# Calculation of screened Coulomb interaction parameters for the charge-
disproportionated insulator CaFeO3

Maximilian E. Merkel<EMAIL_ADDRESS>Materials Theory, ETH
Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland  Claude Ederer
<EMAIL_ADDRESS>Materials Theory, ETH Zürich, Wolfgang-Pauli-Strasse
27, 8093 Zürich, Switzerland

(August 28, 2024)

###### Abstract

We calculate the screened electron-electron interaction for the charge-
disproportionated insulator CaFeO3 using the constrained random-phase
approximation (cRPA). While in many correlated materials, the formation of a
Mott-insulating state is driven by a large local Coulomb repulsion,
represented by the Hubbard $U$, several cases have been identified more
recently where $U$ is strongly screened and instead the Hund’s interaction $J$
dominates the physics. Our results confirm a strong screening of the local
Coulomb repulsion $U$ in CaFeO3 whereas $J$ is much less screened and can thus
stabilize a charge-disproportionated insulating state. This is consistent with
the case of the rare-earth nickelates where similar behavior has been
demonstrated. In addition, we validate some common assumptions used for
parametrizing the local electron-electron interaction in first-principles
calculations based on density-functional theory (DFT), assess the dependence
of the interaction on the choice of correlated orbitals, and discuss the use
of the calculated interaction parameters in DFT+$U$ calculations of CaFeO3.
Our work also highlights some limitations of the cRPA approach for systems
with strong entanglement between the correlated and uncorrelated bands, which
can lead to a strong overscreening of the calculated interaction parameters.

##  I Introduction

Strongly correlated materials exhibit many interesting physical phenomena,
such as high-temperature superconductivity, colossal magnetoresistance, and
metal-insulator transitions [1, 2, 3], which makes them also very attractive
candidates for a variety of technological applications, such as, e.g., Mott-
transistors [4, 5, 6, 7]. Here, the idea is to exploit metal-insulator
transitions for achieving higher carrier densities, larger on-off ratios,
lower switching voltage, and faster switching times. The emergence of a metal-
insulator transition in strongly correlated materials is typically associated
with a large on-site Coulomb repulsion, represented by the Hubbard parameter
$U$, which forces the electrons to localize and thereby produces an insulating
state. A metal-insulator transition can then be triggered by variations in
temperature, strain, doping, applied voltage, etc.

However, for certain transition-metal oxides exhibiting metal-insulator
transitions, it was suggested that the Hubbard $U$ is in fact strongly
screened and therefore rather small, and that instead the metal-insulator
transition observed in these systems is controlled by the strength of the
Hund’s interaction $J$ [8, 9, 10, 11, 12]. In these cases, the metal-insulator
transition is accompanied by a charge disproportionation of the transition-
metal cations, resulting in crystallographically inequivalent sites and a
breathing distortion of the surrounding oxygen polyhedra (see Fig. 1).

Such strong screening of the Coulomb repulsion between the transition-metal
$d$ electrons can be caused by the O-$p$ electrons and is the more efficient
the closer in energy the O-$p$ states are to the correlated $d$ states. This
makes materials with a small or even negative charge-transfer energy good
candidates for this type of behavior [8, 13], such as oxides containing late
transition-metal cations with high oxidation states. Indeed, rare-earth
nickelates [14, 15, 16] and certain ferrites [17, 18, 19] have been identified
as _charge-disproportionated insulators_. For the case of the well-studied
rare-earth nickelates, $R$NiO3, the strong screening of the local Coulomb
repulsion was confirmed by ab initio calculations of the screened interaction
parameters [20, 21].

Figure 1: Experimental crystal structure of CaFeO3 below the metal-insulator
transition [22]. The breathing distortion creates long-bond (light orange) and
short-bond (dark orange) FeO6 octahedra, which are rotated and tilted by other
distortions.

Long before the relevance of charge disproportionation for the metal-insulator
transition in the rare-earth nickelates has been recognized, a metal-insulator
transition around room temperature involving charge disproportionation was
reported for paramagnetic CaFeO3 [17, 23]. Here, the Fe cation has a nominal
average oxidation state of $4+$ and disproportionates according to
$2d^{4}\rightarrow d^{3}+d^{5}$, with a high-spin $d^{5}$ configuration at
ambient pressure [24]. The charge disproportionation in CaFeO3 also couples to
a breathing distortion of the FeO6 octahedra [25, 26, 22], resulting in a
symmetry lowering from $Pbnm$ to $P2_{1}/n$ and a transition from metal to
insulator [23, 27]. The low-temperature $P2_{1}/n$ crystal structure of CaFeO3
is depicted in Fig. 1.

Several computational studies based on density-functional theory (DFT) have
addressed the charge disproportionation and metal-insulator transition in
CaFeO3 in order to clarify the underlying physics [28, 29, 30, 31, 32, 33, 34,
35, 36]. Thereby, to obtain a realistic description of the insulating state,
the local electron-electron interaction is typically explicitly treated within
DFT+$U$ [37] or DFT+dynamical mean-field theory (DMFT) [38, 39]. However,
since the screened interaction parameters for CaFeO3 have not been calculated
yet, different empirically chosen parameters have been used in these
calculations, ranging from $U=$3\text{\,}\mathrm{e}\mathrm{V}$$ to more than
$7\text{\,}\mathrm{e}\mathrm{V}$ for the Hubbard parameter, and from
$J=$0\text{\,}\mathrm{e}\mathrm{V}$$ to as much as
$2\text{\,}\mathrm{e}\mathrm{V}$ for the Hund’s interaction. In our recent
tight-binding+DMFT study [12], we have found a very rich phase diagram for
paramagnetic CaFeO3 as a function of $U$ and $J$, which shows that using
suitable interaction parameters is crucial for obtaining the correct phase. We
also note that different studies often use different definitions of the
correlated local Fe-$d$ orbitals, which hampers a systematic comparison of the
corresponding interaction parameters.

Here, we calculate screened interaction parameters for CaFeO3 using the
constrained random-phase approximation (cRPA) [40, 41, 42]. Our results
confirm a strong screening of the Hubbard parameter $U$, while the Hund’s
interaction $J$ is less affected by the screening. We also compare different
choices of the correlated $d$ orbitals, corresponding either to an atomic-
orbital-like basis, as commonly used in DFT+$U$, or to more molecular-orbital-
like hybridized Fe-$d$/O-$p$ frontier orbitals, which closely resemble the
nominal oxidation states and are more commonly used in model Hamiltonian or
DFT+DMFT studies. Furthermore, we assess the validity of the typically-used
spherical approximation for the local electron-electron interaction and
discuss the applicability of the calculated screened interaction parameters in
DFT+$U$ and DFT+DMFT calculations of CaFeO3. To that end, we also revisit the
$U$ (and $J$) dependence of the relaxed high- and low-temperature structure of
CaFeO3 within DFT+$U$ and compare the interaction parameters used for these
structural relaxations with the ones obtained within cRPA.

##  II Methods

In this section, we introduce the interaction Hamiltonian acting on the
correlated subspace of Fe-$d$ electrons and discuss the determination of the
corresponding screened interaction parameters using cRPA. Then, we describe
the details and all parameters used in our DFT(+$U$) calculations and in the
construction of the Wannier basis representing the correlated orbitals.

###  II.1 Interaction Hamiltonian and constrained random-phase approximation

The general local two-particle interaction Hamiltonian can be written as
follows (see, e.g., [43]):

$\displaystyle
H_{\mathrm{int}}=\frac{1}{2}\sum_{\begin{subarray}{c}mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}\\\
\sigma\sigma^{\prime}\end{subarray}}U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}c^{\dagger}_{m\sigma}c^{\dagger}_{m^{\prime}\sigma^{\prime}}c_{m^{\prime\prime\prime}\sigma^{\prime}}c_{m^{\prime\prime}\sigma},$
(1)

where $c_{m\sigma}$ is the annihilation operator of an electron with spin
$\sigma$ in orbital $m$ and the matrix elements
$U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}$ are defined through

$\displaystyle U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}=\int
d\bm{r}d\bm{r}^{\prime}[$
$\displaystyle\phi_{m}^{*}(\bm{r})\phi_{m^{\prime}}^{*}(\bm{r}^{\prime})v(\bm{r},\bm{r}^{\prime})$
$\displaystyle\times\phi_{m^{\prime\prime\prime}}(\bm{r}^{\prime})\phi_{m^{\prime\prime}}(\bm{r})].$
(2)

Here, $\phi_{m}(\bm{r})$ are the orbitals and $v(\bm{r},\bm{r}^{\prime})$ is
the interaction between these orbitals, which in the following can be either
the bare or the screened Coulomb interaction.

For practical calculations, the full four-index form of
$U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}$ is sometimes reduced
to a two-index form according to $U_{mm^{\prime}}=U_{mm^{\prime}mm^{\prime}}$
and $J_{mm^{\prime}}=U_{mm^{\prime}m^{\prime}m}$. These two-index terms are
assumed to dominate the interaction. The third possible two-index matrix
$I_{mm^{\prime}}=U_{mmm^{\prime}m^{\prime}}$ equals $J_{mm^{\prime}}$ if the
orbitals $\phi_{m}(\bm{r})$ are real-valued, which can be directly seen from
Eq. (2). In the following, we always use real-valued, maximally localized
Wannier functions [44], and thus we only discuss $U_{mm^{\prime}}$ and
$J_{mm^{\prime}}$.

The interpretation of these two-index terms becomes clear when considering
only the density-density terms of the Hamiltonian

$\displaystyle H_{\mathrm{int}}=$
$\displaystyle\frac{1}{2}\sum_{mm^{\prime},\sigma}U_{mm^{\prime}}n_{m\sigma}n_{m^{\prime}\bar{\sigma}}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{m\neq
m^{\prime},\sigma}(U_{mm^{\prime}}-J_{mm^{\prime}})n_{m\sigma}n_{m^{\prime}\sigma},$
(3)

with the density operators $n_{m\sigma}=c_{m\sigma}^{\dagger}c_{m\sigma}$ and
the convention $\bar{\sigma}=-\sigma$. Here,
$U_{mm^{\prime}}=U^{\sigma\bar{\sigma}}_{mm^{\prime}}$ represents the
interaction between electrons with opposite spins and
$U_{mm^{\prime}}-J_{mm^{\prime}}=U^{\sigma\sigma}_{mm^{\prime}}$ the
interaction between electrons with the same spin.

To obtain screened interaction parameters within cRPA [40, 41, 42, 45], the
Hilbert space is divided into the _correlated subspace_ , represented by the
orbitals $\phi_{m}({\bf r})$ for which $H_{\mathrm{int}}$ is defined, and the
rest. Then, the bare electron-electron interaction
$v(\bm{r},\bm{r}^{\prime})=|\bm{r}-\bm{r}^{\prime}|^{-1}$ is screened by the
polarization function $P_{\mathrm{rest}}$ including all electronic transitions
that involve the rest of the Hilbert space, but not those purely within the
correlated subspace itself. The screened Coulomb interaction $U$ is then given
by $U=[1-vP_{\mathrm{rest}}]^{-1}v$ [40].

Typically, $P_{\mathrm{rest}}$ is calculated as
$P_{\mathrm{total}}-P_{\mathrm{corr}}$, where $P_{\mathrm{total}}$ is the
total polarization, which can easily be calculated from the Kohn-Sham wave-
functions, and $P_{\mathrm{corr}}$ is the polarization only within the
correlated subspace. If the correlated bands are not entangled with the rest,
$P_{\mathrm{corr}}$ is well-defined and can also be calculated from the Kohn-
Sham wave-functions of the corresponding bands. If there is entanglement
between the correlated and uncorrelated subspace, different approaches have
been introduced to calculate $P_{\mathrm{corr}}$. One option is to neglect the
hybridization between the correlated subspace and the rest in the Hamiltonian,
which produces unentangled correlated bands, which for small hybridization
only differ slightly from the original Kohn-Sham bands [41]. Alternatively,
$P_{\mathrm{corr}}$ can also be calculated using projectors from the bands
onto the correlated orbitals either through the scheme introduced in Ref. 42,
where each transition is assigned a probability according to the weight of the
correlated subspace in each Bloch function, or using an alternative projector-
based scheme described in Ref. 45. The latter is the default option
implemented in the Vienna Ab-initio Simulation Package (VASP) [46, 47] and is
used throughout this work, except for the explicit comparison between these
three methods presented in Sec. III.3.3.

###  II.2 Spherical parametrization and symmetrization of the interaction

For DFT+$U$ and DFT+DMFT calculations, the interaction matrix
$U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}$ is typically
constructed assuming spherical symmetry, i.e,
$v(\bm{r},\bm{r}^{\prime})=v(|\bm{r}-\bm{r}^{\prime}|)$ and atomic-like
orbitals $\phi_{m}(\bm{r})$, where $m$ is the magnetic quantum number
corresponding to an electronic shell with well-defined orbital momentum $l$.
This allows to expand the interaction $v$ in terms of complex spherical
harmonics $Y_{kq}$ and to arrive at a unique parametrization of the
interaction matrix in terms of $l+1$ independent parameters, the Slater
integrals $F_{k}$, where $k$ is an even integer with $0\leq k\leq 2l$ (see,
e.g., [43, 48, 49]). The resulting interaction matrix in the basis of complex
spherical harmonics is

$\displaystyle
U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}=\sum_{k=0}^{2l}F_{k}\alpha^{(k)}_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}$
(4)

with

$\displaystyle\alpha^{(k)}_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}=$
$\displaystyle\frac{4\pi}{2k+1}\sum_{q=-k}^{k}\langle
lm|Y^{*}_{kq}|lm^{\prime\prime}\rangle\langle
lm^{\prime}|Y_{kq}|lm^{\prime\prime\prime}\rangle$ $\displaystyle=$
$\displaystyle(2l+1)^{2}\begin{pmatrix}l&k&l\\\
0&0&0\end{pmatrix}^{2}(-1)^{m+m^{\prime\prime\prime}}$
$\displaystyle\times\begin{pmatrix}l&k&l\\\
-m&m-m^{\prime\prime}&m^{\prime\prime}\end{pmatrix}$
$\displaystyle\times\begin{pmatrix}l&k&l\\\
-m^{\prime}&m^{\prime}-m^{\prime\prime\prime}&m^{\prime\prime\prime}\end{pmatrix}\delta_{m^{\prime\prime}-m,m^{\prime}-m^{\prime\prime\prime}}$
(5)

where the “$2\times 3$ matrices” are Wigner 3-$j$ symbols.

The Slater integrals are in principle defined as the radial integrals
appearing in this expansion. Conversely, they can be extracted from the
interaction matrix by inverting Eq. (4) and using the orthogonality relation
of the $\alpha$ tensor as specified in Eq. (15):

$\displaystyle F_{k}$
$\displaystyle=c_{lk}^{-1}\sum_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}\alpha^{(k)}_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}$
$\displaystyle c_{lk}$
$\displaystyle=\frac{(2l+1)^{4}}{2k+1}\begin{pmatrix}l&k&l\\\
0&0&0\end{pmatrix}^{4}$ (6)

Eq. (6) allows us to reduce any interaction matrix, e.g., one obtained from
cRPA to a set of Slater parameters, which can then be used in Eq. (4) to
construct a spherically symmetrized interaction matrix 111Note that our
definition of the Slater integrals in Eq. (6) differs from Eq. (B3) of Ref. 48
due to the appearance of the Kronecker
$\delta_{m^{\prime\prime}-m,m^{\prime}-m^{\prime\prime\prime}}$ in Eq. (5).
Both forms are equivalent if the matrix
$U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}$ is already spherically
symmetric, but for a general interaction, e.g., one obtained from cRPA, only
Eq. (6) corresponds to the correct inversion of Eq. (4).. Note that while
these formulas are specified in the basis of complex spherical harmonics, in
Sec. III we represent the interaction matrices in the basis of real cubic
harmonics. These two types of functions are related by a simple linear
transformation.

Often, the parametrization of the interaction is further simplified to only
two parameters $U$ and $J$, which are defined in terms of the two-index
matrices:

$\displaystyle U$
$\displaystyle=\frac{1}{(2l+1)^{2}}\sum_{mm^{\prime}}U_{mm^{\prime}},$
$\displaystyle U-J$
$\displaystyle=\frac{1}{2l(2l+1)}\sum_{mm^{\prime}}(U_{mm^{\prime}}-J_{mm^{\prime}})$
(7)

From Eq. (6) and Eq. (7), it follows directly that $F_{0}\equiv U$ and, for
$l=2$, $J\equiv(F_{2}+F_{4})/14$. (The general relation between Slater
integrals and $J$ for all $l$ is given in Eq. (16)). However, the inverse
operation of obtaining the interaction matrix from only $U$ and $J$ is
underdefined for $l\geq 2$ since this requires knowledge of all $l+1$ Slater
integrals. Therefore, a fixed ratio of $F_{4}/F_{2}\approx 0.63$ is usually
assumed for $d$ electrons (see, e.g., [43]), which is based on calculations of
the unscreened interaction using atomic-orbital-like orbitals [50, 51].

Within cRPA, it is straightforward to not only calculate the local elements of
the interaction matrix between orbitals on the same site but also between
orbitals on different sites in the same unit cell. For practical purposes, it
is often assumed that the intersite terms are not strongly orbitally dependent
[52]. Therefore, we only discuss the orbitally averaged nearest-neighbor (NN)
interaction

$\displaystyle V_{\mathrm{NN}}$
$\displaystyle=\frac{1}{(2l+1)^{2}}\sum_{mm^{\prime}}U_{mm^{\prime}},$
$\displaystyle J_{\mathrm{NN}}$
$\displaystyle=\frac{1}{(2l+1)^{2}}\sum_{mm^{\prime}}J_{mm^{\prime}},$ (8)

where now $m$ and $m^{\prime}$ are on NN sites.

###  II.3 DFT calculations and Wannier orbitals

For all calculations, we use the VASP code (version 6.3.0) [46, 47] in
combination with the PBEsol exchange-correlation functional [53]. We use
standard projector augmented wave (PAW) potentials [54, 55] with valence
states 3$s^{2}$3$p^{6}$4$s^{2}$ for Ca, 3$s^{2}$3$p^{6}$3$d^{7}$4$s^{1}$ for
Fe, and 2$s^{2}$2$p^{4}$ for O. We treat CaFeO3 in a 20-atom unit cell, which
allows us to describe both the $Pbnm$ and $P2_{1}/n$ structures, using a
$7\times 7\times 5$ k-point mesh, a plane-wave energy cutoff of
$600\text{\,}\mathrm{e}\mathrm{V}$, and an energy tolerance of
${10}^{-8}\text{\,}\mathrm{e}\mathrm{V}$ between electronic steps, which leads
to good convergence.

Structural relaxations are performed up to an energy tolerance of
${10}^{-7}\text{\,}\mathrm{e}\mathrm{V}$ between ionic steps. Except where
explicitly noted, relaxations of the unit cell and atomic positions are done
using DFT+$U$ [56] with $U=$4\text{\,}\mathrm{e}\mathrm{V}$$ and
$J=$1\text{\,}\mathrm{e}\mathrm{V}$$, similar to previous studies [29, 32,
34], but we also present results where we systematically vary $U$ over a wider
range. For the structural relaxation, we emulate the complex low-temperature
helical magnetic order in CaFeO3 [22, 57] with the much simpler A-type
antiferromagnetic ordering, which has the lowest energy out of A-, C- and
G-type orderings in our calculations. For the high-temperature $Pbnm$
structure, we also perform calculations where we relax only the atomic
positions using spin-degenerate DFT without +$U$ correction, which we call
“bare DFT” here.

We compare the relaxed to the experimental structures using the mode-
decomposition tool ISODISTORT [58, 59]. All amplitudes are normalized to the
20-atom unit cell, and the symmetry-labeling convention uses a cubic
perovskite parent cell with Fe at the origin. The most important modes are the
$R_{1}^{+}$ breathing mode in the $P2_{1}/n$ structure, which splits the Fe
into two inequivalent sites, and the $M_{2}^{+}$ Jahn-Teller mode in the
$Pbnm$ structure, which locally splits the Fe-${e_{g}}$ orbitals but
experimentally is not relevant in CaFeO3. Also present in $Pbnm$ are the modes
related to octahedral rotations and tilting, $M_{3}^{+}$, $R_{4}^{+}$, and
$X_{5}^{+}$, which mainly decrease the hopping and therefore the band width
[29]. Additionally, there are three more symmetry-allowed modes with very
small amplitudes: $R_{5}^{+}$ ($Pbnm$), $R_{3}^{+}$, and $M_{5}^{+}$ (both
$P2_{1}/n$).

We use Wannier90 (version 3.1.0) [60] to construct the Wannier orbitals
defining the correlated subspace for our cRPA calculations. We start with
initial projections corresponding to $d$-shell cubic harmonics defined in a
local coordinate system aligned along the Fe-Fe distances. Thereby, we employ
a disentanglement (outer) window from $-105\text{\,}\mathrm{e}\mathrm{V}$
relative to the Fermi energy, which is chosen to be large enough to encompass
all 56 O-$p$ and Fe-$d$ derived bands, and, optionally, a frozen (inner)
window from $-1.15\text{\,}\mathrm{e}\mathrm{V}$. We always first perform the
disentanglement, where convergence is defined as a relative change of less
than ${10}^{-9}$ of the gauge-invariant part of the spread, and then the
localization (Wannierization) down to an absolute change of less than
${10}^{-9}\text{\,}\mathrm{\text{\r{A}}}^{2}$ of the total spread. Note that
the resulting Wannier functions still resemble the cubic harmonics, which is
required to map the cRPA results onto Eq. (6).

For the cRPA calculations, we first converge a nonmagnetic DFT calculation
with 512 bands and otherwise the same parameters as listed above. Then we
perform one exact-diagonalization step in DFT to obtain the Kohn-Sham band
energies. We perform cRPA calculations only for the zero frequency component
of the screened interaction and with the correlated subspace defined by the
Wannier orbitals.

##  III Results

###  III.1 Structural relaxations

Table 1: Lattice parameters and distortion mode amplitudes (with respect to the cubic $Pm\bar{3}m$ high symmetry reference structure) obtained from DFT+$U$ structural relaxations within $Pbnm$ and $P2_{1}/n$ symmetries. The third line corresponds to a calculation where lattice parameters are fixed to those obtained from DFT+$U$ and only the atomic positions are relaxed using spin-degenerate DFT without a “$+U$” correction. The last two lines correspond to experimental data from Ref. 22 obtained at 15 K ($P2_{1}/n$) and 300 K ($Pbnm$). The $M_{5}^{+}$, $R_{1}^{+}$ and $R_{3}^{+}$ modes are not allowed in $Pbnm$ symmetry.  | Unit cell | Mode amplitudes ($\mathrm{\text{\r{A}}}$) |
---|---|---|---
|  $a$ ($\mathrm{\text{\r{A}}}$) |  $b$ ($\mathrm{\text{\r{A}}}$) |  $c$ ($\mathrm{\text{\r{A}}}$) |  $\beta$ ($\mathrm{\SIUnitSymbolDegree}$) | $M_{2}^{+}$ | $M_{3}^{+}$ | $R_{4}^{+}$ | $R_{5}^{+}$ | $X_{5}^{+}$ | $M_{5}^{+}$ | $R_{1}^{+}$ | $R_{3}^{+}$
DFT+$U$ $P2_{1}/n$ | 5.29 | 5.33 | 7.43 | 89.85 | 0.01 | 0.75 | 1.12 | 0.09 | 0.45 | 0.01 | 0.16 | 0.05
DFT+$U$ $Pbnm$ | 5.27 | 5.41 | 7.38 | 90.00 | 0.24 | 0.79 | 1.13 | 0.10 | 0.52 |  |  |
bare DFT $Pbnm$ | fixed to DFT+$U$ $Pbnm$ results | 0.03 | 0.89 | 1.10 | 0.12 | 0.53 |  |  |
Exp. $P2_{1}/n$ [22] | 5.31 | 5.35 | 7.52 | 90.07 | 0.04 | 0.82 | 1.05 | 0.11 | 0.40 | 0.01 | 0.18 | 0.05
Exp. $Pbnm$ [22] | 5.33 | 5.35 | 7.54 | 90.00 | 0.01 | 0.77 | 0.99 | 0.05 | 0.36 |  |  |

We first perform full structural relaxations of CaFeO3 within both $Pbnm$ and
$P2_{1}/n$ symmetries using DFT+$U$ and A-type antiferromagnetic order, as
described in Sec. II.3. The corresponding results are listed in Table 1,
together with experimental data from Ref. 22.

It can be seen that the lattice parameters obtained for the $Pbnm$ and
$P2_{1}/n$ structures differ only very little, with only a small deviation
from $\beta=$$$ in the monoclinic case. Furthermore, the agreement with the
experimental lattice parameters is very good (there is only a slight
underestimation of the $c$ lattice parameter by $2.1\text{\,}\mathrm{\char
37\relax}$ and the monoclinic distortion is slightly lower than $None$ rather
than larger).

The calculated mode amplitudes also agree very well with the experimental
data, except for the $M_{2}^{+}$ Jahn-Teller mode in the $Pbnm$ case, which is
essentially zero in the room-temperature structure of Ref. 22, whereas a
noticeable Jahn-Teller distortion develops in the zero temperature DFT+$U$
A-type antiferromagnetic case. This is due to the nominally Jahn-Teller-active
$d^{4}$ electron configuration of the Fe4+ cation, and has also been found
with a very similar amplitude of $0.26\text{\,}\mathrm{\text{\r{A}}}$ in a
previous DFT+$U$ study of CaFeO3 if the symmetry is constrained to $Pbnm$
[35].

In the $P2_{1}/n$ structure, which is about
$11\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ per formula unit lower in energy
than the $Pbnm$ structure, the $M_{2}^{+}$ mode nearly vanishes and instead
the $R_{1}^{+}$ breathing mode appears with a magnitude of
$0.16\text{\,}\mathrm{\text{\r{A}}}$, in very good agreement with the
experimental data. Furthermore, it can be seen that the modes related to
octahedral tilts and rotations, $R_{4}^{+}$ and $M_{3}^{+}$ (and also
$X_{5}^{+}$), do not change significantly between $Pbnm$ and $P2_{1}/n$, and
also agree well with the experimental data.

Figure 2: (a) $R_{1}^{+}$ breathing mode amplitude and (b) local magnetic
moments of the two inequivalent Fe sites obtained from full structural
relaxation within $P2_{1}/n$ symmetry as a function of $U$ at
$J=$1\text{\,}\mathrm{e}\mathrm{V}$$.

Fig. 2 shows the evolution of the relaxed $R_{1}^{+}$ breathing mode amplitude
and the local magnetic moments of the two inequivalent Fe sites as function of
the Hubbard $U$ parameter for fixed $J=1$ eV in the $P2_{1}/n$ structure. The
$R_{1}^{+}$ amplitude becomes nonzero for $U\geq 2$ eV and is accompanied by a
sizeable difference in the local magnetic moments, indicating the
disproportionation of the Fe sites into nominal $d^{3}$ and $d^{5}$ valences.
For $4\,\mathrm{eV}\leq U\leq 7$ eV the system is insulating and the
$R_{1}^{+}$ amplitude stays nearly constant. For larger $U>7$ eV, the
breathing mode amplitude decreases and the system becomes metallic again. This
is similar to what has been observed for the rare-earth nickelates in Ref. 61,
where a large $U$ shifts the transition-metal $d$ levels below the O-$p$
states in energy, which then disfavors the charge disproportionation. Thus,
even though a small or maybe even slightly negative charge-transfer energy
seems to be necessary for this type of charge disproportionation, it appears
unfavorable if the charge-transfer energy becomes too negative.

Overall, our results show that structural parameters in good agreement with
available experimental data can be obtained within DFT+$U$ using interaction
parameters of $U=$4\text{\,}\mathrm{e}\mathrm{V}$$ and
$J=$1\text{\,}\mathrm{e}\mathrm{V}$$ and an appropriate magnetic order,
consistent with previous work [32, 34].

Finally, to asses the effect of the imposed A-type magnetic order on the high-
temperature $Pbnm$ structure, we perform an additional relaxation for the
spin-degenerate case using $U=0$ eV, where we fix the lattice parameters to
the ones obtained from the $Pbnm$ DFT+$U$ relaxation and then relax only the
internal atomic positions. We note that fixing the lattice parameters to the
DFT+$U$ values is necessary, since the spin-degenerate DFT calculation leads
to a low-spin state of the Fe cation which would in turn lead to a
significantly lower unit cell volume. Table 1 shows that, in this case, the
$M_{2}^{+}$ Jahn-Teller mode disappears almost completely, while all other
mode amplitudes are rather similar to the magnetic DFT+$U$ calculation. This
shows that the Jahn-Teller mode is indeed stabilized by the +$U$ correction in
combination with the magnetic order, while the octahedral rotation modes are
unaffected by this. Throughout the remainder of this work, we use the so-
obtained structure as the high-temperature $Pbnm$ structure unless otherwise
noted.

###  III.2 Band structure and the correlated subspace

Figure 3: (a) Total and projected density of states (DOS) and (b) band
structure along high symmetry lines of the orthorhombic Brillouin zone of
CaFeO3, obtained from the spin-degenerate DFT calculation for the $Pbnm$
structure. The green shaded region in (b) indicates the frozen energy window
used to construct the frontier-type Wannier orbitals, which result in the
dispersion shown by the green lines. (c, d) Isosurfaces of the
($x^{2}-y^{2}$)-type Wannier orbital (c) in the frontier basis and (d) in the
localized basis. Brown (red) spheres indicate Fe (O) atoms.

Next, we discuss the electronic structure obtained from spin-degenerate DFT
calculations without a +$U$ correction in the $Pbnm$ structure. Fig. 3 (a) and
(b) show the total density of states (DOS), its projection on different atoms
and orbitals, as well as the band structure around the Fermi level. The DOS
shows three (partially overlapping) groups of bands with mixed atomic/orbital
character due to hybridization. Bands with predominant Fe-${e_{g}}$ character
above $0.5\text{\,}\mathrm{e}\mathrm{V}$ are separated from lower-lying bands
with predominant Fe-${t_{2g}}$ character between about
$-10.5\text{\,}\mathrm{e}\mathrm{V}$ due to a strong octahedral crystal field
splitting. The Fe-${t_{2g}}$ dominated bands overlap slightly with O-$p$
dominated bands below $-1\text{\,}\mathrm{e}\mathrm{V}$. The fact that the
Fe-$d$ dominated bands are higher in energy than the O-$p$ dominated ones but
also overlap slightly indeed indicates a small positive but nearly vanishing
charge-transfer energy within the classification scheme according to Zaanen,
Sawatzky, and Allen [62].

We now construct two different sets of Wannier orbitals, starting from initial
projections on Fe-centered $d$ orbitals. In both cases, we use an outer energy
window comprising all bands shown in Fig. 3 (a) and (b).

For the first set of Wannier orbitals, we use an additional inner (frozen)
energy window for the disentanglement procedure which comprises all of the
Fe-${e_{g}}$ dominated bands as well as the upper part of the Fe-${t_{2g}}$
dominated bands down to $-1.1\text{\,}\mathrm{e}\mathrm{V}$, as indicated in
Fig. 3 (b) by the green shading. The corresponding Wannier bands [green lines
Fig. 3 (b)] remain restricted to the energy range of the Fe-$d$ dominated
bands, and perfectly reproduce the DFT bands [black lines in Fig. 3 (b)]
within the inner energy window. Since this inner energy window contains the
Fermi level and there is no entanglement with higher lying bands, the Wannier
orbitals recover an occupation of exactly four electrons per Fe site,
corresponding to the nominal valence of the Fe4+ cation.

The resulting Wannier orbitals are centered on the Fe sites but also exhibit
strong “tails” on the surrounding oxygen ligands, representing the strongly
hybridized character of the corresponding bands. Fig. 3(c) depicts the
($x^{2}-y^{2}$)-type Wannier function as an example. One can recognize the
antibonding character of the underlying hybridization between Fe-$d$ and O-$p$
orbitals. In the following, we refer to this set of Wannier functions as the
_frontier basis_. Such frontier orbitals are often used in DFT plus dynamical
mean-field theory calculations or other more model-based approaches that
consider only the effective $d$ bands (see, e.g., [12]).

We also construct another set of maximally localized Wannier functions, which
we refer to as the _localized basis_. Here, we do not use a frozen inner
window during the disentanglement. This results in more localized orbitals
that closely resemble atomic orbitals and are therefore similar to the
orbitals used for the projected DOS discussed above and for the +$U$
correction applied in the VASP code [54]. Fig. 3(d) shows the resulting
($x^{2}-y^{2}$)-type Wannier function, which does not exhibit the pronounced
$p$-like tails on the surrounding oxygen ligands (small $s$-like tails remain,
due to a corresponding weak hybridization and the orthogonalization between
Wannier functions centered on different sites).

We note that the additional structural distortions in the low-temperature
$P2_{1}/n$ structure do not lead to substantial changes of the band structure
around the lower bound of the frozen energy window, so that we can use the
same energy window relative to the Fermi energy for both crystal structures.

###  III.3 Screened-interaction parameters from cRPA

####  III.3.1 Spherical symmetrization and effective interaction parameters

We first discuss the effect of the spherical symmetrization of the interaction
parameters by comparing the matrix elements obtained directly from cRPA with
their spherically averaged form, i.e., by using Eq. (6) and Eq. (4). Thereby,
we focus on the two-index interaction matrices
$U^{\sigma\bar{\sigma}}_{mm^{\prime}}$ and $U^{\sigma\sigma}_{mm^{\prime}}$
between electrons with opposite and same spins, respectively. We note that the
largest interaction that is not contained in the two-index matrices is
$U_{yz,z^{2},yz,x^{2}-y^{2}}=$0.19\text{\,}\mathrm{e}\mathrm{V}$$ (and
equivalent matrix elements) in the frontier basis, and
$U_{yz,z^{2},yz,x^{2}-y^{2}}=$0.29\text{\,}\mathrm{e}\mathrm{V}$$ in the
localized basis, which is clearly smaller than the dominant components of the
two-index matrices (see below).

In the frontier basis, the two-index matrices obtained directly from cRPA are

$\displaystyle U_{\mathrm{cRPA}}^{\sigma\sigma}$
$\displaystyle=\begin{pmatrix}[r]0.00&1.09&1.01&1.10&1.61\\\
1.09&0.00&1.47&1.12&1.15\\\ 1.01&1.47&0.00&1.47&1.01\\\
1.10&1.12&1.47&0.00&1.16\\\ 1.61&1.15&1.01&1.16&0.00\\\ \end{pmatrix}$
$\displaystyle U_{\mathrm{cRPA}}^{\sigma\bar{\sigma}}$
$\displaystyle=\begin{pmatrix}[r]2.28&1.47&1.45&1.48&1.87\\\
1.47&2.29&1.78&1.49&1.54\\\ 1.45&1.78&2.54&1.78&1.51\\\
1.48&1.49&1.78&2.30&1.55\\\ 1.87&1.54&1.51&1.55&2.51\\\ \end{pmatrix}$ (9) and
their spherically symmetrized forms are $\displaystyle
U_{\mathrm{symm}}^{\sigma\sigma}$
$\displaystyle=\begin{pmatrix}[r]0.00&1.14&0.99&1.14&1.60\\\
1.14&0.00&1.44&1.14&1.14\\\ 0.99&1.44&0.00&1.44&0.99\\\
1.14&1.14&1.44&0.00&1.14\\\ 1.60&1.14&0.99&1.14&0.00\\\ \end{pmatrix}$
$\displaystyle U_{\mathrm{symm}}^{\sigma\bar{\sigma}}$
$\displaystyle=\begin{pmatrix}[r]2.36&1.55&1.45&1.55&1.85\\\
1.55&2.36&1.75&1.55&1.55\\\ 1.45&1.75&2.36&1.75&1.45\\\
1.55&1.55&1.75&2.36&1.55\\\ 1.85&1.55&1.45&1.55&2.36\\\ \end{pmatrix}.$ (10)

The order of the orbitals in these matrices is $xy$, $yz$, $z^{2}$, $xz$, and
$x^{2}-y^{2}$.

It can be seen that there is a noticeable difference of around
$0.210.26\text{\,}\mathrm{e}\mathrm{V}$ between the intra-orbital terms (on
the diagonal of $U^{\sigma\bar{\sigma}}_{\mathrm{cRPA}}$) corresponding to the
$t_{2g}$-like orbitals and that of the $e_{g}$-like orbitals. However, this
orbital dependence of the intra-orbital interaction is weaker than what has
been found for other materials [48, 63]. The deviations of the various (off-
diagonal) inter-orbital terms from the symmetrized form are all smaller than
$0.1\text{\,}\mathrm{e}\mathrm{V}$ and even slightly smaller for the same-spin
case.

In the localized basis, the corresponding interaction matrices are

$\displaystyle U_{\mathrm{cRPA}}^{\sigma\sigma}$
$\displaystyle=\begin{pmatrix}[r]0.00&0.26&-0.06&0.26&0.93\\\
0.26&0.00&0.69&0.26&0.19\\\ -0.06&0.69&0.00&0.69&-0.12\\\
0.26&0.26&0.69&0.00&0.19\\\ 0.93&0.19&-0.12&0.19&0.00\\\ \end{pmatrix}$
$\displaystyle U_{\mathrm{cRPA}}^{\sigma\bar{\sigma}}$
$\displaystyle=\begin{pmatrix}[r]\hphantom{-}2.30&0.93&\hphantom{-}0.73&0.93&\hphantom{-}1.39\\\
0.93&2.30&1.22&0.94&0.89\\\ 0.73&1.22&2.33&1.22&0.69\\\
0.93&0.94&1.22&2.29&0.89\\\ 1.39&0.89&0.69&0.89&2.32\\\ \end{pmatrix}$ (11)
and $\displaystyle U_{\mathrm{symm}}^{\sigma\sigma}$
$\displaystyle=\begin{pmatrix}[r]0.00&0.21&-0.04&0.21&0.94\\\
0.21&0.00&0.69&0.21&0.21\\\ -0.04&0.69&0.00&0.69&-0.04\\\
0.21&0.21&0.69&0.00&0.21\\\ 0.94&0.21&-0.04&0.21&0.00\\\ \end{pmatrix}$
$\displaystyle U_{\mathrm{symm}}^{\sigma\bar{\sigma}}$
$\displaystyle=\begin{pmatrix}[r]\hphantom{-}2.30&0.90&\hphantom{-}0.74&0.90&\hphantom{-}1.39\\\
0.90&2.30&1.23&0.90&0.90\\\ 0.74&1.23&2.30&1.23&0.74\\\
0.90&0.90&1.23&2.30&0.90\\\ 1.39&0.90&0.74&0.90&2.30\\\ \end{pmatrix}.$ (12)

Here, there is almost no orbital dependence of the intra-orbital entries in
the cRPA matrices, and also the deviations of the inter-orbital matrix
elements from their symmetrized form is rather small. Notably, the interaction
for parallel spins has some relatively small, but negative inter-orbital
entries, both before and after symmetrization. This could be a sign of
overscreening, which is discussed further below.

To better quantify the deviations between the unsymmetrized and symmetrized
two-index matrices, we determine the maximum absolute deviation of a single
matrix element, $\Delta_{\mathrm{max}}$, and we also compute the Frobenius
norm

$\displaystyle\Delta_{\mathrm{Frob}}$
$\displaystyle=\left(\sum_{mm^{\prime}}\left|U^{\mathrm{cRPA}}_{mm^{\prime}}-U^{\mathrm{symm}}_{mm^{\prime}}\right|^{2}\right)^{1/2}.$
(13)

These deviations are summarized in Table 2. Additionally, we also quantify the
deviation between the cRPA values and an alternative spherically symmetrized
interaction matrix, which is obtained by first calculating parameters $U$ and
$J$ according to Eq. (7) and then assuming $F_{4}/F_{2}=0.63$ to reconstruct
the interaction matrix, as done in many DFT+$U$ and DFT+DMFT implementations.

Table 2: Deviations in eV between the cRPA matrices and the spherically symmetrized two-index interaction matrices for same-spin ($\sigma\sigma$) and opposite spin ($\sigma\bar{\sigma}$), both for the frontier and the localized basis. The symmetrization is performed either without or with the additional assumption $F_{4}/F_{2}=0.63$.  |  | $\Delta_{\mathrm{max}}$ | $\Delta_{\mathrm{Frob}}$
---|---|---|---
Basis | $F_{4}/F_{2}=0.63$ | $\sigma\sigma$ | $\sigma\bar{\sigma}$ | $\sigma\sigma$ | $\sigma\bar{\sigma}$
Frontier | no | 0.05 | 0.17 | 0.12 | 0.33
yes | 0.07 | 0.17 | 0.18 | 0.33
Localized | no | 0.08 | 0.05 | 0.18 | 0.11
yes | 0.18 | 0.12 | 0.42 | 0.28

For the frontier basis, both $\Delta_{\mathrm{max}}$ and
$\Delta_{\mathrm{Frob}}$ are noticeably larger for the opposite-spin
interaction than for the same-spin case, due to the larger variations of the
intra-orbital entries, as already discussed above. In the localized basis, the
deviations from spherical symmetry are less pronounced for the opposite-spin
interaction, and even slightly smaller than for the same-spin case. On the
other hand, assuming $F_{4}/F_{2}=0.63$ in the symmetrization has a stronger
effect in the localized basis, while in the frontier basis this barely
increases the overall deviation from spherical symmetry. The effect of the
$F_{4}/F_{2}$ ratio discussed further in Sec. III.3.2.

Compared to the interaction matrices reported in Ref. 63 for LaFeAsO, the
deviations from spherical symmetry are less pronounced in CaFeO3. This could
be due to the different structure or coordination of the Fe cation in both
systems (octahedral in CaFeO3 versus tetrahedral in LaFeAsO), or also due to
different degrees of covalency in the two systems. All in all, a spherical
approximation of the local interaction appears to be relatively unproblematic
for CaFeO3.

Table 3: Screened and unscreened interaction parameters obtained within cRPA for the two different basis sets. The orbitally averaged nearest-neighbor Fe-Fe intersite interactions $V_{\mathrm{NN}}$ and $J_{\mathrm{NN}}$ are also listed.  |  $F_{0}\equiv U$ (eV) |  $F_{2}$ (eV) |  $F_{4}$ (eV) |  $(F_{2}+F_{4})/14\equiv J$ (eV) | $F_{4}/F_{2}$ |  $V_{\mathrm{NN}}$ (eV) |  $J_{\mathrm{NN}}$ (eV)
---|---|---|---|---|---|---|---
Frontier basis | screened | 1.75 | 4.25 | 3.21 | 0.53 | 0.76 | 0.59 | 0.01
unscreened | 14.82 | 5.91 | 3.60 | 0.68 | 0.61 | 3.83 | 0.02
ratio (%) | 12 | 72 | 89 | 78 | – | 16 | 33
Localized basis | screened | 1.25 | 7.16 | 5.73 | 0.92 | 0.80 | 0.03 | 0.00
unscreened | 23.34 | 10.57 | 6.21 | 1.20 | 0.59 | 3.87 | 0.00
ratio (%) | 5 | 68 | 92 | 77 | – | 1 | 47

Next, we discuss the effective parameters obtained for the screened and
unscreened interaction, which are listed in Table 3. We focus on the Hubbard
$U$ and Hund $J$ obtained directly by averaging according to Eq. (7) and start
the discussion with the values for the frontier basis. The comparison of
screened and unscreened values shows that $U$ is strongly screened to
$U=$1.75\text{\,}\mathrm{e}\mathrm{V}$$, which corresponds to only
$12\text{\,}\%$ of its unscreened value, while $J$, as expected, is less
screened to $J=$0.53\text{\,}\mathrm{e}\mathrm{V}$$, i.e., $78\text{\,}\%$ of
its unscreened value. The rather strong screening of $U$ in CaFeO3 is indeed
comparable to what has been found in cRPA calculations for rare-earth
nickelates [20, 21], and can be explained by the close vicinity between the
correlated bands and the screening bands, which even overlap slightly [see
Fig. 3(b)].

A systematic comparison with literature data is hampered by the use of
different types of correlated subspaces (full $d$-shell vs. ${t_{2g}}$\- or
${e_{g}}$-only models, frontier vs. localized orbitals) or different
parametrizations of the interaction Hamiltonian. The values obtained in Ref.
21 for LuNiO3, if transformed from the $e_{g}$-only Kanamori parametrization
used in [21] to the Slater parametrization used here, correspond to
$U=$1.3\text{\,}\mathrm{e}\mathrm{V}$$ and
$J=$0.5\text{\,}\mathrm{e}\mathrm{V}$$, i.e., an even smaller $U$ than in
CaFeO3 and a fairly similar $J$. Note that, due to the use of an $e_{g}$-only
correlated subspace in [21], the $t_{2g}$ bands will also contribute to the
screening. In cRPA calculations for other transition-metal oxides, $U$ is
typically reduced to around 15-25 % of its bare value [48, 64, 65]. The
overall magnitude of the screened $U$ and $J$ in CaFeO3 is also comparable to
values obtained for Fe in LaFeAsO, with
$J=$0.59\text{\,}\mathrm{e}\mathrm{V}$$, almost equal to CaFeO3, and a
slightly larger $U=$2.14\text{\,}\mathrm{e}\mathrm{V}$$ [63].

In the localized basis, the unscreened values of $U$ and $J$ increase
significantly compared to the frontier basis, which reflects the higher
Coulomb repulsion due to the stronger localization of the corresponding
orbitals. However, and perhaps surprisingly, the screened value of
$U=$1.25\text{\,}\mathrm{e}\mathrm{V}$$ is noticeably smaller than for the
frontier basis and corresponds to only $5\text{\,}\%$ of the unscreened value.
In contrast, the screening of $J$ in the localized basis is similar (in
percent) to that in the frontier basis, resulting in a screened
$J=$0.92\text{\,}\mathrm{e}\mathrm{V}$$, i.e., larger than in the frontier
basis.

The extremely strong screening in the localized basis in CaFeO3 can be
understood from the strong band entanglement and hybridization. As a result,
virtually every band inside the energy window has a mixed character, with
contributions from both the correlated subspace and the screening subspace.
This results in many potential screening channels close to the Fermi energy.
To avoid this, some previous studies have used a screening subspace similar to
that for the frontier orbitals also for the localized basis [66, 63, 48].
While this reduces the screening on $U$, it also introduces an inconsistency
between correlated and screening subspace, and thus we have not used this
approach in our work. A small decrease in the screened $U$ (and an increase in
the screened $J$) in a localized basis compared to a frontier basis has also
been observed for SrMoO3 [65]. However, in that case, the effect is much less
extreme than here. Generally, cRPA is known to have a tendency to overscreen,
which becomes more pronounced the less the correlated and screening subspaces
are separated [67, 68]. Consequently, one can expect that the overscreening
affects both basis sets, but is particularly strong in the localized basis.
Finally, we note that our cRPA calculations are based on the low-spin state
obtained within bare DFT, which might also influence the degree of screening.

Since the localized basis is similar to the correlated subspace used in our
DFT+$U$ calculations, the corresponding screened interaction parameters can
also be compared to the values of $U=$4\text{\,}\mathrm{e}\mathrm{V}$$ and
$J=$1\text{\,}\mathrm{e}\mathrm{V}$$ used for our structural relaxations in
Sec. III.1. Using the cRPA value of $U=$1.25\text{\,}\mathrm{e}\mathrm{V}$$ in
these calculations would lead to a rather poor description of the $P2_{1}/n$
phase with a low-spin state of the Fe cations and essentially zero $R_{1}^{+}$
mode, as shown in Fig. 2. This can also be seen as a sign of severe
overscreening of the cRPA value of $U$. On the other hand, the cRPA value of
$J=$0.92\text{\,}\mathrm{e}\mathrm{V}$$ is very close to our initial choice of
$J=$1\text{\,}\mathrm{e}\mathrm{V}$$ and (in combination with an appropriate
$U$ value) appears quite reasonable, indicating that the Hund’s coupling $J$
is generally less affected by overscreening.

Additionally, we compute the orbitally averaged nearest-neighbor intersite
interaction $V_{\mathrm{NN}}$ and $J_{\mathrm{NN}}$, also shown in Table 3.
$V_{\mathrm{NN}}$ assumes a moderate value of
$0.59\text{\,}\mathrm{e}\mathrm{V}$ in the frontier basis, screened to
$16\text{\,}\%$ of its bare value, and comparable to LuNiO3 [20] and LaFeAsO
[63]. In the localized basis, the unscreened $V_{\mathrm{NN}}$ is comparable
to that of the frontier basis, but is then completely screened, to only
$1\text{\,}\%$ of the bare value, and thus becomes negligibly small. However,
it is unclear whether this indeed indicates a highly efficient inter-site
screening, and thus a good quality of the locality assumption for the screened
interaction, or whether this values is also affected by a potential
overscreening. $J_{\mathrm{NN}}$ is negligible in all cases (both screened and
unscreened).

Finally, we also compute screened interaction parameters in the frontier basis
for the relaxed structure with $P2_{1}/n$ symmetry, and obtain only a
negligible difference of $3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ for $U$
and less than $1\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ for $J$ compared to
the $Pbnm$ structure.

####  III.3.2 $F_{4}/F_{2}$ ratio in cRPA

Next, we analyze the ratio $F_{4}/F_{2}$ obtained from the spherically
averaged cRPA interaction parameters. Previous work has indicated that this
ratio can deviate significantly from the value of 0.63 typically used to
compute the three Slater parameters from $U$ and $J$, and can reach up to
$0.86$ [48]. As shown in Table 3, this also applies to the case of CaFeO3. The
unscreened $F_{4}/F_{2}$ ratios are slightly smaller than the “atomic” value
of 0.63, in particular for the localized basis, while in the screened case
this ratio is increased to 0.80 and 0.76 for the localized and frontier basis,
respectively. Thus, while a constant ratio of $F_{4}/F_{2}=0.63$ still appears
as a reasonable approximation for the unscreened interaction, this is no
longer guaranteed in the presence of strong screening.

To assess whether a variation in $F_{4}/F_{2}$ can also affect physical
properties, we estimate the energy difference between the low-spin (LS) and
high-spin (HS) states of the Fe4+ or other $d^{4}$ cations in the zero band-
width limit by evaluating the difference in the local interaction and crystal
field energies for a $t_{2g}^{4}e_{g}^{0}$ and a $t_{2g}^{3}e_{g}^{1}$
configuration (see also the Appendix of Ref. 12)

$\displaystyle E^{\text{LS}}-E^{\text{HS}}$
$\displaystyle=\frac{6}{49}F_{2}+\frac{145}{441}F_{4}-\Delta_{\text{CF}}$ (14)
$\displaystyle\approx\begin{cases}2.83J-\Delta_{\text{CF}}\ \text{for}\
F_{4}/F_{2}=0.63\\\ 3.00J-\Delta_{\text{CF}}\ \text{for}\
F_{4}/F_{2}=0.80\end{cases}$

where $\Delta_{\text{CF}}$ is the local crystal-field splitting between
$e_{g}$ and $t_{2g}$ states. Thus, for the standard value of
$F_{4}/F_{2}=0.63$, a slightly larger $J$ is necessary to obtain a high spin
state. For a realistic crystal-field splitting
$\Delta_{\text{CF}}=$2.5\text{\,}\mathrm{e}\mathrm{V}$$, this corresponds to a
difference of about $0.05\text{\,}\mathrm{e}\mathrm{V}$ in the critical $J$.
While this difference does not appear very large, it could be important for
systems that are very close to a high-spin low-spin transition.

####  III.3.3 Different methods for disentangling the correlated and screening
subspaces

Finally, as discussed in Sec. II.1, for band-structures where the correlated
bands are entangled with the rest, different methods for calculating the
corresponding cRPA polarization functions have been suggested. Here, we
compare the interaction parameters obtained from the “projector method” [45],
which is the default method in this paper, the “weighted method” [42], and the
“disentanglement method” [41], as implemented in VASP. The comparison of
different interaction parameters is shown in Table 4.

Table 4: Screened interaction parameters obtained with different band disentanglement methods for the two different basis sets. We compare the parameters $U$ and $J$, the ratio $F_{4}/F_{2}$, and the averaged intersite interaction $V_{\mathrm{NN}}$.  Interaction params. |  $U$ (eV) |  $J$ (eV) | $F_{4}/F_{2}$ |  $V_{\mathrm{NN}}$ (eV)
---|---|---|---|---
Frontier | proj. | 1.75 | 0.53 | 0.76 | 0.59
weighted | 1.74 | 0.53 | 0.76 | 0.60
disent. | 3.12 | 0.55 | 0.73 | 1.35
Localized | proj. | 1.25 | 0.92 | 0.80 | 0.03
weighted | 0.46 | 0.80 | 0.98 | 0.03
disent. | 5.41 | 1.07 | 0.64 | -0.07

In the frontier basis, the projection and weighted methods give almost
identical results for all interaction parameters, while the disentanglement
method predicts larger values, and thus less screening, especially for the on-
and intersite Coulomb repulsion $U$ and $V_{\mathrm{NN}}$. The former two
methods likely agree well because the Fe-dominated bands captured by the
frontier basis only slightly overlap with the O-dominated bands. This in turn
means that almost all bands clearly belong to either the correlated or
uncorrelated subspace and only a few bands right below the frozen window at
$-1.1\text{\,}\mathrm{e}\mathrm{V}$ partially belong to both subspaces.
Furthermore, both methods work on the unaltered DFT band-structure. In
contrast, neglecting the hybridization between correlated and uncorrelated
subspaces, will slightly affect the band structure, in particular in the
overlapping region below $-1\text{\,}\mathrm{e}\mathrm{V}$, which can explain
the more significant deviation of the disentanglement method compared to the
other two method.

In the localized basis, the differences in interaction parameters, in
particular $U$, are much more pronounced between the different methods. Again,
the disentanglement method results in a significantly higher value of $U$ and
thus less screening. However, also the projection and weighted methods now
differ quite drastically, with a very small
$U=$0.46\text{\,}\mathrm{e}\mathrm{V}$$ obtained from the latter. The large
spread of the calculated interaction parameters for the localized basis is
most likely a result of the strong band entanglement, such that small
differences in the way the two subspaces are separated can lead to rather
large differences in the corresponding polarization functions.

On first view, the interaction parameters obtained from the disentanglement
method appear quite reasonable and compatible with the values used in our
DFT+$U$ calculations in Sec. III.1. However, ignoring the hybridization
between correlated and uncorrelated subspaces in the localized basis reduces
the band width of the correlated bands to only
$0.6\text{\,}\mathrm{e}\mathrm{V}$ for the Fe-${e_{g}}$ and
$0.5\text{\,}\mathrm{e}\mathrm{V}$ for the Fe-${t_{2g}}$ bands, and thus leads
to severe changes compared to the DFT bands. This is very different from the
situation for which this method was introduced [41]. Therefore, it seems
likely that the interaction values from the disentanglement method result
mainly from a cancellation of errors between the cRPA overscreening and the
strong modification of the disentangled band structure.

The intersite terms are always negligibly small for the localized basis.

##  IV Summary and discussion

In this work, we have presented cRPA calculations of the screened interaction
parameters for the charge-disproportionated insulator CaFeO3. Thereby, we have
compared results for two different types of Wannier functions used to describe
the Fe-$d$ states. First, a frontier orbital basis, incorporating the
hybridization with the oxygen ligands, which describes only the bands
immediately around the Fermi level and maps well on simple models involving
only Fe cations with formal charge states. Second, a more localized basis
resembling an atomic orbital-like basis, similar to the basis typically used
in DFT+$U$ calculations.

We have shown that cRPA predicts CaFeO3 to have a strongly screened Hubbard
$U$ and a relatively large Hund’s interaction $J$, consistent with other
charge-disproportionated insulators such as the rare-earth nickelates [20,
21]. This supports the picture that the insulating state in these systems is
driven by the Hund’s interaction rather than a strong Hubbard $U$.
Additionally, the DFT band structure indicates that CaFeO3 has a relatively
small but positive charge-transfer energy.

We have also tested the quality of certain approximations typically applied to
parametrize the interaction matrices in first-principles calculations, such as
the assumption of spherical symmetry and a fixed ratio between the Slater
integrals $F_{2}$ and $F_{4}$. Indeed, the spherical approximation appears
relatively uncritical, both for the more localized and the frontier basis,
with the largest deviations from spherical symmetry observed for the intra-
orbital interactions in the frontier basis, which vary by around
$0.20.25\text{\,}\mathrm{e}\mathrm{V}$. The calculated $F_{4}/F_{2}$ ratio was
found to deviate noticeably from the commonly used fixed ratio of 0.63, in
agreement with previous cRPA calculations for other transition-metal oxides
[48]. While this affects the value of $J$ only slightly, it can potentially
become important, e.g., close to a high-spin low-spin transition.

In addition, we have discussed the choice of interaction parameters in DFT+$U$
calculations, to obtain structural parameters for both the high- and low-
temperature structures that are in good agreement with experimental data. We
have shown that a simple A-type antiferromagnetic order and moderate
electronic interactions suffice to reproduce the experimentally observed low-
temperature structure. By subsequently neglecting the local electron-electron
interactions and magnetic order, a good description of the experimental high-
temperature structure can also be obtained.

However, our study also highlights some problems of cRPA calculations of
interaction parameters for systems with strong entanglement between the
correlated and the uncorrelated subspaces. The cRPA values obtained for $U$ in
the localized basis are clearly too small to obtain a good description of the
charge-disproportionated state of CaFeO3 in DFT+$U$ calculations. Whether this
overscreening of the interaction is mostly related to fundamental deficiencies
of the random-phase approximation, as also indicted by model considerations
[68], or whether it is also, at least to some extent, a result of inadequate
disentanglement between the correlated and uncorrelated subspaces, is an open
question. From our comparison between different methods in Sec. III.3.3 it
becomes clear that, in situations with strong entanglement, the results can
vary widely and none of the methods is likely to produce interaction
parameters that can reliably be used in first-principles calculations.
Furthermore, it would be desirable to perform cRPA calculations for CaFeO3
also in the high-spin state, either by constraining the orbital occupations to
a $t_{2g}^{3}e_{g}^{1}$ configuration or by performing cRPA calculations for a
magnetically ordered state. While a solution of these issues goes beyond the
scope of the present work, our results can serve as starting point for future
attempts aimed at improving the currently available methods for calculating
effective interaction parameters.

###### Acknowledgements.

This research was supported by ETH Zurich and a grant from the Swiss National
Supercomputing Centre (CSCS) under project ID s1128.

*

##  Appendix A Equations for the spherically symmetric interaction

Here, we summarize properties related to the tensors $\alpha^{(k)}$ relevant
for Sec. II.2. The $\alpha^{(k)}$ tensors are orthogonal with the
normalization

$\displaystyle\sum_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}\alpha^{(k_{1})}_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}\alpha^{(k_{2})}_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}=c_{lk_{1}}\delta_{k_{1},k_{2}}$
$\displaystyle\text{with}\quad
c_{lk}:=\frac{(2l+1)^{4}}{2k+1}\begin{pmatrix}l&k&l\\\
0&0&0\end{pmatrix}^{4},$ (15)

which directly allows us to derive Eq. (6). From Eq. (6) and Eq. (7), a
general expression for $J$ for all $l$ follows as

$\displaystyle J=\frac{2l+1}{2l}\sum_{k\geq 2}F_{k}\begin{pmatrix}l&k&l\\\
0&0&0\end{pmatrix}^{2},$ (16)

which gives $J=(F_{2}+F_{4})/14$ for $l=2$.

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