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# The effect of off-diagonal density matrix in DFT+DMFT for Li2MnO3

Alex Taekyung Lee  Department of Chemical engineering, University of Illinois
at Chicago, Chicago, IL 60608, USA  Materials Science Division, Argonne
National laboratory, Lemont, IL 60439, USA  Hyowon Park  Department of
Physics, University of Illinois at Chicago, Chicago, IL 60608, USA  Materials
Science Division, Argonne National laboratory, Lemont, IL 60439, USA  Anh T.
Ngo  Department of Chemical engineering, University of Illinois at Chicago,
Chicago, IL 60608, USA  Materials Science Division, Argonne National
laboratory, Lemont, IL 60439, USA

###### Abstract

Li2MnO3 has garnered much attention as one of the new-generation battery
material, due to the high capacity and low cost. In the present work, we
performed density functional theory (DFT)+$U$ and dynamical mean field theory
(DMFT) calculations with continuous time quantum Monte Carlo impurity solver
to study the electronic properties of Li2MnO3. Due to the nature of monoclinic
$C2/m$ symmetry, the off-diagonal terms in the $d$-orbital block Hamiltonian
(and $d$-orbital density matrix) are large, which results the large
suppression of the energy gap due to the underestimation of the crystal-field
splitting. We diagonalize the Mn $d$ block in the full $p-d$ Hamiltonian by
applying unitary rotation matrix, and obtained an energy gap of 0.8 eV,
although it is still smaller than the experimental gap of 2.1 eV even with the
large $U$. In the $p$-$d$ model, a small double counting energy is essential
to reduce the $p$-$d$ hybridization, thus to obtain the experimental gap. We
show that the low-energy ($d$-only basis) model is efficient to study the
electronic structure of Li2MnO3, since the Wannier basis is the hybridized
state of Mn $d$ and O $p$ orbitals. These results suggest the correct way to
investigate the low-symmetry materials using DFT+ DMFT method and to our
knowledge, there is no systematic study of the effect of the off-diagonal
terms so far. We also find that the antiferromagnetic ground state
$\Gamma_{2u}$ is stable with $U\leq 2$ within DFT+$U$, which is much smaller
than widely used $U$=5 eV.

###### pacs:

##  I Introduction

In the past two decades, the demand for the batteries has been enormously
grown due to the advances in technology of potable devices and electric
vehicles Goodenough and Park (2013). Lithium-ion battery technology has
contributed greatly to this demand, and have been extensively studied.
Traditional cathode materials for Li-battery is transition metal (TM) oxides
with extractable Li ions, where TM atoms are 3$d$ elements Cr, Co, Ni, Fe, or
Mn or 4$d$ elements Mo or Ru Seo et al. (2016); Sathiya et al. (2013); Lee et
al. (2014). Recently, Li2MnO3 attracted much attention as a next generation of
cathode material, due to the high voltage (4.4$-$5 V) and the low cost
Yabuuchi et al. (2011); Seo et al. (2016). Li2MnO3 has a layered structure
with $C/2m$ space group (Fig. 1), and Li atoms are located between MnO3 layer
and at the center of the hexagon. During the charging and discharging process
of the battery, Li ions are removed or added, respectively. Therefore,
studying atomic, electronic, and magnetic properties Li2MnO3 is important to
understand and improve the performance of the batteries.

Li2MnO3 is an insulator with a band gap of 2.1 Tamilarasan et al.
(2015)$-$2.17 Singh and Singh (2020) eV. Mn ion in Li2MnO3 has 4+ charge
states ($d^{3}$), and it has high-spin configuration ($S=3/2$) with magnetic
moment of 2.3$-$2.7 $\mu_{B}$ Lee et al. (2012); Strobel and Lambert-Andron
(1988). In the high-spin state, 3 $d$ electrons fully occupy the spin-up
$t_{2g}$ band, and thus Mn4+ has a nonzero gap due to the crystal field
splitting and Hund coupling. At low temperature, Li2MnO3 is antiferromagnetic
with Néel temperature of $T_{N}$= 36$-$36.5 K Strobel and Lambert-Andron
(1988); Lee et al. (2012). While the antiferromagnetic ground state of
$\Gamma_{3g}$ is reported in early study Strobel and Lambert-Andron (1988),
recent study suggested that $\Gamma_{2u}$ is the ground state, while both
$\Gamma_{3g}$ and $\Gamma_{2u}$ have same magnetic propagation vector
$Q_{m}=(0,0,0.5)$.

There are several studies of Li2MnO3 through the density functional theory
(DFT) Xiao et al. (2012); Seo et al. (2016); Wang et al. (2017); Chen and
Islam (2016); Chen et al. (2019); Hikima et al. (2022). However, detailed
electronic and magnetic structures are not fully understood yet. While the
band gaps are predicted within DFT+$U$ Xiao et al. (2012); Wang et al. (2017),
GW calculation Seo et al. (2016) or using hybrid functionals Seo et al.
(2016); Chen and Islam (2016), previous studies did not focus on the magnetic
ground state, and only consider nonmagnetic Seo et al. (2016); Hikima et al.
(2022) or ferromagnetic Xiao et al. (2012); Chen and Islam (2016); Wang et al.
(2017) configurations. Given that the operating temperature of the battery is
around room temperature and $T_{N}$ of Li2MnO3 is only 36 K Lee et al. (2012),
paramagnetic phase of Li2MnO3 is very important. However, nonmagnetic
calculation within DFT cannot describe the paramagnetic spin order properly,
since the paramagnetic phase has zero averaged spin due to the fluctuations of
the local spin, while there is no local spin in the nonmagnetic phase. To
study the paramagnetic phase of Li2MnO3 at finite temperature, more advanced
theories such as dynamical mean field theory (DMFT) is needed.

Other transition metal oxides Li batteries, such as LiCoO2 Isaacs and
Marianetti (2020) and LiNiO2 Korotin et al. (2019) are also studied within
DFT+DMFT, and it has been shown that the electronic structure and the energy
gap size of LiNiO2 strongly depends on the choice of the different Wannier
basis Korotin et al. (2019). Since the symmetry of Li2MnO3 ($C2/m$), is lower
than the symmetry of LiNiO2 ($R\bar{3}m$), the distortion of TM-O octahedron
is larger in Li2MnO3, and thus the choice of the Wannier basis can be even
more important for Li2MnO3. Therefore, the systematic study of Li2MnO3 with
different basis is needed within DFT+DMFT.

Figure 1:  (a) top and (b)side view for the atomic structure of Li2MnO3. Li
atoms are located between MnO3 layers and the center of the hexagon.

In the present work, we performed DFT+DMFT calculation Singh et al. (2021) to
study the electronic structure of Li2MnO3. Li2MnO3 has $C2/m$ symmetry with
the $C_{2h}$ point group symmetry of Mn ion, and the Mn-O6 octahedron is
squeezed along the [111] direction (see Figure 8 in Appendix A). As a result,
the Wannier Hamiltonian projected along the global coordinate has large off-
diagonal terms, which result the large suppression of the energy gap within
DMFT using $U$=5, while the off-diagonal terms are negelected in the quantum
Monte Carlo impurity solver. Off-diagonal terms are reduced by using the
Wannier basis close to the local coordinate of MnO6, but they are still large
and the energy gap is still suppressed, because of the local symmetry of MnO6
is $C_{2h}$. To reduce the effect of the off-diagonal terms, we diagonalize
the block diagonal part of Mn $d$ Hamiltonian, by applying unitary rotation
matrix. Block-diagonalization of the Hamiltonian enhances the gap, but the gap
is still much smaller than the experimental value, and increasing $U$ cannot
resolve the problem.

Experimental gap is obtained by using nonzero double counting term and
increasing $U$. Since the double counting term represents the $p-d$ covalency,
importance of the double counting term indicates that $p-d$ correlation is
also important on the electronic structure of Li2MnO3. Finally we considered
$d$-only model Hamiltonian for DMFT. Since the Wannier $d$ basis of effective
model is a hybridized state of Mn $d$ and O $p$, $U$ of effective low-energy
model (we refer it as $d$-only model) provides both $d-d$ and $p-d$
correlation. Therefore, the energy gap is sensitive on $U$ with $d$-only
model, and experimental energy gap is obtained with small $U$ value.

##  II Methods

###  II.1 DFT+DMFT

Figure 2:  Schematic diagram of the Wannier Hamiltonian matrix for Mn $d$ and
O $p$ orbitals. We diagonalize the diagonal blocks of each Mn $d$ states
(orange boxes), and control $p-d$ correlation by changing double counting
parameter.

We employ the non-charge-self-consistent DFT+DMFT method Park et al. (2020);
Singh et al. (2021) for relaxed structures obtained from DFT calculations. For
DFT calculation, we use the projector augmented wave (PAW) method Blöchl
(1994) and the revised version of the generalized gradient approximation (GGA)
proposed by Perdew et al. (PBEsol) Perdew et al. (2008) as implemented in the
VASP software Kresse and Joubert (1999). In all cases, the spin-dependent
version of the exchange correlation functional is employed. A plane wave basis
with a kinetic energy cutoff of 500 eV is used. We used 24 atom unit cells
(i.e., $1\times 1\times 2$ unit cells), which contains 4 Mn atoms, and
$\Gamma$-centered k-point meshes of size 10$\times$10$\times$5\. Atomic
positions within the unit cells were relaxed until the residual forces were
less than 0.01 eV/Å, and the stress was relaxed below 0.02 kB.

We solve the many-body problem on the manifold of both Mn 3$d$+O $p$ Wannier
orbitals and Mn 3$d$-only orbitals. The DFT+DMFT calculation has the following
steps. First, we solve the non-spin-polarized Kohn-Sham (KS) equation within
DFT using VASP. Second, we construct a localized-basis Hamiltonian for the Mn
3$d$ bands by generating maximally localized Wannier functions (MLWFs) Marzari
et al. (2012) for the nonmagnetic DFT band structure. Finally, we solve the
DMFT self-consistent equations for the correlated subspace of Mn 3$d$ and O
$p$ Wannier orbitals (or only Mn 3$d$ orbitals) using the continuous time
quantum Monte Carlo (CTQMC) Haule (2007); Gull et al. (2011) impurity solver.
Both Hubbard $U$ and Hund’s couplings $J$ are parameterized by the
($F^{0}$,$F^{2}$,$F^{4}$) Slater integrals , using $U=F^{0}$ and
$J=(F^{2}+F^{4})/14$. Only Coulomb interaction matrix elements of the density-
density types are considered in the CTQMC while the spin-flip and pair-hopping
terms are neglected. For the $pd$ model Hamiltonian, we used $U$ values of 5
and 7 eV and $J$ of 0.9 eV. We used electronic temperatures of 300 K to study
the temperature effect on the spectral function.

We note that applying DMFT method to Li2MnO3 has two issues: (i) there are
large off-diagonal terms in the Wannier Hamiltonian (yellow region in Fig. 2),
(ii) $p-d$ correlation is important due to the strong $p-d$ hybridization
(green region in Fig. 2). To overcome these problems, we (i) diagonalize the
block $d$ Hamiltonian and (ii) use different values of double counting
paramters, as summarized in Fig. 2.

If the off-diagonal terms in the Hamiltonian are large, they can induce large
error within DMFT, since CTQMC only treat the diagonal terms of the
Hamiltonian to avoid the sign problem. The cartesian axes of the Wannier
orbitals and the directions of the Mn-O bonds are not parallel, due to the
$C_{2h}$ point group symmetry of MnO6 octahedron. If the point group symmetry
of TM octahedron is not cubic, e.g., trigonal or monoclinic, then there is
large mixing of $d$ basis ($d_{xy}$, $d_{xz}$, $d_{yz}$, $d_{z^{2}}$,
$d_{x^{2}-y^{2}}$), since this basis is defined by the cubic crystal field. In
this case, the off-diagonal terms of the Wannier Hamiltonian with cubic $d$
orbital basis are large, and results in the error within DMFT. Note that
DFT+$U$ does not have such problem since DFT+$U$ is rotationally invariant
Liechtenstein et al. (1995) and thus can have off-diagonal terms in the
density matrix.

On the other hand, if $p-d$ hybridization is strong as the nickelates Park et
al. (2015, 2012), applying nonzero $U$ on $pd$ Wannier Hamiltonian may not
capture the physics of $pd$ correlation properly. To resolve this issue, we
use the double counting parameter, which control the degree of $p-d$
covalency. In term of double counting corrections for DFT+DMFT, we use a
double counting energy ($E^{\rm{DC}}$) and potential ($V^{\rm{DC}}=\partial
E^{\rm{DC}}/\partial N_{d}$) similar to the conventional fully localized limit
Anisimov et al. (1991); Park et al. (2020):

$E^{\rm{DC}}=\frac{U}{2}N_{d}\cdot\left(N_{d}-1\right)-\frac{J}{4}N_{d}\cdot\left(N_{d}-2\right)\,,$
(1)
$V^{\rm{DC}}=\frac{U}{2}\left(\overline{N}_{d}-\frac{1}{2}\right)-\frac{J}{2}\left(\overline{N}_{d}-1\right)$
(2)

Here $\overline{N}_{d}=N_{d}-\alpha$, where $N_{d}$ is the $d$ occupancy
obtained self-consistently at each Mn site, and $\alpha$ is double counting
parameter. Note that $\alpha=0$ gives the conventional fully localized limit.
From the Eqs. 1 and 2, changing $\alpha$ (or $V^{\rm{DC}}$) can tune the $p-d$
covalency by effectively shifting the $d$ orbital level. If $V^{\rm{DC}}$
potential is smaller than the DC potential of fully localized limit, it will
make $d$ orbital level higher and the covalency effect weaker, with reduced
$N_{d}$.

###  II.2 DFT+$U$ and atomic structures

The GGA+$U$ scheme within the rotationally invariant formalism together with
the fully localized limit double-counting formula Liechtenstein et al. (1995)
is used to study the effect of electron interactions. In this work, we do not
employ a Hund’s $J$ parameter for any atom in our DFT+$U$ calculations. As
explained by prior work, (i) our spin-dependent PBE DFT exchange-correlation
functional already describes a sizable exchange interaction prior to including
of any $+U$ correction, and (ii) further inclusion of a $J$ atop a spin-
dependent functional can lead to unexpected (and/or incorrect) physical
behavior in a number of transition metal oxides Park et al. (2015); Chen et
al. (2015); Chen and Millis (2016). Projected density of states are obtained
by the spherical harmonic projections inside spheres around each atom. Wigner-
Seitz radii of 1.323 Å were used for the projection of Mn atoms, respectively,
as implemented in the VASP-PAW pseudopotential.

##  III  Results and discussion

There are several DFT+$U$ studies of Li2MnO3, while $U$(Mn) =5eV is used Xiao
et al. (2012); Wang et al. (2017); Chen and Islam (2016); Chen et al. (2019);
Hikima et al. (2022). However, previous studies only considered nonmagnetic
Hikima et al. (2022) or ferromagnetic Xiao et al. (2012); Chen and Islam
(2016); Wang et al. (2017) configurations, while Li2MnO3 is antiferromagnetic
with $T_{N}$= 36K in the experiment Lee et al. (2012). Given that the spin
state of single transition metal (TM) ion and their magnetic interactions can
strongly depend on $U$, systematic study is needed within DFT+$U$. In Secion
III.1, we show that $U$(Mn) = 2 is needed to obtain the proper magnetic ground
state. While the antiferromagnetic ground state can be obtained by DFT+$U$,
the paramagnetic phase of Li2MnO3 cannot be described within DFT+$U$. Studying
the paramagnetic phase is essential, since the operating temperature of the
battery is around 300 K. In Secion III.2, we systematically study the
electronic structure of paramagnetic phase within DFT+DMFT.

Figure 3:  Possible magnetic phases of Li2MnO3 at low temperature, (a)
$\Gamma_{2u}$ and (b) $\Gamma_{3g}$. Mn ions with spin-up and spin-down are
represented by different colors (blue and skyblue). (c) the relative energies
of the ferromagnetic, $\Gamma_{2u}$, and $\Gamma_{3g}$ phases, as a function
of $U$(Mn). Energy of the ferromagnetic phase is set to be zero.

###  III.1 DFT+U

We begin the discussion by studying the magnetic phase of Li2MnO3. Li2MnO3 is
antiferromagnetic with Neel temperature of $T_{N}=36$ K, and the magnetic
propagation vector is $Q_{m}=(0,0,0.5)$ Strobel and Lambert-Andron (1988).
Among the possible magnetic configuration with $Q_{m}=(0,0,0.5)$,
$\Gamma_{3g}$ Strobel and Lambert-Andron (1988) and $\Gamma_{2u}$ Lee et al.
(2012) are suggested as the ground state, and the recent powder neutron
diffraction data and single crystal neutron diffraction data at 6 K showed
that the ground state is $\Gamma_{2u}$. Magnetic moment of Mn is 2.3$-$2.7
$\mu_{B}$ Lee et al. (2012); Strobel and Lambert-Andron (1988), indicating
that Mn has high-spin state.

In Figure 3, we compare the energies of the ferromagnetic (FM),
antiferromagnetic $\Gamma_{3g}$ and $\Gamma_{2u}$ phases. $\Gamma_{2u}$ phase
is most stable if $U$(Mn) $\leq 2$, while $\Gamma_{3g}$ becomes more stable
when $U$(Mn) $>2$, while Mn ion always has the high-spin state. This result
shows that $U$(Mn) $\leq 2$ is needed to obtain the experimental ground state.
Previous linear response calculation for spinel Mn4+ suggested that $U$(Mn)=
5.04 eV Zhou et al. (2004), and many DFT+$U$ studies Xiao et al. (2012); Wang
et al. (2017); Chen and Islam (2016); Chen et al. (2019); Hikima et al. (2022)
used $U$(Mn)= 5 eV based on this study. We note that using $U$ values from the
linear response theory within GGA+$U$ overestimate the correlation in the
transition metal oxides, such as nickelates, and the overestimation can be
fixed within DMFT Park et al. (2014). Therefore, we suggest that within
DFT+$U$, $U$(Mn) = 2 eV is reasonable value to study the electronic and
magnetic properties of Li2MnO3. The energy difference between $\Gamma_{2u}$
and other magnetic configuration decreases as a function of $U$. With $U$(Mn)
$=2$ eV, $\Gamma_{2u}$ is more stable than FM and $\Gamma_{3g}$ by 1.8 and 0.8
meV, respectively, consistent with the low $T_{N}=36$ K. Magnetic moment of Mn
in $\Gamma_{2u}$ phase is 2.8 $\mu_{B}$, similar to the experimental value 2.3
$\mu_{B}$.

Next we study the effect of $U$ on the electronic structure and energy gap of
Li2MnO3. Mn ion in Li2MnO3 has 4+ charge state with 3 $d$ electrons, and it
has high-spin state. MnO6 in $C2/m$ phase has $C_{2h}$ point group symmetry,
which splits the $d$ bands into 3 low $A$ bands ($d_{xy}$, $d_{z^{2}}$, and
$d_{x^{2}-y^{2}}$) and 2 high $B$ bands ($d_{xz}$ and $d_{yz}$). In the high-
spin state, 3 electrons of Mn occupy the three spin-up $A$ bands (see Figure 8
in Appendix). Therefore, there are two types of splittings which determine the
energy gap ($E_{g}$): (i) crystal field splitting ($\Delta_{\mathrm{CF}}$)
between $A$ and $B$ bands, and (ii) exchange splitting
($\Delta_{\mathrm{ex}}$) between spin-up $A$ and spin-down $A$ bands. These
splittings are also presented in the projected density of states in Fig. 4.

For the non spin-polarized phase with $U$(Mn) $=0$, $\Delta_{\mathrm{CF}}$ is
around 2.5 eV while $\Delta_{\mathrm{ex}}$ is only $\sim$0.6eV. As a result,
$E_{g}$ is almost zero, as shown in Fig. 4(a). If the spin-polarization is
turned on [Fig. 4(b)], then $\Delta_{\mathrm{ex}}$ is dramatically enhanced to
$\sim$2.2eV, and $\Delta_{\mathrm{CF}}$ is also increased to $\sim$3 eV.
$\Delta_{\mathrm{ex}}$ determines the size of the energy gap since
$\Delta_{\mathrm{ex}}<\Delta_{\mathrm{CF}}$, and the $E_{g}$ becomes 1.19 eV.
If nonzero $U$ is applied, then both $\Delta_{\mathrm{ex}}$ and
$\Delta_{\mathrm{CF}}$ are increased, as shown in Fig. 4(c) With $U$(Mn) $=2$,
we obtain $E_{g}$= 2.0 eV, similar to the experimental gap 2.1$-$2.17eV Singh
and Singh (2020); Tamilarasan et al. (2015). When $U$(Mn)=5, $E_{g}$ is 1.9
eV, similar to the value with $U$(Mn) $=2$. While the occupied Mn $d$ band is
lowered with large $U$, the energy of the valence band maximum is less
sensitive to $U$, and thus $E_{g}$ does not change much.

Figure 4:  Projected density of states (PDOS) onto the Mn $d$ (red) and O $p$
(black) in Li2MnO3. (a) $U$(Mn) $=0$ for non spin-polarized configuration, (b)
$U$(Mn) $=0$ for spin-polarized configuration (ferromagnetic), (c) $U$(Mn)
$=2$ and (d) $U$(Mn) $=5$ for spin-polarized configuration.

###  III.2 DFT+DMFT

Figure 5: DFT+DMFT spectral functions of Mn $d$ Wannier orbitals. (a) global
coordinate is used for Wannier projection, (b) local coordinate obtained by
unitary rotation matrix is used for Wannier projection, and (c) block diagonal
part of Mn $d$ Hamiltonian is diagonalized The calculations use $U=5$ eV,
$J=0.9$ eV, and a temperature of 300 K.

As mentioned, MnO6 has $C2/m$ point group symmetry, which is different from
the cubic ($O_{h}$) symmetry (See Apendix A) In the cubic ($O_{h}$) phase, the
local Mn–O coordinates and the global coordinates based on the symmetry are
parallel, thus the Hamiltonian based on the $d$ orbital does not have off-
diagonal terms. On the other hand, in the $C2/m$ phase, local Mn–O axes and
the global axes are not parallel. Crystal splitting of $d$ bands shown in Fig.
8(b) is based on the global axes, not the local axes. Therefore, if we use the
global coordinates for the Wannier projection, there are large off-diagonal
terms:

$\displaystyle\mathbf{H}_{i}=$ (3)
$\displaystyle\bordermatrix{&d_{z^{2}}&d_{xz}&d_{yz}&d_{x^{2}-y^{2}}&d_{xy}\cr
d_{z^{2}}&3.784&0.002&0&$-$0.007&0\cr
d_{x^{2}-y^{2}}&0.002&4.583&0&\textbf{$-$0.592}&0\cr
d_{yz}&0&0&4.578&0&\textbf{0.584}\cr
d_{x^{2}-y^{2}}&$-$0.007&\textbf{$-$0.592}&0&4.153&0\cr
d_{xy}&0&0&\textbf{0.584}&0&4.152\cr}\qquad$

Using the non-rotated Hamiltonian, we employed DFT+DMFT calculations plot the
spectral functions in Fig. 5(a). Since the battery is used at room
temperature, we use the temperature of 300 K, and consider the paramagnetic
phase since $T_{N}$ = 65 K is much lower than 300 K. Using the non-rotated
basis within DMFT, $E_{g}$ is only 0.4 eV, much smaller than the experimental
value.

Since the large off-diagonal terms give large error of the energy gap, we also
use the local coordinate by applying the unitary rotation matrix to the
Hamiltonian, which gives the minimum values of the off-diagonal terms. As
shown in Eq. 4,

$\displaystyle\mathbf{H}_{rot}=$ (4)
$\displaystyle\bordermatrix{&d_{z^{2}}&d_{xz}&d_{yz}&d_{x^{2}-y^{2}}&d_{xy}\cr
d_{z^{2}}&4.069&0.018&$-$0.009&\textbf{$-$0.530}&0.059\cr
d_{xz}&0.018&3.775&0.064&0.010&0.003\cr
d_{yx}&$-$0.009&0.064&4.997&$-$0.011&0.006\cr
d_{x^{2}-y^{2}}&\textbf{$-$0.530}&0.010&$-$0.011&4.668&$-$0.101\cr
d_{xy}&0.059&0.003&0.006&$-$0.101&3.754\cr}\qquad$

off-diagonal terms are much smaller, and $E_{g}$ is increased to 0.6 eV.
However, $E_{g}$ is still much smaller the experimental gap. As shown in Fig.
5(b), underestimation of $E_{g}$ is due to the small $\Delta_{CF}$.
$\Delta_{CF}$ within DMFT with rotated Wannier axes is around 1.5 eV, which is
much smaller than the value within DFT+$U$ ($\sim$3 eV). On the other hand,
$\Delta_{ex}$ is around 2.7 eV, does not affect to $E_{g}$.

To resolve the underestimation of $E_{g}$ within DFT+DMFT, we diagonalize the
each Mn $d$ block of the Hamiltonian (orange boxes in Fig. 2), which can be
obtained by the unitary rotation matrix. As presented in Fig. 5(b), $E_{g}$
within DMFT is increased to 0.8 eV. Using the diagonalized Hamiltonian, we
also consider various $U$ and double counting parameter $\alpha$ (see Eq. 2)
Interestingly, $\Delta_{CF}$ and resulted $E_{g}$ are sensitive on $U$. As
shown in Fig. 6, $E_{g}$ with $U$=5 and 7 are 0.8 and 0.9eV, respectively.

Figure 6: DFT+DMFT spectral functions of Mn $d$ Wannier orbitals with
different $U$ and $\alpha$. The calculations use temperature of 300 K.

On the other hand, $\Delta_{CF}$ and resulted $E_{g}$ are more sensitive to
$\alpha$. With $U$=5, $E_{g}$ is increased to 1.4 eV when $\alpha$= 0.8. In
addition, when $\alpha$= 0.8, increasing $U$ =5 to 7 enhances $E_{g}$ from 1.4
to 2.0 eV. The different $\alpha$ values effectively shift the energy of the
$d$ bands, $p-d$ covalency can be tuned by $\alpha$, since the energy of the
$d$ bands is effectively shifted by different $\alpha$ values. Therefore,
$E_{g}$ sensitive on $\alpha$ indicate that $p-d$ correlation $U_{pd}$ is
important on $\Delta_{CF}$, and changing $U$(Mn) for the $pd$ model
Hamiltonian is not enough since it only changes the $d-d$ correlation.

Figure 7: DFT+DMFT spectral functions of Mn $d$ Wannier orbitals with $d$-only
model. The calculations use $U=2.2$ eV, $J=0.9$ eV, and a temperature of 300
K.

In this reason, we also consider the $d$-only model Hamiltonian to describe
the electronic structure of Li2MnO3. While the Wannier $d$ orbitals are close
to the Mn $d$ orbital in the $pd$ model, Wannier $d$ orbitals in $d$-only
model are the hybridized Mn $d$ and O $p$ orbitals. Thus, the effect of
$U$(Mn) on $d$-only Wannier Hamiltonian is similar to the effects of applying
both $U_{dd}$ and $U_{pd}$ to the $pd$ model Hamiltonian. Therefore,
$\Delta_{CF}$ and $E_{g}$ are sensitive to $U$(Mn) within the $d$-only model.
Indeed, the value of $E_{g}$ is similar to $U$(Mn) using $d$-only Hamiltonian
within DMFT. With $U$(Mn) = 2.2 eV, we obtain $E_{g}$= 2.1 eV, as presented in
Fig. 7.

##  IV Summary

In this work, we show that the off-diagonal terms of Mn $d$ Hamiltonian and
$p$-$d$ interaction play crucial role on the electronic structure of Li2MnO3,
through DFT+DMFT calculations. We show that when only the diagonal terms are
treated in the CTQMC solver to avoid the sign problem, the off-diagonal terms
of the Mn $d$ block of the Wannier Hamiltonian play an important role in the
electronic structure. Due to the monoclinic symmetry ($C2/m$) of Li2MnO3, the
off-diagonal terms of Mn $d$ are large if the global coordinate is chosen, and
thus the energy gap is greatly suppressed compared to the experimental value.
Choosing local coordinate can reduce the magnitude of the off-diagonal terms
and increase the band gap enormously, but it is much smaller than the
experimental value.

Therefore, we diagonalize the Mn $d$ block of the Wannier Hamiltonian by
multiplying the unitary rotation matrix, and show that the energy gap is
further enhanced, while the gap is still not large enough. We find that since
Mn $d$ \- O $p$ hybridization is strong, increasing $U_{dd}$ is not enough,
and applying $p$-$d$ correlation by changing the double counting parameter is
essential to obtain the experimental gap. To verify this idea, we also
consider $d$-only model Hamiltonian, since the basis of the $d$-only model is
hybridized state of Mn $d$ and O $p$. We confirm that energy gap is sensitive
on $U$, indicating that both $dd$ and $pd$ correlations are important on the
electronic structure of Li2MnO3.

We also show that the magnetic ground state strongly depend on $U$. While the
antiferromagnetic states ($\Gamma_{2u}$ or $\Gamma_{3g}$) are more stable than
the ferromagnetic state in the wide range of $U$ ($U\leq 5$), experimentally
observed ground state $\Gamma_{2u}$ phase is most stable with $U\leq 2$.
Therefore, we suggest that the careful choice of $U$ is needed for future
DFT+$U$ studies of Li2MnO3.

##  V Acknowledgments

This research is supported by the Vehicle Technologies Office (VTO),
Department of Energy (DOE), USA, through the Battery Materials Research (BMR)
program. We also acknowledge financial support from the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, Materials Science
and Engineering Division. We gratefully acknowledge the computing resources
provided on Bebop, a high-performance computing cluster operated by the
Laboratory Computing Resource Center at the Argonne National Laboratory.

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##  Appendix A Crystal symmetry of Li2MnO3

In Figure 8 we present the atomic structure with cubic and trigonal
symmetries, and schematic band pictures.

Figure 8:  Schematic band splitting based on the (a) cubic ($O_{h}$) and (b)
trigonal ($C_{2h}$) crystal fields.