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J. Okamoto, G. Shibata, Yu. S. Ponosov, [H. Hayashi](https://orcid.org/0000-0001-7787-9082), [K. Yamaura](https://orcid.org/0000-0003-0390-8244), H. Y. Huang, A. Singh, C. T. Chen, A. Tanaka, S. V. Streltsov, D. J. Huang, A. Fujimori

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[Spin-orbit-entangled state of Ba2CaOsO6 studied by O K-edge resonant inelastic X-ray scattering and Raman spectroscopy](https://mdr.nims.go.jp/datasets/e93d8291-2f12-43bf-bf45-b5ce367f1275)

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Spin-orbit-entangled state of Ba2CaOsO6 studied by O K-edge resonant inelastic1X-ray scattering and Raman spectroscopy2J. Okamoto,1 G. Shibata,2 Yu. S. Ponosov,3 H. Hayashi,4, 5, ∗ K. Yamaura,4, 5 H. Y. Huang,1 A. Singh,13C. T. Chen,1 A. Tanaka,6 S. V. Streltsov,3, 7, † D. J. Huang,1, 8, 9, ‡ and A. Fujimori1, 2, 10, 11, §41National Synchrotron Radiation Research Center, Hsinchu 300092, Taiwan52Materials Sciences Research Center, Japan Atomic Energy Agency, Sayo, Hyogo 679-5148, Japan63Institute of Metal Physics, 620041 Ekaterinburg GSP-170, Russia74Research Center for Materials Nanoarchitectonics (MANA),8National Institute for Materials Science, Tsukuba, Ibaraki 305-0044,Japan95Graduate School of Chemical Sciences and Engineering,10Hokkaido University, Sapporo, Hokkaido 060-0810, Japan116Department of Quantum Matter, Hiroshima University, Hiroshima 739-8530, Japan127Department of Theoretical Physics and Applied Mathematics,13Ural Federal University, 620002 Ekaterinburg, Russia148Department of Physics, National Tsing Hua University, Hsinchu 300044, Taiwan159Department of Electrophysics, National Yang Ming Chiao Tung University, Hsinchu 300093, Taiwan1610Center for Quantum Science and Technology and Department of Physics,17National Tsing Hua University, Hsinchu 300044, Taiwan1811Department of Physics, The University of Tokyo, Bunkyo-Ku, Tokyo 113-0033, Japan19(Dated: February 26, 2025)20Transition-metal ions with 5d2 electronic configuration in a cubic crystal field are prone to havea vanishing dipolar magnetic moment but finite higher-order multipolar moments, and they areexpected to exhibit exotic physical properties. Through an investigation using resonant inelasticX-ray scattering (RIXS), Raman spectroscopy, and theoretical ligand-field (LF) multiplet and abinitio calculations, we fully characterized the local electronic structure of Ba2CaOsO6, particularly,the crystal-field symmetry of the 5d2 electrons in this anomalous material. The low-energy multipletexcitations from RIXS at the oxygen K edge and Raman-active phonons both show no splitting.These findings are consistent with the ground state of Os ions dominated by magnetic octupoles.Obtained parameters pave the way for further realistic microscopic studies of this highly unusualclass of materials, advancing our understanding of spin-orbit physics in systems with higher-ordermultipoles.I. INTRODUCTION215d transition-metal oxides exhibit various exotic phys-22ical properties arising from the strong spin-orbit cou-23pling (SOC) that competes with Hund’s coupling and24Jahn-Teller effect and strongly influences the exchange25interaction [1–3]. Particularly attractive have been the26Mott-insulating Ir4+ (5d5) compounds with effective an-27gular momentum Jeff = 1/2 and the Kitaev quantum-28spin-liquid candidate of Ru3+ (4d5) honeycomb lattices.29Especially exotic are localized 5d2 electrons in a cubic30crystal field. A d2 ion coordinated by ligand atoms in31the octahedral (Oh) environment is expected to be Jahn-32Teller active, but many cubic crystals with 5d2 ions re-33main undistorted, probably due to the strong SOC of34the 5d electrons [4]. Such a 5d2 ion in the Oh-symmetry35crystal field has a non-Kramers doublet ground state that36supports either an electric quadrupole or a magnetic oc-37∗ present address: Institute for Solid State Physics, The Univer-sity of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581,Japan† email: streltsov@imp.uran.ru‡ email: djhuang@nsrrc.org.tw§ email: fujimori@phys.s.u-tokyo.ac.jptupole [5–9]. This is contrasted with 5d1 systems, where38the 5d ion has an electric quadrupole and Jahn-Teller dis-39tortion is induced [1, 2, 4, 7, 10]. As a staggered octupolar40order has been predicted theoretically for quarter-filled41manganites [11], one may conceive disordered octupoles42or a quantum octupole liquid under particular conditions43in manganites or other materials.44The B-site-ordered double perovskite Ba2AOsO6,45where A is an alkali-earth metal, is one of such 5d2 (Os6+)46systems. The face-centered cubic (fcc) lattice formed47by the Os ions may induce geometrical frustrations be-48tween the multipoles, and may lead to intriguing quan-49tum magnetism predicted theoretically [5–9]. For exam-50ple, Ba2CaOsO6 shows a cusp-like anomaly in the mag-51netic susceptibility at T ∗ ∼ 50 K, signaling a magnetic52transition, and muon-spin rotation (µ-SR) has revealed53a small magnetic moment of ∼ 0.2 µB [12] or ∼ 0.05 µB54per Os ion [13] whereas neutron scattering has detected55no magnetic Bragg peaks below T ∗ [12]. (According to56Ref.[13], the small magnetic moment may be induced by57impurities and may not be intrinsic.) According to X-58ray diffraction, the crystal remains cubic down to the59lowest temperatures [14]. This precludes static electric60quadrupolar order, which should distort the cubic lat-61tice [10], but is consistent with magnetic octupolar order62as the origin of the ‘hidden order’ in Ba2AOsO6. Theo-632Non-Kramersdoublet EgJeff = 2E(b)Jeff = 1Jeff = 2Jeff = 0SOC(a)jeff = 1/2jeff = 3/2t2gegT2gE LF ~ 4 eV ~ 0.7 eV 32c ~ 20 meVLFE ~ 0.4 eV Jeff = 0FIG. 1. Energy diagrams of the Os6+ (5d2) ion in the cubic(Oh) crystal field. (a) One-electron energy diagram. ∆LF isthe t2g-eg splitting. Spin-orbit coupling (SOC) further splitsthe t2g level into jeff = 12and jeff = 32levels separated by 32ζ,and the jeff = 32level is occupied by two electrons. (b) Two-electron energy diagram. The t22g part of the d2 multipletis shown whereas the t2geg part is located at higher energiesseparated by ∼ ∆LF. Due to the ligand field (LF) of cubicsymmetry, the Jeff = 2 ground state is split by a ‘residual’cubic splitting ∆c into the non-Kramers Eg doublet and theT2g triplet. The first Jeff = 0 excited state appears ∆E ∼ 0.4eV above the ground state, and the first Jeff = 1 excited state∼ 0.065 eV above it.retically, exchange coupling between neighboring Os ions64favors ferro-octupolar order in the fcc lattice [6].65In an Oh-symmetry crystal field, the one-electron Os665d level is split into the t2g and eg levels separated by67∆LF [Fig. 1(a)]. The strong SOC splits the t2g level into68the jeff = 12 and jeff = 32 sublevels, and the latter is oc-69cupied by the two electrons of the Os6+ ion. In the two-70electron energy diagram [Fig. 1(b)], the ground state has71the total effective angular momentum of Jeff = 2. Un-72der the Oh symmetry, the ‘residual’ cubic crystal-field73splits the Jeff = 2 quintet into the ground-state Eg dou-74blet and the triply-degenerate T2g excited states with a75separation ∆c [14, 15]. The Eg ground state is a non-76Kramers doublet consisting of |ψg,↑⟩ ≡ |Jzeff = 0⟩ and77|ψg,↓⟩ ≡ 1√2(|Jzeff = 2⟩ + |Jzeff = −2⟩), whereas the78T2g excited states consist of |ψe,±⟩ ≡ |Jzeff = ±1⟩ and79|ψe,0⟩ ≡ 1√2(|Jzeff = 2⟩ − |Jzeff = −2⟩). In the ferro-80octupole-ordered state, all the Os ions are in one of the81two eigenstates, |ψg,±⟩ ≡ 1√2(|ψg,↑⟩ ± i|ψg,↓⟩), of the oc-82tupole operator Txyz ∝ JxJyJz, where the overline de-83notes symmetrization.84While the magnetic properties of Ba2AOsO6 double85perovskites have been widely studied and microscopic86models to explain observed anomalies have been pro-87posed, their electronic structure remains unexplored ex-88perimentally. The present paper aims to fill this gap.89In particular, we studied Ba2CaOsO6 by X-ray absorp-90tion spectroscopy (XAS) and resonant inelastic X-ray91scattering (RIXS) at the O K edge. RIXS studies of925d transition-metal oxides have so far been performed93mainly at the transition-metal L2,3 edge since one can94directly study the spin and orbital excitation of the 5d95states [16–19]. However, owing to the strong SOC of the965d electrons, RIXS at the O K edge can also be used to97study spin excitations [Fig. 1(b)][20–24]. OK-edge RIXS98has the advantage of having higher energy resolution than99transition-metal L2,3-edge RIXS, allowing us to study100low-energy electronic excitation and electron-phonon in-101teraction. We have also utilized Raman scattering to102detect possible local lattice distortion that induces low-103symmetry crystal fields. None of the above measure-104ments have indeed shown evidence for the lowering of the105cubic symmetry, favoring the scenario that the Os ions106have dominantly octupolar moments in Ba2CaOsO6.107II. RESULTS AND ANALYSES108We performed O K-edge RIXS on high-quality poly-109crystalline samples with the energy resolution of ∼30110meV using π polarized X-rays and 90◦ scattering angle111(see Methods). The O K-edge XAS spectrum is shown in112Fig. 2(a). Figure 2(b) shows the RIXS intensity map in113the Ein-Eloss plane, where Ein is the energy of incident X-114ray and Eloss is the energy loss of scattered X-rays. The115same data are plotted in the Ein-Eem plane in Fig. 2(c),116where Eem is the energy of emitted X-ray. RIXS spectra117are plotted in Fig. 2(d). In the figure, above Ein ∼ 529118eV, some spectral features start to shift to higher Eloss119with increasing Ein, indicating a cross-over from Raman-120like to fluorescence-like.121Splitting due to ligand field and spin-orbit coupling122In O K-edge RIXS, the excitation of the O 1s core123electron into empty states followed by the electron tran-124sition from the non-bonding O 2p band to the O 1s core125level leaves a hole in the non-bonding O 2p band and an126electron in the empty states. The resulting final state is127equivalent to that of the O 2p → Os 5d charge-transfer128(CT) excitation, which measures the unoccupied part of129the O 2p partial density of states (PDOS). Because the130energy position of the non-bonding O 2p band is located131near the top of the O 2p band, we assumed it located ∼2132eV below the Fermi level (EF), based on the occupied O1332p PDOS deduced from fluorescence spectra as discussed134below. Thus one finds the eg band 4-6 eV above EF, the135jeff = 12 band ∼1 eV above EF, the empty and occu-136pied parts of the jeff = 32 band just above and below EF,137respectively.138The occupied part of the O 2p PDOS can be measured139by the fluorescence component of the O K-edge RIXS.140We have taken spectrum k in Fig. 2(d) as the represen-141tative fluorescence spectrum. The EF position for the1423FIG. 2. X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS) spectra of Ba2CaOsO6 at theoxygen K edge recorded at 25 K. (a) XAS spectra. Top: XAS spectrum measured using the total fluorescence-yield method.The broad absorption band at 532-536 eV consists of transitions to the empty eg states (blue dashed curve) as well as to theBa 5d- and Ca 3d-derived conduction-band states. The blue dashed curve is the partial fluorescence-yield spectrum measuredat Eem = 528 eV. Bottom: Spectrum calculated using ligand-field (LF) multiplet theory. See Methods. (b) & (c) Coloredintensity maps of scattered X-rays: (b) as a function of incident X-ray energy Ein and energy loss Eloss. Solid lines mark thepositions of constant Eloss. Dashed lines indicate those of constant Eem, (c) Scattered X-ray intensity map plotted against Einand emission energy Eem. Dashed lines indicate constant Eem’s, corresponding to fluorescence from occupied states to the O1s core level. (d) RIXS spectra measured for various Ein’s indicated by vertical bars at the top spectrum of (a). The shadedparts mark transitions from O 2p → Os 5d charge-transfer (CT) excitation. (e) LF multiplet calculation to simulate the RIXSspectra. The red curves show spectra arising from d-d excitation for a series of Ein’s indicated by vertical bars in the calculatedO K-edge XAS spectrum shown at the bottom of (a). The dashed blue curves show O 2p → Os 5d CT excitation simulatedby 5d2 → 5d3 multiplet calculation. The black curve is the measured RIXS spectrum for Ein = 528.5 eV [spectrum d in panel(d)].4fluorescence spectrum has been fixed under the assump-143tion that Eem = 528.2 eV is the excitation threshold of144the O K-edge XAS [photon energy c in Fig. 2(a)]. The145combined occupied and unoccupied parts of the O 2p146PDOS thus derived are plotted in Fig. 3(a).147The obtained O 2p PDOS is compared with the DOS148calculated by DFT (see Methods) in Fig. 3(b). One can149see good one-to-one correspondence between the exper-150imental and calculated structures in the O 2p PDOS.151In particular, the assumed non-bonding O 2p-band posi-152tion well agrees with the peak position of the calculated153non-bonding O 2p band. Nevertheless, the conclusion of154the present paper will not altered by the magnitude of155the band gap unless it collapses and the system becomes156metallic.157Low-energy multiplet and phonon satellites158In order to examine the effect of electron-phonon cou-159pling and possible low-symmetry crystal field, an en-160larged plot of the RIXS spectra in the low-energy region161is shown in Fig. 4(a). The “elastic” peak at Eloss = 0 eV162is a superposition of the genuine elastic scattering (which163should be very weak for the present π scattering geome-164try) and low-energy (0–20 meV) elastic and quasi-elastic165scattering between the nearly degenerate five components166of the Jeff = 2 ground state [see Fig. 1(b)]. The sharp167peak at Eloss ∼ 0.4 eV is due to excitation from the Jeff =1682 ground state to the Jeff = 0 and 1 excited states. The169latter excitation is also observed by Raman scattering as170described below. The weak peak at Eloss ∼ 0.8 eV is171an excited state of the t22g multiplet having the quantum172number Jeff = 2 [Fig. 1(b)]. Unfortunately, the residual173cubic splitting ∆c of the Jeff = 2 state [Fig. 1(b)] is too174small to be resolved in the RIXS spectra. Each of the175quasi-elastic, Eloss ∼ 0.4 eV, and ∼ 0.8 eV peaks are176accompanied by sub-peaks with ∼80 meV intervals on177the higher-energy side. These sub-peaks are attributed178to phonon replicas, as described below.179Analyses using ligand-field multiplet theory180The magnitude of the SOC of the Os 5d states can181be estimated from the O K-edge XAS [Fig. 2(a)]. X-ray182absorption into the empty t2g state observed at 528–530183eV is split into double peaks separated by 32ζ ∼ 0.7 eV.184Note that for the effective angular momentum operator185leff , the SOC constant ζ ′, defined through the SOC en-186ergy ζ ′leff ·s, is given by ζ ′ ≡ −ζ because leff ≡ −l for the187t2g electrons [26], where l is the bare angular momentum188operator. The cubic ligand-field (LF) splitting ∆LF of the189Os 5d level into t2g and eg [Fig. 1(a)] can also be esti-190mated from the O K-edge XAS. From the broad absorp-191tion feature at 532–536 eV, the eg component could be192isolated by monitoring the RIXS intensity of the Jeff = 2193(t2g2)→ t2geg energy-loss feature at Eloss ∼5 eV [marked194FIG. 3. O 2p partial density of states (PDOS) derived fromexperiment and theory. (a) Experimental O 2p PDOS. Theempty part is derived from the O 2p → Os 5d CT excitationin the RIXS spectrum [Ein = 528.2 eV, c in Fig. 2(d)], and theoccupied part from the fluorescence component of the RIXSspectrum [Ein = 534.6 eV, k in Fig. 2(d)]. The shaded partmarked by “d → d” arises from t22g → t2geg transition, and isunrelated to the O 2p PDOS. (b) PDOS of the nonmagneticstate obtained by the GGA+U+SOC calculation with U −JH = 2.5 eV.by a solid line in Fig. 2(b)] as a function of Ein: Thus195obtained intensity plotted by the blue dashed curve at196the bottom of Fig. 2(a) gives the eg component, allowing197us to obtain ∆LF ≃ 4 eV.198To interpret the RIXS spectra quantitatively, LF mul-199tiplet calculations were performed and their results are200shown in Fig. 2(e). (For details, see Methods.) The cal-201culated d2 multiplet (red curves) reproduces the observed202loss peaks at Eloss ∼0.4 and 0.8 eV (t22g part), and those203at Eloss ∼4 eV (t2geg part). The RIXS spectra in the blue204shaded parts, Eloss ≃ 2–4 and 6–7 eV, cannot be repro-205duced by the d2 multiplet. We attribute these features to206O 2p→ empty Os 5d CT excitation, and simulate the CT207excitation spectrum by d2 → d3 multiplet calculation of208inverse-photoemission leaving a hole in the non-bonding209O 2p band. By assuming that the non-bonding O 2p band210is located ∼ 2 eV below EF, as indicated by the DFT cal-211culation, and by ignoring the p-band width, we could re-212produce the CT spectrum from the Jeff = 2 ground state213and plotted it by dashed blue curves in Fig. 2(e). The214Eloss ≃ 2–4 eV region arises from O 2p → t2g excitation215and the Eloss ≃ 6–7 eV region arises from O 2p→ eg CT216excitation.217Note that there are no features in the RIXS spectra218that indicate the lowering of the cubic symmetry: If the219LF symmetry were lower than the cubic one, the Jeff = 2220ground state, which is split into the doublet Eg and the221triplet T2g [Fig. 1(b)], would be further split into multiple222states as shown in Fig. 4(b) for cubic (Oh)-to-tetragonal223(D4h) symmetry lowering, and the low-energy part of the224RIXS spectra would be significantly different from the ex-2255FIG. 4. Low-energy excitations measured by O K-edge RIXS.∆c is the ‘residual’ cubic splitting of the Jeff = 2 ground statedefined in Fig. 1(b). (a) RIXS spectra in the low energy-lossregion at different temperatures across the magnetic transi-tion at T ∗ ∼ 50 K for the incident photon energy of 528.5 eV.The d-d excitation shown in Fig. 1(b) is seen. The energy-losspeaks as well as the quasi-elastic peaks are accompanied bymultiple phonon satellites. The splitting ∆c ∼ 20 meV is notresolved in the spectra. Details of the line-shape analysis aregiven in Supplementary Note 1 with Supplementary Figure 1and Supplementary Table 1. (b) Energies of low-lying excitedstates of the Os6+ (5d2) ion as functions of the low-symmetryD4h LF parameter 6Cp [25], the splitting of the t2g level. TheOh LF parameter ∆LF is set to 4.1 eV. |6Cp| ≃ 0.15 repro-duces the low-energy excitation in the O-K RIXS spectra ofthe 5d1 double perovskites [22, 23].perimental ones in Fig. 4(a). If the tetragonal distortion226were comparable to that of the 5d1 double perovskites227Ba2NaOsO6 [22] and Ba2MgReO6 [23], which show peak228at Eloss ∼0.1 eV in the O K-edge RIXS spectra, the229tetragonal crystal field 6Cp, which is equal to the tetrag-230onal splitting of the t2g level [25] should be as large as231±0.15 eV, and the quasi-elastic Jeff = 2 → Jeff = 2 peak232would be split into a few peaks over an energy range233of ∼0.1 eV in addition to the phonon satellites. The234absence of temperature dependence in the spectral line235shapes across the magnetic transition at T ∗ ∼ 50 K sug-236gests that the magnetic transition does not involve any237appreciable structural change. Furthermore, there are no238spectral features that can be attributed to magnons nor239bi-magnons, consistent with the absence of spin order in240Ba2CaOsO6.241The octupolar nature of the ground state of the Os6+242ion in the Oh field can be demonstrated by LF multi-243plet calculation with a weak magnetic field in the (1,1,1)244direction as a time-reversal symmetry-breaking perturba-245tion that splits the non-Kramers Eg doublet into |ψg,+⟩246and |ψg,−⟩. For B = 10 T, we obtained octupolar mo-247ment Txyz ≡ ⟨JxJyJz⟩ ≃ 1.2 with a tiny dipole mag-248netic moment of ∼ 8× 10−3µB induced along the (1,1,1)249direction on top of the octupolar ground state. For a250smaller field of B = 1 T, the induced dipolar moment251was as small as ∼ 7 × 10−4µB. Generally, for a small252field B ≪ ∆c/µB, the induced magnetic dipole moment253along the direction of B (∼ µBB/∆c) is proportional to254B. While ordering with the finite magnetic octupolar255or electric quadrupolar moment can occur within the Eg256ground state, the appearance of a finite magnetic dipolar257moment requires hybridization between the Eg and T2g258states. Thus, to obtain a stable magnetic dipole order by259the exchange interaction, a molecular field of µBB > ∆c260at least is required.261Phonon Raman scattering262To further confirm the absence of low-symmetry crys-263tal field, we employed Raman scattering spectroscopy, a264sensitive probe of lattice symmetry. Figure 5(a) shows265one-phonon Raman spectra of a Ba2CaOsO6 polycrystal266taken at 80 K with two polarization geometries (∥ and267⊥; for technical details, see Methods). There must be268phonons of A1g +Eg + T1g + 2T2g + 5T1u + T2u symme-269tries at the Brillouin-zone center in case of cubic Fm3̄m270structure, out of which four (A1g, Eg, and 2T2g) phonons271are Raman-active.272To obtain information about the symmetry of the ob-273served excitation, polarization measurements were per-274formed in two geometries - with parallel (∥) and with275mutually perpendicular (⊥) polarizations of the incident276and scattered light. We utilized the rules that in isotropic277or cubic systems the depolarization ratio ρ = I⊥/I∥ does278not exceed 0.75 for totally symmetric modes, while it is279close to 0.75 for non-totally symmetric ones [27]. One can280see that the line at ∼796.5 cm−1 obviously dominates in281the ∥ spectrum and can be assigned to the A1g mode,282while the phonons at frequencies 102.5, 375, and 495283cm−1 are observed in both polarized ∥ and depolarized284⊥ spectra and are assigned to T2g, T2g, and Eg modes,285respectively, with the help of the non-magnetic DFT cal-286culation of phonon modes as described in Supplementary287Note 2 with Supplementary Figure 2. The broad peak at288720 cm−1 has a fairly low depolarization ratio (∼0.2),289which suggests its A1g symmetry. While its origin is not290clear, the symmetry lowering cannot split the A1g mode291without the increase of the unit cell. (Note also that29262 0 0 4 0 0 6 0 0 8 0 00 , 111 0 0 0 2 0 0 0 3 0 0 0 4 0 0 00 , 113 0 0 0 4 0 0 0 5 0 0 00 , 40 , 60 , 8A 1 gT| |A 1 gA 1 gE g  T 2 gIntensity (104 * counts)R a m a n  s h i f t  ( c m - 1 )B a 2 C a O s O 6  5 3 2  n m  8 0 KT 2 g( a )Intensity (104 *counts)R a m a n  s h i f t  ( c m - 1 )( b )| |TIntensity (103 *countsR a m a n  s h i f t  ( c m - 1 )| |5 3 2  n m  3 0 0 KTFIG. 5. Raman spectra in two scattering symmetries: ∥ (red)and ⊥ (black) are presented. (a) Frequency range with aone-phonon excitations. (b) Extended frequency range wherethe high-order phonon processes (blue arrows show featuresoriginating from the A1g branch, black - another weak two-phonon features) and electronic excitation are seen. Latestare seen in the range from 3000 to 5000 cm−1, as shown inthe inset, most probably due to the Jeff = 2 ground state tothe Jeff = 1, 0 excited states seen by RIXS (Fig. 4).the second intensive T2g mode remains unsplit.) The ap-293pearance of this low intensity A1g peak can be, e.g., due294to imperfections of the crystal structure (e.g. there are295indications of anti-phase boundary defects with disorder296at the B sites and other types of defects [28–31]) or a297two-phonon repetition of 375 cm−1 vibration. Thus, our298Raman experiments do not detect any direct evidence of299the symmetry lowering in the cubic Ba2CaOsO6.300To examine the possible lattice distortions across T ∗ ∼30150 K, we measured spectra at 10 K with better resolu-302tion and found only minor changes in the spectrum, such303as further hardening and narrowing of the A1g mode at304∼796.5 cm−1. Interestingly, the frequency and unique305line width of the low-frequency T2g mode did not change306in the whole temperature range of 10-300 K.307The higher frequency range from 800 to 3000 cm−1 pre-308sented in Fig. 5(b) shows several peaks dominating in the309∥ geometry and hence of the A1g symmetry. The weak310peaks at 870 and 1055 cm−1 indicated by black arrows311are probably two-phonon features. The peaks at 1450,3121540 and 2310 cm−1 shown by blue arrows can be as-313sociated with double and triple phonon scattering from314the A1g phonon branch. At higher frequencies (>3000315cm−1), two broad peaks are observed. They are clearly316seen in both polarization geometries (∥ and ⊥), as well317as upon excitation by both used laser lines (532 and 633318nm, see Methods), which indicates Raman scattering by319electron excitations of T2g or Eg symmetry. Their en-320ergies agree well with the Jeff = 2 → Jeff = 0, 1 RIXS321peaks (Fig. 4).322Typically, in the case of 5d2 double perovskites, the323effect of tetragonal distortions on the ground state is324considered due to their stronger coupling with electronic325structure [32]. Interestingly, Rayyan et al. [33] included326the trigonal distortions in their analysis and demon-327strated that, while these distortions are unable to split328the Eg doublet due to symmetry constraints, they can329lift the degeneracy of the higher-lying T2g triplet. Under330trigonal distortions, some of these T2g states go to a lower331energy. However, as shown in [33] there is a rather wide332range of trigonal distortion under which the non-Kramers333Eg doublet remains the ground state. This doublet may,334therefore, survive as the ground state under a small lat-335tice distortion. It is also possible that distortion is sub-336stantially reduced due to dynamical Jahn-Teller effect.337Whether the Eg doublet hosts an electric quadrupole or a338magnetic octupole or both depends on the type of broken339symmetry (spatial symmetry or time-reversal symmetry)340and the strength of the mean field from neighboring Os341ions.342Electron-phonon coupling effects on RIXS343In the low-energy RIXS spectra shown in Fig. 4, the344quasi-elastic peak and the peak at Eloss ≃ 0.4 eV are ac-345companied by sub-peaks separated by ∼80, 160, and 240346meV with decreasing intensities. We attribute the sub-347peaks to phonon replicas created by the simultaneous348excitation of optical phonons. The one-phonon energy of349∼80 meV is somewhat lower but in a similar range as350the Raman A1g mode energies 720 cm−1 = 88 meV and351796.5 cm−1 = 99 meV. The replica energies are close to352those observed in the RIXS spectra of Ba2NaOsO6 [22].353From the replica intensities, the dimensionless electron-354phonon coupling constant is estimated to be M/ω0>∼ 1,355where M is the average electron-phonon coupling matrix356element and ω0 is the phonon energy [34]. In spite of357the moderately strong coupling, Jahn-Teller distortion is358suppressed in Ba2CaOsO6 due to the strong SOC and359dynamical Jahn-Teller effect, suggesting that Os-Os ex-360change interaction is strong enough to stabilize the mag-361netic octupole over the electric quadrupole. Here, it362should be noted that sub-peaks similar to the phonon363replicas may appear in the RIXS spectra if dynamical364Jahn-Teller effect [35] exists, as reported for the 5d1 sys-3657TABLE I. Parameter values for the Os 5d electrons hybridizedwith O 2p orbitals in Ba2CaOsO6 derived in the present XASand RIXS spectra.Parameter Symbol Value (eV)ligand-field splitting ∆LF 4.1Spin-orbit coupling for the 5d shell ζ 0.47Spin-orbit coupling for the t2g shell ζ′ -0.47Hund’s coupling for the t2g shell JH 0.27tem Ba2CaReO6 [18, 19]. Whether such an effect also366exists in 5d2 systems or not is an interesting question to367be pursued in future studies. Considering the different368time scales of RIXS ( on the order of 0.01 fs) and Raman369scattering (> 10 fs), it is possible that dynamical Jahn-370Teller effect was seen in RIXS as the “phonon replicas”371but not in Raman scattering.372III. DISCUSSION373We have investigated the electronic structure of374Ba2CaOsO6 by XAS, RIXS, and Raman scattering ex-375periment as well as DFT calculation, focusing on ex-376tracting reliable parameters characterizing the systems,377as summarized in Table I, and on the confirmation of the378local cubic symmetry of the Os ions that favors the oc-379tupolar state as the origin of its ‘hidden order’. We have380also confirmed the octupolar nature of the non-Kramers381Eg doublet ground state (|Sz| = 0 and |Lz| = 0 under382an infinitesimally small magnetic field) by LF multiplet383calculation.384Owing to the hybridization between the O 2p and Os3855d orbitals, electronic excitation within the 5d2 multiplet386and charge-transfer excitation from the occupied O 2p to387the empty Os 5d states could be identified by the O K-388edge RIXS. From comparison of the XAS and RIXS line389shapes with the LF multiplet calculation, the absence of390splitting of low-energy RIXS peaks as well as the lack of391additional lines in Raman scattering spectra we conclude392that no crystal field lower than the cubic one can be iden-393tified, consistent with the small (∼20 meV) residual cubic394splitting of the Jeff = 2 ground state. The present results395obtained by different types of X-ray and optical spec-396troscopy, which are typically very sensitive to a local en-397vironment of transition metals, substantially strengthen398previous findings, in particular diffraction data demon-399strating the absence of non-cubic distortions [14].400There are two possible mechanisms working hand-in-401hand in suppressing the Jahn-Teller distortion expected402for the Os6+ ion with the d2 configuration. In both403mechanisms, the strong SOC is involved. One is an404on-site effect related to the stabilization of electrons405not at cubic harmonics as the crystal field (i.e. Jahn-406Teller effect) would prefer, but rather on entangled spin-407orbitals [4, 36]. Our RIXS measurements clearly resolved408phonon replicas of the Jeff = 2 → 1, 0, 2 excitation peaks.409This allowed us to estimate the electron-phonon cou-410pling strength, which turns out to be moderately strong,411M/ω0>∼ 1 and, therefore, may not be sufficiently strong412to recover the Jahn-Teller distortion but might induce413dynamical Jahn-Teller effect. On the other hand, there414is also inter-site effect – the energy gain due to exchange415interaction between the octupoles, which are formed by416SOC. Further spectroscopic and theoretical studies are417necessary to identify the octupolar order and its micro-418scopic origin.419METHODS420Materials preparation421Polycrystalline Ba2CaOsO6 was synthesized through422a solid-state reaction using fine powders of BaO2 (99%423purity, Kojundo Chemical Laboratory Co., Ltd.), CaO2424(prepared in the laboratory [37]), and Os (99.95% pu-425rity, Nanjing Dongrui Platinum Co. Ltd.) in a ratio426of 2:1:1.1. Approximately 200 mg of the mixed materi-427als were placed into an alumina crucible. The mixture428was then heated in air to 1000°C for 7 hours, followed429by a 1-hour dwell time, and subsequent cooling to room430temperature over a span of 7 hours. After re-mixing and431pressing, the sample was annealed at 1000°C for 24 hours.432The resulting product is a grey sintered pellet, possessing433sufficient solidity to be manipulated with tweezers. Pow-434der X-ray diffraction analysis was performed using Cu435Kα radiation within the 5◦ ≤ 2θ ≤65◦ range at 293 K.436The measurements were conducted with a MiniFlex600437diffractometer (Rigaku, Tokyo, Japan). The acquired438data, shown in Supplementary Figure 3, exhibited good439agreement with simulations based on the crystallographic440data of Ba2CaOsO6 [12], confirming the single-phase na-441ture of the product.442Resonant inelastic X-ray scattering443All resonant inelastic X-ray scattering (RIXS) and X-444ray absorption spectroscopy (XAS) measurements at the445O K edge were performed using the AGM-AGS spec-446trometer of beamline 41A at Taiwan Photon Source447of National Synchrotron Radiation Research Center448(NSRRC) [38]. This beamline is based on the energy449compensation principle of grating dispersion [39]. The450energy bandwidth of incident X-ray was 0.2 eV (0.1 eV451for XAS measurement) while keeping the total energy452resolution of RIXS as 30 meV at the incident photon453energy of 528.5 eV. The sample surface was cleaned by454scraping with a diamond file in the Ar glove box before455the measurement and was transferred into the measure-456ment chamber without exposure to the air. The base457pressure of the measurement chamber was ≤ 1 × 10−8458Torr. The sample was cooled down to 25 K with liquid459helium during the measurements. Both RIXS and XAS4608measurements were carried out using linear horizontally461(π) polarized X-rays. The XAS spectra were measured462with a normal-incident X-ray in the total fluorescence463yield mode. For the RIXS measurement, the incidence464angle was fixed at 20◦, and the scattering angle was fixed465at 90◦. The combination of the π-polarized X-rays and466the 90◦ scattering angle makes the RIXS signals purely467magnetic. The same geometry also allowed us to reduce468the elastic peak and to study low-energy excitation effec-469tively.470Ligand-field multiplet calculation471Ligand-field multiplet (LF) calculations were per-472formed by using the XTLS 8.5 package [40]. In the calcu-473lation of the O K-edge RIXS spectra, we assumed that474the excited states of the 5d2 multiplet can be reached475by O K-edge RIXS through the strong Os 5d-O 2p hy-476bridization and could be simulated by the calculation of477Os L2,3-edge RIXS by setting the 2p-5d Slater integrals478and the Os 2p core-level SOC to zero. While this simu-479lation would give the energy positions of RIXS features480correctly, it would not give correct intensities because481relevant transition-matrix elements are not used. The482O K-edge XAS [Fig. 2(a)] was also simulated by the Os483L2,3-edge XAS in the same manner.484In general, the Slater integrals F ’s and G’s (anisotropy485of Coulomb interaction) and the SOC coupling constant486ζ in solids are smaller than those of isolated atoms, be-487cause the wavefunctions are more spatially extended due488to hybridization. In order to model this effect, the atomic489Slater integrals and ζ, deduced from Hartree-Fock cal-490culations [41, 42], were multiplied by constant factors491RSlater and RSOC (0 ≤ RSlater < 1, 0 ≤ RSOC < 1),492respectively. These factors RSlater and RSOC and the493cubic LF splitting ∆LF and were treated as adjustable494parameters. For O K-edge RIXS, ζ = 0.50 eV and495∆LF = 4.1 eV were used, and the Slater integrals be-496tween the Os 5d orbitals were reduced to 35% of the497atomic Hartree-Fock values. Hund’s coupling JH between498two d electrons (Table I) is related to Slater integrals499through JH = 349F2 + 20441F4 [43]. The value JH = 0.27500eV in the table is smaller than JH = 0.5 eV used for the501DFT+U+SOC calculation because the former is for the502Os 5d-O 2p anti-bonding orbitals while the latter for the503Os 5d atomic orbitals. The reduction of JH from 0.5 eV504to 0.27 eV suggests that the atomic orbitals consisting of505the antibonding t2g band have the weight Os 5d : O 2p ∼50670%: 30%.507In the calculation of the RIXS spectra, the same geom-508etry as the experiment was adopted: The incident and509scattered X-rays were set parallel to the cubic [001] and510[100] directions, respectively. Taking the [001], [100], and511[010] directions as the z, x, y axes, respectively, the linear512polarizations of the incident and scattered X-rays were513set to be (x, y) and (x, z), and the spectra for these two514polarization sets were averaged.515The calculated spectra were broadened by a Voigt func-516tion, which is the convolution of a Lorentz function and517a Gauss function. The widths (half width at half maxi-518mum, HWHM) of the Lorentz functions were determined519from the natural lifetime of the core holes: 0.05 eV for520the O K-edge RIXS [44]. The widths (standard devia-521tion) of the Gauss functions were assumed to be 0.01 eV.522The XAS and RIXS spectra were calculated for the five523lowest states [the lowest Jeff = 2 state in Fig. 1(b)] as524the initial state and were summed up according to the525Boltzmann distribution of the initial states.526Raman spectroscopy527Raman measurements in the 10-300K range were per-528formed in backscattering geometry from the polycrys-529talline sample using an RM1000 Renishaw microspec-530trometer equipped with a 532 nm solid-state laser and531633 helium–neon laser. Very low power (up to 1 mW)532was used to avoid local heating of the sample. A pair of533notch filters with a cut-off at 60 cm−1 were used to sup-534press light from the 633 nm laser line. To reach as close535to the zero frequency as possible, we used a set of three536volume Bragg gratings (VBG) at 532 nm excitation to537analyze the scattered light. The resolution of our Raman538spectrometer was estimated to be 2–3 cm−1.539The temperature dependence of the two narrow lines in540the spectrum [Fig. 5 (a)] turned out to be opposite. The541fully symmetric line softened from 797.5 to 788.5 cm−1542with an increase in the temperature range from 10 to 300543K, and its width increased from 6.5 to 12 cm−1, which544can be explained by anharmonicity effects. In contrast to545this behavior, the energy and width of the low-frequency546T2g phonon line remains constant within the measure-547ment error (ω ∼102.5 cm−1 and Γ ∼1.5 cm−1) when548heated from 10 to 300 K. Unfortunately, the tempera-549ture behavior of the two broad lines - T2g at 375 cm−1550(Γ ∼ 20 cm−1) and Eg at 495 cm−1 (Γ ∼ 60 cm−1) - is551difficult to study due to their weak intensity. However,552despite the large width of the lines, we did not find any553signs of their splitting.554Density-functional-theory calculation555The generalized gradient approximation (GGA) in the556form proposed by Perdew, Burke, and Ernzerhof [45]557as realized in VASP code [46] was used for the density558functional theory calculations. Phonon spectra shown in559Supplementary Figure 2 were calculated by the frozen560phonon method [47] with 5 × 5 × 5 mesh of the Bril-561louin zone of the 2 × 2 × 2 supercell in non-magnetic562GGA. Planewave cut-off was set up to 500 eV. The struc-563ture was relaxed until convergence in energy of 10−6 eV564in electronic subsystem and 10−5 eV in ionic one was565achieved.5669DATA AVAILABILITY567All data generated or analysed during this study are568available from the corresponding authors upon reason-569able request.570[1] T. Takayama, J. Chaloupka, A. Smerald, G. Khaliullin,571and H. Takagi, Spin–orbit-entangled electronic phases in5724d and 5d transition-metal compounds, J. Phys. Soc. Jpn57390, 062001 (2021).574[2] D. I. Khomskii and S. V. Streltsov, Orbital effects in575solids: Basics, recent progress, and opportunities, Chem.576Rev. 121, 2992 (2021).577[3] D. I. Khomskii and S. V. Streltsov, Magnetic oxides, in578Encyclopedia of Condensed Matter Physics (Second Edi-579tion), edited by T. Chakraborty (Academic Press, Ox-580ford, 2024) second edition ed., pp. 98–111.581[4] S. V. Streltsov and D. I. Khomskii, Jahn-Teller effect and582spin-orbit coupling: Friends or foes?, Phys. Rev. X 10,583031043 (2020).584[5] G. Chen and L. Balents, Spin-orbit coupling in d2 ordered585double perovskites, Phys. Rev. B 84, 094420 (2011).586[6] A. Paramekanti, D. D. Maharaj, and B. D. Gaulin, Oc-587tupolar order in d-orbital Mott insulators, Phys. Rev. B588101, 054439 (2020).589[7] C. Svoboda, W. Zhang, M. Randeria, and N. Trivedi,590Orbital order drives magnetic order in 5d1 and 5d2 double591perovskite Mott insulators, Phys. Rev. B 104, 024437592(2020).593[8] G. Khaliullin, D. Churchill, P. P. Stavropoulos, and H.-594Y. Kee, Exchange interactions, Jahn-Teller coupling, and595multipole orders in pseudospin one-half 5d2 Mott insula-596tors, Phys. Rev. Res. 3, 033163 (2021).597[9] S. Voleti, K. Pradhan, S. Bhattacharjee, T. Saha-598Dasgupta, and A. Paramekanti, Probing octupolar hid-599den order via janus impurities, npj Quantum Mater. 8,60042 (2023).601[10] D. Hirai, H. Sagayama, S. Gao, H. Ohsumi, G. Chen,602T. Arima, , and Z. Hiroi, Detection of multipolar orders603in the spin-orbit-coupled 5d Mott insulator Ba2MgReO6,604Phys. Rev. Res. 2, 022063(R) (2020).605[11] A. Takahashi and H. Shiba, Possible orbital orderings606in a model of metallic double-exchange ferromagnets, J.607Phys. Soc. Jpn. 69, 3328 (2000).608[12] C. M. Thompson, J. P. Carlo, R. Flacau, T. Aharen,609I. A. Leahy, J. R. Pollichemi, T. J. S. Munsie, T. Medina,610G. M. Luke, J. Munevar, S. Cheung, T. Goko, Y. J. Ue-611mura, and J. E. Greedan, Long-range magnetic order in612the 5d2 double perovskite Ba2CaOsO6: comparison with613spin-disordered Ba2YReO6, J. Phys.: Condens. Matter61426, 306003 (2014).615[13] R. Cong, E. Garcia, P. C. Forino, A. Tassetti, G. Al-616lodi, A. P. Reyes, P. M. T. P. M. Woodward, C. Fran-617chini, S. Sanna, and V. F. Mitrović, Effects of charge618doping on mott insulator with strong spin-orbit coupling,619Ba2Na1−xCaxOsO6, Phy. Rev. Mater. 7, 084409 (2023).620[14] D. D. Maharaj, G. Sala, M. B. Stone, E. Kermarrec,621C. Ritter, F. Fauth, C. A. Marjerrison, J. E. Greedan,622A. Paramekanti, and B. D. Gaulin, Octupolar versus Néel623order in cubic 5d2 double perovskites, Phys. Rev. Lett.624124, 087206 (2020).625[15] S. Voleti, D. D. Maharaj, B. D. Gaulin, G. Luke, and626A. Paramekanti, Multipolar magnetism in d-orbital sys-627tems: Crystal field levels, octupolar order, and orbital628loop currents, Phys. Rev. B 101, 155118 (2020).629[16] D. G. Mazzone, Y. Shen, H. Suwa, G. Fabbris, J. Yang,630S.-S. Zhang, H. Miao, J. Sears, K. Jia, Y. G. Shi, M. H.631Upton, D. M. Casa, X. Liu, J. Liu, C. D. Batista, and632M. P. M. Dean, Antiferromagnetic excitonic insulator633state in Sr3Ir2O7, Nat. Commun. 13, 913 (2022).634[17] B. W. Lebert, S. Kim, B. H. Kim, S. H. Chun, D. Casa,635J. Choi, S. Agrestini, K. Zhou, M. Garcia-Fernandez, and636Y.-J. Kim, Nonlocal features of the spin-orbit exciton in637Kitaev materials, Phys. Rev. B 108, 155122 (2023).638[18] F. I. Frontini, G. H. Johnstone, N. Iwahara, P. Bhat-639tacharyya, N. A. Bogdanov, L. Hozoi, M. H. Upton,640D. M. Casa, D. Hirai, , and Y.-J. Kim, Spin-orbit-lattice641entangled state in A2MgReO6 (A = Ca, Sr, Ba) revealed642by resonant inelastic X-ray scattering, Phys. Rev. Lett.643133, 036501 (2024).644[19] N. Iwahara, J.-R. Soh, D. Hirai, I. Živković, Y. Wei,645W. Zhang, C. Galdino, T. Yu, K. Ishii, F. Pisani,646O. Malanyuk, T. Schmitt, and H. M. Rønnow, Persistent647quantum vibronic dynamics in a 5d1 double perovskite648oxide, arXiv:2409.08095 (2024).649[20] X. Lu, P. Olalde-Velasco, Y. Huang, V. Bisogni, J. Pel-650liciari, S. Fatale, M. Dantz, J. G. Vale, E. C. Hunter,651J. Chang, V. N. Strocov, R. S. Perry, M. Grioni, D. F.652McMorrow, H. M. Rønnow, and T. Schmitt, Dispersive653magnetic and electronic excitations in iridate perovskites654probed by oxygen K-edge resonant inelastic x-ray scat-655tering, Phys. Rev. B 97, 041102(R) (2018).656[21] M. M. Sala, M. Rossi, S. Boseggia, J. Akimitsu, N. B.657Brookes, M. Isobe, M. Minola, H. Okabe, H. M. Rønnow,658L. Simonelli, D. F. McMorrow, and G. Monaco, Orbital659occupancies and the putative jeff = 1/2 ground state in660Ba2IrO4: A combined oxygen K-edge XAS and RIXS661study, Phys. Rev. B 89, 121101(R) (2014).662[22] S. Agrestini, F. Borgatti, P. Florio, J. Frassineti, D. F.663Mosca, Q. Faure, B. Detlefs, C. J. Sahle, S. Fran-664coua, J. Choi, M. Garcia-Fernandez, K.-J. Zhou, V. F.665Mitrović, P. M. Woodward, G. Ghiringhelli, C. Franchini,666F. Boscherini, S. Sanna, and M. M. Sala, Origin of mag-667netism in a supposedly nonmagnetic osmium oxide, Phys.668Rev. Lett. 133, 066501 (2024).669[23] I. Živković, J.-R. Soh, O. Malanyuk, R. Yadav, F. Pisani,670A. M. Tehrani, D. Tolj, J. Pasztorova, D. Hirai, YuanWei,671W. Zhang, C. Galdino, T. Yu, K. Ishii, A. Demuer, O. V.672Yazyev, T. Schmitt, and H. M. Rønnow, Dynamic Jahn-673Teller effect in the strong spin-orbit coupling regime, Nat.674Commun. 15, 8587 (2024).675[24] V. Zimmermann, A. K. Yogi, D. Wong, C. Schulz,676M. Bartkowiak, K. Habicht, L. Wang, M. Isobe, M. Mi-677nola, G. Khaliullin, B. Keimer, and M. Hepting, Coherent67810propagation of spin-orbit excitons in a correlated metal,679npj Quantum Mater. 8, 53 (2023).680[25] G. van der Laan, R. Chopdekar, Y. Suzuki, and E. Aren-681holz, Strain-induced changes in the electronic structure682of MnCr2O4 thin films probed by x-ray magnetic circular683dichroism, Phys. Rev. Lett. 105, 067405 (2010).684[26] J. Kanamori, Theory of the magnetic properties of fer-685rous and cobaltous oxides I, Prog. Theor. Phys. 17, 177686(1957).687[27] D. P. Strommen, Specific values of the depolarization688ratio in Raman spectroscopy: Their origins and signif-689icance, J. Chem. Educ. 69, 803 (1992).690[28] P. Woodward, R. Hoffmann, and A. Sleight, Order-691disorder in A2M3+M5+O6 perovskites, J. Mater. Res.6929, 2118 (1994).693[29] V. Ting, Y. Liu, R. Withers, L. Norén, M. James, and694J. F. Gerald, A structure and phase analysis investigation695of the “1:1” ordered A2InNbO6 perovskites (A = Ca2+,696Sr2+, Ba2+), J. Solid State Chem. 179, 551 (2006).697[30] C. Bernuy-Lopez, M. Allix, C. A. Bridges, J. B. Clar-698idge, and M. J. Rosseinsky, Sr2MgMoO6−δ: structure,699phase stability, and cation site order control of reduc-700tion, Chem. Mat. 19, 1035 (2007).701[31] S. Vasala and M. Karppinen, A2B′B′′O6 perovskites: A702review, Prog. Solid State Chem. 43, 1 (2015).703[32] I. B. Bersuker, The Jahn-Teller Effect (Cambridge Uni-704versity Press, Cambridge, 2006).705[33] A. Rayyan, X. Liu, and H.-Y. Kee, Fate of multipo-706lar physics in 5d2 double perovskites, Phy. Rev. B 108,707045149 (2023).708[34] L. J. P. Ament, M. van Veenendaal, and J. van den Brink,709Determining the electron-phonon coupling strength from710resonant inelastic X-ray scattering at transition metal L-711edges, Europhys. Lett. 95, 27008 (2011).712[35] N. Iwahara andW. Furukawa, Vibronic effect on resonant713inelastic x-ray scattering in cubic iridium hexahalides,714Phys. Rev. B 108, 075136 (2023).715[36] S. V. Streltsov, F. V. Temnikov, K. I. Kugel, and D. I.716Khomskii, Interplay of the Jahn-Teller effect and spin-717orbit coupling: The case of trigonal vibrations, Phys.718Rev. B 105, 205142 (2022).719[37] Y. Shi, Y. Guo, S. Yu, M. Arai, A. Sato, A. A. Be-720lik, K. Yamaura, and E. Takayama-Muromachi, Crystal721growth and structure and magnetic properties of the 5d722oxide Ca3LiOsO6: extended superexchange magnetic in-723teraction in oxide, J. Am. Chem. Soc. 132, 8474 (2010).724[38] A. Singh, H. Y. Huang, Y. Y. Chu, C. Y. Hua, S. W. Lin,725H. S. Fung, H. W. Shiu, J. Chang, J. H. Li, J. Okamoto,726C. C. Chiu, C. H. Chang, W. B. Wu, S. Y. Perng, S. C.727Chung, K. Y. Kao, S. C. Yeh, H. Y. Chao, J. H. Chen,728D. J. H. Huang, and C. T. Chen, Development of the Soft729X-ray AGM-AGS RIXS beamline at the Taiwan Photon730Source, J. Synchrotron Rad. 28, 977 (2021).731[39] C. H. Lai, H. S. Fung, W. B. Wu, H. Y. Huang, H. W.732Fu, S. W. Lin, S. W. Huang, C. C. Chiu, D. J. Wang,733L. J. Huang, T. C. Tseng, S. C. Chung, C. T. Chen, and734D. J. Huang, Highly efficient beamline and spectrometer735for inelastic soft x-ray scattering at high resolution, J.736Synchrotron Rad. 21, 325 (2014).737[40] A. Tanaka and T. Jo, Resonant 3d, 3p and 3s photoemis-738sion in transition metal oxides predicted at 2p threshold,739J. Phys. Soc. Jpn. 63, 2788 (1994).740[41] J. B. Mann, Atomic Structure Calculations. I. Hartree-741Fock Energy Results for the Elements Hydrogen to742Lawrencium, Tech. Rep. (Los Alamos National Lab., Los743Alamos, New Mexico, 1967).744[42] F. Herman and S. Skillman, Atomic structure calcula-745tions, in Atomic Structure Calculations (Prentice-Hall746Inc., Englewood Cliffs, New Jersey, 1963) Chap. 2, pp.7471–17.748[43] A. Georges, L. de’ Medici, and J. Mravlje, Strong correla-749tions from Hund’s coupling, Annu. Rev. Condens. Matter750Phys. 4, 137 (2013).751[44] M. O. Krause and J. H. Oliver, Natural widths of atomic752K and L levels, Kα X-ray lines and several KLL Auger753lines, J. Phys. Chem. Ref. Data 8, 329 (1979).754[45] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized755gradient approximation made simple, Phys. Rev. Lett.75677, 3865 (1996).757[46] G. Kresse and J. Furthmüller, Efficient iterative schemes758for ab initio total-energy calculations using a plane-wave759basis set, Phys. Rev. B 54, 11169 (1996).760[47] A. Togo, First-principles phonon calculations with761phonopy and phono3py, J. Phys. Soc. Jpn. 92, 012001762(2023).763ACKNOWLEDGEMENTS764We are grateful to M. Haverkort, A. Paramekanti, C.765Franchini, and A. Hariki for useful discussions. The soft766X-ray measurements were conducted at beam line 41A767of Taiwan Photon Source. This work was partly sup-768ported by the National Science and Technology Coun-769cil of Taiwan under Grant Nos. 103-2112-M-213-008-770MY3, 108-2923-M-548 213-001, and 113-2112-M-007-033771and by the Japan Society for the Promotion of Science772under Grant Nos. JP20K14416 and JP22K03535. We773thank the Ministry of Science and Higher Education of774the Russian Federation for supporting Raman experi-775ments and their interpretation through funding the Insti-776tute of Metal Physics, while theoretical calculations were777performed with aid of the Russian Science Foundation778(project RSF 23-42-00069). A.F. acknowledges the sup-779port from the Yushan Fellow Program under the Ministry780of Education of Taiwan.781AUTHOR CONTRIBUTIONS782A.F., S.V.S. and D.J.H. coordinated the project. J.O.,783H.Y.H., A.S., D.J.H. and C.T.C. developed the RIXS in-784struments and conducted the RIXS experiments. Y.S.P.785performed Raman experiments. H.H. and K.Y. synthe-786sized and characterized the sample. G.S. and A.T. per-787formed multiplet calculations. J.O., D.J.H., S.V.S. and788A.F. analyzed the data and wrote the paper with inputs789from other authors.790COMPETING INTERESTS791The authors declare that there are no competing inter-792ests.793