Proposed ordering of textured spin singlets in a bulk infinite layer nickelate

The infinite-layer structure nickelate Ba$_2$NiO$_2$(AgSe)$_2$ (BNOAS) with $d^8$ Ni ions and a peculiar susceptibility $\chi(T)$ is studied with correlated density functional methods. The overriding feature of the calculations is violation of Hund's rule coupled with complete but unconventional spin-orbital polarization, leading to an unexpected low spin $^1B_1$, ``off-diagonal singlet'' (ODS) textured by an internal orbital structure of compensating $d_{x^2-y^2}^{\uparrow}$ and $d_{z^2}^{\downarrow}$ spins. This unconventional configuration has lower energy than conventional high-spin or low-spin alternatives. An electronic transition is obtained at a critical Ni-O separation $d_c^{Ni-O}=$2.03~\AA, above which Ni becomes magnetic in square planar NiO$_2$ compounds. We propose scenarios for the signature of magnetic reconstruction in $\chi(T)$ at $T_{m}$=130 K without any Curie-Weiss background (no moment) that invoke ordering of Ni $d^8$ moieties that are largely this generalized Kondo singlet. The underlying physics of this system is modeled by a Kondo sieve model (2D Kondo necklace) of a ``Kondo'' $d_{z^2}$ spin on each site, coupled to a $d_{x^2-y^2}$ spin that is itself strongly coupled to neighboring like-spins within the layer. The observed magnetic order places BNOAS below the quantum critical point of the Kondo sieve model, providing a realization of the previously unreported long-range ordered near-singlet weak antiferromagnetic phase. We propose electron doping experiments that would drive the system toward a $d^{9-\delta}$ configuration and possible superconductivity with similarity to the recently reported Ba$_2$CuO$_{3.2}$ that superconducts at 73 K.

Layered nickelate materials have attracted attention for some decades primarily due to the similarities they provide to the layered cuprates that display high temperature superconductivity. [1,2] A number of layered nickelates with non-integer formal valences have been synthesized, showing a great variety of magnetic and spin/charge ordering behavior that is now understood, [3][4][5] but no superconductivity.
However, the recent discovery [6] and verification [7] of superconductivity in thin films of strained, hole-doped NdNiO 2 has reinvigorated interest in layered nickelates. The focus has been on d 9 , spin-half layers, due to their similarity to the cuprates, which may be (hole or electron) doped to produce high temperature superconducting (HTS) phases. The complementary route to superconductivity, not formerly taken seriously, may arise in compounds with a Ni d 8 configuration that approach the same d 9 regime upon electron doping. The d 8 configuration brings with it the competition between low-spin and high-spin configurations, which is expected to depend on strain applied to the NiO 2 layer by other components of the compound as well as crystal field splittings, and may or may not promote HTS.
The report of synthesis of a new layered nickelate sys- * Electronic address: wepickett@ucdavis.edu † Electronic address: mckwan@korea.ac.kr FIG. 1: (a) bct structure of Ba2NiO2(AgSe)2. The layered square lattice of Ni ions without apical oxygen ions forms a single "infinite layer" structure of the NiO2 layer analogous to the bulk layers of CaCuO2 and NdNiO2. Subsequent layers have a large interlayer 'spacer layer' distance of about 10Å, leading to strong two dimensionality in many respects. (b) Spin density plot of the spin singlet state at an isovalue of 0.015 e/Å (U = 7 eV with wien2k). Red (blue) denotes spin-up (-down) character. This combination comprises the orbitally off-diagonal singlet state d 1↓ z 2 -d 1↑ x 2 −y 2 . Hybridization with the planar oxygens is apparent.
tem indicates further anomalous properties of layered nickelates. Matsumoto and collaborators have reported the synthesis, [8] under 7 GPa pressure at 850 • C, of Ba 2 NiO 2 Ag 2 Se 2 (BNOAS) containing an 'infinite layer' NiO 2 sublattice without apical oxygen sites but with an anomalously large Ni-O separation of 2.10Å. The susceptibility χ(T ) is peculiar and unexplained. In spite of no Curie-Weiss term in the otherwise constant χ(T ), a peak occurs at T m =130 K with 30 K full width before returning to its original value. Field cooling below T m increases the susceptibility below T m , and weak magnetic peaks in powder neutron diffraction [8] at 5K have been interpreted as indicating ( 1 2 , 1 2 , 0) antiferromagnetic (AFM) order, with best fit to weak structure in diffraction data suggesting S=1 Ni spins. This interpretation however seems inconsistent with the peculiar 'spinless' behavior of χ(T ). BNOAS was discussed as insulating but no conductivity data on the powder samples was presented.
A non-magnetic ground state is observed in other square planar d 8 nickelates, [8,9] viz. BaNiO 2 [10] and monovalent LaNiO 2 (see references in [11]) so BNOAS is an anomalous case. The isovalent sister strontium compound SNOAS with smaller lattice constant is more conventional, showing Curie-Weiss behavior above 150 K characteristic of an S=1 moment and strong AFM magnetic coupling (Weiss θ= -158 K). χ(T ) indicates that SNOAS undergoes some type of magnetic order around 50 K. Matsumoto et al. ascribe the differences between the two compounds to the different Ni-O bond lengths (that of BNOAS being unusually large), which could tip the balance between high spin and low spin states.
Formal valence counting, confirmed by calculations presented below, proceeds as Ba giving Ni a d 8 , two hole configuration. A major issue is the relative importance of low-spin S=0 and high-spin S=1 states of the ion, and (we will propose) the nature of the S=0 ion. As mentioned, most known layered d 8 nickelates are non-magnetic [8,9] S=0, so magnetic behavior (even without its strangeness) of these new nickelates presents new physics. Building on the underlying magnetism, one can anticipate that electron doping, moving the Ni formal valence toward the d 9 configuration, might induce superconductivity as has been found in the (Nd,Sr)NiO 2 system.
Our main result here is the discovery of a distinct ionic configuration, an "off-diagonal singlet" with internal magnetic and orbital textures in spite of vanishing magnetic moment. A secondary finding is a phase boundary at a critical Ni-O separation of 2.03Å that concurs with the separation of low spin and high spin Ni ions in square planar NiO 2 layered compounds. After presenting procedures in Sec. II, in Sec. III we analyze the resulting electronic and magnetic structures of BNOAS in Sec. III, and briefly provide some results for applied strain in Sec. IV. In Sec. V we introduce a minimal spin model for this compound which we feel provides a platform for accounting for the peculiar behavior of χ(T ). A discussion and wrap-up is given in Sec. VI.
An intriguing feature of the structure is that the (AgSe) 2 substructure is the same as that of the FeSe-type superconductors. The overall structure is reminiscent of many cuprate and Fe-based superconductors: electronically active (NiO 2 ) layers separated by inactive "spacer" layers. The strain supplied by the (AgSe) 2 layers results in an anomalous elongation of the Ni-O bond length to 2.10 (2.05)Å for BNOAS (SNOAS), about 10 (6)% larger than in CaCuO 2 and also other layered d 8 nickelates. The (AgSe) 2 layers however are insulating and relatively inert in these compounds.
Our calculations were performed with the experimental crystal structure, using the BNOAS lattice constants a =4.2095Å and c =19.8883Å, obtained from the neutron diffraction measurement at 5 K [8], except for a few cases where smaller planar lattice constants a were used to probe the dependence of the Ni-O separation and identify a critical separation. We have compared results from two all-electron, full potential density functional theory (DFT) codes, fplo-14 [15] and wien2k [16], the former being an atomic orbital basis code. The latter is a linearized augmented plane wave code that is considered to give the most precise solutions of the quantities that arise in DFT. In many tests these two codes have produced results that are equivalent within the accuracy of the underlying DFT exchange-correlation functionals.
fplo is a local orbital based DFT code with builtin atomic orbitals, particularly adapted to characterizing the electronic and magnetic structures of transition metal compounds. [23] In wien2k, the basis size was determined by R mt K max = 7 with augmented-plane-wave sphere radii R mt (in a.u.): Ni, 2.13; O, 1.83; and 2.50 for Ag, Se, and Ba. For both codes, the Brillouin zone was sampled by a dense k−mesh of 20 × 20 × 20 to check the energetics carefully.
We utilize the generalized gradient approximation (GGA) [17] plus Hubbard U method (GGA+U ) [18], with the double counting term of the so-called around mean field scheme [18,19] as implemented in rotationally invariant fashion in the two codes we use. We note that, since U and J are applied to the 'atomic orbital' (which is not uniquely defined) in different ways by the two codes, results differ somewhat. An important point of our work is that these two codes give qualitatively similar results for which the quantitative differences are unimportant. To probe correlation effects in the GGA+U calculations, we have at times varied U in the range 1 − 9 eV, always with the fixed Hund's spin coupling J = 0.7 eV. Varying U is accepted practice. There is no specific value to use for Ni (or any specific atom), because the large screening of the atomic value depends on the charge state, on the local environment, and on the conducting (or not) behavior of the system, the latter two providing screening. Small values are more appropriate for conducting LaNiO 3 , larger values for highly insulating NiO. Values of U ≈ 4 − 8 eV have been applied in different contexts, [2,[20][21][22] (and see references in [11] for LaNiO 2 ). For BNOAS, which undergoes a metalinsulator transition in our calculations as U increases, values in the 5-7 eV range seem most physical. It is common practice to check the sensitivity of DFT+U results for sensitivity to the value of U . Our results are not sensitive to U within the 5-7 eV range, or even higher.

III. RESULTS FOR BNOAS
We begin by announcing that our results are unexpected and rather startling. The formal d 8 configuration conventionally suggests that the possibilities are (i) a high spin S=1 Ni ion, with holes in both the d x 2 −y 2 and d z 2 minority orbitals, or (ii) a non-magnetic low spin S=0 state with d x 2 −y 2 of both spins unoccupied. We find another, (iii) an off-diagonal singlet (ODS), with holes of opposite spins in the d x 2 −y 2 and d z 2 orbitals ( 1 B 1 symmetry), with schematic energy level diagrams at the GGA level presented in Fig. 2(c). (The t 2g states are nearly degenerate at the GGA level.) The ODS is described and analyzed in Sec. III.D.
There are unusual features of the spin density, pictured in Fig. 1(b). Distinctive features are the following.
(1) While the net moment is exactly zero (singlet S=0), there remains full spin polarization of the two orbitals, with substantial magnetic energy benefit. This specifies a singlet [24] with internal texture, which is orbital polarization. (2) There is full spin-orbital polarization, unlike what occurs in some insulating nickelates. (3) This singlet is a pure spin analog of the non-magnetic Eu 3+ f 6 ion with S=3, L=3, J = |L−S| = 0, a singlet which nevertheless has large spin and orbital polarization. [25][26][27] (4) It violates Hund's first rule, without apparent cause (such as crystal field splitting). (5) It gains Coulomb energy by remaining maximally anisotropic (fully orbitally polarized) within the two-orbital subspace. Properties related to (1,2,5) were discovered in DFT+U studies [28] of MnO under pressure. We will return to this comparison in Sec. VI.

A. Energetics and Moments
The gap values of ODS, FM, and AFM states versus U are shown in Fig. 2(a). Not surprisingly, the gap opens for much smaller U = 1eV for AFM order due to its smaller bandwidth; for the ODS the gap opens only at U =5 eV. In all cases the gap remains small for a transition metal oxide, even for U =9 eV. Next, we compare energies of the FM and AFM states relative to the ODS state in Fig. 2(b). NM (trivial S = 0 1 A singlet) lacks spin polarization energy and can be neglected. Both FM and AFM ordered states have S = 1 Ni ions with their Hund's rule energy gains, with AFM being slightly favored over FM with a difference that decreases with increasing U . The ODS state, which is spin-orbital polarized but S=0, above U c =4 eV becomes favored over both states having S=1 ions, with the energy difference growing rapidly as U increases. The Hund's exchange energy is overcome by other energy changes, see the Supplementary Material (SM). [29] The values of the FM and AFM moments at GGA level are ∼1.1µ B , as shown in the large lattice constant a region of the upper panel of Fig 3. Including U , these grow (not shown) to around 1.9µ B , a value that is characteristic of S=1 reduced somewhat by hybridization. To emphasize: for the ODS ions, the orbital moment is around 0.16µ B while the net spin moment vanishes.

B. Effect of strain
Matsumoto and coauthors [8] have drawn a connection between the Ni-O bond length and the low-spin and highspin states in layered d 8 compounds, identifying a critical separation in the 2.00-2.05Å range and ascribing the transition to crystal field splitting. This critical distance led us to study the lattice constant dependence, taking the FM state as an example. We performed the corresponding calculations, with results shown in Fig. 3 using fplo within GGA (wien2k results are similar). As a is decreased from 4.2Å to 3.9Å with the cell volume fixed and internal parameters of Ba and Se being optimized (the observed value is a=4.21Å), an electronic transition at Ni-O is observed. The Ni moment initially decreases (∆M/∆a ≈ 1.6µ B /Å). When the Ni-O separation is lowered below the critical value of a c /2 ≡ d N i−O c =2.03Å, the rate of decrease of the Ni moment doubles.
At this same separation, the (magnetic) energy undergoes a dramatic change. The energy increases slowly and quadratically down to 4.06Å. Below a = 4.06Å, where Matsumoto and collaborators suggest a crossing of magnetic and nonmagnetic Ni ions, the energy difference rises steeply with a slope of -300 meV/Å. This electronic transition involves a reconfiguration of how orbital occupations change with volume.
This behavior is consistent with the critical Ni-O separation proposed by Matsumoto et al. They observed that most NiO 2 compounds with only square planar Ni (no apical oxygens) are nonmagnetic, but if the lattice constant is pushed above 2.0Å, the Ni ion becomes magnetic. For example Sr 2 NiO 2 (AgSe) 2 , with a=4.094Å, is magnetic; BaNiO 2 and LaNiO 2.5 are not. The significant new aspect is that while our Ni ion is spin-polarized, i.e. magnetism must be considered in the electronic structure, it forms a singlet and is not magnetic to most experimental probes.

C. Electronic structure at the GGA level
The band structure at the GGA level, which forms the basis for most beyond-GGA studies, is presented in Fig. 4 from the atom-projected density of states (PDOS). The d z 2 band is degenerate with the t 2g bands, with a similar 1 eV bandwidth. The Ni d x 2 −y 2 dispersion is 3 eV (see the Γ − M line) with center of weight around 0.5 eV, giving an e g crystal field splitting of 1-1.5 eV. The lowest conduction band at Γ has Ag s and Se p z characters, which become mixed with Ni d z 2 as that orbital becomes unoccupied with application of U . This mixed character will have implications for electron-doped BNOAS.
Dimensionality of the band structure is one central point of interest in nickelates as well as cuprates. k z dispersion along Γ − Z of valence bands is small except for the Ni d z 2 band just below E F , with dispersion of roughly 0.5 eV, accommodated by hybridization with the Se p orbitals, i.e. through the electronic polarization of the blocking layer. The two 3d holes are mostly in the d x 2 −y 2 orbitals with a minor amount in the small Fermi surfaces with mostly d z 2 character. These differences are central factors in how the Coulomb repulsion U will produce changes in the spectral positions. In the conduction band, the first band with minimum at Γ at 1.5 eV shows in-plane dispersion of 1 eV and k z dispersion from Γ of nearly 0.5 eV.

D. Correlation and spin polarization
The PDOS displayed in the top panel of Fig. 5 reveals the paradigm-shifting effect of U . The ODS state is obtained already for U =1 eV, though not yet favored energetically. The gap opens for U in the 4-5 eV range, beyond which the singlet is increasingly strongly favored (see Figs. 2(a,b)). The spin density shown in Fig. 1(b) is strongly anisotropic (maximally so), similar to that obtained previously [2] for d 9 LaNiO 2 , where large U drove the system toward an unexpected d 8 configuration. U =6-8 eV was judged to be too large for conducting LaNiO 2 , which is observed to be conducting and would have a more strongly screened (smaller) value of U . [30]  The noticeable difference is that the occupied d z 2 band is 1.5 eV lower for the FM case at U =2 eV. A close look indicates that the t 2g PDOS is lower for FM than for ODS. At U =4 eV and above, the d state positions have become very similar.

E. Analysis of the ODS
The band structure of the ODS state is shown in Fig. 6, with the corresponding PDOS of relevant orbitals pictured in Fig. S3 of the SM. [29] Both e g orbitals are split by roughly U =7 eV (the Mott-Hubbard gap), with unoccupied d x 2 −y 2 bands 1.5 eV higher in energy as expected. A substantial effect of U is to remove the t 2g orbitals (both spins occupied) from just below the GGA Fermi level to -4 to -5 eV below the gap, for U =7 eV. This displacement serves to leave the Se 4p states at the bottom of the gap at Γ. Note: with zero net spin, we have chosen that 'up' is the direction of the d x 2 −y 2 spin, 'down' is of course the opposite.
The gap opening in both spin directions by U c ≈ 5 eV signals a metal-insulator transition and ensures that the net moment vanishes, giving the ODS state. This state violates Hund's first rule, signaling that other (intraatomic or interatomic [environmental]) contributions to the energy are compensating. We propose that one factor arises from the differences in crystal field splittings from neighboring divalent cations (Ba or Sr) versus the tripositive ions in d 9 LaNiO 2 and NdNiO 2 . More discussion is given in SM. [29] The unoccupied Ni d z 2 hole band shows dispersion with a minimum at Γ, about 1 eV from Γ to the zone edge, and 0.5 eV width along k z . This orbital is mixed with the Ag s, Se p z orbitals mentioned in Sec. III.B. The gap increases to 0.5 eV for U =8 eV, as occupancies attain essentially integer values. The spin density shown in Fig. 1(b) vividly illustrates the spin polarization that is a combination of d z 2 and d x 2 −y 2 contributions of opposite spins, with d x 2 −y 2 orbitals mixed with oxygen p σ orbitals. We return in Sec. V to further implications of this ODS.
The electron-hole asymmetry in the U =4-7 eV range is anomalous for a transition metal oxide, rendering the usual Mott insulator versus charge transfer insulator inapplicable. Oxygen 2p states do hybridize with d x 2 −y 2 but not as strongly as conventionally, since they lie 2-4 eV below the gap. Hole states, see Fig. 6, are instead Se 4p states in the down spin bands of the Ag 2 Se 2 spacer layer and are highly itinerant. Electron states also present new behavior: electrons go into the weakly k z dispersive d z 2 band at Γ hybridized with Se p states, thus disturbing the singlet and enabling moments to emerge. The consequences of doping will be addressed elsewhere.

A. The model
The insulating ground state ODS that is discussed above provides a system in which the elementary excitations are spin waves. We suggest here a minimal model of the spin system. Let the d x 2 −y 2 and d z 2 spin operators be denoted by the Pauli matrices σ and τ respectively. Factors of 1 2 will be folded into the constants. The minimal spin Hamiltonian for the insulating state contains three parameters: (1) J ≡ J σ is the positive in-plane near-neighbor (< i, j >) superexchange constant between d x 2 −y 2 σ spins, and K is the on-site spin coupling between σ and "Kondo-like" τ spins, with contributions from Hund's coupling, Hubbard U correlation, and environmental influences. The on-site singlet-triplet splitting is 2K, and the sign of interest here (making it Kondo-like) being positive K. J z is the (spacer layer assisted) coupling between τ -spin neighbors [i, j] alongẑ, and the total on-site spin is S i = σ i + τ i couples to the magnetic field B. For discussion in this section we set units such that J=1.
H KS , which we refer to as the Kondo sieve model, is an extension to 2D of the Kondo necklace model introduced by Doniach. [32] It was derived from the 1D Kondo Hamiltonian by applying the Luther-Pelcher transformation [33] from 1D spinless itinerant particles to on-site spins. Quantum Monte Carlo calculations indicate that the 1D Kondo necklace does not display magnetic order at any finite value of the exchange coupling constant. [34] The extension to 2D by Brenig [35] has been known (somewhat contradictorily) as the 2D Kondo necklace. This two spins per site model can also be viewed as the large Hubbard U regime of the Kondo-Hubbard model at half filling. [36] We picture BNOAS, in first approximation, as a realization of the Kondo sieve model, then with small and mostly frustrated interlayer coupling that may become important in emergence of ordered phases.
Quite a bit is known from Brenig about the Kondo sieve model. For J=0, H KS consists of isolated singlets; for K=0 it is the exhaustively studied spin-half 2D AFM Heisenberg model, which does not order at finite T (Mermin-Wagner theorem). Using stochastic series expansion methods, Brenig found that weak AFM order exists for H KS at low T=0.05 up to a quantum critical point (QCP) [35,36] K c =1.4. Below K c the τ -projected order parameter is around 35% larger than that of the neighbor-coupled σ (d x 2 −y 2 ) spins.
The ordering reveals that the Kondo spin promotes long range order even in 2D, and becomes a central influence even for a small Kondo coupling, reflecting apparent smaller quantum fluctuations than the σ spin. The uniform susceptibility decreases with increasing K, vanishing at K c above which the singlets control the physics. Site dilution of the "Kondo" moments, as by non-stoichiometry or doping, results in structure in both χ(Q AF M , T ) and S(Q AF M , T ) around T=0.1 or below, depending on concentration. The QCP vanishes into a crossover, i.e. weak AFM order extends to larger K, for which an order-by-disorder mechanism was proposed. [36] These results will also impact the doping experiments discussed below.

B. Relation to Ba2NiO2(AgSe)2
Due to the body-centered stacking of NiO 2 layers both the τ and also much smaller σ couplings between AFM layers will be frustrated. The 0.5 eV k z dispersion of the d z 2 band imbues importance to this coupling, since it is the largest coupling available to enable 3D magnetic correlations and potential long-range 3D order. An inplane term <i,j> τ i · τ j and interlayer coupling [i,j] σ i · σ j are symmetry-allowed, but both should be small.
Of special interest, insightful for the large K that our calculations indicate, is that the on-site (i) term can be diagonalized in total spin S i = σ i + τ i space, with singlet S i =0 and triplet S i =1 sectors. For the isolated ion K is negative, giving the familiar high-spin Hund's coupling. In BNOAS we find instead from total energy results that the low-spin ODS configuration is favored (rather strongly) for physical values of U [see Fig. 2(b)]. The non-magnetic 1 A state with two holes in the d x 2 −y 2 orbital is highly disfavored and is not included in the Kondo sieve model.
Given that the low-spin ODS state is favored in BNOAS, in zero-th approximation configurations are confined to the ODS sector S i = 0 for all sites i. Then the second term in H becomes diagonal and the fourth term in the Hamiltonian vanishes (and also hence the linear susceptibility as in experiment), and τ i = S i − σ i can be eliminated to give (up to a constant) The in-plane superexchange J is ubiquitously calculated to be positive and large in nickelates, so the first term is the one that is commonly applied to the AFM insulating phase of cuprates. Since the σ spins are antialigned (or at least strongly correlated), the body-centering of the BNOAS structure determines that interlayer τ i · τ j -here transformed to σ i · σ j -coupling is frustrated. The result from the Kondo sieve model without considering further symmetry breaking (such as single ion anisotropy or structural distortion) is that the σ spins are strongly correlated in-plane and become ordered below K c /J = 1.4, with experimental properties becoming nonstandard for a magnetically ordered system. The substantial Kondo coupling indicates that weak AFM should be considered in the interpretation of magnetization data. Ordering across the layers due to additional interactions may be a lower temperature possibility that is left for further work. For comparison with experimental data, the sensitivity of the QCP to defects should be kept in mind, since the synthesis methods of this class of compounds have been found to allow the incorporation of impurities and produce some minority phases. [37] V. DISCUSSION Our results suggest that the non-Curie-Weiss form and the peculiar structure in χ(T ) of Ba 2 NiO 2 (AgSe) 2 are couched in formation of a Ni on-site 'off-diagonal singlet' in which both e g orbitals are singly occupied but with Kondo-like oppositely spin-directed singlets. This singlet forming tendency was obtained previously [2] for infinitelayer LaNiO 2 at large interaction U -very strange for a supposed d 9 metal -but seemed not to be reflected in measured properties. Among the numerous theoretical viewpoints of NdNiO 2 , the undoped parent of the newly discovered superconducting nickelate, Zhang et al. have suggested that Kondo singlets play a part in the underlying electron structure. [38] We propose that BNOAS may comprise the first realization of Kondo sieve physics, and that it may provide important insight into layered nickelate physics which now includes superconductivity when appropriately doped. [6,7] We now turn to consideration of χ(T ).
Scenario #1: Integrity of the Singlet. Above the characteristic temperature T m = 130 K of BNOAS the S=0 singlets are magnetically inert, which naturally accounts for the lack of a Curie-Weiss term. The internal structure presumably correlated due to the d x 2 −y 2 coupling, but contributing only a little to χ(T ) due to mixing of the S=1 triplet, consistent with experiment. In the language of the Kondo sieve model, the ordering below T m means that K < K c = 1.4J, i.e. below the QCP. As long range ordering of the d x 2 −y 2 internal structure emerges at and below T m , magnetic behavior emerges apparently requiring involvement of the S=1 sector, only to return (in zero field cooled data) to invisibility in χ in the ordered state. This long-range correlation and ordering of weak moments (including a small orbital moment, which we calculate to be 0.16µ B parallel to the d x 2 −y 2 moment) should be detectable, though possibly challenging, by neutron diffraction, by utilizing the magnetic (spin + orbital) structure factor that contributes to the magnetic density correlation function. Muon spin resonance is also a particularly sensitive method to detect magnetic order.
Probably more telling is that the susceptibility will be distinct from that of truly non-magnetic ions, with some T-dependence arising from a van Vleck contribution from the proximity in energy of the S=1 states. The fieldcooled susceptibility below T m may arise from domain structure in which S=1 moments at the domain boundaries come into play. To repeat from above: extrinsic phases may interfere in the data with intrinsic behavior.
Scenario #2: Quenching of the Singlet. The preceding discussion has assumed the integrity of the quantum ODS singlet. This is textbook behavior as long as the two spins remain in a coherent singlet, which conventionally assumes degenerate orbitals and negligible coupling to the environment. Neither of these conditions applies strictly to BNOAS. At elevated temperature these "perturbations" may average out so the singlet survives, but may become more influential as temperature is lowered. Similarly to the quenching of atomic orbital angular momentum by environmental effects (non-spherical potential), the ODS may revert to the non-quantum pair of individual but coupled σ and τ spins (as described by DFT). Coupling then will strictly antialign the individual orbital spins, which can then separately respond to an applied magnetic field, and have separate (and unequal) orbital contributions and provide a distinct susceptibility though probably still small. The magnetic transition at T m would, in this scenario, correspond to the quenching of the singlet, with consequences that may not have been studied previously.
Analogous materials. Related behavior was observed in DFT+U studies of MnO under pressure, which both experiment and theory identify as a high spin to low spin (but not zero spin) transition [31] in the vicinity of an insulator-metal transition. In DFT+U calculations [28], the low pressure S = 5 2 Mn ion first undergoes, un-der pressure, an insulator-to-insulator transition in which two Mn 3d orbitals flip spin direction whilst each orbital remains fully spin polarized, i.e. an S = 5 2 → S = 1 2 spin collapse without change in orbital occupation. This spin collapse amounts to an even more drastic violation of Hund's first rule than in BNOAS that is discussed here. The origin of the energy gain was traced to the persisting strong (maximal) anisotropy in spin and charge, and to a change in pdσ hybridization. In that system MnO was also in proximity to a critical metal-oxygen separation. Both of these characteristics (anisotropy and a critical bond length) are also apparent in BNOAS.
A prime interest in BNOAS will be how it may inform the electronic structure and magnetic behavior in superconducting (Nd,Sr)NiO 2 . As mentioned, the ODS tendency was seen fifteen years ago in LaNiO 2 , which is commonly used as a stand-in for NdNiO 2 in correlated electron calculations. It should be mentioned that others have not reported obtaining this singlet tendency in LaNiO 2 . However, in BNOAS we obtain it with two different codes and GGA+U implementations. We obtain the magnetic states others report but they have higher energies (for physical values of U ). There are still things to be learned about the peculiar behavior of the d z 2 orbital in infinite layer nickelates. It is of considerable interest to try to electron-dope BNOAS toward the d 9−δ regime where superconductivity emerges in (Nd,Sr)NiO 2 .
Another related system is Ba 2 CuO 3.2 , which superconducts at 73K [39] after having been synthesized at 18 GPa and 1,000C. Although not described as such, this compound is an infinite layer cuprate that is hole-doped with 0.2 apical oxygen atoms per copper, making it formally somewhat overdoped. The (average) apical Cu-O distance is small at 1.86Å, whereas the in-plane Cu-O separation is 2.00Å, larger than other superconducting cuprates. The interpretation of spectral data suggested that the crystal field has exchanged the positions of the d z 2 and d x 2 −y 2 levels, which would provide a new paradigm for cuprate superconductivity. The relationship of this material to BNOAS suggests the usefulness of further study of both systems.
While BNOAS is not simple to synthesize due to the need of high pressure, it is nevertheless available (after synthesis) to experimental probes that are not available to MnO at 100 GPa pressure. We encourage further synthesis and study of BNOAS by thermodynamic and spectroscopic (electron, neutron, and muon) probes to illuminate the peculiar magnetic behavior observed in this infinite layer nickelate.

VI. SUMMARY
Our density functional theory with correlation effects study of Ba 2 NiO 2 (AgSe) 2 has uncovered the likelihood that an off-diagonal singlet arises in the subspace of the two e g holes (or electrons) in its d 8 ionic configuration. This "low spin" singlet however has orbital polarization, i.e. orbital texture, with one orbital's spin being coupled to neighbors while the other is not. Moreover, a critical Ni-O separation of 2.03Å has been obtained, which seems to be associated with the distinction between low spin and high spin Ni d 8 ions in square planar coordination, occurring at the same Ni-O distance. Two scenarios have been presented, either of which might account for the peculiar measured susceptibility: a magnetic transition at T m =130K but non-magnetic behavior above T m .

VII. ACKNOWLEDGMENTS
We acknowledge assistance from Young-Joon Song and Mi-Young Choi in the early stages of this work. The spin model aspects of this paper have benefited from feedback and references from R. T. Scalettar and comments from R. R. P. Singh. We appreciate extensive comments on the manuscript from M. D. Johannes and thank A. B. Shick for the use of his code that calculates the anisotropic U and J matrices reported in the Supplemental Material. H.S.J and K.W.L were supported by National Research Foundation of Korea Grant No. NRF2019R1A2C1009588. W.E.P. was supported by National Science Foundation Grant DMR 1607139.