties from experimental data can in
general be reduced by the inclusion of exact exchange contributions.
Neglecting the long-range exact exchange terms is proven to be a very good
approximation. The quality of the HSE06 results is very similar to that of the
PBE0 results or even better in the case of band gaps. However, the average
computation time per self-consistency cycle of the benchmark systems is
reduced by a factor of 0.8 which makes the HSE06 functional preferable to the
PBE0 functional for the given types of systems. The experimental formation
energies, intercalation potentials, bulk moduli, and band gaps of the studied
(lithium) manganese oxides are in between the PBE and PBE0 results.
Consequently, an intermediate HF mixing factor as used in the PBE0r functional
can further decrease the deviation in many cases. In PBE0r an individual HF
mixing factor is assigned to each element. They were determined in a
systematic search on a grid of values minimizing the differences to given
reference data. Due to the restriction of using only on-site HF exchange terms
in PBE0r its average computation time per iteration of the benchmark systems
is only increased by a factor of 1.3 compared to PBEPAW, which is
substantially less expensive than the PBE0 and HSE06 functionals. However,
this approximation is not well-suited for strongly covalently bonded or
metallic systems while it provides reliable results for highly correlated
systems. The admixture of on-site HF exchange terms yields most of the
improvement from GGA to hybrid functionals in systems with narrow, partially
filled d and f shells as they are present in many transition metal oxides.
Consequently, the accuracy of the results for the (lithium) manganese oxides
is greatly improved compared to PBE. We conclude that PBE0r has the optimum
cost-benefit ratio for these types of systems. In summary, the hybrid
functionals PBE0, HSE06, and PBE0r agree well with experiment.
Theoretical calculations for the lithium manganese oxide spinel LixMn2O4, with
$0\leq x\leq 2$, agree well with experimental results if hybrid functionals
are used. The calculated formation energies, the trend of the intercalation
potentials, the equilibrium lattice constants, the bulk moduli, and the band
gaps are in accordance with experimental data. Several essentially degenerate
antiferromagnetic states exist close to the energetically lowest configuration
of LixMn2O4. A formation of two oxidation states, +III and +IV, is observed
for Mn. The Mn${}^{\text{III}}$ ions are in the hs-state and the corresponding
MnO6 octahedra are JT distorted.

###### Acknowledgements.

We thank the DFG (SFB 1073 project B03 and C03, project 217133147, project
405832858) for financial support. We gratefully acknowledge the funding of
this project by computing time provided by the Paderborn Center for Parallel
Computing (PC2). JB is grateful for a DFG Heisenberg professorship
(BE3264/11-2, project 329898176).

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