# Distinct electride-like nature of infinite-layer nickelates and the
resulting theoretical challenges to calculate their electronic structure

Kateryna Foyevtsova  Ilya Elfimov  George A. Sawatzky  Department of Physics &
Astronomy, University of British Columbia, Vancouver, British Columbia V6T
1Z1, Canada  Stewart Blusson Quantum Matter Institute, University of British
Columbia, Vancouver, British Columbia V6T 1Z4, Canada

###### Abstract

We demonstrate in this paper that the recently discovered infinite-layer (IL)
nickelates have much in common with a class of materials known as electrides.
Oxide based electrides are compounds in which topotactic removal of loosely
bound oxygens leaves behind voids with a landscape of attractive potentials
for electrons. We show that this is also what happens in the IL nickelates,
where one of the two electrons (per formula unit) freed during the topotactic
synthesis is to a large degree located in the oxygen vacancy position,
occupying partially a local $s$-symmetry interstitial orbital, rather than
taking part alongside the other electon in converting Ni from 3+ to a full 1+
oxidation state. On the other hand, the involvement of the rare-earth $5d$
states is found to be rather indirect. Our study offers a new and detailed
perspective on the mechanisms through which the presence of the interstitial
charge can determine the superconducting and other properties of the IL
nickelates. To this end, we demonstrate that the interstitial orbital in
question, referred to by us as the zeronium $s$ or Z $s$ orbital, forms strong
covalent bonds with neighboring Ni $3d_{3z^{2}-r^{2}}$ orbitals, which in turn
facilitates the one-dimensional-like dispersion of the Ni $3d_{3z^{2}-r^{2}}$
band along the $c$-axis direction, leading also to a possible large out-of-
plane coupling between Ni magnetic moments. This finding, reinforced by our
electron localization function analysis, points to a fundamental distinction
between the nickelates and the structurally analogous cuprates, may explain
the absence of superconductivity in hydrogen-poor samples, and is certainly in
agreement with the observed large $z$-polarized component in the Ni
$L_{3}$-edge x-ray absorption spectra. In addition, by using DFT+U
calculations as an illustration, we show that the electride-like nature of the
IL nickelates is one of the main reasons for the theoretical difficulty in
determining the much debated elusive Fermi surface of these novel
superconductors and aslo in exploring the possibility of them becoming
excitonic insulators at low temperatures.

##  I Introduction

The study of the recently discovered Ni-based high transition temperature
($T_{\text{c}}$) superconductors with an infinite-layer (IL) crystal structure
Li et al. (2019); Osada et al. (2020a); Zeng et al. (2022); Pan et al. (2022)
promises to provide important clues to our understanding of unconventional
superconductivity in complex oxides. Such optimistic expectations are
primarily fueled by the intriguing similarity that the nickelates bear with
the famous superconducting cuprates Bednorz and Müller (1986); Keimer et al.
(2015). In both, the two-dimensional NiO2 or CuO2 layers constitute an
essential part of the crystal structure [see Fig. 1 (a)], the transition metal
(TM) ion is found in, formally, the same $d^{9}$ configuration, and a
superconducting dome develops upon hole doping within similar doping ranges.
As, however, intense research efforts rapidly expand our knowledge of the
nickelate superconductors, important differences are becoming apparent, too
(see reviews Chow and Ariando (2022); Nomura and Arita (2022); Gu and Wen
(2022); Zhou et al. (2022a); Botana et al. (2021) and references therein).
Thus, unlike the cuprates, the parent compounds of the IL nickelates behave as
bad metals and, despite the evidenced strong short-range magnetic correlations
Fowlie et al. (2022); Ortiz et al. (2022); Zhou et al. (2022b); Lu et al.
(2021), seem to have a charge Rossi et al. (2021a) rather than spin Ortiz et
al. (2022) ordered low-temperature state. Also, as the most recent
experimental study reveals, superconductivity occurs only in samples with
substantial hydrogen concentrations within a narrow range of 22 to 28
percentDing et al. (2023).

These observations are in line with the conclusion from an early theoretical
study of the nickelates’ electronic structure Lee and Pickett (2004) that,
notwithstanding their identical formal $d$ electron count, “Ni+ is not Cu2+”,
implying also a possibly different superconducting pairing mechanism. One of
the major distinctions originates from the fact that the charge-transfer
energy between the TM $3d$ and oxygen $2p$ states is considerably larger in
the nickelates Jiang et al. (2020), which, seemingly, justifies placing these
systems in the Mott-Hubbard rather than charge-transfer regime of the Zaanen-
Sawatzky-Allen (ZSA) classification scheme Zaanen et al. (1985). Yet, a simple
Ni $3d_{x^{2}-y^{2}}$ orbital based Mott-Hubbard picture is inconsistent with
many of the key experimental features of the parent IL nickelates either. One
obvious such feature is their metallic behavior that is likely to be related
with the presence of two small electron pockets of mixed orbital characters,
one at the Brillouin zone center and one at its corner, which are routinely
seen in electronic structure calculations Nomura and Arita (2022); Been et al.
(2021). It should be noted, though, that a full and precise determination of
the nickelates’ Fermi surface remains challenging to both theory and
experiment. Also at odds with the single orbital scenario is the fact that the
x-ray absorption spectra (XAS) at the Ni $L_{3}$-edge have a strong
$z$-polarized component, clearly indicating a presence of Ni
$3d_{3z^{2}-r^{2}}$ holes in the ground state of the parent compounds Rossi et
al. (2021b). Such observed partial occupation of the $3z^{2}-r^{2}$-symmetric
Ni $3d$ orbital in addition to the $x^{2}-y^{2}$ one makes is rather
challenging to describe the IL nickelates using the ZSA classification scheme,
since, in order to keep the unit cell charge neutral, there must be electrons
in formerly unoccupied states of local symmetry that can mix with the
$3z^{2}-r^{2}$ symmetry at the Ni site, which, however, are not provided by
the atomic orbitals of the $R$NiO2 crystal structure.

Given therefore that the IL nickelates seem to transcend the standard ZSA
paradigm and also motivated by the recently suggested critical and intimately
intertwined roles of the hydrogen and the interstitial charge for the
nickelates’ superconductivity, in this paper we take an alternative look at
their electronic structure by considering the electride concept frequently
encountered in oxides prepared by a topotactic removal of selected oxygen ions
Zhang et al. (2015); Matsuishi et al. (2003). Scientific interest in electride
materials has recently been on a rise Liu et al. (2020a) and is driven by
their exotic nature and a range of interesting physical properties they
display, including superconductivity Zhang et al. (2017). Electrides consist
of structural voids coordinated by mainly positive cations where unbound
electrons resulting from the oxygen removal can get trapped by a strongly
attractive potential. Rather than occupying the cation bands, these electrons
are relatively localized in the voids and act like anions of charge $-1$ and
spin $1/2$. An analogy with the IL nickelates can readily be drawn by
considering the topotactic synthesis of the latter from the precursor rare-
earth nickel perovskites $R$NiO3 ($R=$ La, Nd, or Pr), in which Ni ions assume
a formal valence of $3+$. The extracted apical oxygens leave behind two
electrons per formula unit which should, according to conventional thinking,
convert Ni3+ into Ni1+. However, it is quite feasible that whereas one of the
two doped electrons is indeed on Ni, forming a very stable $d^{8}$ state with
spin 1 as in NiO, the second electron is localized in the void at the O
vacancy site which indeed is surrounded by four Nd3+ and two Ni ionic
neighbors, with the actual ionic charges of the latter being closer to 2+ than
to 1+ in our proposed electride scenario.

In this paper, we use density functional theory (DFT) calculations to study
the possibility of this actually occurring in NdNiO2, a representative parent
IL nickelate, by explicitly introducing a fictitious atom without a nucleus at
the oxygen vacancy site, which is represented by a large white sphere in Fig.
1 (a), and centering a set of basis functions around it. Without changing any
physics, this allows us to analyze the electronic structure of NdNiO2 in terms
of local $s$, $p$, and $d$ symmetries. We will call this fictitious atom
“zeronium” becasue of its effective zero nuclear charge, a term coined in the
late 1970s Rompa et al. (1984), and use a chemical symbol “Z” for it. We note
that a variety of alternative terminologies exist. Thus, the electron
occupying (either fully or partially) an interstitial orbital inside the void
is also referred to in the literature as an interstitial, anionic, or
electride electron Liu et al. (2020a); Dale and Johnson (2018) or as an
interstitial quasiatom Miao and Hoffmann (2015). As will be shown, the charge
density of the Ni $d_{3z^{2}-r^{2}}$ character extends well into the
interstitial region in the $c$ direction, and by including Z we can quantify
this in the same way that we can quantify the covalent mixing of the Ni
$d_{x^{2}-y^{2}}$ orbital with the linear combination of the neighboring O
$2p$ orbitals of the $x^{2}-y^{2}$ symmetry in the Zhang-Rice singletZhang and
Rice (1988). It will be emphasized that the hybridization between the Ni
$d_{3z^{2}-r^{2}}$ and the Z $s$ orbitals is by far more important in the
interpretation of the low-energy electronic structure than the symmetry-
restricted involvement of the Nd $5d$ and O $2p_{\pi}$/$2p_{z}$ states, which
have been at the focus of most previous theoretical studies.

The structure of the paper is as follows. In Section II, we will describe our
computational methods and provide details of our electronic structure
calculations. In Section III, we will present our computational results, first
demonstrating the importance of the hybridization between the zeronium $s$
orbital and its two neighboring Ni $3d_{3z^{2}-r^{2}}$ orbitals along the $c$
axis. The relatively secondary roles of the Nd $5d$ and O $2p_{\pi}$/$2p_{z}$
states will be illustrated by means of a computational experiment involving
elimination of the Z $s$ states from the vicinity of the Fermi level. We will
then use the electron localization function (ELF) to show the real space
localization of the electride (i.e., zeronium $s$) electron and compare NdNiO2
with the iso-strictural CaCuO2, finding the two of them to approach
qualitatively very different electronic ground states in the limit of an
exact-exchange (i. e., Hartee-Fock) treatment. Further, we will show that the
popular DFT+U method is seriously challenged in the presence of the strong
covalent bonding between the zeronium $s$ orbital and the Ni
$3d_{3z^{2}-r^{2}}$ orbital, by highlighting qualitative inconsistencies
between the results of DFT+U calculations obtained from different DFT codes.
It will be emphasized that, since there is no doubt that these materials are
strongly correlated, in the end it will be model Hamiltonians with methods
like dynamical mean-field theory (DMFT) or exact diagonalization or impurity
like approximations that are needed to describe the experimental data. In
Section IV, drawing support from our computational results and from the recent
impurity calculations by Jiang et al.Jiang et al. (2022), we will discuss a
realistic electronic configuration for Ni in the parent IL nickelates, in
which the zeronium states play one of the key roles giving rise to such
strongly contributing configurations as $d^{8,\,S=1}_{x^{2}-y^{2},\,z^{2}}\,s$
and $d^{9}_{z^{2}}\underline{L}_{x^{2}-y^{2}}\,s$, where $s$ denotes an
electron in Z $s$ and $\underline{L}$ a hole in O $2p$. This will be followed
by a short discussion of the possibility for the nickelates to transition into
an excitonic insulating state at low temperatures. Conclusions will be offered
in Section V.

##  II Methods

We consider primitive unit cells of NdNiO2, with the lattice constants
$a=3.91$ Å and $c=3.37$ Å, and of CaCuO2, with the lattice constants $a=3.86$
Å and $c=3.20$ Å. When not specified otherwise, our DFT calculations are
performed using the augmented plane wave all-electron package WIEN2kBlaha et
al. (2020) and choosing the gradient-corrected local density approximation
(GGA)Perdew et al. (1996) for the energy functional to treat the exchange and
correlation effects. We use $RK_{\text{max}}=7.0$ and a $12\times 12\times 14$
$\Gamma$-centered $k$-point grid for bandstructure calculations but increase
the $RK_{\text{max}}$ value to 9.0 for calculating ELF. The atomic muffin-tin
radii $R$ are set as $R$(Nd)=2.5, $R$(Ni)=1.57, $R$(O)=2.12, and $R$(Z)=1.61
in bohr [$R$(Nd)=1.32, $R$(Ni)=0.83, $R$(O)=1.12, and $R$(Z)=0.85 in Å]. The
three Nd $4f$ electrons are treated as core electrons. For the two-formula-
unit supercell with the $G$-type ordering of Ni spins considered in Section
III C, we use a $10\times 10\times 10$ $\Gamma$-centered $k$-point grid and
keep the same values for $RK_{\text{max}}=7.0$ and for the muffin-tin radii.
We find that it is important to use the tetrahedron rather than smearing
method for integrating over the Brillouin zoneSantos and Marzari (2022), since
otherwise the Fermi level can be determined incorrectly resulting in a loss of
the hole Fermi pockets, which would obviously violate the charge neutrality
principle.

In addition to WIEN2k, two pseudo-potential codes, VASPKresse and Furthmüller
(1996) and Quantum EspressoGiannozzi et al. (2009, 2017), are also used in
order to calculate the ELF of NdNiO2 and CaCuO2 within the hybrid functional
method and/or the Fermi surface of NdNiO2 within the DFT+U method. In these
calculations, the energy cut-offs are set to ENCUT=320 eV and ecutwfc=60 Ry in
VASP and Quantum Espresso, respectively. In the case of hybrid functional
calculations on the primitive unit cells of NdNiO2 and CaCuO2, the size of the
$k$-point grid is reduced down to $6\times 6\times 6$.

The Wannier functions based analysis of the bandstructure of NdNiO2 is
performed by using the maximally localized Wannier functions (MLWF)
methodMarzari and Vanderbilt (1997) as implemented in the WANNIER90 codePizzi
et al. (2020).

##  III Results

###  III.1 Zeronium $s$ orbital without magnetism and correlations

In Figs. 1 (b) and (c) we present a typical result of a non-spin-polarized GGA
calculation on NdNiO2, in a form of atomic orbital projected band dispersion
diagrams. This should give us information about the directional dependence of
the dispersion of the various atomic like states, which is the first step in
the evolution of a model Hamiltonian approach to the problem. As already
mentioned, two electron pocket bands cross the Fermi level beside the cuprate-
like $3d_{x^{2}-y^{2}}$ band, one with a strong Nd $5d_{3z^{2}-r^{2}}$
character at $\Gamma$ and one with mixed Nd $5d_{xy}$ and Ni
$3d_{3z^{2}-r^{2}}$ characters at $A$. Most interesting, though, is the fact
that the Ni $3d_{3z^{2}-r^{2}}$ states are found to behave in a rather non-
trivial wayChoi et al. (2020). Thus, while along the $Z-R-A-Z$ $k$-path, which
belongs to the $k_{z}=\pi$ Brillouin zone boundary, the Ni $3d_{3z^{2}-r^{2}}$
band is flat and lies fully below the Fermi level at around $-1$ eV, it then
strongly disperses along the $c$ direction going from $Z$ to $\Gamma$ where it
stops at around $-3$ eV and roughly stays there in the entire $k_{z}=0$ plane
(the $\Gamma-X-M-\Gamma$ sector). Moreover, at all the $k$-vectors with
$k_{z}\neq\pi$ the Ni $3d_{3z^{2}-r^{2}}$ states become scattered over
multiple bands in a wide energy range both below and above the Fermi level,
which signals their partial occupation and ensuing active involvement in low-
energy electronic excitations.

Figure 1: (a) Crystal structure of infinite-layer nickelates $R$NiO2
($R$=rare-earth atom). Different atoms are represented by spheres as follows:
$R$ = large red, Ni = small red, O = blue, O vacancy (zeronium) = large white
at the center of the unit cell. (b) and (c) Band structure of NdNiO2 from a
non-spin-polarized GGA calculation projected onto atomic (b) Nd $5d$ and (c)
Ni $3d$ orbitals. Orbital character in a given state is proportional to the
area of the circle of a respective color. (d) Same NdNiO2 band structure as in
(c) and (b) but projected onto the zeronium orbitals with $s$, $p$, and $d$
symmetries. The yellow rectangle highlights the bonding combination of the Ni
$3d_{3z^{2}-r^{3}}$ and Z $s$ orbitals at $k$-vectors with $k_{z}=0$. The zero
of energy is at the Fermi energy. (e) Electron localization function (ELF) of
NdNiO2 in the $(110)$ plane crossing Nd, Ni, and oxygen vacancy sites as shown
in (a).

The near flat band behaviour in the variation of the in-plane $k$ for
$k_{z}=0$ or $k_{z}=\pi$ suggests that the band has a strong one-dimensional
(1D) character. Remarkably, it becomes clear from inspecting the zeronium
local orbital projections in Fig. 1 (d) that such 1D like behaviour is a
result of the strong mixing of the Ni $3d_{3z^{2}-r^{2}}$ states with the
interstitial states in the $c$ axis direction. Indeed, one finds that, on one
hand, the zeronium $s$ orbital has a large contribution to the occupied band
at around $-3$ eV, which is highlighted by a yellow rectangle in the figure,
at $k$-points with $k_{z}=0$ (the $\Gamma-X-M-\Gamma$ sector). This is also
the band that has a strong Ni $3d_{3z^{2}-r^{2}}$ character in Fig. 1 (c). On
the other hand, at the $k$-points with $k_{z}=\pi$ (the $Z-R-A-Z$ sector), the
Z $s$ orbital character is mostly above the Fermi level, while the Ni
$3d_{3z^{2}-r^{2}}$ orbital character is fully concentrated in the occupied
band at around $-1$ eV. This is consistent with the fact that, by symmetry, an
$s$ orbital and a $3z^{2}-r^{2}$ orbital that alternate along the $z$
direction have the strongest coupling when $k_{z}=0$ and zero coupling when
$k_{z}=\pi$. Hence, the band at $-3$ eV enclosed in the yellow rectangle
corresponds to the bonding combination of the Ni $3d_{3z^{2}-r^{2}}$ and Z $s$
orbitals, whereas their anti-bonding combination is pushed above the Fermi
level where it hybridizes with empty Nd $5d$ orbitals. This also explains why
in the $Z-R-A-Z$ sector the Ni $3d_{3z^{2}-r^{2}}$ band at $-1$ eV appears
flat and well disentangled from other bands, because this is where $k_{z}=\pi$
and its coupling with the Z $s$ states is canceled out. Moreover, because of
the large radial extend of the Ni $3d_{3z^{2}-r^{2}}$ orbital in the $z$
directions and small extent in the plane, the hybridization in the in-plane
directions will be relatively small. Note that coupling of Z $s$ to any of the
other four Ni $3d$ orbitals is symmetry forbidden at any $k$-vector. As shown
in Supplementary Materialsm , very similar band structures with the same
zeronium features are obtained for PrNiO2Osada et al. (2020b) and LaNiO2Osada
et al. (2021), two other parent IL nickelates demonstrating superconductivity
upon doping.

Figure 2: (a) GGA (solid grey) and TB model (dashed blue) band dispersions of NdNiO2. On the right-hand side also shown is the energy-level diagram with the on-site energies of the 17 atomic and electride orbitals entering our TB model. The Fermi energy is at zero. (b) The zeronium $s$ Wannier function. Table 1:  The on-site energies in eV of the 17 orbitals entering our MLWF based TB model. The Fermi energy is at zero.  |  | Z $s$ | 1.84 |  |
---|---|---|---|---|---
Ni |  |  | Nd
$3d_{xy}$ | -1.74 |  |  | $5d_{xy}$ | 2.15
$3d_{xz,yz}$ | -1.61 |  |  | $5d_{xz,yz}$ | 3.34
$3d_{x^{2}-y^{2}}$ | -1.01 |  |  | $5d_{x^{2}-y^{2}}$ | 2.93
$3d_{3z^{2}-r^{2}}$ | -1.08 |  |  | $5d_{3z^{2}-r^{2}}$ | 2.33
|  | O |  |
|  | $2p_{\sigma}$ | -4.96 |  |
|  | $2p_{\pi}$ | -3.74 |  |
|  | $2p_{z}$ | -3.64 |  |
Table 2:  The nearest-neighbor electron hopping integrals in eV in our MLWF based TB model. Only those with the absolute values larger than 0.3 eV are listed. The $\sigma$ and $\pi$ designations of the O $2p$ orbitals are done considering their position relative to a neighboring Ni atom. Note that there is no Nd $5d$—Ni $3d$ hybridization in this list, which indicates that the dominant interaction path between these two orbitals is through Z $s$. Z $s$—Ni $3d_{3z^{2}-r^{2}}$ | -1.14 |  | Ni $3d_{x^{2}-y^{2}}$—O $2p_{\sigma}$ | 1.32
---|---|---|---|---
Z $s$—Nd $5d_{xy}$ | -1.04 |  | Ni $3d_{3z^{2}-r^{2}}$—Z $s$ | -1.14
Z $s$—Nd $5d_{3z^{2}-r^{2}}$ | 0.66 |  | Ni $3d_{xz}$—O $2p_{z}$ | 0.78
Z $s$—O $2p_{\sigma}$ | 0.66 |  | Ni $3d_{xy}$—O $2p_{\pi}$ | 0.71
Z $s$—O $2p_{z}$ | 0.65 |  | Ni $3d_{3z^{2}-r^{2}}$—O $2p_{\sigma}$ | 0.67
Nd $5d_{xz,yz}$—O $2p_{\pi}$ | 1.22 |  | O $2p_{\sigma}$—O $2p_{\sigma}$ | 0.40
Nd $5d_{xy}$—Z $s$ | -1.04 |  | O $2p_{\pi}$—O $2p_{\pi}$ | -0.36
Nd $5d_{xz,yz}$—O $2p_{z}$ | 0.96 |  |  |
Nd $5d_{x^{2}-y^{2}}$—O $2p_{z}$ | 0.88 |  |  |
Nd $5d_{xy}$—O $2p_{\sigma}$ | 0.56 |  |  |
Nd $5d_{3z^{2}-r^{2}}$—O $2p_{\pi}$ | 0.53 |  |  |
Nd $5d_{xz,yz}$—O $2p_{\sigma}$ | 0.50 |  |  |
Nd $5d_{x^{2}-y^{2}}$—O $2p_{\pi}$ | 0.41 |  |  |
Nd $5d_{3z^{2}-r^{2}}$—O $2p_{z}$ | -0.32 |  |  |

We also perform an MLWF analysis of the NdNiO2 bandstructure using the five Nd
$5d$ orbitals, the five Ni $3d$ orbitals, the three $2p$ orbitals of each one
of the two oxygens in the primitive unit cell, and the Z $s$ orbital as a
basis of the corresponding effective tight-binding (TB) model. We note that
several other studies have been reported that also consider an empty
$s$-symmetry interstitial orbital centered at the O vacancy sitesGu et al.
(2020); Nomura et al. (2019); Xie et al. (2022); Ding et al. (2023) but here
we focus on discussing the important ways it plays a role in the physical
interpretation of the band structure of NdNiO2 and especially of the unusual
dispersion of the Ni $3d_{3z^{2}-r^{2}}$ band. The quality of the agreement
between our TB model’s and the GGA energy bands is very good, as one can see
in Fig. 2 (a). Figure 2 (b) shows the Z $s$ Wannier function in real space,
and Tables 1 and 2 provide the on-site energies and the largest nearest-
neighbor hopping integrals in the model. These parameter values can be used in
model Hamiltonians by transforming them into a configuration interactions like
picture and adding in the on-site Coulomb and exchange interactions as done in
the recent study by Jiang et al.Jiang et al. (2022). As one can see in Tables
1 and 2, the on-site energies of the Ni $3d_{3z^{2}-r^{2}}$ and Z $s$ orbitals
are $-1.08$ eV and $1.84$ eV, respectively, and the nearest-neighbor electron
hopping integral between them is -1.14 eV, which is almost as large as the
$dp\sigma$ hopping of 1.32 eV between the Ni $3d_{x^{2}-y^{2}}$ and O
$2p_{\sigma}$ orbitals. Interesting also is that the on-site energies of the
Nd $5d$ orbitals are all higher than that of the Z $s$ orbital, as we also
demonstrate in the energy-level diagram of Fig. 2 (a). The hybridization of
the Z $s$ orbital with the Nd $5d_{xy}$ states at higher energies pushes the
bonding part down and it is this that causes the highly mixed band crossing
the Fermi energy and forming the electron pocket at $A$.

It is important to discuss the large nearest-neighbor Z $s$—Ni
$3d_{3z^{2}-r^{2}}$ hopping in the light of the recent hydrogen
experimentsDing et al. (2023) that indicate that superconductivity in the Sr
substituted NdNiO2 is only present if there is a substantial amount of H in
the structure. As other theoretical studies have shownSi et al. (2020), that
hydrogen assumes a close to H- ionic state and favours to be in the O vacancy
position. If this is correct, then we note that the presence of H- would block
the Z $s$—Ni $3d_{3z^{2}-r^{2}}$ hybridization since the vacancy position is
already occupied by a negative charge. This would put the Ni electronic
structure close to that of Cu in the cuprates. One could also speculate that
the inclusion of some F- in the vacancy position could do the same and perhaps
more efficiently. This is worth a try but one should also realize that the
relatively large size of F- could cause a substantial $c$ axis expansion which
may indeed act to destabilize the structure.

Figure 3: The result of a computational experiment where the zeronium $s$
orbital is forcefully depopulated yielding significant changes in the NdNiO2
electronic structure around the Fermi level: (a) and (b) Atomic orbital
projections of the altered band structure. (c) A comparison between the Ni
$3d_{3z^{2}-r^{2}}$ projected densities of states (PDOS) of NdNiO2 before
(“GGA“) and after (“GGA, no Z $s$“) depopulating the Z $s$ orbital. The Fermi
energy is at zero.

###  III.2 Computational experiment

In contrast with the emphasis on the Nd $5d$ states that prevails in the
literature, our results from Table 2 indicate that their direct hybridization
with the Ni $3d_{3z^{2}-r^{2}}$ and Ni $3d_{x^{2}-y^{2}}$ orbitals is small,
but it is rather the Z $s$ orbital that mediates the involvement of these
empty states with the Ni states.

To better illustrate this important observation, we consider here results from
a computational experiment where the Z $s$ orbital is intentionally removed
from playing any role in the low-energy electronic properties. This can be
achieved via a charge self-consistent calculation where electrons are made to
pay a high energy cost for occupying this orbital. The resulting bandstructure
shown in Figs. 3 (a) and (b) displays significant differences with the
original one in Figs. 1 (b) and (c). Most notably, the Ni $3d_{3z^{2}-r^{2}}$
band becomes essentially flat throughout the entire Brillouin zone. Its
dramatic transformation is emphasized in Fig. 3 (c) showing the Ni
$3d_{3z^{2}-r^{2}}$ projected densities of states before and after the removal
of the Z $s$ orbital. Furthermore, instead of two electron pocket bands, we
now have only one at $\Gamma$, of mainly Nd $5d_{3z^{2}-r^{2}}$ character. It
crosses and hybridizes with the Ni $3d_{3z^{2}-r^{2}}$ band, but this
hybridization is much weaker compared to that between the Ni
$3d_{3z^{2}-r^{2}}$ and Z $s$ bands, in accordance with Table 2. In essence,
the Ni $3d_{3z^{2}-r^{2}}$ orbital becomes much more atomic-like here, while
it clearly cannot be treated as such in reality because of its strong covalent
bonding with the zeronium $s$ orbital.

###  III.3 ELF (electron localization function)

We find a total of 0.36 electride electrons inside a sphere of radius 0.85 Å
around the vacancy site, out of which 0.24 occupy the local $s$ orbital. This
is large enough to support the electride scenario, but (as it is often the
case in studies on electrides Dale and Johnson (2018)) not sufficient to
produce a significantly visible maximum in the real-space electron density
distribution, due to the strong background density of other electrons. There
is, however, a better way of measuring electron localization in an electride
material which is by using the electron localization function (ELF) Becke and
Edgecombe (1990); Savin et al. (1992). ELF has been established as one of the
best descriptors for identifying electride systems Dale and Johnson (2018).
The definition of ELF is tailored such that its value can vary between 0 and
1, with 1 corresponding to maximal electron localization. As one can see in
Fig. 1 (e), the computed ELF of NdNiO2 has a clear maximum at the oxygen
vacancy (i. e., zeronium) site. A maximum of a similar strength was taken as a
sign of an electride-like nature of the prototypical electride compound
[Ca24Al28O64]4+($4e^{-}$) Matsuishi et al. (2003).

There is also a clear minimum in the ELF of NdNiO2 at a distance of about 1 Å
going from the Ni site towards the Z site. This further indicates that the
electron density in that region cannot be derived solely from the Ni $3d$ wave
function but rather from a combination of the Ni $3d$ and Z $s$ wave
functions, because a pure atomic $3d$ wave function does not have a radial
node. An alternative interpretation would be to say that this minimum is a
result of an effective mixing of the Ni $3d$ states with the $4d$ states, a
point of view adopted, for instance, by Katukuri et al. in Ref. Katukuri et
al., 2020. The $4d$ wave function of an isolated Ni+ ion has its node indeed
close to 1 Å from the Ni nucleus. This observation is of great importance in
the development of model Hamiltonians to describe these correlated electron
systems and, as we will argue below, might also help us understand better the
results of DFT+U calculations.

Figure 4: Electron localization functions of NdNiO2 and CaCuO2 in the (110)
crystal plane, calculated using the GGA, PBE0, and Hartree-Fock (HF) methods
and the pseudo-potential code Quantum Espresso.

It is interesting to compare the ELF of NdNiO2 with that of the iso-structural
cuprate system CaCuO2. The two top panels of Fig. 4 show their ELFs obtained
from ferromagnetically spin-polarized GGA calculations using the pseudo-
potential code Quantum Espresso. We will explain the need to switch to both a
new code and a new magnetic configuration below, but for now let us note that
this does not change the ELFs of the considered oxides in the region around
the zeronium, as one can see by comparing with Fig. 1 (e). An important result
is that, opposite to NdNiO2, the ELF of CaCuO2 has a minimum rather than a
maximum at the oxygen vacancy site. This indicates that the electride-like
electronic behavior of the IL nickelates is a unique feature that makes them
distinctly different from the cuprates.

Moreover, this difference becomes more and more pronounced as one switches to
using a hybrid functional with some amount of an exact [i. e., Hartee-Fock
(HF)] exchange in it, a strategy known to help improve the description of
electronic correlations and electron localization, particularly, in the case
of electride electrons (see Ref. Dale and Johnson, 2018, page 9373). We
consider the PBE0 hybrid functional method Perdew et al. (1996) which is based
on the following simple expression for the exchange (”x“) and correlation
(”c“) energy functional:

$E^{\text{PBE0}}_{\text{xc}}=\frac{1}{4}E^{\text{HF}}_{\text{x}}+\frac{3}{4}E^{\text{GGA}}_{\text{x}}+E^{\text{GGA}}_{\text{c}},$

in which a quarter of the exchange term of the original GGA functional is
proportionally replaced by an HF contribution. As one can see in the two
middle panels of Fig. 4, introducing a quarter of an exact exchange results in
an enhanced localization of the electride electron in NdNiO2 but does little
in the case of CaCuO2, which reinforces our previously stated conclusions.
Finally, performing a pure HF calculation (the bottom panels of Fig. 4) is
found to drive the electride electron of NdNiO2 towards full localization.

Before finishing this section, let us explain our choices regarding the
magnetic configuration and the DFT code used to produce the results in Fig. 4.
First, it becomes very challenging to stabilize a non-magnetic solution as
soon as one adds some amount of an exact exchange into the energy functional,
and therefore all the calculations presented in Fig. 4 are performed keeping
consistently the same ferromagnetically ordered Ni spin configuration, which
is the only possible configuration one can have in a primitive unit cell.
Second, in the case of hybrid functional calculations, computational
efficiency becomes a serious issue which can be better dealt with in a pseudo-
potential code like Quantum Espresso. For the same reason, we did not choose
to consider here some kind of an antiferromagnetic order, as this would
require using a computationally more demanding multiple formula unit
supercell.

###  III.4 Can one use the DFT+U method to unveil the Fermi surface of the IL
nickelates?

Figure 5: The band structure of NdNiO2 in the $G$-type antiferromagnetically
ordered state obtained from WIEN2k with Ni atomic and electride orbital
projections as a function of $U$ in the DFT+U method. The Fermi energy is at
zero.

We now explore further how magnetism and electronic correlations might affect
the behavior of the electride electrons in the parent IL nickelates. Our
hybrid functional calculations already indicate that correlations can strongly
enhance their electride-like properties, but a detailed study would require us
to use a more computationally flexible method. Therefore, we will consider
here the popular DFT+U approximation Anisimov et al. (1991), in which local
electronic correlation effects are treated in a static mean-field fashion.
This is a method that has extensively been used in computational studies of
various TM compounds with strong on-site electronic correlations, including
the IL nickelatesLiu et al. (2020b); Been et al. (2021); Hepting et al.
(2020). Its efficiency, however, comes at a price. First, with DFT+U one is
bound to deal with a magnetically long-range ordered state, which is different
from the experimentally observed ground state of the parent IL nickelates
where localized magnetic moments, although indeed present, have no long-range
correlationsFowlie et al. (2022). Second, in DFT+U the electron interaction
parameters appear as adjustable parameters, introducing additional
uncertainty. There are also other, more subtle yet potentially very important,
differences in how a DFT+U calculation can be set up, which will be discussed
later.

Figure 5 shows the evolution of the DFT+U band structure of NdNiO2 as a
function of $U$, the on-site Coulomb repulsion between Ni $3d$ electrons. The
Hund’s rule exchange coupling, $J_{\text{H}}$, is set to zero for simplicity.
We note, however, that this does not mean that the net $J_{\text{H}}$ is zero,
because in DFT full exchange is included already at the GGA level but with a
Hamiltonian involving only $S_{z}\cdot S_{z}$ terms. In other words, the
Hund’s rule exchange is treated as Ising-like since DFT methods with $\vec{k}$
as a good quantum number have only $S_{z}$ and not the local $S$ as a good
quantum number. This is very important since the singlet energy in
$J\vec{S}\cdot\vec{S}$ is $-3/4J$ for spin 1/2 and the triplet energy is
$+1/4J$, while for the Ising-like interaction the spin parallel and
antiparallel energies are $\mp 1/4J$.

We order the Ni spin magnetic moments antiferromagnetically with a
$Q=(\pi,\pi,\pi)$ (i. e., $G$-type) ordering vector, which causes doubling of
the unit cell and, consequently, of the number of bands compared with Figs. 1
(b)-(d). As mentioned earlier, the need to have a long-range magnetic order is
an unavoidable component of DFT+U, reflecting the static nature of this
approach. However, as for example discussed by Trimarchi and co-
workersTrimarchi et al. (2018), the influence of the kind of magnetic order
usually is very small especially if there are resulting strongly localized
spins in the ordered state.

Several comments are to be made regarding the antiferromagnetic GGA results
presented in the left-most column of Fig. 5. First, like in the cuprates, spin
polarization occurs involving mainly the Ni $3d_{x^{2}-y^{2}}$ bands. Second,
this has almost no effect on the Z $s$ bands, including the occupied ones at
-3 eV, which means that the electride-like nature of the compound stays
intact. The small electron pockets at $\Gamma$ and $A$ (now folded on top of
each other) are also preserved. But now, because of the antiferromagnetic
splitting of the Ni $3d_{x^{2}-y^{2}}$ bands, the compensating positive charge
is located in the similarly small hole pockets at half way between $M$ and
$\Gamma$ (and, equivalently, $A$ and $Z$) and has a spin majority Ni
$3d_{x^{2}-y^{2}}$ orbital character. This suggests that, within the rigid
band shift approximation, in the chemically hole-doped superconducting
nickelates like SrxNd1-xNiO2, doped holes should bind with the stoichiometric
holes into local singlets with the $x^{2}-y^{2}$ symmetry, a situation similar
to what happens in the cupratesJiang et al. (2020, 2022). Note that the
occupied $x^{2}-y^{2}$ states are strongly hybridized with the same symmetry
molecular combination of O $2p$ orbitals as in the Zhang-Rice picture, which
is obvious from the $x^{2}-y^{2}$ symmetry states forming another band at
between 5 and 6 eV of mainly O $2p$ molecular orbital character.

This picture, however, changes into a qualitatively very different one as the
value of $U$ starts to increase. First, the Ni $3d_{3z^{2}-r^{2}}$ electrons
become spin-polarized too and get coupled ferromagnetically with the Ni
$3d_{x^{2}-y^{2}}$ electrons. Second, only one electron pocket survives. Also,
instead of the spin majority Ni $3d_{x^{2}-y^{2}}$ hole pocket between $M$ and
$\Gamma$, we have now for $U>4$ eV a shallow hole pocket at $X$ (and,
equivalently, $R$) with a spin minority Ni $3d_{3z^{2}-r^{2}}$ orbital
character. In this situation, the chemically doped holes and the
stoichiometric holes would be found forming local triplets rather than
singlets, which could possibly lead to a superconducting state with a
completely different symmetry and perhaps even of a fundamentally different
nature. Interestingly, the Z $s$ orbital will play an important role in this
scenario as at large $U$ it acquires a substantial weight in the hole pocket
band at $X$ resulting from its strong hybridization with the Ni
$3d_{3z^{2}-r^{2}}$ orbital.

Figure 6: Fermi surface of NdNiO2 calculated in DFT+U using different DFT
codes for a range of on-site Hubbard interaction parameters $U=$ 0, 2, 4, 6,
and 8 eV. The Hund’s exchange coupling is set to zero. The Fermi surface color
represents the Fermi velocity.

We have noticed that some of the published DFT+U based studiesLiu et al.
(2020b); Been et al. (2021); Hepting et al. (2020) of the IL nickelates report
a qualitatively different $U$-dependence of the electronic structure, finding
in particular that there is a charge gap opening at $U=8$ eV, whereas in our
DFT+U calculations NdNiO2 remains metallic at all $U$. Suspecting that the
difference in the choice of a DFT code might be the issue, we repeat our
calculations using also the pseudo-potential codes VASP and Quantum Espresso.
The obtained results as a function of $U$ and a DFT code are summarized in
Fig. 6 in a form of Fermi surfaces. Indeed, the three considered DFT codes are
found to give qualitatively different $U$-dependence for the Fermi surface of
NdNiO2 and, as such, for its electronic structure overall.

We stress that within each code we have intentionally chosen to use nominally
the same rotationally invariant formulation of the DFT+U method by
Liechtenstein et al. Liechtenstein et al. (1995) (which is often referred to
as a ”self-interaction correction“ method). Therefore, the main source for the
discrepancy has to be the different ways that the codes use to define local Ni
$3d$ orbitals for calculating occupation matrices in DFT+U expressions. By
default, atomic and Lowdin orthogonalized atomic wave functions are used in QE
and VASP, respectively, whereas WIEN2k considers atomic-like wave functions
that go to zero outside muffin tin spheres. These differences have minor
effect when the correlated orbitals in question are well-localized (see, for
instance, corresponding results for CaCuO2 in Supplementary Materialsm ), but
become critically important in the presence of strong hybridization with
uncorrelated subset. In the case of the IL nickelates, such is primarily the
hybridization between the Ni $3d_{3z^{2}-r^{2}}$ and Z $s$ orbitals. One way
to see this is by noticing that it is the hole pocket band of exactly that
mixed Ni $3d_{3z^{2}-r^{2}}$ and Z $s$ orbital character that gets pinned to
the Fermi level at $U>4$ eV in WIEN2k and whose absence results in the charge
gap opening in VASP. This strong hybridization is all but absent in the
cuprates, while both have relatively strong hybridization of the O $2p$ and
the TM $3d_{x^{2}-y^{2}}$ orbitals although weaker for the nickelates because
of the larger charge-transfer energy.

It is also worth mentioning that in WIEN2k projections onto local states of a
certain symmetry are done without considering the radial dependence, i. e.,
without differentiating between principal quantum numbers, while in VASP and
Quantum Espresso this is not the case, which, in the light of our earlier
argument about the possible mixing of Ni $3d$ and $4d$ states, can be another
important factor contributing to the discrepancy.

Summarizing this section, Fig. 6 provides yet another demonstration of the
important role of the Z $s$ orbital coupled with the Ni $3d_{3z^{2}-r^{2}}$
orbital in a strong covalent bond. Such unique property of the IL nickelates
combined with the discussed limitations of the DFT+U method causes an
uncertainty in the nature of the symmetry of the hole pocket and the states
that are closest to the Fermi level.

##  IV Discussion

###  IV.1 Ni electronic configuration: $d^{9}$, $d^{8}s$, or
$d^{9}s\underline{L}$?

Our presented DFT based analysis of the parent IL nickelates points to a
microscopic picture where the oxygen vacancy centered $s$ orbital hybridizes
very strongly with the Ni $3d_{3z^{2}-r^{2}}$ orbital, pulling a large portion
of its atomic electron density into the interstitial region. As a result,
these two orbitals form a bonding and an anti-boning states, with the bonding
state having larger Ni $3d$ than Z $s$ character and being totally occupied
and with the anti-bonding state having more Z $s$ character and being totally
empty. This leads to a substantial hole concentration in the Ni
$3d_{3z^{2}-r^{2}}$ band driving Ni+ towards Ni2+. Noteworthily, a series of
recent DFT+DMFT studies have arrived at a similar conclusion about the Ni
ionisation state Lechermann (2020a, b); Leonov et al. (2020); Xie et al.
(2022); Petocchi et al. (2020); Wang et al. (2020); Pickett (2021); Werner and
Hoshino (2020); Ortiz et al. (2021); Karp et al. (2021). Such deviation from
the nominal valency is reminiscent of that encountered in negative charge-
transfer compounds Zaanen et al. (1985); Saw , but here the role of ligand
states pulling in holes from the transition metal states is delegated to the
zeronium s states pulling in electrons. Many materials with rich physics
belong to this class, including the high-$T_{\text{c}}$ superconductors
bismuth oxides Sleight et al. (1975); Foyevtsova et al. (2015), the battery
material LiNiO2 with a proposed electronic high-entropy state Goodenough et
al. (1958); Foyevtsova et al. (2019), and also the previously mentioned
precursor compounds $R$NiO3 Medarde (1997). In $R$NiO3, for instance, a
negative value of the charge-transfer energy between the Ni $3d$ and O $2p$
states is known to result in a $d^{8}\underline{L}$ (Ni2+) rather than the
formal $d^{7}$ (Ni3+) configuration for Ni ions, where $\underline{L}$ stands
for an O $2p$ hole coupled strongly in a covalent bond with a Ni $3d$ hole. By
analogy, we can say that in the IL nickelates a state like $d^{8}s$, with $s$
denoting an electron in the Z $s$ orbital bound in a strong covalent bond with
neighboring Ni $3d_{3z^{2}-r^{2}}$ orbitals, should have a substantial weight
in the ground state Ni configuration.

It is important to keep in mind that in order to fully reveal the true ground
state configuration of Ni, including the local spin and multiplet structure,
one needs to go beyond DFT and use advanced many-body techniques. In this
regard, recent configuration interaction based impurity calculations by Jiang
et al.Jiang et al. (2022) show that the ground state is a mixture of similarly
strong contributions from the $d^{9}_{x^{2}-y^{2}}$,
$d^{8,\,S=1}_{x^{2}-y^{2},\,z^{2}}\,s$,
$d^{9}_{z^{2}}\,\underline{L}_{x^{2}-y^{2}}\,s$, and
$d^{10}\underline{L}_{x^{2}-y^{2}}$ configurations, all having a net spin of
$1/2$. This is very different from the cuprates where the ground state
electronic structure is dominated by $d^{9}_{x^{2}-y^{2}}$ and
$d^{10}\underline{L}_{x^{2}-y^{2}}$.

Our findings complement the ideas from Ref. Ding et al., 2023 about the
possibility of the interstitial charge suppressing superconductivity in
hydrogen-poor samples. We emphasize, however, that it is the nearest-neighbor
Z $s$—Ni $3d_{3z^{2}-r^{2}}$ hybridization that is primarily responsible for
turning the stoichiometric nickelates into effectively three-dimensional
systems and thus driving them away from the two-dimensional regime that is
known to favor superconductivity in cuprates. This is supported by the
calculations for the 20 percent hole doped LaNiO2 system presented in
Supplementary Materialsm , in which the electride-like nature and the
associated strong out-of-plane electronic coupling are found to be well
preserved.

Important implications regarding magnetism are also in place. The on-site
Hund’s rule coupling dictates that the Ni $3d_{3z^{2}-r^{2}}$ holes have the
same spin as the $d_{x^{2}-y^{2}}$ holes, again pointing to a strong admixture
of the $d^{8}$ ($S=1$) state in the net ground state configuration of Ni. This
is indeed what our DFT+U calculations in Fig. 5 suggest and what the mentioned
DFT+DMFT studies report and is in line with the conclusions of the impurity
calculations including all multiplets by Jiang et al.Jiang et al. (2022) We
can also think of this as a large part of the Ni+ $3d_{3z^{2}-r^{2}}$ electron
finding itself in the zeronium site with spin parallel to the local
$d_{x^{2}-y^{2}}$ remaining electron. This would result in a very strong anti-
ferromagnetic coupling between the Ni spins in the one-dimensional chains
along the $c$ direction, whose consequences require a detailed theoretical
exploration, especially, in connection with the low-energy excitations seen in
the recent resonant inelastic x-ray scattering (RIXS) experiments Rossi et al.
(2021b); Lu et al. (2021), but this kind of a study is beyond the scope of the
present work.

###  IV.2 Possible excitonic insulating state at low temperatures

Keeping in mind that, strictly speaking, the standard DFT+U approach is not
applicable to the IL nickelates, let us nevertheless note that all the three
DFT codes are consistent in that they find a strongly dispersive electron
pocket band and a much flatter hole pocket band at physically reasonable $U$
values around 6 eV. Furthermore, the number of charge carriers occupying these
electron and hole pockets is extremely small. In the specific case of the
WIEN2k solution with $U=6$ eV (Fig. 5), we find that this number corresponds
to a charge carrier concentration of only $0.031$ electrons and holes per
formula unit. This is a situation where we have a diluted nearly-free-
electron-like gas of electrons interacting with an equally diluted pool of
holes that are however much more localized and residing primarily on Ni $3d$
orbitals. At low enough temperatures, such a system will have a strong
tendency to form spatially bound excitonic pairs, which, since the number of
electrons and holes has to be equal because of the charge neutrality
condition, can possibly result in an excitonic insulating ground state for the
parent IL nickelates of the kind similar to that observed in some transition
metal dichalcogenidesWilson (1977). This scenario, which is in line with the
Kondo-like physics in the limit of the Nozieres exhaustion principle
considered in Refs. feng Yang and Zhang, 2022 and Shao et al., 2022, would
naturally explain the measured resistivity up-turn. Also, in light of this, it
is not impossible that the low-energy dispersive excitations seen in the
recent RIXS measurements Rossi et al. (2021a); Lu et al. (2021) are of an
excitonic origin rather than being driven by short-range magnetic order
excitations.

##  V Conclusion

In conclusion, the present study elucidates the electride-like nature of the
IL nickelates and reestablishes that in these materials “Ni+ is not Cu2+” but
rather in a state closer to Ni2+ with substantial
$d^{8,\,S=1}_{x^{2}-y^{2},\,z^{2}}\,s$ and
$d^{9}_{z^{2}}\,\underline{L}_{x^{2}-y^{2}}\,s$ contributions, in addition to
the cuprate-like $d^{9}_{x^{2}-y^{2}}$ and
$d^{10}\,\underline{L}_{x^{2}-y^{2}}$ ones. This results from the strong
attractive potential for electrons left behind by the missing O in the NdO
layer leading to a considerable charge density attracted to that region in the
ground state of the undoped material. We label this region of the attractive
potential with a fictitious atom with zero nuclear charge (Z) and find in DFT
that its lowest-energy orbital is an $s$ symmetry orbital.

In the language of tight-binding modeling, such peculiar charge density
redistribution can be described in terms of the strong hybridization of the Ni
$3d_{3z^{2}-r^{2}}$ orbital with the Z $s$ orbital and their resulting
bonding/anti-bonding like splitting that pulls some of the Ni
$3d_{3z^{2}-r^{2}}$ electron density into the vacancy region in the ground
state. Our TB analysis indicates also that there is a strong involvement of
the Nd $5d_{xy}$ and O $2p_{z}$ orbitals, which, however, have only an
indirect effect on the Ni $3d$ electronic structure. We also pointed out that
all the theoretical methods considered so far, like DFT, the impurity
calculations, and DMFT, lead to the conclusion that the $d^{8}$ states
involved in the ground state are in a triplet configuration and the total
configuration has spin 1/2 per formula unit. However, the propagation of this
spin 1/2 composite object is much more complicated than that presented by the
corresponding spin 1/2 composite object in the cuprates, which only involves
the $x^{2}-y^{2}$ holes of Cu $3d$ and a linear combination of O $2p$ orbitals
of the same symmetry. Such distinct difference in the electronic behaviors of
the nickelates and the cuprates is further emphasized in our calculations of
their electron localization functions, which is a well documented measure of
the electride characteristics.

Our study strongly suggests that the Z $s$—Ni $3d_{3z^{2}-r^{2}}$ covalent
bonding is a necessary component to be explicitly considered in both
electronic structure and model calculations that might eventually help us
unravel the nickelates’ elusive Fermi surface as well as understand the
striking sensitivity of their superconducting properties to hydrogen content.
In the latter regard, we argued that if the O vacancy orbital is compensated
by H- as recently suggestedDing et al. (2023), this would tend to remove the
need for the Ni $3d_{3z^{2}-r^{2}}$ hybridization with the Z $s$ state to
develop a negative charge in the O vacancy position and so the system would
tend to be much more similar to the cuprate case. This suggests that perhaps
the resulting superconductivity by the substitution of Nd with divalent Sr and
with the substantial vacancy compensation by H- would indeed be very similar
to that of the cuprates driven mainly by the Ni $3d_{x^{2}-y^{2}}$ orbitals.

Going back to the undoped parent compound though, we also argued that because
of the condition of charge neutrality of the unit cell and the material as a
whole in the ground state the much discussed electron pocket is perfectly
compensated by a hole pocket of equal size but occurring in the bands of
mainly Ni $3d_{x^{2}-y^{2}}$ or $3d_{3z^{2}-r^{2}}$ character. This, in
addition to the missing $d$ electron charge in the Ni+ formal configuration,
means that we must have a finite density of Ni $d^{8}$ states, be it with two
holes in the $x^{2}-y^{2}$ orbital, which would have to form a singlet as in
the Zhang-Rice scenario, or with one hole in the $x^{2}-y^{2}$ and one hole in
the $3z^{2}-r^{2}$ orbital, which locally would form an energetically
favourable triplet state but for charge neutrality reasons there must also be
another electron around, i. e., Z $s$, which we find has spin opposite to the
$d^{8}$ triplet so the net spin is again 1/2.

Concerning the relative likelihoods of the singlet versus triplet scenarios, a
Lifschitz like transition from a Fermi surface hole pocket of the
$x^{2}-y^{2}$ symmetry into that of the $3z^{2}-r^{2}$ symmetry is found to
occur as a function of $U$ for $U$ values larger than 4 eV in the DFT+U method
and using Wien2k. This is, however, where we also observed significant
qualitative inconsistencies between different DFT codes, such as Wien2k, VASP,
and Quantum Espresso, which can be traced down to the complications that the
DFT+U method faces in the presence of a strong hybridization of correlated
orbitals with their uncorrelated surroundings, which in the present case is
exactly the hybridization between the interstitial Z $s$ and Ni
$3d_{3z^{2}-r^{2}}$ orbitals.

We also pointed out that the increased charge density at the oxygen vacancy
position relative to the usually found rather uniform electron density in the
interstitial region can be described in a DFT calculation in several different
ways depending on the basis set of wave functions used. In the end, as density
functional theory dictates it is only the electron density that matters. So,
the extra density in the Z region can also be obtained by introducing for
example $4d$ orbitals in the Ni atomic basis, which have a radial node and
have their maximum amplitude in the vicinity of the O vacancy. It is the
choice of the basis set and the nature of the projections in determining the
state that one wants to apply $U$ to that causes the difference in the Fermi
surface for the various codes when applying $U$. Although beyond the scope of
this work, these observations hopefully will stimulate a more careful look at
what causes this strong code dependent Fermi surface in the DFT+U methods.

From the experimental point of view, it would be of great interest to have a
detailed study of the magnetic field and temperature dependent magnetic
susceptibility done using either x-ray magnetic circular dichroism
spectroscopy or neutron scattering in order to gain insight into the magnetic
properties of the IL nickelates. With regard to the Fermi surface, an angle-
resolved photoemission spectroscopy study on high-quality stoichiometric
crystals would be essential and could in principle at least distinguish
between the $x^{2}-y^{2}$ and $3z^{2}-r^{2}$ character at the Fermi level in
the doped materials.

## Acknowledgments

This research was undertaken thanks, in part, to fund- ing from the Max
Planck-UBC-UTokyo Center for Quantum Materials and the Canada First Research
Excellence Fund, Quantum Materials and Future Technologies Program as well as
from CIFAR and NSERC.

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