\section{Introduction}
In superconducting cuprates, the CuO$_2$ square lattice has been widely considered
a crucial ingredient in producing high $T_c$ due to the profuse number of
examples \cite{wep1988}.
However, some more structurally complex cuprates include
Cu-O chains. An example of one class is the `telephone number compounds'
exemplified by Sr$_{14}$Cu$_{24}$O$_{41}$ with a mean copper valence of
Cu$^{2.25+}$, which is in the same doping regime as the square lattice
cuprates. For issues and references, see {\it e.g.}
Schmidt {\it et al.} \cite{schmidt2007} and Armstrong {\it et al.} \cite{armstr1991}.
The Cu-O chains in these compounds were found
to be plagued (or, depending on viewpoint, blessed) with inhomogeneities,
likely related to one dimensionality and spin dimerization. What is
necessary, versus what is peripheral,
for high T$_c$ superconductivity in cuprates remains unsolved.

Over 30 years ago \cite{abb1988,armstr1991,zhang1990,forget1997,cwchu1994}
Ba$_2$CuO$_{3+\delta}$~ was synthesized in the range of $\delta=0-0.4$
and refined based on the K$_2$NiF$_4$ structure
of the $\delta=1$ phase, {\it i.e.} a Ba$_2$CuO$_{4-\alpha}$ picture.
At that time no superconductivity in the system was observed.
Recently, Li {\it et al.} successfully obtained a bulk superconducting
sample with a high critical temperature $T_c\approx70$ K
at $\delta\approx0.2$ \cite{pnas2019,jsnm2020}. Their synthesis technique
used  a very high pressure of 18 GPa and high temperature of 1000 C$^\circ$.
Through x-ray absorption spectroscopy  and resonant inelastic
x-ray scattering measurements
on a superconducting powder sample \cite{sala2021},
Fumagalli {\it et al.} inferred two nonequivalent Cu ions and an
in-plane nearest neighbor superexchange $J_s\sim 0.12-0.18$ eV.
Theoretical studies \cite{scalapino2019,lieb2020,jinang2021} based on two-band models
proposed unconventional superconductivity of either $d$-wave or $s_\pm$-wave pairing.

Compared with other superconducting cuprates,
this  extraordinarily heavily doped superconductor
(Cu$^{p}$, $p\sim 0.4-0.6$)
also shows notable distinctions  in structure.
It has a substantially stretched Cu-planar O$_{\rm P}$ distance $d_{\rm O_P}$
of 2.00 \AA~(normally around 1.92 \AA),
which may partially due to the large Ba$^{2+}$ ionic radius,
and concomitantly occurring, yet very significantly compressed
Cu-apical O$_{\rm A}$ distance $d_{\rm O_A}$ of 1.86 \AA~
(compare 2.4 \AA~ in La$_2$CuO$_4$).
These values are substantially different from
the isovalent (and isostructural) superconductor Sr$_2$CuO$_{3+\delta}$~ at
up to $T_c$=95 K \cite{hiroi1993,liu2006,liu2014,pnas2020},
where $d_{\rm O_P}$ and $d_{\rm O_A}$ are nearly identical at 1.95 \AA.
Additionally, in Ba$_2$CuO$_{3+\delta}$~  reduction of oxygen concentration leads to
a high fraction of vacancies in the  O$_{\rm P}$ sites,
not the O$_{\rm A}$ sites \cite{abb1988,zhang1990,pnas2019},
whereas the superconducting system Sr$_2$CuO$_{3+\delta}$~ has vacancies in both
O$_{\rm P}$ and O$_{\rm P}$ sites \cite{pnas2020}.

Much of these observations have been confirmed by theoretical calculations \cite{ladder2019}.
Oxygen vacancies break up the CuO$_2$ square lattices,
leading to a 1-dimensional (1D) Cu-O$_{\rm P}$ chain
lattice \cite{abb1988,zhang1990,c.jin2015}.
Stoichiometric Ba$_2$CuO$_3$, {\it i.e.} at the $\delta=0$ phase, would have a
pure 1D Cu-O$_{\rm P}$ chain lattice,
but that stoichiometry has not been well investigated yet.
The chain lattice in the sister compound Sr$_2$CuO$_3$~ has been intensively investigated
as an ideal 1D Heisenberg $S=\frac{1}{2}$ antiferromagnetic (AFM) system
for over 20 years \cite{ami1995,thurber2001,schlappa2012,schlappa2018,serge2020}.

In this paper, we will investigate the $\delta=0$ and $\delta=1/4$ phases through {\it ab initio} calculations.
The latter one approximates the superconducting phase around $\delta\approx0.2$.
For the $\delta=0$ phase of Cu$^{2+}$, compared with those of Sr$_2$CuO$_3$,
the half-filled $d_{X^2-Y^2}$ orbital (notation to be established later)
leads to half the interchain hopping strength
but nearly identical intrachain hopping strength,
suggesting this compound provides a more nearly ideal 1D Heisenberg
$S=\frac{1}{2}$ AFM system, as will be discussed.
Considering AFM order on the chain, the system is
gapped by a 1D spin-Peierls instability at $q=k_F$.
In contrast to the previous suggestion that the substantially compressed $d_{\rm O_A}$
leads to inverted  energies between the $e_g$ orbitals \cite{pnas2019},
our calculations indicate that this system is little affected by the
unusual Cu-O separations.
The energy gap, however, is substantially changed by variation of
Cu-O$_{\rm P}$ separation and O$_{\rm A}$ phonon modes,
likely related to the spin-Peierls instability.

Based on the inferred vacancies in the planar oxygens in Ba$_2$CuO$_{3+\delta}$,
several lattices \cite{lieb2020,ladder2019,ladder2020,brick} with
ordered planar oxygens have been suggested
to approximate the superconducting phase.
Among them, we have investigated structures with brickwall \cite{brick}
and two-type ladder plus chain lattices \cite{ladder2019,ladder2020},
which have Cu ions in up to three types of coordination.
At $\delta=1/4$ doping,
the bilayer structure of a ladder layer plus a chain layer is found to be
the ground state among the three structures.
This structure contains all three Cu coordinations, displaying both
(roughly) half-filled and quarter-filled Cu orbitals.
Copper oxidation states, $e_g$ orbital characteristics, and characteristics
of Cu coordination (four, five, and six) are compared to identify
distinctive aspects of the favored bilayer structure.



\begin{figure}[tbp]
{\resizebox{8cm}{6cm}{\includegraphics{Fig1.eps}}}
\caption{(a) Idealized crystal structure of Ba$_2$CuO$_3$, based on the 1D Cu-O$_{\rm P}$ chain.
The apical oxygens O$_{\rm A}$ and Ba ions sit at nearly identical heights.
The Cu-O$_{\rm P}$ chains are well separated by the insulating BaO$_{\rm A}$ layer, 
by 6.57 \AA~ along the $\hat{c}$-direction.
(b) Spin density plot (GGA) of the  Ba$_2$CuO$_3$~ AFM state, 
showing the strong $pd\sigma$ hybridization.
The isovalue is 0.015 e/\AA$^3$. Each spin character is described 
by the different colors.
}
\label{str}
\end{figure}

\section{Methods and Structures} 
\subsection{Methods of calculation}
We initially carried out first principles calculations using the generalized 
gradient approximation (GGA)\cite{gga} for exchange and correlation,
which is implemented in the all-electron full-potential code {\sc wien2k} \cite{wien2k}.
The results for the AFM phase at $\delta=0$
were confirmed by another all-electron full-potential code {\sc fplo-18} \cite{fplo}.

In {\sc wien2k}, the basis size was determined by $R_{mt}K_{max}=7$
with augmented-plane-wave sphere radii $R_{mt}$ (in $a.u.$): Cu, 1.71;
O, 1.47; and 2.50 for Ba.
For {\sc fplo} the default orbitals were used.
For both codes, the Brillouin zone was sampled by a dense mesh
containing up to 8,000 $k$-points 
to check the energetics and fine band structures (specifically, small
energy gaps) carefully.

Structural optimization for the doped cases was carried out with the {\sc VASP} code \cite{vasp}, 
using an energy cutoff of 600 eV. This code is efficient for the supercells, which 
contain up to 50 atoms. The lattice parameters were optimized, while the
internal structural parameters were relaxed until all forces were
less than 0.01 eV/\AA. 


\subsection{Structures for $\delta=0$ and $\delta=1/4$}
The available structural information is ambiguous, and it seems stoichiometric
and ordered Ba$_2$CuO$_3$~(Cu$^{2+}$) has not been achieved. Powder x-ray diffraction
data on the T$_c$=70 K multi-phase sample was refined within the $I4/mmm$ space
group (as for La$_2$CuO$_4$), obtaining \cite{pnas2019} $a=4.00$ \AA, 
$c=12.94$ \AA, with 40\% vacancies randomly placed on the planar O sites,
obtaining the Ba$_2$CuO$_{3.2}$ composition. The in-plane lattice constant is
unexpectedly large compared  to most layered cuprates, and an O$_{\rm A}$-Cu
separation of 1.86 \AA~ was emphasized as being exceptionally small, viz. 
2.4 \AA~ in La$_2$CuO$_4$. A cluster
expansion study \cite{ladder2020} of compositions around $\delta\sim 0.2$
found that ordered supercells containing Cu-O$_{\rm P}$ chains and ladders are 
favored up to 900 K. Our choices are guided by the chain structure
of isovalent Sr$_2$CuO$_3$ for $\delta$=0, and by the simplest choices of
added in-plane oxygen atoms (described below) for $\delta$=1/4. These choices
allow the assessment of differences due to coordination (there are four-,
five-, and six-fold Cu ions), which seems to be a more useful viewpoint
than crystal field splitting.

{\it $\delta=0$: Ba$_2$CuO$_3$.}
Our first calculations are based on the lattice parameters of $a=b=4.003$ \AA~
and $c=12.942$ \AA~\cite{pnas2019},
recently refined in a superconducting sample at room temperature as noted above.
These values are close to the early reported
values \cite{abb1988,zhang1990,c.jin2015} also on multi-phase samples.
For Ba$_2$CuO$_3$~ with 50\% ordered vacancies in the planar oxygens
(space group: $Immm$, No. 71),
shown in Fig. \ref{str} (a), the Cu and O$_{\rm P}$ ions sit at $2a$ (0,0,0)
and $2d$ (0,$\frac{1}{2}$,0) sites,  respectively.
The Ba and O$_{\rm A}$ ions lie on $4i$ (0,0,$\xi$) sites
with internal parameters $\xi_{\rm Ba}=0.1437$ and $\xi_{\rm O_A}=0.1438$,
at effectively identical heights.
As discussed below, calculations for Ba$_2$CuO$_3$~ show a substantial 
orthorhombic ($a\not=b$) distortion,
reflecting the chain structure.
This difference is consistent with the experimental observations in Sr$_2$CuO$_{3+\delta}$~
that show a rapidly reduced difference in the planar lattice parameters,
as oxygen concentration in Sr$_2$CuO$_{3+\delta}$~ is increased from $\delta=0$ \cite{pnas2020}.

We began our investigations from the $\delta=0$ Cu$^{2+}$ phase that 
serves as the closest stoichiometric parent compound of the Ba$_2$CuO$_{3+\delta}$. We
use the chain structure originally proposed by Armstrong {\it et al.} \cite{armstr1991}. 
In contrast to the above mentioned $I4/mmm$ optimization of the 70 K sample,
other data on Ba$_2$CuO$_{3+\delta}$~  has indicated a difference of in-plane lattice parameters,
less than  1\% (0.04 \AA) in a high temperature, possibly strongly site 
disordered, sample \cite{zhang1990,forget1997,c.jin2015}
but a large 0.3  \AA~ difference in a low temperature sample \cite{abb1988}.
This distortion is consistent with ordering of O vacancies, likely in
a 1D Cu-O$_{\rm P}$ chain (sub)structure.
Due to the differing reports, for the $\delta=0$ phase we fully optimized 
all structural parameters in the orthorhombic space group $Immm$.


\begin{figure}[tbp]
{\resizebox{8cm}{3.5cm}{\includegraphics{Fig2.eps}}}
\caption{Structures of the $z=\frac{c}{2}$ layer, investigated here, 
for the $\delta=1/4$ phase.
For simplicity, only Cu (larger, red spheres) and planar O 
(smaller, green spheres) ions are shown.
 Both the bilayer (a) and brickwall (c) structures have 1D Cu-O chains 
 extending horizontally in the $z=0$ layer,
 whereas the monolayer (b) phase  has the same structure in both the $z=0$ 
 and the $z=\frac{c}{2}$ layers. Each structure is derived from the Ba$_2$CuO$_3$~chain
 structure, by (i) inserting O ions linking the chains, and (ii) moving the planar
 O on alternating chains to complete the ladder rung, or the long edge of the
 brick.
 We used supercells of  $2\times1\times1$ for (a), $4\times1\times1$ for (b), 
 and $2\times2\times1$ for (c).
}
\label{str_sc}
\end{figure}

{\it $\delta=\frac{1}{4}$: Ba$_2$CuO$_{3.25}$}.
To describe models of ordered planar oxygens in the superconducting doped regime,
we investigated three structures of doubled or quadrupled
supercells (as necessary) having two or three Cu sites, 
with the $z=c/2$ layers pictured in
Fig. \ref{str_sc} \cite{ladder2019,ladder2020,brick}.
All structures are derived from the Ba$_2$CuO$_3$~layers of 
Cu-O$_{\rm P}$ chains in which each Cu
has an apical O above and below. The structures differ in where the additional
planar O, necessarily bridging chains, per four Cu sites is distributed.
The additional ions produce five-fold coordinated Cu sites 
(denoted Cu5) along the chains,
octahedral six-fold (Cu6) sites in the bilayer structure,  
and new four-fold sites (Cu4$^{\ast}$) on the rungs in addition to 
the Cu-O$_{\rm P}$ chain site (Cu4), which is the only site for $\delta$=0.
We describe the structures because the coordination has strong effects
on the Cu $e_g$ orbitals.

\noindent 
{\it Bilayer structure.} 
Figure \ref{str_sc}(a) illustrates the oxygen
ladder ordering, which consists of chains bridged by O-Cu-O units.
In this bilayer case the $z=\frac{c}{2}$ and  $z=0$ layers consist
of the Cu-O ladders and Ba$_2$CuO$_3$-type chain lattices, respectively. 
In this structure two bridging O ions are inserted in alternating layers.
The $z=\frac{c}{2}$ layers can be viewed as connected ladders. 
There are one six-fold coordinated Cu ion (Cu6) and three four-fold 
Cu ions, one along each
chain (Cu4) and one along the rungs (Cu4$^{\ast}$). 


\noindent 
{\it Monolayer structure.} 
In this structure the two layers are identical (hence {\it monolayer}),
but displaced body-centering operation. The additional O ions (with
respect to Ba$_2$CuO$_3$) are shown in Fig.~\ref{str_sc}(b) to lead to a chain and a ladder. 
There are two 
five-fold coordinated Cu5 site, and two distinct four-fold sites, Cu4
along the chains and Cu4$^{\ast}$ forming the rung of the ladder.

\noindent 
{\it Brickwall structure.} 
This structure \cite{brick} consists of one layer of
Cu-O chains (Cu4 sites), two per primitive supercell. The other
layer has a brickwall ordering of oxygen vacancies,
as illustrated in Fig. \ref{str_sc}(c). 
Planar oxygen vacancies occur every two squares along one in-plane direction,
being staggered by ($a,a,0)$ in the other direction to give the brickwall
motif. Each copper site in this layer is bonded to two apical O ions and
three planar O ions, thus being five-fold coordinated Cu5 sites.


\section{Results for parent $\delta=0$ phase}
Structural differences between nonmagnetic (NM) and AFM states 
were on the order of 10$^{-2}$\AA.
The optimized NM lattice parameters are $a=4.144$, 
$b=3.808$, $c=13.142$ (in units of \AA),
with the internal parameters of $\xi_{\rm Ba}=0.1459$, 
$\xi_{\rm O_A}=0.1481$.
The corresponding Cu-O$_{\rm P}$ and Cu-O$_{\rm A}$  distances 
of 2.07 \AA~ and 1.95 \AA~respectively, a difference of $\pm$6\%.
The lattice parameters are consistent with the early experimental 
low temperature values \cite{abb1988}.
As expected, the Cu-O$_{\rm P}$ chains lie along the 
longer $\hat{a}$-direction.
Compared with the experimental value of the superconducting sample \cite{pnas2019}, 
the Cu-O distances are larger by 0.07-0.08 \AA, but the 
interchain distance is smaller by 0.2 \AA~ in the layer, very significant
differences.
For our optimized structures with the same cell volumes \cite{optimize},
the $a\not=b$ case has a lower energy by 30 meV/f.u. than for  $a=b$.
Thus for the $\delta=0$ we will focus on results for the optimized 
$a\not=b$ structure, unless specified otherwise.


In both $a=b$ and $a\not=b$ cases, 
the AFM state with alternating spin alignment along the Cu-O$_{\rm P}$ chain 
has a lower energy than in the NM state,  
by 4 meV/f.u. for the $a\not=b$ case.
This difference is remarkably small, but comparable with the 
energy gain of about 7 meV/f.u. 
from a simple Stoner picture $I_{st}M^2/4$ with Stoner $I_{st}=0.9$ 
eV \cite{erik} and the Cu moment of 0.18 $\mu_B$ (see below).
This indifference of the AFM energy gain to the $a/b$ ratio is
another property that is insensitive to the distortion 
in the planar lattice parameters.
Our attempts to obtain a ferromagnetic state always reverted to NM states.


\begin{figure}[tbp]
{\resizebox{8cm}{6cm}{\includegraphics{Fig3.eps}}}
\caption{ (a) GGA NM band structure for the optimized structure,
with fatband representation of the Cu $d_{X^2-Y^2}$ and $d_{Z^2}$ characters
in the range of --6 eV to 3 eV, containing the Cu $3d$ and O $2p$ orbitals Ba$_2$CuO$_3$. 
The $X$ and $Y$ points are chosen as the zone boundaries 
parallel and perpendicular to the Cu-O$_{\rm P}$ chain, respectively.
(b) The corresponding Fermi surfaces, 
showing very strong 1D character arising from the Cu-O$_{\rm P}$ chain band.
} 
\label{nmband}
\end{figure}

\begin{figure}[tbp]
{\resizebox{8cm}{8cm}{\includegraphics{Fig4.eps}}}
\caption{ For NM  Ba$_2$CuO$_3$,
(top) total and atom-projected densities of states,
 and (bottom) orbital-projected densities of states for each atom, within GGA.
The total Fermi level DOS $N(E_F)$ = 0.72 eV$^{-1}$ per f.u. for both spins.
The total DOS, yellowish-shaded in the top panel, varies little around $E_F$, 
reflecting the mid-band 1D character.
Due to the chain structure, the mostly filled O $p_\pi$ orbitals 
show differences between the two sites.
} 
\label{nmdos}
\end{figure}


\subsection{Nonmagnetic electronic structure}
Now we address the NM state to establish the underlying 
(spin-symmetric) electronic structure.
The bands and corresponding atom and orbital projected-densities of states (PDOSs) 
are provided in Figs. \ref{nmband}(a) and \ref{nmdos}, respectively.
These are similar to the previously reported ones in the $a=b$ case \cite{jinang2021}.
To analyze this, we focus on the CuO$_4$ unit and choose the $X$-axis 
along the Cu-O$_{\rm P}$ chain and the $Y$-axis toward the apical oxygen ${\rm O_A}$.
Consistent with the Cu$^{2+}$ $d^9$ configuration,
the $d_{X^2-Y^2}$ antibonding state with a band width of 2.3 eV 
crosses the Fermi energy $E_F$ exactly at half filling.
The less dispersive $d_{Z^2}$ orbital (perpendicular to the CuO$_4$ unit)
is occupied at --3 eV. 
The half-filled $d_{X^2-Y^2}$ band leads to a strongly 1D Fermi surface (FS), 
displayed in Fig. \ref{nmband}(b), suggesting strong 1D instabilities 
at $q=2k_F$ (see below).

Using a simple single band model, 
the $d_{X^2-Y^2}$ band leads to  the following nearest neighbor (NN) and next NN hopping parameters 
with on-site energy of $-0.146$ eV:
intrachain $t_a=0.532$ eV, $t'_a=0.083$ eV,
interchain $t_b=0.044$ eV, and small $t_{ab}=0.015$ eV,  interlayer $t_c=-0.010$ eV.
Compared with our calculated hopping parameters of Sr$_2$CuO$_3$~ with the 
experimental structure parameters of Ami {\it et al.} \cite{ami1995},
$t_b$ is reduced by a factor of $\frac{1}{2}$, whereas changes in the 
other parameters are unimportant.
This comparison indicates that Ba$_2$CuO$_3$~ is a more ideal 1D Heisenberg 
AFM spin-half chain system (see below).
The dominant intrachain hopping $t_a$ gives a superexchange strength 
$J_s=4t^2/U\approx0.14$ eV ({\it i.e.}, about 1640 K)
with the Coulomb repulsion $U=8$ eV, consistent with an experimental 
estimate \cite{sala2021}.
This value is similar to values of the other superconducting cuprates.

As mentioned above, 
effects of the distortion of the planar ($a-b$) lattice parameters on the 
electronic structures and magnetic characters are not substantial.
However, a few distinctions can be observed.
The distortion decreases $t_{a}$ by 10\% and increases $t_{b}$ 
by a factor of $\frac{3}{2}$,
reflecting the increased intrachain and reduced interchain Cu-O$_{\rm P}$ separations.
In both NM and AFM states,
a clearly distinguishable change is to increase a gap between 
the $E_F$-crossing $d_{X^2-Y^2}$ band 
and the bottom of the unfilled bands (the $M$ point) at 1 eV.


\begin{figure}[tbp]
{\resizebox{8cm}{6cm}{\includegraphics{Fig5.eps}}}
\caption{AFM band structure of optimized Ba$_2$CuO$_3$~ within GGA,  
highlighting the spin-resolved characters by fatband plots 
of the Cu $d_{X^2-Y^2}$ and $d_{Z^2}$ characters with a positive moment.
The AFM-induced gap indicates insulating character.
The $S$ point is the zone boundary in the (110) direction, 
as described in Fig. \ref{fs_sc}.
} 
\label{afmband}
\end{figure}

\subsection{Antiferromagnetic state: spin-Peierls instability}
The AFM state, which is the expected and calculated ground state of Ba$_2$CuO$_3$~\cite{ladder2019}, 
was investigated using a supercell doubled  along the Cu-O$_{\rm P}$ chain direction.
The band structure in GGA is displayed in Fig. \ref{afmband}, showing 
an AFM-induced gap \cite{gap} of 0.20 eV.
The half-filled $d_{X^2-Y^2}$ band lies in the range of --1.5 eV to 1 eV,
whereas the full-filled $d_{Z^2}$ orbital lies around --2.5 eV to --3 eV.
The corresponding orbital-PDOSs are given in Fig. \ref{afmdos}, 
indicating  the strong $pd\sigma$ hybridization, 
consistent with the spin density plot of Fig. \ref{str}(b).
The Cu spin moment, ideally a spin-half value of 1 $\mu_B$, is strongly
reduced to 0.18 $\mu_B$ by the $pd\sigma$ hybridization.
The O$_{\rm P}$ ions are thereby strongly polarized, but have zero 
net moment due to the symmetry of AFM order.

A noteworthy aspect in the AFM state is a gap of 0.2 eV opening  
at the GGA level without the assistance of a Hubbard $U$ repulsion, which is
required in square-lattice cuprates. This result seems to place Ba$_2$CuO$_3$~ 
in the Slater insulator, versus Mott insulator regime.
The zone-folding due to doubling of the cell along the 
$\hat{a}^\ast$-direction ($a^\ast=2a$)
leads to overlapped bands along the $X'-S$ line, {\it i.e.} 
on the $k_x=\pi/a^\ast$ plane. 
Then nesting of the 1D FS, evident in Fig. \ref{nmband}(b)
that it is near perfect, 
leads to a 1D spin-Peierls instability, analogous to the 
lattice-Peierls instability -- symmetry-breaking along the chain
is the underlying mechanism in both cases. 
As shown in the band structure of Fig. \ref{afmband},
along the $X'-S$ line the $d_{X^2-Y^2}$ bands bordering the gap have 
opposite spin character.

Inclusion of the Hubbard $U$ to the Cu ions simply 
increases the energy gap and the magnetic moment,
since the gap is already open in the GGA level.
At $U=7$ eV, the moment of the Cu ion reaches 0.64 $\mu_B$ and the gap is about 1.5 eV.


\begin{figure}[tbp]
{\resizebox{8cm}{6cm}{\includegraphics{Fig6.eps}}}
\caption{GGA orbital-PDOSs of Cu, planar and apical oxygens in AFM  Ba$_2$CuO$_3$.
The Cu ion with a positive moment is chosen.
 Most spin-polarized holes in the oxygens are in O$_{\rm P}$, nearly none in O$_{\rm A}$.
} 
\label{afmdos}
\end{figure}


\subsection{Effects of oxygen mode displacements}
We have performed frozen phonon calculations for a few O displacements,
two involving O$_A$ motion and  one involving O$_P$ motion.
One is the beating $A_g$ mode of O$_{\rm A}$  ions against the Cu-O$_{\rm P}$ chain 
with amplitudes up to $\pm$0.06 \AA. 
The change in energy versus displacement (not shown here) 
is fit well with a simple harmonic form,
with frequency of 603 cm$^{-1}$, which is
within 7\% of the observed Raman frequency \cite{cwchu1994} of 563 cm$^{-1}$ 
in a multiphase sample of Ba$_2$CuO$_{3+\delta}$~with $\delta$ in the 0.09-0.17 range.

We have studied changes with increase of the displacement in the IR active $B_{1u}$ mode
of the O$_A$-Cu-O$_A$ unit.
The energy gap linearly reduces with distortion,  reaching 
$\Delta E_g$= --0.1 eV at the amplitude of $\pm0.06$ \AA~
that is a typical rms displacement of an oxygen ion.
The frequency of the $B_{1u}$ mode is 560 cm$^{-1}$.

The O$_{\rm P}$ breathing mode along the chain direction was studied for
amplitudes $u_{\rm O}$ up to 0.07 \AA.
This broken symmetry results on two Cu sites alternating along the chain,
but the allowed spin disproportion is negligible, {\it i.e.}
spin-lattice coupling is small.
This oxygen optic mode has an anomalously low energy frequency of roughly 
43 cm$^{-1}$,  probably a remnant of the Peierls instability.
The electron-lattice coupling shows up in the modulation of energy gap,
with  $dE_g/du_{\rm O}\approx$3.0 eV/\AA. 
The gap vanishes at 0.07 \AA; since this is a
typical oxygen rms vibrational amplitude, electron-phonon coupling in Ba$_2$CuO$_3$~
may be worth further study 


\section{Results for the superconducting phase}
Before considering structures of ordered planar oxygens in the 
superconducting phase around $\delta\approx0.2$,
we carried out calculations of the virtual crystal approximation 
(VCA) up to $\delta=0.25$
by increasing the number of valence electrons from the $\delta=0$ phase.
As might be expected from the AFM DOS given in Fig. \ref{afmdos} 
and the FS $2k_F=\pi/a$,
once hole-doped the system moves away from the spin-Peierls 
instability and the gap and the small moment vanish.
In this region, the single-band feature at $\delta=0$ 
remains unchanged from the NM character (not shown here).
Above $\delta\approx0.25$, a small AFM moment re-appears as the
increased hole density begins to favor polarization.

\subsection{Supercells}
We now turn to the supercell results at $\delta=1/4$.
The bilayer structure is favored,
with the monolayer cell being 70 meV higher and the
brickwall structure being 250 meV higher than the bilayer.\cite{energy}
These are of course ground state
(zero temperature) values. For varying high pressure and high temperature
synthesis conditions,
samples might contain regions with the metastable monolayer structure interspersed
with a majority amount of bilayer domains.
(Our band structure of the monolayer structure, not shown here, is similar
to that published previously \cite{ladder2020}.)

Regularities across these structures based on coordination can be identified. The
PDOS spectra of the various Cu sites are provided in Fig.~\ref{pdos_sc} and the Appendix. 
In all three structures,\\
$\bullet$ the \underline{four-fold coordinated Cu4} chain 
sites display filled and strongly bound $Z^2$ orbitals while
the $X^2-Y^2$ bands are roughly half-filled.
This filling is
characteristic of a Cu$^{2+}$ ion with large crystal field splitting. 
Isolated chains in each structure
are likely to have strong AFM correlations, which (in the
absence of magnetic ordering, which is not yet reported) will reduce but not
eliminate conduction along the chain. See Sec. V for further description.\\
$\bullet$ the \underline{five-fold Cu5} sites in the two disfavored structures 
(see the Appendix) show 
holes distributed between both $e_g$ orbitals, reflecting the vast reduction
in the $e_g$ crystal field splitting compared to Cu4 sites. 
In the brickwall structure, the hole count 
is  dominated by $x^2-y^2$ weight ($z^2$ weight is minor); 
in the monolayer structure, surprisingly, $z^2$ hole weight dominates.
In each case the 
amount of Cu5 holes seems roughly characteristic of a formal valence of 
2.5+. \\
$\bullet$ for the \underline{four-fold rung Cu4$^{\ast}$} site, 
the hole(s) lies entirely in the
roughly half-filled $X^2-Y^2$ band, thus with one hole of each spin,
again close to a nominally Cu$^{2+}$ ion, as for the chain Cu4 ion. 
In the monolayer structure (see the Appendix), the Cu4$^{\ast}$ DOS
is gapped and clearly 2+.  Differing somewhat
from the brickwall structure, both Cu4 and Cu4$^{\ast}$ ions 
in the bilayer case have strongly
bound $Z^2$ states, with metallic but different from half-filled $X^2-Y^2$ 
bands. \\
$\bullet$ with two (chain layer) Cu4$^{2+}$ ions in the bilayer primitive cell, 
the average formal valence of the Cu6 and Cu4$^{\ast}$ ions must be Cu$^{2.5+}$.
The PDOSs in Fig.~\ref{pdos_sc} reveal that the \underline{six-fold Cu6} ion  
has {\it more $z^2$ holes than $x^2-y^2$ holes,} making it  
very highly oxidized (viz. approaching Cu$^{2.75+}$) and, given the overall highly
hole-doped character,  a likely candidate to be heavily involved
in superconducting pairing and in hosting the superconducting carriers.
Supporting this viewpoint is the expectation that strong spin correlations 
along the chain will reduce the participation of the Cu4 ion.
An interesting parallel is that $z^2$ holes have also been 
implicated \cite{bnoas,pickett2021} in the long sought and recently discovered 
superconductivity in infinite-layer hole-doped NdNiO$_2$ \cite{H.Hwang2019}.


\subsection{The Bilayer Oxygen Ordering}
We now focus on this energetically favored bilayer structure, 
whose electronic features have not been investigated previously.
In this phase, our calculations in {\sc wien2k} were performed 
with the experimental lattice and internal parameters given in
Ref.~[\onlinecite{pnas2019}].


\begin{figure}[tbp]
{\resizebox{8cm}{6cm}{\includegraphics{Fig7.eps}}}
\caption{ The Cu $e_g$-orbitals' projected DOSs for the NM $\delta=1/4$  
bilayer structure pictured in Fig. \ref{str_sc}(a), within GGA.
For the differing types of the local coordination, see the text.
} 
\label{pdos_sc}
\end{figure}

In the bilayer structure of Fig. \ref{str_sc}(a),
the supercell, containing 4 formula units, 
has two Cu4 (chain) sites in the $z$=0 plane, and one planar Cu4$^{\ast}$
(rung) CuO$_4$ unit and one six-fold coordinated Cu6 in the $z=\frac{c}{2}$ plane.
In this structure, our attempts to obtain an AFM state (after all, in one layer
two well isolated chains remain intact) in GGA 
always reverted to a NM state. This result might be thought to be
consistent with the previous calculation,\cite{ladder2019} 
and our VCA results, if charge transfer between layers occurs.
Since the material is (super)conducting, we did not pursue possible magnetic
and insulating states using correlated electronic structure methods.

Figure \ref{pdos_sc} shows the $e_g$-orbitals' PDOSs for all three Cu sites.
Although the electronic structure is metallic, a formal valence 
viewpoint might still be instructive.
Since the Cu4 and Cu4$^{\ast}$ $Z^2$ states are both localized near --3 eV, 
their nominal valences (from $X^2-Y^2$ hole count) should be nearly the same.
On the contrary, the Cu6 sites contain hole(s) from both orbitals, 
with (surprisingly) somewhat more $z^2$ weight than $x^2-y^2$ weight. 
Figure~\ref{pdos_sc} indicates
that their crystal fields are drastically different: that of Cu6 is close
to an octahedral $e_g$ degeneracy. 



\begin{figure}[tbp]
{\resizebox{7.5cm}{7.5cm}{\includegraphics{Fig8.eps}}}
\caption{ Band structure, enlarged in the range of --2 eV to 2 eV, 
of the NM $\delta=1/4$ bilayer phase, within GGA.
The characters of the $e_g$ orbitals of (a) Cu6, 
(b) Cu4$^\ast$, and (c) Cu4 are highlighted
by magenta ($d_{x^2-y^2}$ or $d_{X^2-Y^2}$) and green ($d_{z^2}$) colors.
}
\label{band_sc}
\end{figure}

\subsection{Charge, Crystal Fields, Bands}
{\it Charge density.} 
Bader charges can sometimes provide insight. They
do depend on environment and thus are not true atomic properties, and often
do not correspond closely to formal valence (atomic sphere charges also can vary
greatly from formal valences). Differences in Bader charge for inequivalent
ions give the most useful information.
Bader charges of the Cu ions are +1.31 (Cu6), +0.89 (Cu4$^\ast$), and +0.86 (Cu4).
The similarity of the latter two supports equal formal valence, and the 
difference of 0.42 to 0.45 relative to Cu6 indicates a 
substantially different valence for
Cu, again in agreement with analysis above based on PDOSs.
The charges of the apical oxygens are $-1.12$ to $-1.28$, 
while these of the planar oxygens are  $-1.08$ to $-1.11$. These differences
of 0.03 to 0.2 likely reflect primarily differences in environment (bonding).
  


{\it Crystal fields.}
As expected from a simple crystal field concept, 
the site energy of the $d_{z^2}$ orbital (equal to that of $d_{x^2-y^2}$
for an ideal octahedron $e_g$ partner) changes 
with variation of the Cu-O$_{\rm A}$ distance $d_{\rm O_A}$.
We find that the $d_{z^2}$ orbital remains partially unfilled in the 
range of $d_{\rm O_A}$ distances from 1.86 \AA~ to 2.10 \AA.
The center of the $d_{z^2}$ band shifts by at most 0.1 eV in this range.
These results indicate that the effects of the shorter 
Cu-O$_{\rm A}$ distance on the electronic structure
are not substantial, consistent with the results of Ba$_2$CuO$_3$.
In addition, as seen above in Ba$_2$CuO$_3$, 
the corresponding variation of the total DOS around $E_F$ is negligible (not shown here).
This indifference to $d_{\rm O_A}$ suggests that this bondlength variation with doping
has little effect on $T_c$. \color{black}
At $E_F$ the total DOS is low and nearly identical to that of the NM $\delta=0$ phase, 
$N(E_F)=$0.73 eV$^{-1}$ per f.u. for both spins.



{\it Band structures.}
The corresponding band structure enlarged in the --2 eV to 2 eV 
region is shown in Fig. \ref{band_sc},
with the highlighted fatband plots of Cu6 $d_{x^2-y^2}$ and $d_{z^2}$ orbitals,
and Cu4$^\ast$ and Cu4 $d_{X^2-Y^2}$ orbitals. Note first the lack of
dispersion along $\Gamma-Z$ -- the layers remain uncoupled,
except for possible charge transfer.
The partially filled $d_{X^2-Y^2}$ orbitals of Cu4$^\ast$ and Cu4 ions
show very similar 1D dispersion, with widths of $\sim$2.4 eV, 
leading to $t_a\approx$0.6 eV.
The Cu4$^\ast$ orbital shows  
$d-d$ hybridization with Cu6 $e_g$ orbitals near $E_F$
along the $S-\Gamma$ line, due to accidental degeneracy and some small coupling.
On the other hand, the Cu4 orbital is negligibly hybridized 
with the other $d$ orbitals of Cu6 and Cu4$^\ast$, 
reflecting the 1D character of the chain.

Both (anti-bonding) Cu6 $d_{x^2-y^2}$ and $d_{z^2}$ 
orbitals of width 2.3 eV 
cross $E_F$  and are energetically nearly degenerate.
These Cu6 bands show two-dimensional (2D) character with $t_a\approx t_b\approx$0.56 eV.
As a result, within a formal charge picture, the Cu4 and Cu4$^\ast$ ions 
are Cu$^{2+}$ with half-filled $d_{X^2-Y^2}^1$ bands.
As noted, for formal charge balance, the Cu6 ion must be over-doped Cu$^{2.5+}$. 
Formal charges can be questionable for metallic systems, of course,
but the consistency of identification is striking.
This difference in Cu sites indicates that the ordered planar oxygen 
leads to a mixture of single-band and two-band features
that could significantly complicate determination of the superconducting pairing symmetry
and  mechanism.



\begin{figure}[tb]
\vskip 4mm
{\resizebox{6cm}{3cm}{\includegraphics{Fig9.eps}}}
\caption{Fermi surfaces within GGA of the NM $\delta=1/4$ phase 
in the bilayer structure, with several flat parallel sheets tending to
support charge, spin, or lattice instabilities.
} 
\label{fs_sc}
\end{figure}

\section{Fermiology of the bilayer structure}
Figure \ref{fs_sc} shows the Fermi surfaces of the NM bilayer $\delta=1/4$ phase.
All FSs show strongly reduced dimensionality.
They contain a large rounded-square barrel surface centered at $\Gamma$, 
with negligible dispersion 
in the third direction, analogous to several doped cuprate 
compounds. In  addition there are quasi-1D sheets near the $X'-S$ zone
edges, one extending along the entire edge length, the other cut off
midway by small in-plane dispersion. These surfaces are indicative of
strong 2D and 1D characters in the respective bands. 
The character of the barrel FS is a mixture of the Cu6 $e_g$ orbitals
with the Cu4$^\ast$ $d_{X^2-Y^2}$ orbital. 
The 1D FSs  have  mostly chain Cu4 and Cu4$^\ast$ $d_{X^2-Y^2}$ character, 
with minor admixture of Cu6 $d_{x^2-y^2}$.

The rms Fermi velocities over all sheets are 
$$v_{F,a}=24.7,~~ 
  v_{F,b}=4.5,~~
  v_{F,c}=0.05$$
 in units of 10$^7$ cm/sec,
a factor of 5 in planar anisotropy and likely at most hopping conductivity
in the perpendicular direction. These large velocities account
for the rather small Fermi level DOS in spite of large FSs.
This strong anisotropy is also reflected in the Drude plasma energies (in eV) of 
$$\Omega_{a}=3.9,~~  
  \Omega_{b}=1.66,~~  
  \Omega_{c}=0.17,$$
but smaller in magnitude.
In contrast to the two 1D FSs containing hole carriers with 
$v_{F,a}^{1D}\approx 34.50\times10^7$ cm/sec,
the $\Gamma$-centered barrel FS is less anisotropic in-plane:
$$v_{F,a}^b=7.2,~~
  v_{F,b}^b=11.1,~~ 
  v_{F,c}^b=0.08$$
 in units of 10$^7$ cm/sec.
This corner-rounded square FS with a side of 0.35$\frac{\pi}{a}$
possesses roughly 0.50 electrons, while hole carriers near the zone
boundary are contained within the 1D FSs.

One can note that this superconductor has a low $N(E_F)$ and 
low carrier density, a characteristic shared with other superconducting cuprates.
This is however different from the superconducting Fe-pnictides with their
higher $N(E_F)$ together with low carrier density \cite{singh2008}.
However, several superconducting cuprates show (within GGA) low $N(E_F)$ along with 
higher carrier density \cite{wep1992}.


\section{Discussion}
Bonner and Fisher demonstrated that quantum fluctuations preclude magnetic ordering
of the (isolated) spin-half Heisenberg chain at finite temperature \cite{bonner1964}, 
although strong spin-spin correlations lead to a vanishing susceptibility as
zero temperature is approached. Nevertheless, in the presence of even very small
interchain coupling, magnetic ordering is observed to occur in several
spin-chain materials. We have quantified the interchain coupling in Sec. III.
 
Schulz treated these intra-chain correlations exactly and interchain coupling
in mean field, and provided a formula for the staggered magnetization 
$m_0=\langle S^z_i \rangle$ at the $i$-th site and for the  N\'eel temperature $T_N$
of the 1D Heisenberg AFM spin-half chain \cite{schulz1996}.
For Ba$_2$CuO$_3$, the staggered magnetization is
\begin{eqnarray} 
m_0\approx g\mu_B\times1.02\frac{t_b}{t_a}\approx0.17\mu_B,
\end{eqnarray}
with $g$ being the Land\'e $g$-factor. This result indicates the degree to
which $1\mu_B$ is strongly reduced by quantum fluctuations. Our calculations indicate
that  Cu-O $pd\sigma$ hybridization reduces the {\it static} ordered moment to 0.2$\mu_B$
to 0.6$\mu_B$ depending on the size of Hubbard $U$, which will be further
reduced by fluctuations. The magnetic susceptibility should contain signatures
of the strength of spin correlations as well as magnetic ordering.

Schulz also provided an implicit equation 
for ordering temperature T$_N$ in terms of simple interchain coupling $J_{\perp}$
and two constants $A$=0.32, $\Lambda$=5.8 from numerical calculations:
\begin{eqnarray}
|J_{\perp}|=\frac{T_N}{4A\sqrt{{\rm ln}(\Lambda J/T_N)}}.
\end{eqnarray}
With our estimates from Sec. III $J$=1640 K, $J_{\perp}\sim 10$ K, 
this expression gives $T_N$ in the neighborhood of 25 K.
His expression with a single interchain coupling does not fit closely 
with the Ba$_2$CuO$_3$~structure with several (small) interchain couplings that
we obtain, so we have used the largest. Within the 
uncertainty of the exchange couplings and the logarithmic dependence
of the expression, this is similar to but somewhat larger than the case
of Sr$_2$CuO$_3$ \cite{serge2020}, which means that  Ba$_2$CuO$_3$~ is also
 near the regime of the 1D Luttinger-liquid quantum-critical phase.\\

In the superconducting sample \cite{pnas2019}, the linear specific 
heat coefficient $\gamma\sim14$ mJ$\cdot$mol/K$^2$
was experimentally measured.
Comparison of this value to our calculated band value of 
$\gamma_0=$1.72 mJ$\cdot$mol/K$^2$
leads to  a factor of 8 due to dynamic correlation effects, implying a 
strong correlation strength in the doped material.

Our calculations indicate that this system is little affected 
by the unusual Cu-O separations.
This behavior can be contrasted with the infinite layer Ni$^{2+}$ 
system of Ba$_2$NiO$_2$(AgSe)$_2$, 
in which the strained Ni-O separation leads to  a calculated
`off-diagonal singlet' state \cite{bnoas},
consistent with previous findings by some of the current authors 
that Cu ions are different from Ni ions even if both have the same 
ionic configuration \cite{lanio2,nno2}.

The bilayer FSs suggest instabilities of the ordered structure that may
help in understanding the paucity of single phase, well ordered samples.
Flat (or parallel -- nesting) regions of FSs provide a strong tendency
toward electronic instability that manifests as charge-, spin-, or
lattice-instabilities, each of which break the symmetry of the underlying phase. 
The barrel FS shows nesting at $2k_F^b\sim 0.70\frac{\pi}{a}$ in both of the
planar directions.  The 1D FSs suggest instabilities at a small value of
$2k_F^{1D}$ in the $\hat{a}$-direction ({\it i.e.} a long wavelength disturbance),
and a large value $2k_F^{1D}$=$2\frac{\pi}{a} - 2k_F^{1D}$, which corresponds to a
short wavelength disturbance. These potential or real instabilities may 
help to account for the complex structures observed in samples. 

\section{Summary}
Our first principles studies of the electronic and magnetic
structures of Ba$_2$CuO$_{3+\delta}$, $\delta=0$ or $\frac{1}{4}$, provide important information
about this unusual superconductor. Our studies indicate that interchain 
coupling in NM (hence metallic) Ba$_2$CuO$_3$~is quite small, thus not surprisingly,
subject to Peierls-type instability of spin, and probably also of lattice,
and possibly charge ordering. Considering other cuprates, we have
focused on the spin (AFM) instability. AFM ordering alleviates the
Peierls instability, however the magnetism is weak without the assistance of
intra-atomic Cu $3d$ repulsion and correlation. Charge and lattice instabilities
would compete with AFM ordering. If AFM ordering occurs, it will be weak
due to the fact that interchain coupling is very small, and isolated chains
do not order -- the Bonner-Fisher result.
Considering the very stable Cu$^{2+}$ ion it is surprising that
stoichiometric, ordered Ba$_2$CuO$_3$~ has not been obtained, even in samples synthesized
with varying pressure and temperature.

Isolated 1D spin-half chains do not order due to strong fluctuations, however
even small interchain coupling does lead to order. The large intrachain and
small interchain couplings that we estimate, using the theory of
Schulz, suggest an ordering temperature of the order of 25 K. This value
puts Ba$_2$CuO$_3$~in the regime of the 1D Luttinger-liquid quantum critical phase,
for which further experimental study will be useful. 

The types of ordering of the planar oxygen dopants play a 
crucial role in determining the crystal and electronic structures of Ba$_2$CuO$_{3+\delta}$, and thereby
the value of $T_c$ and pairing mechanism.
This observation is consistent with experimental data that show 
substantial change of $T_c$ 
related to the modulation of apical oxygen positions and concentration
in superconducting Sr$_2$CuO$_{3+\delta}$~\cite{liu2014,pnas2020}. The doping level of
$\delta$=1/4 has led to the picture of ordered approximants being modeled
as addition O ions being inserted, seemingly necessarily bridging Cu-O$_{\rm P}$ chains,
rather than removing a much larger fraction of O ions from Ba$_2$CuO$_4$.

Our study of three proposed ordered structures of Ba$_2$CuO$_{3+\delta}$~has indicated that a
{\it bilayer} structure (one with two structurally different layers) is 
most stable, although a {\it monolayer} ordered structure
is only modestly higher in energy. In the bilayer structure, all three Cu
ions have $e_g$ states at the Fermi energy and participate in conductivity,
hence presumably also in superconductivity. There is however strong site
 differences: the two four-fold coordinated sites have the $z^2$ orbital
strongly bound and localized, leaving only more or less half-filled
$x^2-y^2$ holes as occurs in most cuprates. 

The octahedrally coordinated site, which occurs only in
the bilayer structure, has a large fraction of holes in both $e_g$ orbitals,
with more in the $z^2$ orbital. This marks this site as very highly
oxidized (viz. 2.75+), accounting for the perplexing high mean formal
valence of Ba$_2$CuO$_{3+\delta}$.  A quasi-degeneracy of the $e_g$ orbitals is restored
by the octahedral coordination, and implies that Ba$_2$CuO$_{3+\delta}$~ is not only strongly
overdoped but also a two-band superconductor. 

In addition, this energetically preferred bilayer structure 
displays intense Fermi surface nesting  of both 1D and 2D nature that will 
drive the system toward instabilities of a few types (wavelengths) of 
charge-, spin-, and lattice-symmetry breaking. The fluctuation associated
with these instabilities may help to account for Ba$_2$CuO$_{3+\delta}$~not having been 
prepared in a fully ordered, single phase sample even for a variety of
synthesis conditions.  

The current work has not dealt specifically with correlation effects due to
strong repulsion in the $3d$ shell of the Cu ion. Crystal fields separating
the two $e_g$ orbitals on the four-fold and six-fold coordinated Cu sites have
been identified and analyzed, and formal valence identifications have been 
discussed. Specifically, four-fold sites qualify as 2+, with the others 
better characterized as 2.5+ or even somewhat more highly oxidized. 
Our results provide a roadmap for
further analysis and experimental exploration of the electronic structure of Ba$_2$CuO$_{3+\delta}$.



\section{Acknowledgments}
H.S.J. and K.W.L. were supported by National Research Foundation of Korea
Grant No. NRF-2019R1A2C1009588.
W.E.P. was supported by NSF Grant No. DMR 1607139.


\section{Appendix}
Since the PDOSs and associated crystal field splitting of the Cu $e_g$ orbitals
play a strong role in our identification of the Cu sites and orbital character that
provide the superconducting carriers, we provide here, for 
the $\delta=1/4$ phase, the Cu $e_g$-orbitals' PDOSs 
for the energetically unfavored structures in Figs. \ref{appendix}.
Strong differences can be seen in the $d_{z^2}$ spectra: for the Cu5 site,
there is strong character at and above $E_F$, while that orbital is localized
at --3 eV in the two four-fold sites, 
while great similarities are evident. 
It is not surprising that the ``isolated'' chain site Cu4 spectrum is relatively
undisturbed by doping on other layers as in Ba$_2$CuO$_3$. There is however a strong
difference between Cu4 and Cu4$^{\ast}$: the former is half-filled, while
the latter is gapped around $E_F$, seemingly approaching a 3+ formal valence.
The brickwall Cu5 site (the right panel of Fig. \ref{appendix}) 
likewise approaches this same high formal valence. The main text provides 
additional discussion and comparison of all three structures.

\begin{figure*}[tbp]
{\resizebox{8cm}{6cm}{\includegraphics{Fig10a.eps}}}
\hskip 4mm
{\resizebox{8cm}{6cm}{\includegraphics{Fig10b.eps}}}
\caption{ The Cu $e_g$-orbitals' PDOSs for the NM $\delta=1/4$  
(left panel) monolayer structure pictured in Fig. \ref{str_sc}(b)
and (right panel)  brickwall structure pictured in Fig. \ref{str_sc}(c), within GGA.
} 
\label{appendix}
\end{figure*}



