# Theory of x-ray absorption spectroscopy for ferrites

Felix Sorgenfrei  Department of Physics and Astronomy, Uppsala University,
Box-516,Uppsala SE-751 20 Sweden  Mébarek Alouani  Université de Strasbourg,
IPCMS, CNRS-UNISTRA UMR 7504, 67034 Strasbourg, France  Johan Schött
Department of Physics and Astronomy, Uppsala University, Box-516,Uppsala
SE-751 20 Sweden  Orexplore AB, Torshamngatan 30B, 164 40 Kista, Stockholm,
Sweden  H. Johan M. Jönsson  Department of Physics and Astronomy, Uppsala
University, Box-516,Uppsala SE-751 20 Sweden  School of Computational
Sciences, Korea Institute for Advanced Study, Seoul 02455, South Korea  Olle
Eriksson  Department of Physics and Astronomy, Uppsala University,
Box-516,Uppsala SE-751 20 Sweden  Wallenberg Initiative Materials Science,
Uppsala University, Box-516,Uppsala SE-751 20 Sweden  Patrik Thunström
<EMAIL_ADDRESS>Department of Physics and Astronomy, Uppsala
University, Box-516,Uppsala SE-751 20 Sweden

###### Abstract

The theoretical calculation of the interaction of electromagnetic radiation
with matter remains a challenging problem for contemporary ab initio
electronic structure methods, in particular for x-ray spectroscopies. This is
not only due to the strong interaction between the core-hole and the photo-
excited electron, but also due to the elusive multiplet effects that arise
from the Coulomb interaction among the valence electrons. In this work we
report a method based on density-functional theory in conjunction with
multiplet ligand-field theory to investigate various core-level
spectroscopies, in particular x-ray absorption spectroscopy (XAS) and x-ray
magnetic circular dichroism (XMCD). The developed computational scheme is
applied to the $L_{2,3}$ XAS edges of magnetite (Fe3O4) as well as cobalt
ferrite (CoFe2O4) and nickel ferrite (NiFe2O4) and the corresponding XMCD
spectra. The results are in overall good agreement with experimental
observations, both regarding the XAS $L_{2}$/$L_{3}$ branching ratio, the peak
positions as well as the relative intensities. The agreement between theory
and experiment is equally good for XAS and the XMCD spectra, for all studied
systems. The results are analyzed in terms of $e_{g}$ and $t_{2g}$ orbitals
contributions and the importance of optimizing the Slater parameters. The
analysis also highlights the strong effect of the $2p$-$3d$ interaction in
x-ray spectroscopy.

##  I Introduction

Core-level spectroscopy is one of the most powerful techniques for probing the
electronic structure of strongly correlated electron systems, as the
interaction between x-rays and matter gives element-specific insights into the
magnetic and electronic properties of materials, as well as their structural
and chemical properties. In particular, the development of synchrotron
radiation facilities and recently, x-ray free electron lasers, have made the
analysis efficient and the resolution very high in energy and in specific
cases, in time [1]. This increase in the experimental resolution has made it
possible to detect fine features and structures in the x-ray spectra [2, 3].
This development of experimental tools and techniques calls for a novel
theoretical approach to compute x-rays absorption spectra, most notably, for
systems with complex or correlated electronic structures, like magnetites as
well as cobalt- and nickel ferrites. In this work, we focus specifically on
these materials and on new insights for the theoretical description of x-ray
absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD).

For the theoretical modelling of core-level and x-ray absorption spectroscopy,
there have been two main distinct methodologies that are primarily used today.
The first one is based on density functional theory (DFT) and the Kubo formula
[4] while the second focuses on multi configuration effects [5]. Both
methodologies are well reviewed in the literature (see for example Ref. [6,
7]). The DFT-based methods are essentially parameter-free and have, as a main
advantage, the description of magnetism, structural properties and chemical
bonding in general. These methods are designed for the ground-state properties
of materials, and hence their description of the excited states probed with
x-ray spectroscopy is very limited. This is due to an incomplete treatment of
electronic correlations, in particular the multiconfigurational effects.
Overall, these methods work best for widely dispersive electron systems. In
contrast, atomic multiplet theory is essentially intended for a better
description of the multiconfigurational effects originating from on-site
electron-electron interaction [8]. For practical computational efficiency, the
multiplet theory is often implemented for an isolated ion and thus neglects
hybridization effects. It is therefore not surprising that this implementation
of the theory has been found to work best for localized electronic states,
e.g., the $4f$ orbitals of rare-earth elements. However, effects like crystal-
field splitting and orbital hybridization can be treated by adding suitable
model terms to the Hamiltonian of the system. The resulting theory is often
referred to as the multiplet ligand-field theory (MLFT) [8]. This theory is
very powerful in reproducing experimental spectra when the electrons have a
tendency to be localized, but has the obvious drawback in the ambiguity of
choosing and justifying the various parameters of the model Hamiltonian (see
e.g. Ref. [9]). One way to reduce the ambiguity is to obtain the hybridization
strength parameters and the crystal-field splitting from a tight-binding
parameterization of a DFT calculation [10]. For the $4f$-orbitals of rare-
earth systems it is also possible to neglect the hybridization and extract the
remaining MLFT parameters completely from DFT [11].

To describe both on-site electron-electron interaction and hybridization
effects, many methods have been proposed, for example DFT+$U$ [12] and DFT in
conjunction with dynamical mean-field theory (DFT+DMFT) [13]. Since DFT+DMFT
takes into account the multiconfigurational effects, it was shown to provide a
better description of valence-band spectral function of, e.g., the 3$d$
transition-metal oxides (TMOs) [14, 15].

In the x-ray absorption process, the excitation involves the promotion of an
electron in a core orbital to an empty orbital, according to the selection
rules of the electric dipole operator [16]. The resulting core-hole interacts
with the valence electrons and the photo-electron. For example, for the
$L_{2,3}$-edges of transition metal complexes the presence of a $2p$ core-hole
in the final state tends to further localize the valence states. In the final-
state approximation, the core hole is treated as an attractive static
potential, and the valence electrons and the photo-electron are allowed to
relax in the presence of this potential. However, it was shown that applying
the final state approximation in DFT+DMFT does not improve the calculated XAS
spectra of Fe and Co, and in the case of Ni dramatically worsens the
comparison with the experiment [17]. Several propositions to include the
dynamics of the core-hole within a single computational framework for
calculating $L$-edges of transition metal (TM) systems are available in the
literature, such as the multiple scattering approach [18] with dynamical
screening from time-dependent DFT [19, 20], the application of configuration
interaction to a set of molecular orbitals [21] and the Bethe-Salpeter
equation [22, 23]. In spite of a high level of theoretical complexity, most of
the methods mentioned above do not deliver a sufficiently accurate description
of XAS of a wide range of materials, exemplified by the transition metal
oxides. In recent years, approaches based on extending the impurity model in
DFT+DMFT with core states have also been used to simulate core level x-ray
photoemission spectroscopy [24] and resonant x-ray emission spectroscopy [25].
A good overview of most methods for calculating x-ray absorption spectra is
presented in [26].

In the present study we use MLFT in conjunction with DFT+$U$ and DFT+DMFT to
describe on a equal footing the multiconfigurational effects as well as the
dynamics of the core-hole photo-electron interaction and its effect on XAS and
XMCD. The method described here has key components that are identical for MLFT
combined with DFT+$U$ and DFT+DMFT, and so for simplicity we have chosen to
give most numerical examples from the MLFT+DFT+$U$ approach. The aim of the
present theory is to rely on as few free parameters as possible. For example,
the hybridization of the orbitals is directly extracted from the electronic
structure calculation and the screening of the local electron-electron
interaction (Hubbard $U$) is estimated using constrained DFT (cDFT). However,
there currently is no reliable way to calculate the screening of the zeroth
order valence-core interaction ($F_{pd}^{0}$), so it is treated as a free
parameter. We also investigate how sensitive the spectra are to the
description of the screening of the higher-order Slater parameters. In this
work transition selection rules have been included within the electric-dipole
approximation. We present the details of the implementation and the results
from an application of the developed method to the description of the
$L_{2,3}$-edges and XMCD spectra of magnetite, Fe3O4, as well as cobalt-
(CoFe2O4) and nickel ferrite (NiFe2O4) at room temperature in the inverse
spinel structure. We show that our theoretical results compare favorably well
with experimental observations. In particular, we show that both the
$L_{3}$/$L_{2}$ experimental XAS branching ratio, as well as fine structures
in the XAS and XMCD spectra, are well reproduced by our calculations, contrary
to previous DFT calculations.

##  II Theory

###  II.1 DFT+MLFT

In the following, we describe the DFT+MLFT approach, and show how to use this
theory to compute x-ray absorption spectra and x-ray magnetic circular
dichroism of the magnetite Fe3O4 as well as the cobalt and nickel ferrites.
The theoretical model can in a simplified way be described as a transfer of
parameters obtained from DFT (or DFT+$U$) based methods, to a MLFT level of
theory. In this way, hybridization effects and crystal-field splittings do not
enter the theory as fitting parameters but rather evaluated from ab initio
theory. We describe the essential aspects of this mapping below.

####  II.1.1 Projection

The starting point of the theory involves the one-particle DFT+$U$ Green’s
function of the lattice, which encodes the contributions from hybridization
and crystal-field splittings of the transition metal $3d$ orbitals. It is
defined as the resolvent of the lattice-momentum dependent Hamiltonian
$\tilde{H}_{k}^{\text{DFT}}$ and the static DFT+$U$ self-energy
$\tilde{\Sigma}_{k}$;

$\displaystyle\tilde{G}_{k}(\omega)=((\omega+\mu)\tilde{1}-\tilde{H}_{k}^{\text{DFT}}-\tilde{\Sigma}_{k})^{-1},$
(1)

where $\mu$ is the chemical potential and $\tilde{1}$ is the identity
operator. The self-energy $\tilde{\Sigma}_{k}$ is obtained from the local
self-energy $\tilde{\Sigma}^{\mathrm{loc}}$ and a double-counting correction
$\tilde{\Sigma}^{\mathrm{loc}}_{\mathrm{DC}}$ through a projection from a set
of impurity orbitals to the lattice orbitals:

$\displaystyle\tilde{\Sigma}_{k}=\tilde{P}_{k}(\tilde{\Sigma}^{\mathrm{loc}}-\tilde{\Sigma}^{\mathrm{loc}}_{\mathrm{DC}})\tilde{P}_{k}^{\dagger},$
(2)

where $\tilde{P}_{k}$ is the projection operator from the orthonormal orbitals
on the impurity site to the lattice orbitals at lattice momentum $k$. The
impurity orbitals are defined through the construction of the projection as
detailed in Refs. [27, 15, 28]. In the construction of the projection operator
we only use the DFT bands within an energy window $[-16,10]$ eV around the
Fermi level, in order to preserve the TM 3$d$ character of the impurity
orbitals. The lower bound of the energy window prevents the impurity orbitals
to obtain any oxygen 2$s$ character, which adversely affects the crystal-field
splitting in the impurity orbitals. The upper bound prevents the impurity
orbitals to include high-energy TM 4$d$ and TM 5$d$ characters.

The local Green’s function is obtained from $\tilde{G}_{k}(\omega)$ through
the reverse projection:

$\displaystyle\tilde{G}^{\mathrm{loc}}(\omega)=\frac{1}{N_{k}}\sum_{k}^{N_{k}}\tilde{P}_{k}^{\dagger}\tilde{G}_{k}(\omega)\tilde{P}_{k},$
(3)

where $N_{k}$ is the number of $k$-points in the first Brillouin zone. The
projection of the lattice Green’s function to the local Green’s function is
identical to that in dynamical mean-field theory (DMFT). The hybridization
function

$\displaystyle\tilde{\Delta}(\omega)=(\omega+\mu)\tilde{1}-\tilde{H}_{0}^{\mathrm{loc}}-\tilde{\Sigma}^{\mathrm{loc}}+\tilde{\Sigma}^{\mathrm{loc}}_{\mathrm{DC}}-\tilde{G}^{\mathrm{loc}}(\omega)^{-1}$
(4)

describes the hybridization of the $3d$-TM orbitals with the orbitals of the
rest of the material. The imaginary part of the hybridization function is
similar to the density of states but has peaks at the energies of the
hybridising orbitals. The intensity of these peaks corresponds to the strength
of the hybridization. The role of the ligand orbitals is to mimic this $ab-
initio$ hybridization i.e. to allow ligand electrons to enter (and leave) the
$3d$-TM orbitals with the appropriate transition energy. In Eq. (4) the local
Hamiltonian $\tilde{H}^{\mathrm{loc}}$ is calculated from
$\tilde{H}_{k}^{\text{DFT}}$ by using the same projection procedure as in Eq.
(3).

####  II.1.2 Discretization of the hybridization function $\Delta(\omega)$

The hybridization function operator in Eq. (4) can be approximated as

$\displaystyle\tilde{\Delta}^{\mathrm{ED}}(\omega)=\tilde{V}\Big{[}\omega\tilde{1}-\tilde{H}^{\mathrm{bath}}\Big{]}^{-1}\tilde{V}^{\dagger},$
(5)

where $\tilde{H}^{\mathrm{bath}}$ is an effective Hamiltonian for a finite
(usually small) set of auxiliary bath orbitals which hybridize with the
impurity orbitals according to the impurity-bath hopping operator $\tilde{V}$.
This approximation is routinely done in the exact diagonalization (ED) solver
in DMFT [29]. The matrix elements of the operators $\tilde{V}$ and
$\tilde{H}^{\mathrm{bath}}$ are parameterized and the parameters are set to
reproduce the main peaks of the imaginary part of the exact hybridization
function $\tilde{\Delta}(\omega)$. In our implementation several energy
windows are selected after an inspection of $\Delta(\omega)$, and within each
window the parameters for one bath orbital is fitted to minimize the
difference between $\Delta^{\mathrm{ED}}$ and $\Delta(\omega)$. The matrix
representation $\Delta(\omega)$ of $\tilde{\Delta}(\omega)$ depends on the
choice of impurity orbitals. $\Delta(\omega)$ becomes block diagonal if the
impurity orbitals are symmetrized such that they transform according to the
irreducible representations of the system. Even in the absence of global
symmetries it is often advantageous to symmetrize the impurity orbitals
according to their local environment to minimize the off-diagonal elements in
the fitting of $\Delta(\omega)$.

####  II.1.3 Impurity Hamiltonian

The total impurity Hamiltonian for the TM $3d$ and $2p$ shells is summarized
as

$\displaystyle\hat{H}=$
$\displaystyle\sum_{ij}H^{\textrm{3d}}_{ij}\hat{c}_{i}^{\dagger}\hat{c}_{j}+\sum_{bb^{\prime}}H_{bb^{\prime}}^{\mathrm{bath}}\hat{c}_{b}^{\dagger}\hat{c}_{b^{\prime}}+\sum_{i,b}(V_{i,b}\hat{c}^{\dagger}_{i}\hat{c}_{b}+\textrm{h.c.})$
$\displaystyle+\sum_{ij}H^{\textrm{2p}}_{ij}\hat{c}_{i}^{\dagger}\hat{c}_{j}+\hat{H}_{U},$
(6)

with

$\displaystyle H^{\textrm{3d}}=$ $\displaystyle
H^{\mathrm{loc}}-\Sigma^{\mathrm{loc}}_{\mathrm{DC}}+H^{\textrm{SOC}}_{3d},$
(7) $\displaystyle H^{\textrm{2p}}=$
$\displaystyle\epsilon_{p}\tilde{1}+H^{\textrm{SOC}}_{2p},$ (8)

where $H^{\textrm{3d}}$ and $H^{\textrm{2p}}$ contains the $3d$ and $2p$ on-
site energies including the spin-orbit coupling (SOC) (see Eq. (22)),
respectively. $H^{\mathrm{bath}}$ and $V$ describe the bath and its coupling
to the impurity, and $\hat{H}_{U}$ is the Coulomb interaction (see Eq. (15) of
the appendix).

The diagonalization of Eq. (II.1.3) results in the set of many-body
eigenstates $\ket{n}$, each expressed by a sum of product states, and the
corresponding eigenenergies $E_{n}$:

$\displaystyle\hat{H}\ket{n}=E_{n}\ket{n}.$ (9)

Once the eigenstates with eigenenergies at most a few $k_{B}T$ above the
lowest eigenenergy are found using a Lanczos algorithm, we use these states to
calculate spectra and other observables such as occupation numbers and spin
moments.

###  II.2 Spectra

Using the eigenstates $\ket{n}$ and eigenenergies $E_{n}$ from the Hamiltonian
$\hat{H}$ in Eq. (II.1.3), we calculate the XAS and XMCD spectra at finite
temperature (300 K) according to

$\displaystyle
I(\omega)=\frac{1}{Z}\sum_{n}-\imaginary(G_{n}(\omega))\exp(-\beta E_{n}),$
(10)

with

$\displaystyle
G_{n}(\omega)=\bra{n}\hat{T}^{\dagger}\frac{1}{(\omega+E_{n}+i\Gamma_{c})\hat{1}-\hat{H}}\hat{T}\ket{n},$
(11)

and where $\omega$ is the incoming photon energy, $\hat{1}$ is the identity
operator, $Z$ the partition function and $\beta$ the inverse temperature. Here
$\Gamma_{c}$ models the core-hole decay rate and defines the energy resolution
in terms of the half-width at half-maximum (HWHM). The transition operator
$\hat{T}$ is, within the dipole approximation, equal to
$\hat{D}=\bm{\epsilon}\cdot\hat{\bm{p}}$, where $\bm{\epsilon}$ is the light
polarization vector and $\hat{\bm{p}}$ is the momentum operator. We have
implemented Eqns. (10) and (11) using a Lanczos algorithm in an open-source
software [30, 31].

##  III Details of the calculations

The Hamiltonian in Eq. (II.1.3) contains several parameters that either can be
calculated or have been estimated. From the electronic structure calculation,
we determine the local impurity Hamiltonian from a direct projection and also
construct the hybridization function. The hybridization strength ($V$) and the
bath state energies ($e_{b}$) are set to reproduce the hybridization function.
The average Coulomb repulsion between the electrons within the 3$d$ and 2$p$
orbitals of the TM atom can be expressed in terms of the Slater-Condon
parameters $F^{k}_{l_{i}l_{i}}$, $G^{k}_{l_{i}l_{j}}$ and $F^{k}_{l_{i}l_{j}}$
and the Wigner 3$j$ symbols as [5]

$\displaystyle U_{l_{i}l_{i}}$
$\displaystyle=F^{0}_{l_{i}l_{i}}-\frac{2l_{i}+1}{4l_{i}+1}\sum_{k}\begin{pmatrix}l_{i}&k&l_{i}\\\
0&0&0\end{pmatrix}^{2}F^{k}_{l_{i}l_{i}},$ (12) $\displaystyle U_{l_{i}l_{j}}$
$\displaystyle=F^{0}_{l_{i}l_{j}}-\frac{1}{2}\sum_{k}\begin{pmatrix}l_{i}&k&l_{j}\\\
0&0&0\end{pmatrix}^{2}G_{l_{i}l_{j}}^{k},$ (13)

where $l$ denotes the angular momentum of the orbitals. Notice that in these
expressions $k$ is not a crystal momentum (as it is in Section II.1.1) but an
angular momentum. The bare higher-order Slater parameters ($k\geq 1$) are
calculated by solving the Slater-Condon integrals. To account for screening
effects the parameters are then individually reduced to between 70% and 90% of
their bare values. The values within this interval are chosen to best fit the
experiment. For comparison, we also performed calculations where the higher-
order Slater parameters are kept at a fixed ratio of 80% of their bare values,
to show the impact of the screening. The screening of the zeroth-order
parameters, $F_{dd}^{0}$ and $F_{pd}^{0}$, are typically so high that they
cannot be estimated by this approach. Therefore, we have calculated
$F_{dd}^{0}$ using cDFT for NiFe2O4 using the electronic structure code Wien2K
[32]. There is currently no accurate method for calculating the screening of
$F_{pd}^{0}$, so it was varied in the empirically motivated interval
$F_{dd}^{0}+1\,$eV$\leq F_{pd}^{0}<1.4F_{dd}^{0}$.

The MLFT double counting of the $d$-orbitals is given by,

$\displaystyle\Sigma_{DC}=n_{d}U_{dd}-n_{p}U_{dp}-\delta_{DC},$ (14)

where $n_{d}$ is the occupation of the $d$ orbitals, $n_{p}$ is the occupation
of the $p$ core states and $\delta_{DC}$ is the charge transfer potential. The
value of $\delta_{DC}$ was taken from Ref. [33].

Compared to the previous work the $p$-orbital potential $\Delta\epsilon_{p}$
was introduced to reproduce the relative difference in the core-level binding
energies of the different Fe sites in the DFT+$U$ calculation. The values of
$\Delta\epsilon_{p}$ are within $0.15$ eV of the DFT calculated values. The
2$p$ and 3$d$ spin-orbit splittings were calculated using a relativistic
DFT+$U$ calculation. All parameters used here are compiled in Table 1.

Table 1: Summary of the double-counting correction, Slater-Condon integrals, and spin-orbit coupling parameters used in the MLFT calculations (unit in eV). F${}^{0}_{pd}$ is treated as a free parameter. An additional potential $\Delta\epsilon_{p}$ is introduced to shift the calculated $2p$ core energies to coincide with experimental values. | $\delta_{DC}$ |  | F${}_{dd}^{0}$ | F${}_{dd}^{2}$ | F${}_{dd}^{4}$ |  | F${}_{pd}^{0}$ | F${}_{pd}^{2}$ |  | G${}_{pd}^{1}$ | G${}_{pd}^{3}$ |  | $\zeta_{p}$ | $\zeta_{d}$ |  | $\Delta\epsilon_{p}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
CoFe2O4 Co | 1.5 |  | 6.6 | 9.23 | 5.73 |  | 7.60 | 5.18 |  | 4.22 | 2.49 |  | 9.85 | 0.078 |  | 0
CoFe2O4 Feoct | 1.5 |  | 6.4 | 9.35 | 5.85 |  | 8.12 | 4.89 |  | 3.21 | 2.05 |  | 8.3 | 0.064 |  | 0
CoFe2O4 Fetet | 1.5 |  | 6.3 | 9.35 | 5.85 |  | 8.12 | 5.19 |  | 3.51 | 2.05 |  | 8.3 | 0.074 |  | 0.25
NiFe2O4 Ni | 1.5 |  | 6.9 | 10.53 | 6.94 |  | 7.90 | 5.79 |  | 4.28 | 2.42 |  | 11.6 | 0.096 |  | 0
NiFe2O4 Feoct | 1.5 |  | 6.4 | 9.35 | 5.85 |  | 7.87 | 5.80 |  | 3.28 | 2.40 |  | 8.3 | 0.063 |  | 0
NiFe2O4 Fetet | 1.5 |  | 6.3 | 9.35 | 5.85 |  | 7.87 | 5.80 |  | 3.28 | 2.40 |  | 8.3 | 0.072 |  | 0.35
Fe3O4 Fe${}^{2+}_{\mathrm{oct}}$ | 1.5 |  | 6.4 | 8.63 | 5.36 |  | 8.45 | 4.69 |  | 3.18 | 1.92 |  | 8.3 | 0.062 |  | 4.02
Fe3O4 Fe${}^{3+}_{\mathrm{oct}}$ | 1.5 |  | 6.4 | 8.88 | 5.52 |  | 8.55 | 4.88 |  | 3.34 | 2.00 |  | 8.3 | 0.064 |  | 0
Fe3O4 Fe${}^{3+}_{\mathrm{tet}}$ | 1.5 |  | 6.3 | 9.28 | 6.26 |  | 7.82 | 5.21 |  | 3.57 | 2.14 |  | 8.3 | 0.076 |  | 1.35

##  IV Crystal and magnetic structure

Figure 1: Conventional unit cell of XFe2O4, where X is Fe, Co or Ni. The
silver-colored octahedral sites represent the Fe2+ sites that are occupied by
Co or Ni in the other ferrites . The blue octahedral sites and the purple
tetragonal are occupied by Fe3+.

Magnetite (Fe3O4) above the well known Verwey transition, has an inverse
spinel structure, with an fcc Bravais lattice. For this structure, DFT+$U$
calculations, as presented here (for details, see Appendix A), result in a
half-metallic electronic structure. The Fe cations are either octahedrally or
tetrahedrally coordinated to the oxygen ions. Above the Verwey temperature,
the six Fe ions in the unit cell are separated into two tetrahedral and four
octahedral sites. The two tetrahedral sites and two of the octahedral sites
are occupied by nominally Fe3+ ions, while the remaining two octahedral sites
are occupied by nominally Fe2+ ions. When we go from magnetite to CoFe2O4 or
NiFe2O4, almost all Co/Ni atoms occupy the Fe atom at the octahedral +2 sites.
Therefore the structures can be expressed as (Fe3+)[X2+Fe3+]O4, where the X is
Fe/Co/Ni and the parentheses indicate tetrahedral sites and the square
brackets denote the octahedral sites. A schematic overview of the inverse
spinel structure and the difference between the compounds is shown in Fig. 1.
The Fe3O4 and the other ferrites are ferrimagnetic spinels. The magnetic
ordering results from the exchange interactions between the cations. The Fe
3$d$ orbitals of the tetrahedral and octahedral sites overlap with the
intermediate 2$p$ oxygen at an angle of $\approx$ 125∘, which according to the
Goodenough-Kanamori-Anderson rules leads to antiferromagnetic exchange between
the two sublattices. The octahedral sites couple ferromagnetically to each
other due to a 90∘ superexchange. The exchange interaction between the
tetrahedral sites is antiferromagnetic, however it does not result in an
antiferromagnetic ordering in the tetrahedral sites as the other exchange
interactions are stronger [34]. In Table 2 we compare the calculated spin and
orbital moments with a previous calculation from Ref. [35]. It can be seen
that the two sets of calculations compare rather well with each other. In
Table 3 the calculated sizes of the total magnetic moments of the tetrahedral
sites and the average moment of the octahedral sites are compared to neutron
scattering experiments [36] for Fe3O4, CoFe2O4 and NiFe2O4 [37] taken at a
temperature of 300 K. It is clear from the table that the theoretical results
are in good agreement with experiment. From Table 3 it is also clear that the
results of magnetic moments from DFT and MLFT are in very good agreement with
each other. Note that the values obtained from MLFT are obtained as
expectation values of the spin and orbital angular momentum operators, using
the ground-state wavefunction (without core hole excitation). The fact that
the two sets of calculations give similar results is rewarding since it
reflects on the accuracy on the mapping to the local Hamiltonian.

Table 2: DFT+$U$ calculated absolute value of the spin and orbital (in parenthesis) magnetic moments, in units of Bohr magneton ($\mu_{B}$) per atom compared with DFT+$U$ calculation of Ref. [35]. | CoFe2O4 |  | NiFe2O4 |  | Fe3O4
---|---|---|---|---|---
Calculation | Co | Feoct | Fetet |  | Ni | Feoct | Fetet |  | Fe${}^{2+}_{\mathrm{oct}}$ | Fe${}^{3+}_{\mathrm{oct}}$ | Fe${}^{3+}_{\mathrm{tet}}$
Present | 2.53(0.00) | 4.09(0.00) | 3.93(0.00) |  | 1.66 (0.34) | 4.07 (0.03) | 3.92 (0.03) |  | 3.59 (0.02) | 4.08 (0.02) | 3.82 (0.02)
Ref. [35] | 2.57 (0.01) | 4.04 (0.04) | 3.90 (0.03) |  | 1.54 (0.27) | 4.09 (0.03) | 3.99 (0.02) |  | 3.54 (0.02) | 4.00 (0.02) | 3.84 (0.02)
Table 3: Calculated DFT+$U$ total magnetic moments, in unit of Bohr magneton ($\mu_{B}$) per atom, of the different sublattices compared with the experimental results for CoFe2O4 and NiFe2O4 of Ref. [36] and for Fe3O4 of Ref. [37]. In the table $\mu_{\mathrm{oct}}$ is the average moment on the octahedral sites.  | Fe3O4 |  | CoFe2O4 |  | NiFe2O4
---|---|---|---|---|---
| $\mu_{\mathrm{tet}}$ | $\mu_{\mathrm{oct}}$ |  | $\mu_{\mathrm{tet}}$ | $\mu_{\mathrm{oct}}$ |  | $\mu_{\mathrm{tet}}$ | $\mu_{\mathrm{oct}}$
DFT+$U$ | 4.23 | 3.85 |  | 3.93 | 3.31 |  | 3.95 | 3.05
MLFT ground state | 4.25 | 4.03 |  | 4.23 | 3.11 |  | 4.03 | 3.43
Expt. | 4.0 | 3.6 |  | 3.5 | 3.24 |  | 3.72 | 2.90

##  V Results and discussion

In this section, the theoretical calculations of XAS and XMCD are compared
with the experimental results (black curves), for all compounds of this
investigation. The theoretical calculation is obtained by varying the higher-
order Slater parameters ($k\geq 1$) in Eq. (A.2) individually between 70% and
90% of their calculated values while making the best fit to the experimental
data. These parameters are shown in Table 1 and the theoretical spectra based
on them are labelled ”theory” in Figs. 2\- 8, and are shown as a red line. In
Figs. 2\- 6 we also show results from an alternative theoretical calculation,
where all the higher-order Slater parameters are screened to 80% of their bare
values (blue dashed line in Figs. 2\- 6). Note also that in each subsection,
we start with an analysis of XAS which is followed by the results of the XMCD.
In the case of CoFe2O4 and NiFe2O4 the spectral features from the Fe sites
will be presented first.

###  V.1 Spectra of Fe3O4

XAS. The dipole allowed Fe $L_{2/3}$ XAS edges of Fe3O4 are displayed in Fig.
2. The spectrum is composed of two main peaks, the $L_{3}$\- and
$L_{2}$-edges, due to the SOC $\Delta_{SO}=(3/2)\zeta_{p}$ in the TM 2$p$
state. This results in a split of the core-levels of 2$p$ $J$ =3/2 and 2$p$
$J$ =1/2 by $\Delta_{SO}$. When photons excite the core electrons in the
higher $J$ =3/2 level and lower $J$ =1/2 level they respectively produce
electronic transitions to the $L_{3}$-edge and the $L_{2}$-edge. The SOC of
the core states is strong and almost independent of the atomic environment.

The experimentally observed spectrum has a broad $L_{3}$-edge (between 705 and
715 eV), which is composed of contributions from all three different Fe sites.
The $L_{2}$-edge (between 718 and 728 eV) has a significantly lower intensity
and shows a double-peak feature. Both edges are well reproduced by the theory
with the $L_{3}$-edge being a bit broader on the low energy side in the
experiment. The theoretical results from the $L_{2}$-edge do not show the same
double-peak feature as in the experiment with the second peak missing,
however, the relative intensity to the $L_{3}$-edge is more or less the same
in theory and in the experiment. In a pure one-electron picture, and with
uniform matrix elements for the excitation, the intensities of the $L_{3}$\-
and the $L_{2}$-edges would reflect the number of electrons available of the
corresponding core levels, i.e., 4 to 2. The data in Fig. 2 show that this is
not the case, and the so called branching ratio deviates significantly from
the 4/2 ratio. It is gratifying that the theory used here reproduces this
aspect of the measured spectrum. Furthermore, we can see that the calculation
is robust towards the choice of Slater parameters as the curves match nearly
perfectly.

XMCD. The measured XMCD spectrum (also shown in Fig. 2) has a characteristic
down-up-down peak structure at the $L_{3}$-edge. Here, the first peak has the
largest intensity and the second peak the lowest. This structure is caused by
the ferrimagnetic ordering and the spinel structure. In a perfect
antiferromagnet no XMCD can be observed as the contributions vanish. The
antiferromangnetically aligned Fe cations occupy inequivalent lattice sites
and have therefore slightly different excitation energies. This results in a
dichroic response from the +3 cations, which here adds to the response from
the Fe2+ cations. At energies just above these peaks one can observe a small
shoulder. The $L_{2}$-edge shows two different peaks with the first feature
having a larger intensity and a distinct two-peak feature. The theoretical
calculation reproduces the behaviour at the $L_{3}$-edge very well. Even the
small shoulder at energies above the three-peak feature is reproduced and the
oscillations right below of the $L_{2}$-edge are reproduced. The only major
difference is that a negative peak appears in the calculation between the
$L_{3}$\- and $L_{2}$-edge which is not in the experiment. Also the
intensities of the $L_{2}$-edge is overestimated on all peaks except the first
peak of the double-peak feature. This could be due to an overestimation of the
3$d$ SOC of the tetrahedral sites. Here, we can see that our choice of the
higher order Slater parameters enhances the signal at the middle peak and
suppresses that of the third peak and the multiplet effects.

###  V.2 Spectra of CoFe2O4

Fe-projected results

XAS. The measured XAS of the Fe sites (Fig.3, upper panel) shows a main peak
with a low energy shoulder at the $L_{3}$-edge and a double peak at the
$L_{2}$-edge. The XAS of the Fe sites shows the same features in experiment
and theory. However, the shoulder of the $L_{3}$-edge is smaller in intensity
for the calculated spectrum than in the experiment. This shoulder is caused by
the octahedral sites in the calculation, therefore a higher ratio of
octahedral to tetrahedral Fe sites than the equal amount that the calculation
used could explain this effect. A reason for this could be Co cations
occupying one or both tetrahedral sites in some cells, causing the Fe sites to
occupy three or four octahedral sites in that cell, which would lead to a
higher contribution of the octahedral Fe sites in the signal. Other point
defects in the experimental sample could cause different oxidation states,
that also would influence smaller details in the observed spectra, that would
not be picked up by calculations of a defect-free system. For the $L_{2}$-edge
the theory clearly puts more intensity on the higher energy peak, while they
have the same intensity in the experiment. It is also notable that the
branching ratio is very similar for theory and experiment. Furthermore, one
can see that the calculation is robust to the choice of Slater parameters as
the two curves match.

XMCD. In the XMCD spectrum (Fig.3, lower panel), one can observe the same
characteristic down-up-down structure as for Fe3O4 in the $L_{3}$ channel,
however due to the absence of Fe2+ the first peak is no longer the largest in
intensity, rather it is the smallest while the third peak in the $L_{3}$
channel has the largest intensity, with the second peak having similar
intensity to the third. The details of these peaks are quite similar if one
compares theory and experiment. The measured $L_{2}$-edge shows two low
intensity peaks with the second one being flat and lower in intensity. In the
calculation the general structure of the $L_{3}$-edge is well reproduced.
However, the intensity for the first peak is underestimated and the second one
is overestimated, this could also be linked to some Co occupying tetrahedral
sites, because the second peak is caused by tetrahedral Fe and the first by
octahedral. Here, the fine structure is not reproduced by the calculation as
it points in the opposite direction compared to the experiment. The two
distinct $L_{2}$ peaks are well reproduced by the theory. However, the second
peak in the experiment is broader than in the theory. Furthermore, one can see
that the calculation is robust to the choice of Slater parameters as the two
curves match.

Co-projected results

XAS. The XAS of Co (Fig.4, upper panel) shows a lot of features, both in
theory and experiment. The $L_{3}$-edge has a double peak, where the peak on
the left is slightly higher in intensity. The peak is accompanied by a small
shoulder on the left, a high intensity shoulder and a small shoulder to the
right. The $L_{2}$-edge is characterized by a single peak. The calculated XAS
of the Co site agrees very well with the experiment. The double-peak feature
of the $L_{3}$ is well reproduced, however in the experiment the first is
larger. The left shoulder is located at lower energies in theory compared to
the experiment and it has also a lower intensity. The right shoulder matches
the experiment perfectly, even the small shoulder at the end is reproduced. At
the $L_{2}$-edge, experiment and theory mostly lie on top of one another,
however the theory shows a slightly higher intensity at the maximum of the
peak. Here, one can see that the two theoretical spectra look very similar and
that the main difference is a shift of energy. This is caused by the fact that
we are aligning the peaks of the largest intensity in the calculation to the
largest peak in the experiment and the intensities of the two peaks in the
double peak feature changes due to the choice of the Slater parameters.

XMCD. The measured $L_{3}$-edge XMCD of the Co site (Fig.4, lower panel) shows
a small down-pointing peak followed by a thin up-pointing peak in front of the
main peak, which has one shoulder on the left and two on the right. After the
main peak, there is a small fine structure peak. At the $L_{2}$-edge, there is
a single flat peak. The theory reproduces the shape of the $L_{3}$-edge main
peak and its shoulders well, even though the shoulders on the right have less
intensity than in the experiment. The peaks before the main peak are not well
reproduced, with the first peak missing and the second peak being broader and
less intense than in the experiment. This could be because these peaks are
caused by Co occupying the tetrahedral site, where it would be closer to a +3
state. Alternatively, other impurities in the sample could play a role. The
small fine structure after the main peak is very well resolved by the
calculation. The $L_{2}$-edge in the theory shows a very clear peak with much
higher intensity than the flat peak in the experiment. This could again be
caused by Co in tetrahedral sites as those would have the opposite spin and
therefore would reduce the signal from the peak. Here, we can see that the
choice of Slater parameters only affects the two shoulders closest to the main
peak, where the intensity is shifted slightly from the left shoulder to the
one on the right.

###  V.3 Spectra of NiFe2O4

Here, the experiment uses a superposition of 77% circular polarized and 23%
linear polarized light [38]. This was explicitly taken into account in the
transition dipole operator.

Fe-projected results

XAS. The measured $L_{3}$-edge of the Fe sites (Fig.5, upper panel) has a main
peak with a shoulder to the left and a smaller shoulder to the right. The
$L_{2}$-edge has a clear main peak with a distinct shoulder to the left. In
the calculations of the XAS of the Fe sites the left shoulder of the
$L_{3}$-edge is not present. However, the right shoulder is nearly perfectly
reproduced by the theory. The observed relative intensity of the $L_{2}$-edge
compared to the $L_{3}$-edge is slightly overestimated by our theory and the
calculated shoulder of the $L_{2}$-edge could be more distinct from the main
peak. Furthermore, the calculated energy difference between the $L_{3}$\- and
$L_{2}$-edges, due to the SOC of the $p$-electrons, is slightly overestimated.
Here, the blue and red lines perfectly match as we have used the same
parameters.

XMCD. The measured XMCD (Fig.5, lower panel) shows the same down-up-down
structure as the other Fe sites. However, here, the up-pointing peak has the
highest intensity and the first peak has the lowest due to the missing Fe2+
sites. After the three peaks, one can see two fine structure features. The
$L_{2}$-edge is characterized by an up-down-up structure with the down-
pointing peak having the highest intensity and the last peak the lowest and
most elongated. The down-up-down peak structure of the $L_{3}$-edge is
reproduced very well by the theory with a slight shift towards higher energies
for the first two peaks and an overestimation of the middle peak. The two
fine-structure peaks after the third peak are also reproduced, however the
intensity of the first peak is overestimated and of the second underestimated.
In the $L_{2}$ channel, one can notice the same structure in the first two
peaks from the calculations, but for the third we see a double-peak structure
with higher intensities. Just like for the XAS, the SOC is overestimated and
therefore, the energy difference between the $L_{3}$\- and $L_{2}$-edges are
slightly larger than in the experiment.

Ni-projected results

XAS. The observed $L_{3}$-edge of Ni (Fig.6, upper panel) is characterized by
a main peak with a large shoulder on the high energy side. The $L_{2}$-edge
has a peak with a shoulder on the high energy side. Overall, the XAS is well
reproduced by the theory. The main peak and shoulder of the $L_{3}$-edge are
located at the correct positions, however the width and intensity are somewhat
underestimated. The shape and relative intensity of the $L_{2}$-edge compared
to the $L_{3}$-edge are very close to the experimental results. However, the
intensity is slightly lower than in the experiment. Additionally the energy
difference between the $L_{3}$ and $L_{2}$ peaks, which is determined by the
SOC of the $p$-states, is slightly overestimated by the theory. From Fig.6 one
can note that the calculated result is robust with respect to an uncertainty
in Slater parameters, as the intensity of the shoulders increases only
slightly when the paramaters are varied.

To further check the sensitivity of the theoretical results, we performed
additional calculations that demonstrate how the theoretical spectra depend on
the Coulomb core-valence (CV) interaction. This is shown in Figs. 7 and 8, for
the Fe respectively Ni sites in NiFe2O4. As the figure shows, it is crucial to
include the CV interaction in order to get a good XAS and XMCD signal and the
effect is most clearly demonstrated for the Fe atom.

XMCD. The XMCD (Fig.6, lower panel) has a large, up-pointing shoulder at the
$L_{3}$-edge. At the $L_{2}$-edge, we observe a large peak with a shoulder to
the left. The calculation nearly perfectly matches the experimental results.
The shoulder of the $L_{3}$-edge is slightly too low in intensity, whereas the
energy difference between the $L_{3}$\- and $L_{2}$-edges is slightly
overestimated like in the XAS. Here, we can see that the calculation is just
as robust to the choice of Slater parameters as the XAS.

To summarize the previous discussion, we observe that all XAS excitations can
be explained in terms of electric-dipole-allowed electronic transitions and
are composed of two strong peaks in the $L_{3}$ and $L_{2}$ edges, due to the
large SOC of the 2$p$ states into $p_{3/2}$ and $p_{1/2}$ relativistic states.
The photon excitations of the core electrons in the higher $J$ =3/2 and low
$J=1/2$ levels produce electronic transitions essentially towards the
unoccupied 3$d$ states, producing respectively the $L_{3}$\- and
$L_{2}$-edges. The features within each of those edges are mainly due to the
3$d$ electron-electron interactions as well as the Coulomb interaction between
the 2$p$ core hole and the photo-electron in the 3$d$ unoccupied states. It is
therefore a good idea to decompose such excitations in terms of the 3$d$
$t_{2g}$ and $e_{g}$ states to determine the various contributions to the XAS
and XMCD spectra. For this we have inspected the partial density of states
(PDOS) to investigate which orbitals resemble $e_{g}$ and $t_{2g}$ the most.
Then we constructed operators that only excite the 2$p$ core electrons into
either the $e_{g}$ or $t_{2g}$ orbitals. The difference between the total
spectra and the sum of the signals from $e_{g}$ and $t_{2g}$, corresponds to
the interference between the excitations into $t_{2g}$ and $e_{g}$ that
produce the same final state. We refer to this as the off-diagonal
contribution in the following.

Figure 9 shows for Fe3O4 the contributions from excitations between $2p_{1/2}$
as well as the $2p_{3/2}$ core levels to the $t_{2g}$ and $e_{g}$ states,
resulting in symmetry projections of the final states reached by the light
matter interaction. Note from Fig. 9 that we show both data for XAS as well as
XMCD. Similar figures for the analysis of the XAS and XMCD spectra of CoFe2O4
and NiFe2O4 are shown in the supplementary information. It is clear from Fig.
9 that the intensity of the two symmetry components is not quite a 3/2 ratio
that would be expected from the occupancy of the $t_{2g}$/$e_{g}$ levels, but
only about 1.4. One may also see that the $t_{2g}$ and $e_{g}$ energy
splitting is not noticeably conspicuous in these spectra. It is interesting to
note that both states contribute almost equally to the $L_{3}$ and $L_{2}$
main peaks. For the XMCD spectrum, the difference between $t_{2g}$ and $e_{g}$
projected levels is much more pronounced compared to the XAS signal. In
particular we note that for the $L_{2}$-edge the $t_{2g}$ and $e_{g}$ have
opposite contributions at most of the energies where this peak has large
intensity. In general, however, Fig. 9 shows that an interpretation of the XAS
and XMCD signals in terms of transitions to $t_{2g}$ or $e_{g}$ projections is
not trivial, since the off-diagonal contributions are significant. Similar
conclusions can be drawn for symmetry decomposed spectra of all the here
investigated materials (see Appendix B.1).

As a final comment to this section we note that the XAS and XMCD spectra of
NiFe2O4 are very similar for calculations where the parameters of the MLFT
Hamiltonian are evaluated from LDA$+U$ and from LDA+DMFT. This is shown in
Fig.14 for the Fe L-ege spectra and in Fig.15 for the Ni spectra. Note that
the two levels of theory give almost identical results for the Fe XAS signal,
but that the XMCD signal agrees sightly better with experiment for the DMFT
based theory. For the Ni signal the two theory curves are almost identical
both for the XAS and XMCD signal.

Table 4: Site electron occupation $N_{3d}$ of the tetrahedral and the octahedral transition metal ion in different ferrites. | Fe3O4 |  | CoFe2O4 |  | NiFe2O4
---|---|---|---|---|---
| Fe${}^{2+}_{\mathrm{oct}}$ | Fe${}^{3+}_{\mathrm{oct}}$ | Fe${}^{3+}_{\mathrm{tet}}$ |  | Co | Feoct | Fetet |  | Ni | Feoct | Fetet
N3d | 6.267 | 5.901 | 5.768 |  | 7.129 | 5.148 | 5.110 |  | 8.291 | 5.177 | 5.123
---
Figure 2: Top: Calculated XAS of the Fe $L_{2,3}$ edges for Fe3O4 compared to
experimental data (black curve). Bottom: Calculated XMCD of Fe3O4 compared to
experimental data[39] (black curve). The theory was carried out with the
higher order Slater parameters screened between 70%-90% (red curve) and to 80%
(blue dashed curve) of the calculated values, for details see text.
---
Figure 3: Same as Fig. 2 but for Fe sites in CoFe2O4. The experimental spectra
are reproduced from [40].
---
Figure 4: Same as Fig. 2 but for Co site in CoFe2O4. The experimental spectra
is reproduced from [40].
---
Figure 5: Same as Fig. 2 but for Fe sites in NiFe2O4. The experimental spectra
is reproduced from [41].
---
Figure 6: Same as Fig. 2 but for Ni site in NiFe2O4. The experimental spectra
is reproduced from [41].

##  VI Conclusion

We have used multiplet ligand-field theory in conjunction with DFT+$U$ or
DFT+DMFT electronic structure theory to describe on an equal footing the
multiconfigurational effects as well as the presence of the core-hole photo-
electron Coulomb interaction and its effect on XAS and XMCD spectra. The
electronic transition selection rules have been included within the electric-
dipole approximation. We have presented the details of the implementation and
its application to the $L_{2,3}$-edges and XMCD spectra of the ferrites, Fe3O4
(magnetite), cobalt ferrite, CoFe2O4, and nickel ferrite, NiFe2O4. We find
that the results are in overall agreement with available experimental spectra.
In particular, we find that the experimental $L_{3}$/$L_{2}$ branching ratio
of the XAS signal is well reproduced by the here described calculations, in
contrast to an approximate 2:1 ratio produced by DFT calculations. In
addition, most of the details of the spectral features of the measured XAS and
XMCD signals are reproduced by the theory. In addition, and as detailed in
Appendix B.2, our analysis suggests that calculations with parameters obtained
from DFT+$U$ and DFT+DMFT give very similar results both for the XAS and XMCD
spectra. If this can be demonstrated to hold for a wider range of compounds,
it would in general simplify the theoretical description of XAS and XMCD
spectra of materials with complex electronic structure, since the
computational effort of DFT+$U$ is significantly smaller than DFT+DMFT.

The aim of the present theory is to rely on as few free parameters as possible
as detailed in the method section III. We explored the renormalization of the
higher-order Slater parameters due to screening by either renormalizing their
values to a fixed 80% of their bare value or varying them between 70% to 90%.
We found that the spectra were only marginally affected by different
descriptions of the screening within this range. In this work, only the
zeroth-order Slater parameter $F_{pd}^{0}$ and the double-counting correction
$\delta_{DC}$ were chosen as free parameters. We kept the value of
$\delta_{DC}$ fixed to 1.5 eV for all investigated compounds. There is
currently no accurate method for calculating the screening of $F_{pd}^{0}$, so
we varied it in the interval $F_{dd}^{0}<F_{pd}^{0}<1.4F_{dd}^{0}$. We are
currently investigating how to obtain screening of $F_{pd}^{0}$ from cDFT in a
reliable way.

We have attempted to make an interpretation in terms of excitations from
$2p_{1/2}$ and $2p_{3/2}$ to the $t_{2g}$ and $e_{g}$ irreducible
representations of the 3d valence electron states. However, due to large
inteference terms between the $e_{g}$ and $t_{2g}$ excitations, such an
analysis was shown to have a convoluted physical interpretation.

###### Acknowledgements.

We acknowledge support from the Swedish Research Council, Wallenberg
Initiative Materials Science (WISE) funded by the Knut and Alice Wallenberg
Foundation, the ERC (FASTCORR project), eSSENCE and STandUPP. Calculations
provided by SNIC/NAISS. We are grateful to Dario Arena, University of South
Florida, for allowing us to show his experimental data of NiFe2O4, prior to
publication. Valuable discussions with Dr. Chin Shen Ong, Dr. Yaroslav
Kvashnin and Prof. Mauritz Haverkort are acknowledged.

##  Appendix A Computational details

The XAS and XMCD spectra were calculated for NiFe2O4, CoFe2O4 and Fe3O4, which
all share the same inverse-spinel structure with Ni and Co occupying the
octahedral sites.

###  A.1 DFT

Self-consistent spin-polarized DFT+$U$ calculations, where the spin
polarization was introduced via the local Hartree-Fock potential (+$U$), have
been performed with $8\times 8\times 8$ $k$-point mesh sampling in the
Brillouin zone for NiFe2O4 and CoFe2O4 and $10\times 10\times 10$ for Fe3O4.
We used a full-potential linear muffin-tin orbital method (LMTO) as
implemented in the “RSPt” code[42, 43] to solve the DFT problem. The set of
localized impurity orbitals is constructed by projecting the total electron
density on a set of Löwdin orthogonalized LMTOs for the TM $3d$ orbitals,
denoted as “ORT” in Ref. [27, 15, 28].

###  A.2 Coulomb $U$

The interacting part of the impurity Hamiltonian for a multi-orbital system
reads

$\hat{H}_{\textrm{int.}}=\frac{1}{2}\sum_{abcd,\sigma\sigma^{\prime}}U_{abcd}\hat{c}_{b,\sigma}^{\dagger}\hat{c}_{a,\sigma^{\prime}}^{\dagger}\hat{c}_{c,\sigma^{\prime}}\hat{c}_{d,\sigma},$
(15)

where each term describes a process and $U_{abcd}$ is given by

$\displaystyle U_{ijkl}$ $\displaystyle=\int\int
d^{3}rd^{3}r^{\prime}\psi_{i}^{*}(\bm{r}^{\prime})\psi_{j}^{*}(\bm{r})\frac{1}{|\bm{r}-\bm{r}^{\prime}|}\psi_{k}(\bm{r}^{\prime})\psi_{l}(\bm{r}).$
(16)

By expanding the Coulomb interaction $1/|\bm{r}-\bm{r}^{\prime}|$ in terms of
spherical harmonics and with basic functions of the form

$\psi_{i}(\bm{r})=f_{n_{i},l_{i}}(r)Y_{l_{i},m_{i}}(\theta,\phi),$ (17)

the Coulomb interaction tensor becomes [44]

$\displaystyle
U_{abcd}=\delta_{m_{a}+m_{b},m_{c}+m_{d}}\sum_{k=0}^{k_{\textrm{max}}}$
$\displaystyle c^{k}(l_{b},m_{b};l_{d},m_{d})c^{k}(l_{c},m_{c};l_{a},m_{a})$
$\displaystyle\times R^{k}(n_{a}l_{a},n_{b}l_{b},n_{c}l_{c},n_{d}l_{d}).$ (18)

The Gaunt coefficients,

$\displaystyle c^{k}(l,m;l^{\prime},m^{\prime})=\sqrt{\frac{4\pi}{2k+1}}$
$\displaystyle\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin{\theta}Y^{*}_{l,m}(\theta,\phi)$
$\displaystyle\times
Y_{k,m-m^{\prime}}(\theta,\phi)Y_{l^{\prime},m^{\prime}}(\theta,\phi),$ (19)

take care of the angular integrals in Eq. (16) and are easily evaluated. By
considering the parity of the spherical harmonics in Eq. (A.2), only Gaunt
coefficients with $l+l^{\prime}+k$ being an even number can be non-zero. The
two Gaunt coefficients in Eq. (A.2) constrain the $k$-expansion to a maximum
of $k_{\textrm{max}}=\textrm{min}(|l_{b}+l_{d}|,|l_{c}+l_{a}|)$. The last
factor to discuss in Eq. (A.2) is the parameter,

$\displaystyle R^{k}(n_{a}l_{a},n_{b}l_{b},n_{c}l_{c},n_{d}l_{d})$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}drdr^{\prime}r^{2}r^{\prime
2}f_{n_{a},l_{a}}(r^{\prime})$ $\displaystyle\times
f_{n_{b},l_{b}}(r)\frac{r_{<}^{k}}{r_{>}^{k+1}}f_{n_{c},l_{c}}(r^{\prime})f_{n_{d},l_{d}}(r),$
(20)

where $r_{<}$ ($r_{>}$) indicates min$(r,r^{\prime})$ (max$(r,r^{\prime})$).
It is customary to define the Slater-Condon parameters,

$\displaystyle F^{k}(nl,n^{\prime}l^{\prime})$
$\displaystyle=R^{k}(nl,n^{\prime}l^{\prime},nl,n^{\prime}l^{\prime})$
$\displaystyle G^{k}(nl,n^{\prime}l^{\prime})$
$\displaystyle=R^{k}(nl,n^{\prime}l^{\prime},n^{\prime}l^{\prime},nl),$ (21)

where $F$ and $G$ describe the Coulomb and exchange integrals, respectively.
For the Coulomb interaction between $d$-orbitals, for any given principal
quantum number, $F^{k}=G^{k}$, and only the three parameters $F^{0},F^{2}$ and
$F^{4}$ are relevant, due to the constrains mentioned above. The bare Slater-
Condon integrals are calculated using the projected $3d$ and $2p$ wave
functions within the muffin-tin sphere. However, the screened value of
$F_{pd}^{0}$ is difficult to calculate due to the strong screening effects
from uncorrelated electrons and is treated as a tunable parameter.

The core-valence interaction ($F^{k}_{pd}$ and $G^{k}_{pd}$) gives crucial
contributions to the spectra. In Fig. 7 and Fig. 8 we compare the calculated
spectra of NiFe2O4 using the full Hamiltonian and neglecting the core-valence
interaction. The L2,3-edges in the spectra without core-valence interaction
are characterized by having a single Lorenzian peak, except the XMCD of Fe
where we have a single peak per site.

Figure 7: Calculated Fe $L_{2,3}$ XAS (top panel) and XMCD (bottom panel)
edges in NiFe2O4 using the full Coulomb interaction (red solid) and neglecting
the $p-d$ interaction (no CV, blue dashed) compared to experiments (black
solid).

Figure 8: Calculated Ni $L_{2,3}$ XAS (top panel) and XMCD (bottom panel)
edges in NiFe2O4 using the full Coulomb interaction (red solid) and neglecting
the $p-d$ interaction (no CV, blue dashed) compared to experiments (black,
solid).

###  A.3 Spin-orbit coupling

The spin-orbit coupling (SOC) Hamiltonian is first quantized form for a
$(n,l)$-shell with $N$ electrons and SOC parameter $\zeta$ is

$\tilde{H}_{\textrm{SOC}}=\zeta\sum_{i=1}^{N}\tilde{\bm{l}}_{i}\cdot\tilde{\bm{s}}_{i}=\zeta\sum_{i=1}^{N}\left(\tilde{l}_{i}^{z}\tilde{s}_{i}^{z}+\frac{1}{2}(\tilde{l}_{i}^{+}\tilde{s}_{i}^{-}+\tilde{l}_{i}^{-}\tilde{s}_{i}^{+})\right),$
(22)

where for particle $i$, $\tilde{\bm{l}}_{i}$ ($\tilde{\bm{s}}_{i}$) is the
orbital (spin) angular momentum vector operator, $\tilde{l}_{i}^{z}$
($\tilde{s}_{i}^{z}$) the $z$-projected orbital (spin) angular momentum
operator and $\tilde{l}_{i}^{\pm}$ ($\tilde{s}_{i}^{\pm}$) the raising and
lowering orbital (spin) angular momentum operators. In the last expression in
Eq. (22), the first term is diagonal in the $(l,m,\sigma)$ basis and in second
quantized form becomes

$\zeta\sum_{m=-l}^{l}\sum_{\sigma\in\\{-\frac{1}{2},\frac{1}{2}\\}}\sigma
m\hat{c}_{l,m,\sigma}^{\dagger}\hat{c}_{l,m,\sigma}.$ (23)

The other terms flip the spin and can be written as

$\zeta\frac{1}{2}\sum_{m=-l}^{l-1}\sqrt{(l-m)(l+m+1)}(\hat{c}_{l,m+1,\downarrow}^{\dagger}\hat{c}_{l,m,\uparrow}+\hat{c}_{l,m,\uparrow}^{\dagger}\hat{c}_{l,m+1,\downarrow}).$
(24)

For $3d$ orbitals of the 3$d$ elements, $\zeta$ is rather small (less than
$\sim$ 100 meV) in comparison to other relevant energies, e.g., the bandwidth.
But for core $2p$ orbitals of the 3$d$ transition metals $\zeta$ is of the
order of several eV and thus absolutely necessary to include in the
calculation.

###  A.4 Double counting

Another important aspect is the double counting (DC) correction $\delta_{DC}$,
which has to be subtracted from the DFT-derived Hamiltonian. This is done in
order to remove the contribution of the Coulomb repulsion that is already
taken into account at the DFT level.

In this work, we apply a DC that is often used in MLFT by considering the
relative energy for different configurations [33].

##  Appendix B Supporting Results

###  B.1 Symmetry-decomposed spectra

In this section, we describe the symmetry-decomposed spectra of all compounds
investigated here. The symmetry decomposition is made onto $e_{g}$ and
$t_{2g}$ orbitals of the 3d states of the transition metal atom, as described
in the main part of this paper. The results of the $e_{g}$ and $t_{2g}$
projections of the XAS and XMCD for Fe3O4 are shown in Fig. 9. In the XAS, we
can see that the $t_{2g}$ has a pronounced shoulder on the right of the main
peak at the $L_{3}$-edge, which in the total spectrum is enhanced due to the
off-diagonal elements, while the intensity of the main peak is suppressed. The
$e_{g}$ signal shows a broad peak that begins at the same energy as the
$t_{2g}$, but peaks at its shoulder with approximately twice the intensity. At
the $L_{2}$-edge, the contributions from $e_{g}$ and $t_{2g}$ both show a
single peak, where the $t_{2g}$ peak is located at higher energies and keeps
the intensity ratio, while the off-diagonal suppresses the signal at the
$e_{g}$ peak and enhances the signal at the $t_{2g}$ peak. The $t_{2g}$ peak
appears broader because they are the lower-lying orbitals in the lower energy
octahedral Fe sites and the higher in the higher energy tetragonal Fe sites.
The intensity ratio can be explained by the fact that we can excite into three
$t_{2g}$ and two $e_{g}$ per site. In the XMCD we see that the first down-
pointing peak of the $L_{3}$-edge is caused by the lower lying $t_{2g}$
orbitals from the octahedral Fe sites that have lower binding energy than the
tetragonal site. This signal is further enhanced by the off-diagonal elements.
The middle up-pointing peak is not caused by the $e_{g}$ peaks of the
tetragonal site, which are completely compensated by the $t_{2g}$ down-
pointing peak, instead, it is caused by the off-diagonal elements. The $e_{g}$
and $t_{2g}$ of the two octahedral sites equally contribute to the third peak,
which is slightly enhanced by the off-diagonals. The small multiplet peaks
between the $L_{3}$ and $L_{2}$-edge are caused by a competition between the
$t_{2g}$ of the tetragonal and the octahedral sites. The signal at the at the
$L_{2}$-edge is mostly comprised of $t_{2g}$ signal with the $e_{g}$ and off-
diagonals reducing the signal causing the split into two different peaks in
the total signal.

The $e_{g}$ and $t_{2g}$ projected spectra of Co in CoFe2O4 can be seen in
Fig. 10. The first peak in the XAS comes from both the $e_{g}$ and the
$t_{2g}$ even though the $t_{2g}$ orbital should be at lower energies,
however, they are nearly completely compensated by the off-diagonal elements.
After this the $t_{2g}$ drops in value and has three more small peaks, which
are overshadowed by the off-diagonals and the $e_{g}$ signal. The double-peak
structure is mainly formed by the off-diagonals and the $e_{g}$ signals with
similar intensities, just that the off-diagonals have a higher intensity at
the first peak, while the $e_{g}$ has the highest intensity at the second
peak. The signal at the signal of the $L_{2}$-edge is dominated by the
$t_{2g}$, which is lowered by the off-diagonals. In the XMCD the first peak of
the $L_{2}$-edge is dominated by the $e_{g}$-signal, while the main peak is
mostly caused by the off-diagonal elements followed by the $t_{2g}$. The
signal of the $L_{2}$-edge is dominated by the $t_{2g}$ which is suppressed by
the off-diagonals.

The Fe-projected $e_{g}$ and $t_{2g}$ spectra can be seen in Fig. 11. The
beginning of the $L_{3}$-edge of the XAS is characterized by the competition
of the off-diagonals and the $t_{2g}$ signals, where the $t_{2g}$ signal is
slightly bigger than the signal by the off-diagonal terms. The main peak is
mainly caused by the $e_{g}$ contributions, but the $t_{2g}$ also shows a
shoulder here, which has the same intensity as the peak of the off-diagonals.
At the $L_{2}$-edge, most of the signal is generated by the $t_{2g}$. In the
XMCD, we can see that all the peaks of the $L_{3}$-edge have different
origins. The first down peak is caused by the excitations into the $t_{2g}$,
the up-pointing peak by the off-diagonals and the last peak by the $e_{g}$.

In Fig. 12 we see the eg\- and $t_{2g}$-projected results of the Ni site in
NiFe2O4. Here, nearly all the signal comes from the $e_{g}$, because in Ni2+
the $t_{2g}$ is fully occupied and one can therefore only make electronic
transitions to the $e_{g}$.

Finally, the result for the Fe sites in NiFe2O4 are shown in Fig. 13. In the
XAS, we see that the lowest energy peak is caused mostly by the $t_{2g}$,
however, the eg signal also contributes and the off-diagonal elements reduce
the intensity. The main peak is caused by a combination of all three signal
sources with the biggest contribution coming from the $t_{2g}$. At the
$L_{2}$-edge, one can see that the two peak signals come mostly from the
$e_{g}$ as the $t_{2g}$ is too broad the easily distinguish between the peaks
even though it has more intensity. While the off-diagonal reduces the
intensity of the first peak and slightly increases the intensity of the second
peak. The XMCD shows that the first down-pointing peak comes from excitations
into the $t_{2g}$ orbitals which corresponds to the lowest lying states of the
octahedral sites. At this position, we also have the contributions from the
$e_{g}$ from the tetrahedral site but the intensity is so small that is not
visible in the total spectrum. The middle peak that is caused by the
tetrahedral site, is instead caused by off-diagonal elements and the $t_{2g}$
contributions. The second down-pointing peak is just as expected caused by the
higher lying $e_{g}$ states in the octahedral Fe site. At the $L_{2}$-edge, we
can see that the initial dip in the total spectra is caused by the $e_{g}$ and
the off-diagonal signal and the peak after that is caused by the same.

---
Figure 9: Calculated $e_{g}$ and $t_{2g}$ $L_{2,3}$ XAS (top) and XMCD
(bottom) of Fe3O4 (for details see text).
---
Figure 10: Same for Co in CoFe2O4.
---
Figure 11: Calculated $e_{g}$ and $t_{2g}$ projection of the Fe $L_{2,3}$ XAS
(top) and XMCD (bottom) of CoFe2O4 (for details see text).
---
Figure 12: Calculated $e_{g}$ and $t_{2g}$ projection of the Ni $L_{2,3}$ XAS
(top) and XMCD (bottom) of NiFe2O4 (for details see text).
---
Figure 13: Calculated $e_{g}$ and $t_{2g}$ projection of the Fe $L_{2,3}$ XAS
(top) and XMCD (bottom) of NiFe2O4 (for details see text).

###  B.2 DMFT

In this section, we investigate the effects of using the local Hamiltonian and
the hybridization function obtained from a DFT+DMFT calculation, instead of a
DFT$+U$ calculation to construct the impurity Hamiltonian of NiFe2O4, while
using the same Slater and SOC parameters as before. We used the method
described in Ref. [15], with one bath state per correlated orbital and the ED
impurity solver. Figs. 14 and 15 show the XAS and XMCD spectra of the Fe and
Ni sites, respectively, calculated using the same parameters as Figs. 5 and 6
except that the relative corelevel shift $\Delta\epsilon_{p}$ was adjusted to
the values extracted from the DFT+DMFT calculation. Here, we can see that the
spectra are very similar. In the case of the Fe-projected XAS, we can see that
the spectrum looks broader, which causes the $L_{2}$-edge to be even more
overestimated. In the XMCD, we can see that the second peak and the
oscillations after the third peak of the $L_{3}$-edge are better reproduced in
the DFT+DMFT calculation. Similarly, we can see improvements at the beginning
of the $L_{2}$-edge, while the end is even more overestimated than in the
DFT+$U$ calculation. In the Ni-projected XAS, we can see that the $L_{3}$-edge
shoulder is more strongly pronounced in the DFT+DMFT calculation. At the
$L_{2}$-edge however, we can see a double peak structure in the DFT+DMFT
solution instead of a pronounced first peak which we see in the experiment and
DFT+$U$ calculation. In the XMCD, we can see that the DFT+DMFT calculation
reproduces the experiment even better than the DFT+$U$ calculation with a more
pronounced $L_{3}$-edge shoulder and a better relative intensity of the peaks
in the $L_{2}$-edge. The similarities between the results from DFT$+U$ and
DFT+DMFT are expected, because the sites mostly hybridize to the O sites,
which are only treated using DFT. Another reason for the similarities is that
we included all exchange interaction in the $+U$-term of the Hamiltonian (the
so-called LDA$+U$ approximation, which is not identical to the LSDA$+U$
approximation, where exchange splitting is also included in the density
functional), that guarantees that exchange interactions are treated in the
same way in the two approaches.

Figure 14:  Calculated Fe $L_{2,3}$ XAS (top panel) and XMCD (bottom panel)
edges in NiFe2O4 starting from a converged DMFT calculation (red solid) and
LDA$+U$ (blue dashed) compared to experiments (black solid).

Figure 15: Calculated Ni $L_{2,3}$ XAS (top panel) and XMCD (bottom panel)
edges in NiFe2O4 starting from a converged DMFT calculation (red solid) and
LDA$+U$ (blue dashed) compared to experiments (black solid).

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