Partial molecular orbitals in face-sharing 3 d manganese trimer: Comparative studies on Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12

We present a new molecular orbital candidate Ba 4 TaMn 3 O 12 with a face-sharing octahedra trimer, by comparing it with a related compound Ba 4 NbMn 3 O 12 . The synthesis of the polycrystalline powder is optimized by suppressing the secondary impurity phase via X-ray diﬀraction. Magnetic susceptibility measurements on the best samples reveal the intrinsic magnetic hysteresis with magnetic transitions consistent with heat capacity results. The eﬀective magnetic moments from susceptibility indicate a strongly coupled S = 2 antiferromagnetic trimer at around room temperature, whereas the estimated magnetic entropy from heat capacity suggests the localized S = 3 / 2 timer. These results can be explainable by a partial molecular orbital state, in which three t 2 g electrons are localized in each Mn ion and one e g electron is delocalized over two-end Mn ions of the trimer based on density functional theory calculations. This new 3 d orbital state is comprehended as a consequence of competition between the hybrid interatomic orbitals within the Mn trimer and the local moment formation by on-site Coulomb correlations. We further ﬁnd the potential manipulation of the band gap in the 3 d trimer material platform in searching for the unconventional insulator-metal transition.


I. INTRODUCTION
A Mott state [1] has been studied extensively for understanding the electronic state of correlated electron systems stabilized by the competition between the electron correlation energy and other energy scales, such as spinorbit coupling and crystalline electric fields.Quantum magnets containing transition metal (TM) ions can be generally understood in the localized electron scheme by learning the properties of the single magnetic ion and the exchange interactions with its surrounding magnetic ions in a given structural motif.
More recently, new electronic states have attracted significant interest in discovering novel electronic properties.Such an example is a molecular orbital state (MO) [2], where electrons are delocalized and shared by several magnetic ions nearby, which cannot be described by a simple single-ion localized electron picture.It has been demonstrated that this unconventional phase can be realized in heavy TM-based compounds, such as the 4d trimer Ba 4 Ru 3 O 10 [3,4] and 5d trimer Ba 5 CuIr 3 O 12 [5] by the interplay between spin-orbit coupling and extended d orbitals.In these compounds, the TM octahedra form a linear trimer via a face-sharing geometry, which enables an unusually short distance between TM ions, allowing a sizable direct overlap of d orbitals.As a result, they can stabilize the MO state.
However, little is known about the MO state from 3d TM-based compounds.It is generally recognized that characters of 3d orbitals are very different from those of 4d and 5d orbitals.For instance, the spin-orbit coupling is reduced in 3d orbitals, and electrons are significantly more localized, leading to large electronic Coulomb interactions.In contrast, 3d materials have a higher Hund's coupling energy, which is associated with an intra-atomic exchange in multiorbital systems [6].This distinct energy hierarchy in 3d TM ions compared to those in 4d and 5d counterparts could render the MO state from 3d materials unusual.Therefore, the interplay between spin-orbit and Hund's coupling of 3d compounds could provide the opportunity to discover novel characters of the MO state.
In this regard, a recently reported hexagonal perovskite Ba 4 NbMn 3 O 12 [7] is intriguing.This compound also contains the linear trimer with a three face-sharing MnO 6 octahedra, which allows a much closer Mn-Mn distance (∼2.47 Å) than the one encountered in common corner-sharing geometries [8].It is comparable to that of an Mn metal (∼2.48 Å [9]).Thus this structural motif could enable a strong direct overlap between 3d orbitals.Nonetheless, Ba 4 NbMn 3 O 12 is an insulator, which suggests an unconventional electronic picture in 3d orbitals.Indeed, magnetic susceptibility measurements uncovered the intriguing magnetic properties of Ba 4 NbMn 3 O 12 [7], such as that the effective magnetic moment extracted from high temperatures corresponded to only S = 2 from the trimer, which is expected to be a combination of two Mn 4+ ions and one Mn 3+ ion.Based on this, a magnetically ordered state within the trimer was suggested even at 300 K [7] even if the long-range magnetic ordering temperature can be lower than 42.4 K.
To understand the magnetic ground state of Ba 4 NbMn 3 O 12 based on the expected mixed valent state of Mn 3+ and Mn 4+ , two hypothetical models [7] in the localized electron scheme are proposed: (i) two parallel moments from Mn 4+ ions and one antiparallel moment from Mn 3+ ion (3d 4 , S = 1 for low spin); (ii) two antiparallel moments from Mn 4+ ions and one moment from Mn 3+ ion (3d 4 , S = 2 for high spin).Note that the spin of Mn 4+ ions in both models is the same (3d 3 , S = 3/2).However, the reported heat capacity [7] disagrees with both models and reveals a significantly lower magnetic entropy for the S = 2 system than expected.Thus the consistent characterization of carefully grown samples needs to be done, compared to detailed theoretical calculations using the suggested magnetic structures.In addition, it is crucial to conduct a comparative study to better understand the peculiar electronic properties of Ba 4 NbMn 3 O 12 by synthesizing and examining a related compound if possible.
Herein, we present the synthesis and characterization of a 3d trimer Ba 4 TaMn 3 O 12 and an improved sample quality of its related compound Ba 4 NbMn 3 O 12 .The growth optimization is made by controlling an amount of volatile MnO 2 powder, monitored by x-ray diffraction refinements.Sharp phonon peaks observed by Raman scattering corroborate the high quality of our samples.We confirm tiny magnetic hysteresis anomalies and find signatures of magnetic transitions in both compounds in magnetic susceptibility measurements, compatible with heat capacity experiments.The effective magnetic moments from susceptibility indicate a strongly coupled S = 2 antiferromagnetic trimer at around room temperature, whereas the estimated magnetic entropy from heat capacity suggests the localized S = 3/2 moments in the timer.Ab initio calculations find that the localized electrons at Mn t 2g orbitals and the delocalized electron at Mn e g orbitals spread over two-end Mn ions of the trimer, which explains both susceptibility and heat capacity results.Hence we propose a partial molecular orbital, which is the coexistent state of localized electrons and molecular orbitals in Ba 4 MMn 3 O 12 (M = Ta, Nb).This peculiar state is interpreted by the result of the competition between the hybrid interatomic orbitals within the Mn trimer and the local moment formation by on-site Coulomb correlations.Further, we discuss the possible relevance of this trimer-based material platform in searching for the insulator-metal transition that might be useful in developing innovative magnetic devices.
The rest of the paper is organized as follows.The experimental and calculational details are shown in Secs.II and III, respectively.The optimization of the polycrystalline powder is explained in Sec.IV A. Refined crystal structures by x-ray diffraction are presented in Sec.IV B. Sections IV C and IV D present magnetic susceptibility and heat capacity results, respectively.Resistivity results are given in Sec.IV E. Raman spectra are compared in Sec.IV F, followed by ab initio calculations in Sec.V. Our results are discussed based on reported references in Sec.VI, and finally, conclusions are given in Sec.VII.The Appendix provides results from nonmagnetic DFT calculations.

II. EXPERIMENTAL DETAILS
The polycrystalline samples of Ba 4 MMn 3 O 12 (M = Ta, Nb) were synthesized using a solid-state reaction.The starting precursors of BaCO 3 (Alfa Aesar, 99.95 %), M 2 O 5 (M = Ta, Nb) (Kojundo, 99.9 %), and MnO 2 (Kojundo, 99.99 %) were taken in stoichiometric ratio and ground well in an agate mortar using a pestle.Mixed powders were calcined at 900°C for 24 h.After the calcination, the powders were repeatedly reground, pelletized, and sintered for 24 hrs at 1100°C, 1300°C, and 1420°C.Slightly modified conditions were also used-the calcination at 1100°C, followed by the subsequent sintering at 1200°C and 1420°C for 24 h at each temperature.Note that our optimized batches were synthesized using the modified conditions.The excess of starting MnO 2 powder was also added to compensate for its evaporation, effectively suppressing the impurity phase formation.
The phase purity of polycrystalline powder was confirmed using x-ray diffraction measurements at room temperature using a Rigaku SmartLab X-ray Diffractometer.The high-quality XRD data were collected from a Rigaku D8 advance high-resolution x-ray diffractometer using a Bragg-Brentano geometry with a Cu Kα 1,2 source, presented in this paper.A jana2006 software [10] was used for Rietveld refinements to determine the crystal structures.
Magnetic susceptibility experiments for zero-fieldcooled (ZFC) and field-cooled (FC) measurements were performed with a Physical Property Measurement System (PPMS, Dynacool, Quantum Design) equipped with a vibrating sample magnetometry module between 300 and 2 K with a magnetic field of 0.01 T or 0.1 T. Magnetization measurements were done in the magnetic field from −7 to 7 T at 2, 30, 50, 70, and 100 K.
Heat capacity measurements were carried out with a PPMS at H = 0 and 9 T from 160 to 2 K. Before each measurement, the addenda was measured using the same temperature range by the thermal relaxation method.Addenda heat capacities were subtracted from total heat capacities to obtain the sample heat capacity as usual.
Electrical resistance was measured on sintered rectangular pellets using a PPMS between 260 and 390 K.The rectangular pellets of 3 × 6 mm 2 dimensions were prepared using a rectangular die set.A four-probe method was utilized, where the electrical contacts were made using gold wires and silver epoxy.Measured resistance was converted to resistivity using the length and area of the cross-section.
Raman measurements were performed using the 514.5 nm line of an Ar-ion laser as the excitation source at room temperature.The laser beam was focused onto the sample through a 50× objective lens (numerical aperture = 0.8) in a backscattering geometry.The laser power was kept at 0.5 mW to minimize the local beam heating.The Raman signals were recorded with a liquid-nitrogencooled charge-coupled-device detector after dispersion by a Horiba iHR550 spectrometer (2400 grooves/mm).

III. COMPUTATIONAL DETAILS
The Vienna ab initio simulation package (vasp) [11,12] was employed for electronic structure calculations, with 400 eV of plane-wave energy cutoff and 5 × 5 × 2 k-point sampling for conventional hexagonal unit cells.A revised Perdew-Burke-Ernzherof generalized gradient approximation for solids (PBEsol) was used to approximate the exchange-correlation functional.The strong Coulomb repulsion within the Mn sites was further treated with a simplified rotationally invariant flavor of density functional theory (DFT)+U eff [13], where the effective on-site Coulomb repulsion U eff ≡ U − J was set to 4 eV for the Mn d-orbital.Note that the value of U eff = 4 eV was widely adopted for a range of Mn compounds [14][15][16][17] and reported to produce reasonable results.

IV. EXPERIMENTS A. Optimization of polycrystalline growths
We found the unwelcome impurity phase, BaM 0.5 Mn 0.5 O 3 (M = Ta, Nb), by x-ray diffraction during the synthesis of Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 polycrystalline samples.Thus we optimized our growths by controlling the amount of the excess MnO 2 starting powder to compensate for its evaporation during the reaction.
Figure 1 presents the mass fraction of the impurity phase extracted by the structural refinements using x-ray diffraction measurements, followed by magnetic susceptibility measurements.As shown in Fig. 1(a), the mass fraction of the impurity phase linearly decreases with an excess of MnO 2 , reaching a minimum value when the 9% (10%) of excessive MnO 2 powder is initially added for Ba 4 TaMn 3 O 12 (Ba 4 NbMn 3 O 12 ) before the first sintering.We found that the magnetic susceptibility splitting ( χ FC -χ ZFC ) at 2 K is bigger for such batches, as shown in Fig. 1(c) and 1(d) for Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 , respectively.We studied the structural, magnetic, thermal, electric, and optic properties using the optimized batch of each compound in this paper.We also found a tiny mass fraction of the secondary impurity phase Mn 3 O 4 (∼2%) towards a higher excess of MnO 2 , as shown in Figs.1(a) and 1(b).

B. Crystal structure
To determine the crystal structure of Ba 4 MMn 3 O 12 (M = Ta, Nb) compounds, powder x-ray diffraction was performed, revealing the high quality of the sample.Consequently, these batches were used for the subsequent characterization measurements presented in this paper.Figure 2 [7] on Ba 4 NbMn 3 O 12 .A minor secondary phase [∼3% of disordered perovskite BaM 0.5 Mn 0.5 O 3 (M = Ta, Nb) ] was identified in both compounds, which was also reported previously for Ba 4 NbMn 3 O 12 [7].Figures 2(c I and Table II, respectively.Note that we found only a few percentages of the site mixing between Mn and Ta (Nb) sites.suggesting that more measurements would be needed at higher magnetic fields.We found additional anomalies at 29.77 K and 23.5 K for Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 , respectively, which were also visible in d( χ T )/dT (Fig. 6).The χ −1 (T ) data between 100 and 370 K were fitted using the Curie-Weiss (CW) equation, χ (T ) = χ 0 + C/(T − θ), where χ 0 is temperature-independent susceptibility, C is the Curie constant, and θ is the CW temperature, as summarized in Tables III and IV.The effective magnetic moment was calculated using µ eff = 3k B C/N A [18], where N A the Avogadro constant and k B is the Boltzmann constant.Our fitting results in the temperature range between 100 and 275 K showed that µ eff   III.Parameters obtained from CW fits using the inverse magnetic susceptibility χ −1 (T ) of Ba4TaMn3O12.θ is the CW temperature.C (emu K Oe −1 mol −1 ) is the Curie constant.χ 0 (emu Oe −1 mol −1 ) is the temperatureindependent magnetic susceptibility.µ eff is the effective magnetic moment.[7], reminiscent of the van Vleck paramagnetism [19].These two observations are compatible with an idea of locally-aligned magnetic moments even at high tem- peratures with remnant susceptibility [7] after the local moments are established within the trimer.It could be contributed from the mixing between the ground and excited states [20] within the Mn trimer in Ba 4 MMn 3 O 12 .Thus, all moments are not freely fluctuating, but the local antiferromagnetic moment within the trimer is already formed even at room temperature, which will not contribute to the Curie-Weiss temperature.This means that the positive Curie temperature reflects the subdominant ferromagnetic exchange interaction.In this view, the extracted parameters can be interpreted with the trimer unity rather than individual spins of Mn.This picture is also consistent with the localized part (S = 3/2) of the extracted effective moment.The delocalized part (S = 1/2) will be discussed soon.In Ba 4 MMn 3 O 12 , the Curie-Weiss temperature is positive, which indicates the effective ferromagnetic intertrimer coupling.We note that positive values of χ 0 are also observed in other related compounds, such as Ba 4 NbMn 3 O 12 [7], Ba 4 NbRu 3 O 12 [21], and Ba 4 Nb 0.8 Ir 3.2 O 12 [22].We also tried the Curie-Weiss fit without the χ 0 term (not shown), but this gave completely nonphysical results, such as wrong effective magnetic moments.Thus, the sizable temperature-independent paramagnetic term is essential for reliable fits to extract the physically sensible parameters for Ba 4 MMn 3 O 12 .
For a more systematic analysis, we also performed the CW fits with a different range of temperatures, as summarized in Table III  .This means that the antiferromagnetically coupled moments within the trimer are robust up to 370 K for Ba 4 TaMn 3 O 12 by giving the compatible, effective magnetic moment with that of Ba 4 NbMn 3 O 12 from Ref. [7].
On the other hand, the reported CW parameters in the literature [7] are somewhat different from our parameters from Ba 4 NbMn 3 O 12 , as compared in Table IV.The effective moments extracted from fits in Ba 4 NbMn 3 O 12 show an increasing trend as the lower bound of the fitted temperature becomes higher.Also, the CW temperatures become more negative simultaneously, consistent with the trend of χ 0 values.These behaviors of χ 0 and µ eff signal that the three Mn spins behave more independently towards higher temperatures in Ba 4 NbMn 3 O 12 , unlike Ba 4 TaMn 3 O 12 .The fitting parameters χ 0 and µ eff for both Ba 4 MMn 3 O 12 are consistent with results from Ref. [7] when the fitting was done in the identical temperature range (between 100 and 275 K).Interestingly, the effective magnetic moment extracted from Ba 4 NbMn 3 O 12 at the higher temperature range is comparable to the values expected when all moments freely respond to the external magnetic field in the trimer unit having two Mn +4 ions and one Mn +3 ion; for instance, 6.16 (for the low-spin model) or 7.35 (for the high-spin model) µ B /f.u [7].As the temperature is lowered, the three Mn spins form the antiferromagnetic trimer even above the long-range magnetic ordering temperature and yield the effective magnetic moment, comparable to the S = 2 system.These results suggest that the bonding of Mn spins in the trimer is stronger for Ba 4 TaMn 3 O 12 than Ba 4 NbMn 3 O 12 .Accordingly, a similar behavior for χ 0 and µ eff is expected for Ba 4 TaMn 3 O 12 at higher temperatures above 370 K.We note that Ref. [7] proposed the ferrimagnetic transition behavior from χ (T ) with a negative θ value for Ba 4 NbMn 3 O 12 , which means the dominant antiferromagnetic intertrimer interaction.However, we did not observe such a trend in our experiments.θ val- ues from our results are positive in both materials when using the same range of the fitted temperatures.
In Figs.

D. Heat capacity
We performed heat capacity measurements on the same samples to understand the nature of the anomalous features found in the magnetic susceptibility data.Figures 4(a Both anomalies at T 1 and T 2 were also seen in d( χ T )/dT (Fig. 6).In the heat capacity data at 9 T, both anomalies are suppressed, indicating their magnetic origins.
Heat capacity data of magnetic insulators typically consist of a phononic (C ph ) and magnetic part (C mag ).A normal way to separate these parts is to subtract the heat capacity data of a nonmagnetic compound having the same crystal structure.However, in the absence of such a nonmagnetic analog, we fitted our heat capacity data with the Debye-Einstein model [23][24][25], a phenomenological approach to capture the primary characteristic of the complex lattice dynamics.To estimate the lattice contribution, we used a combined model with one Debye term and two Einstein terms, as given by The first term in Eq. ( 1) is the Debye term (responsible for the acoustic modes), which is given by where n is the number of moles, R is the universal gas constant, θ D is the characteristic Debye temperature, and x = ℏω k B T , where ω is the vibrational frequency.The second term contains two Einstein terms (optical modes) and is expressed as where θ Ei is the characteristic Einstein temperature.Since one formula unit of Ba    2) and ( 3) was not used in our fits.Figure 5 exhibits the evolution of the magnetic entropy (∆S mag ) between 2 and 70 K.The change in magnetic entropy was calculated by integrating C mag /T with respect to temperature between 2 and 70 K, which yielded the saturated value of ∆S mag = 11.52 J mol −1 K −1 for Ba 4 TaMn 3 O 12 and ∆S mag = 11.94J mol −1 K −1 for Ba 4 NbMn 3 O 12 .Both values are smaller than the expected entropy for the S = 2 magnet (dashed horizontal lines in Fig. 5); ∆S mag = Rln(5) ∼ 13.38 J mol −1 K −1 , where R is the gas constant.However, these values are nearly the same as that of the unexpected S = 3/2 an- tiferromagnetic trimer (horizontal solid lines in Fig. 5), ∆S mag = Rln(4) ∼ 11.53 J mol −1 K −1 .Thus our heat capacity analysis suggests the S = 3/2 trimer in both compounds, which could not be determined in a previous report [7].With the uncertainty of the empirically estimated phonons, the agreement of the saturated moments with the expected one for the S = 3/2 trimer is satisfactory.The reduced magnetic entropy from that of the S = 2 trimer to the S = 3/2 trimer can be interpreted by one delocalized electron within the trimer in a partial molecular orbital state (to be explained in Sec.V).
To determine the anomalous temperature more systematically, we compared d( χ T )/dT and dC p /dT , finding that the extracted anomalous temperatures (gray dashed vertical lines) from both measurements agreed very well, as shown in Fig. 6.The additional anomaly (T 2 ) in the magnetic system could appear for various reasons, such as an additional magnetic transition, or a (continuous) spin reorientation [27].It is also possible that a spinglass transition [28] from a finite disorder occurs at T 1 , followed by the long-range magnetic order at T 2 .Their origins can be examined via other complementary measurements, such as neutron diffraction, in future studies.

E. Resistivity
Figure 7 shows the resistivity data of Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 between 260 and 390 K. Data below 260 K were not obtained due to the exceedingly large resistivity that prevented any measurements in that region in our resistivity option of PPMS.Consequently, we fitted the data between 300 and 390 K with the resistivity equation ρ = ρ o e Ea k B T to obtain activation energy of transport, E a , which was found to be 0.383 eV for Ba 4 TaMn 3 O 12 and E a = 0.365 eV for Ba 4 NbMn 3 O 12 .Thus, it was Red and blue lines depict spin-up and -down bands, respectively.Note that the band of Ba4NbMn3O12 is qualitatively the same as that of Ba4TaMn3O12 (not shown).Conventions for high-symmetry points follow the ones described in Ref. [26].
confirmed that both materials exhibited semiconductor properties.

F. Raman scattering
Raman spectroscopy is a suitable technique for examining phonons sensitive to crystal structure variations.We performed Raman measurements on both compounds at room temperature to confirm the sample quality and explore lattice dynamics.Figure 8 compares similar phonons of two compounds.Although we included measurements up to 1100 cm −1 , multiple phonon peaks are merely observed beyond 800 cm −1 .The sharp phonon peaks observed indicate the good qualities of our polycrystalline samples.A phonon of Ba 4 TaMn 3 O 12 at approximately 180 cm −1 are slightly softened, possibly because Ta ions are heavier than Nb ions.In contrast, Raman phonon spectra at higher energies are more similar, which could be explained by the lighter ions Mn and O. Raman measurements at lower temperatures (i.e., lower than the anomalous temperatures obtained from susceptibility and heat capacity measurements) might be sensitive in detecting bond angle changes and distances in the magnetically ordered state by spin-lattice coupling [29].These measurements may also provide information on magnetic ground states and exchange interaction with respect to molecular orbital states [5].

V. AB INITIO CALCULATIONS
To understand the electronic structure and magnetism of both compounds, we performed ab initio density functional theory (DFT) calculations.We initially assessed the magnetic nature of the Mn trimer and Fig. 9(a) shows its two simplest magnetic configurations-a ferromagnetic and a ferrimagnetic order in the primitive unit cell.Note that the ferrimagnetic order in this section means the ferrimagnetic trimer in the primitive unit cell.We found that the total energy per formula unit of the ferrimagnetic order was lower than that of the ferromagnetic case for Ba 4 NbMn 3 O 12 and Ba 4 TaMn 3 O 12 by 62.7 meV and 55.9 meV, respectively.This could be attributed to the antiferromagnetic coupling between Mn-Mn in the face-sharing MnO 6 octahedra originating from a strong direct orbital overlap between Mn 3d orbitals discussed in previous literature [15].
On the one hand, a molecular orbital (MO) state is proposed in a 4d [4] and 5d [5] transition-metal compound with a similar crystal structure (the trimer with the face-sharing octahedra) owing to the strong direct overlap of orbitals.On the other hand, we identified that the direct overlap between the face-sharing Mn a 1g orbitals was significant but not strong enough to break the local moment scheme in both compounds.For instance, Fig. 9 t 2g shell was half filled with a robust S = 3/2 local moment at each Mn site in the antiferromagnetic trimer.In Fig. 9(b), this is evident from six electrons occupying the t 2g down-spin orbital and three electrons occupying the t 2g up-spin orbitals.
Interestingly, unlike the t 2g shell, one electron having the e g character in an Mn 3+ ion was strikingly delocalized and preferred a metallic phase in the ferrimagnetic state.The e g orbital at the central Mn site was almost empty, whereas one electron is equally shared by the e g orbitals at the top and bottom Mn site, as illustrated in Fig. 9(b).This is reminiscent of a similar 4d analog for Ba 4 Ru 3 O 10 [3,4], where the middle ion of the Ru trimer was nonmagnetic.Hence, the total spin moment at the Mn trimer became S = 2, which is decomposed into the localized t 2g (S = 3/2) and delocalized e g (S = 1/2) components.Note that this observation was also consistent with our magnetic susceptibility data (Fig. 3) and heat capacity data (Fig. 5).
Figure 9(c) shows the calculated band structure, revealing that the partially filled e g band (solid blue lines) exhibits a strong two-dimensional character of an almost flat dispersion along with the Z-Γ line parallel to the Mn trimer along the c axis.In contrast, our resistivity data in Fig. 7 strongly suggest that both Ba 4 NbMn 3 O 12 and Ba 4 TaMn 3 O 12 are gapped.This is due to a well known flaw of the DFT method, which cannot capture the paramagnetic Mott phase as it cannot describe fluctuating magnetic moments.However, the magnetic fluctuation diminishes in the long-range magnetic order below the transition temperature, where the DFT method can provide a qualitatively valid picture.The nature of the band gap can be examined via dynamical mean-field theory calculations in further study, by properly considering fluctuating magnetic moments.
To determine a possible magnetic ground state, we used the two simplest antiferromagnetic configurations (denoted as AF1 and AF2) by adopting the ferrimagnetic Mn trimer as the magnetic building block [see Figs.10(a) and 10(b)].The AF1 state contains a magnetic unit cell that is twice enlarged along the c axis compared to the parent nuclear unit cell with a wave vector, q = (0, 0, 1/2), whereas the AF2 state has a doubled unit cell along the a axis in an orthorhombic magnetic unit cell based on the in-plane antiferromagnetic order, q = (1/2, 0, 0).We set up the AF1 state based on information provided in Ref. [15], where the antiferromagnetic order within the trimer is coupled ferromagnetically in the ab plane and antiferromagnetically along the c axis.In our calculations, we detected that the total energy of the orthorhombic AF2 configuration is lower by 178.9 meV and 85.0 meV per formula unit than that of the AF1 state for Ba 4 NbMn 3 O 12 and Ba 4 TaMn 3 O 12 , respectively.We also found a similar tendency of the MO state in both compounds.Note that our calculated band gap of Ba 4 TaMn 3 O 12 is vanishingly small even in the orthorhombic configuration with U eff = 4 eV for both compounds.We point out that the AF2 state is a collinear antiferromagnetic order, which can serve as a good approximate magnetic structure.

VI. DISCUSSION
We are in a position to discuss the nature of the Mn trimer in Ba 4 MMn 3 O 12 (M = Ta, Nb).Our firstprinciple calculations predict an unusual combination of localized and delocalized electrons; that is, the localized antiferromagnetic magnetic moments (S = 3/2) within the trimer and one electron delocalized in the two-end Mn ions of the trimer.Our picture can be compatible with both magnetic susceptibility (Sec.IV C) and heat capacity results (Sec.IV D).
In magnetic susceptibility experiments, we found that the effective magnetic moments from a paramagnetic temperature indicate the S = 2 trimer in both compounds.They are compatible with the effective magnetic moments reported in Ref. [7] for Ba 4 NbMn 3 O 12 .At first, our results might be counterintuitive because of the localized S = 3/2 trimer in Ba 4 MMn 3 O 12 .However, they can be explained by the additional contribution from one delocalized electron in the trimer.For instance, the effective magnetic moments above the transition temperature reported in weakly ferromagnetic itinerant metals, such as ZrZn 2 [30] and Sc 3 In [30], are much bigger than those expected from the small ordered magnetic moment; i.e., in ZrZn 2 , the saturated magnetic moment is only about 0.12 µ B [30,31], whereas the effective magnetic moment obtained from the Curie-Weisslike law is about 1.4 µ B [31,32], close to the effective magnetic moment of spin-1/2, 1.73 µ B .These experimental observations are understood by the interaction of spatially extended modes of spin fluctuations, giving the Curie-Weiss-like behavior with a large effective magnetic moment [30,31,33].Since the experimental observations in itinerant magnetic systems are consistent with our observations, a similar microscopic mechanism could be applied to the delocalized electron in Ba 4 MMn 3 O 12 .
Possible origins of the weak hysteresis in the magnetization data could be associated with various reasons, such as the ferrimagnetic moments deviating from the collinear antiferromagnetic moments and a spin glass feature from the finite disorder.Further magnetic susceptibility measurements could reveal the nature of the mag-netic ground state.For instance, such states could be studied via ac magnetic susceptibility measurements, by probing domain walls of ferrimagnetic order, or by confirming freezing temperatures with the shift of the ac data by different oscillating frequencies [34].These measurements will reveal the existence of finite disorder, which might result in the weak hysteric behavior in the magnetization data [Figs.3(g) and 3(h)].
In addition, we checked the possible existence of the "unpaired" S = 2 moment in both compounds using the susceptibility data [Figs.3(a) and 3(b)].If one e g electron is localized in one of the two-end Mn ions of the trimer without forming a partial molecular orbital, the fully localized S = 2 moment will be possible in the sample.Since the unpaired S = 2 will not have a presumed longranged spin-spin correlation with other normal trimers, we tabulated it with the paramagnetic term of C u /T in the revised Curie-Weiss fit, where C u means the Curie constant of the unpaired moment.The revised fits did not work and gave diverging results, meaning no significant fraction of the impurity state.This outcome is consistent with our additional DFT + U eff calculations (not shown), which confirm that such a state is unstable in the realistic range of U eff (3 < U eff < 5 eV).Instead, we found that all initial configurations that started with the unpaired state converged to the partial molecular orbital state in our calculations.
In heat capacity measurements, we clearly showed that the magnetic entropy between 2 and 70 K is only S = 3/2, not S = 2 (Fig. 5) in both compounds.One might suggest that the reduction of the magnetic entropy in Fig. 5 is led by the strong quantum fluctuations due to the frustrated magnetic exchanges [35].However, this seems less likely as the frustration indexes (f = |θ|/T 1 ) [36] are tiny in both compounds, although we cannot completely rule out a possibility of a sizable magnetic entropy below 2 K.Note that highly frustrated magnetic materials usually reveal a large frustration factor, f [37].
As a more feasible explanation, we point out that the magnetic entropy of the itinerant magnet [38] is much smaller than that of the localized magnets.A tiny fraction of the magnetic entropy can be released via the magnetic transition in the itinerant magnet; for instance, only 2% of the S = 1/2 magnetic entropy (∆S mag = 0.02Rln2) was reported for the weakly ferromagnetic metal ZrZn 2 [38,39].Hence it will be sensible to interpret that our heat capacity measurements detect the magnetic entropy dominantly released by the long-range magnetic order from the localized magnetic moment in both compounds.We note that Ref. [7] reported a puzzling magnetic entropy of Ba 4 NbMn 3 O 12 , which is much smaller than the values expected for both the S = 2 and S = 3/2 trimer.In our heat capacity measurements, we resolved this issue by using optimally grown polycrystalline samples.
Based on our results on Ba 4 MMn 3 O 12 (M = Ta, Nb) presented in this paper, further investigation of these peculiar magnetic properties would be highly desired.Per- taining to Ba 4 NbMn 3 O 12 , Ref. [7] suggested the presence of ordered magnetic moments inside the trimer even at 300 K; however, the long-range magnetic order stabilized by intertrimer exchange interactions appeared only at a lower temperature [7].Thus neutron diffraction as a bulk probe will be useful to determine the magnetic structure and its evolution with temperature, which are related to the magnetic anomalies found in susceptibility measurements (Fig. 3).On the other hand, muon spin relaxation as a local probe can measure the local magnetic field sensitively; for instance, it is suitable for studying static and dynamic magnetic properties of Ba 4 MMn 3 O 12 (M = Ta, Nb).Thus the nature of the unusual paramagnetic state and the coexistence of localized and delocalized magnetism might be investigated by muons.These measurements could be sensitive to confirm the S = 3/2 trimer in the localized moment picture.
Moreover, the underlying spin Hamiltonian for these compounds needs to be studied experimentally.In these compounds, spin dynamics can be determined by the inter-and intratrimer interaction of the Mn 3 O 12 trimer because only Mn ions are magnetic (i.e., nonmagnetic Ba 2+ and Nb 5+ ions).A recent theory [15] suggested a ferromagnetic exchange in the ab plane that is antiferromagnetically coupled along the c axis.Consequently, inelastic neutron scattering (INS) will be crucial to assess the number of effectively dominant exchange couplings and their characteristics by testing various models, including the proposed one.As we found a strong coupling between Mn spins within the trimer from extracted effec-tive moments at high temperatures [CW fits in Figs.3(c Unlike the more studied 4d and 5d trimer-based materials [21,22,40], our results on Ba 4 MMn 3 O 12 (M = Ta, Nb) suggest an unconventional character-a partial molecular orbital state.Figure 11  This noteworthy difference in their natures of MO states implies a further research opportunity to examine the crossover between atomic and molecular descriptions of magnetism by mixing 3d ions with heavier 4d or 5d ions; in particular, Ba 4 NbRu 3 O 12 [21] and Ba 4 NbIr 3 O 12 [40] can be synthesized.We note that the electronic structure and magnetism of Ba 4 MMn 3 O 12 resemble those of the metallic double exchange in La x Sr 1−x MnO 3 [41,42], indicating that a possible ferromagnetic metal phase in Ba 4 MMn 3 O 12 might be realized via an external magnetic field, pressure, or doping, which can be potentially useful in spintronics and magnetic devices.
Finally, in addition to Ba 4 MMn 3 O 12 (M = Ta, Nb), a wide range of different compositions is possible in the similar hexagonal perovskite, Ba 4 MM ′ 3 O 12 (M = Nb, Ta, Ce, Pr and M ′ = Mn, Ru, Ir) [21,[43][44][45][46] with face-sharing trimer octahedra.Therefore, this compound family is versatile but a less-studied platform for searching for unconventional magnetic properties, including molecular orbital-based magnetism [47,48], calling for both experimental and theoretical studies in the future.

VII. CONCLUSIONS
We reported an unconventional molecular orbital candidate Ba 4 TaMn 3 O 12 by comparing it to a recently reported compound Ba 4 NbMn 3 O 12 .We synthesized the polycrystalline sample by optimizing the amount of the MnO 2 starting powder monitored by x-ray diffraction measurements.Both magnetic susceptibility and heat capacity measurements revealed two compatible anomalies.Susceptibility showed the dominance of ferromagnetic intertrimer interaction.The effective magnetic moments indicated the strong coupling between spins within the trimer above the magnetic ordering temperature.The estimated magnetic entropy from heat capacity measurements is consistent with the antiferromagnetic S = 3/2 trimer with localized moments.These results indicate the combination of the localized S = 3/2 and one delocalized electron in the trimer.Ab initio calculations found that three localized electrons are present in each of the three Mn ions and one delocalized electron is spread over two-end Mn ions of the trimer, consistent with both susceptibility and heat capacity results.The magnetic ordering wave vector is predicted to be q = (1/2, 0, 0).Thus our comprehensive results propose the partial molecular orbital in 3d trimer-based materials-an unconventional electronic state with delocalized and localized electrons in a single compound.This could be understood by the competition between the hybrid interatomic orbitals within the Mn trimer and the local moment formation by on-site Coulomb correlations.In searching for novel electronic states, the possibility of finding the insulatormetal transition in this compound family by an external magnetic field, pressure, or doping warrants active further investigations in the future.To understand the physics above the magnetic transition temperature, we performed additional DFT cal-culations. Figure 12 shows Mn-site-projected densities of states from a nonmagnetic DFT calculation without U eff .Therein it can be seen that the t 2g -e g splitting in central and side Mn sites are about 2.5 and 1.8 eV, respectively, whereas the strength of intersite hybridization within the Mn trimer (∼ the bandwidth of t 2g and e g orbitals) is shown to be about 1 eV.We note that Fig. 9(b) presents that the size of the exchange splitting is about 4 eV when U eff = 4 eV in our DFT+ U eff calculations.A separate calculation employing a parameter-free r2SCAN meta-GGA functional [49] reveals the size of the exchange splitting to be about 4 eV, which is consistent with results in Fig. 9(b).Overall, the hierarchy of on-site energy scales at Mn sites is as follows; Coulomb interaction (U ) > exchange splitting (Hund's coupling) > cubic crystal fields (t 2g -e g splitting) > intersite hybridization within Mn trimer.This relation stabilizes the partial molecular orbital state in Ba 4 MMn 3 O 12 (M = Ta, Nb).
) and 2(d) illustrate the refined crystal structures, where three MnO 6 octahedra are connected by their faces and form a Mn 3 O 12 trimer.The Mn trimers are alternatively connected by corner-sharing nonmagnetic TaO 6 or NbO 6 octahedra by forming 12 hexagonal layers in total within the unit cell.The extracted structural parameters of Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 are provided in Table

FIG. 2 .
FIG. 2. (a), (b) Structural refinements using the powder x-ray diffraction data collected at room temperature from Ba4TaMn3O12 and Ba4NbMn3O12, respectively.(c), (d) Refined crystal structures.The Mn trimer is indicated by shaded purple polyhedra along the crystallographic c axis.
3(e)-3(h) the magnetization M is plotted against the magnetic field H (between −70 and 70 kOe) with temperatures.At 100 K, the M (H) varies linearly and behaves similarly to a paramagnetic system.The M (H) curves form a hysteresis loop at lower temperatures, compared in Figs.3(g)-3(h), where a hysteresis loop in the M (H) data at 2 K disappears at 100 K.This is consistent with the bifurcation anomaly seen in magnetic susceptibility in Figs.3(a) and 3(b).
) and 4(a) present the heat capacity (C p ) of Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 , measured between 160 and 2 K at 0 and 9 T. The normalized heat capacity measurements (C p /T ) of Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 are shown in Figs.4(c) and 4(d) as a function of temperature.Small lambda-like anomalies are observed around T 1 for both Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 at 0 T. As the temperature was lowered further, heat capacity exhibited additional anomalies at T 2 ∼ 29.77K for Ba 4 TaMn 3 O 12 and T 2 ∼ 23.5 K for Ba 4 NbMn 3 O 12 .

Figs. 4
Figs. 4(e) and 4(f)].We also tested the fit using a Debye term and a Debye plus Einstein term, but the fits did not work.Thus we used a Debye term and two Einstein terms as a minimal set of terms in the representative fit in this work.We estimated the magnetic contribution to heat capacity (C mag ) by subtracting the phonon contribution (C ph ) from the measured heat capacity (C p ) and we extrapolated the fitted curve for temperatures between 160 and 2 K [see solid red lines in Figs.4(e) and 4(f)].Figures 4(g) and 4(h) show the values of C mag /T between 2 and 70 K, which reveal a sharp peak around T 1 and a broad hump centered at approximately 20 K in both compounds, capturing the magnetic contribution.Note that the number of moles (n) in Eqs.(2) and (3) was not used in our fits.Figure5exhibits the evolution of the magnetic entropy (∆S mag ) between 2 and 70 K.The change in magnetic entropy was calculated by integrating C mag /T with respect to temperature between 2 and 70 K, which yielded the saturated value of ∆S mag = 11.52 J mol −1 K −1 for Ba 4 TaMn 3 O 12 and ∆S mag = 11.94J mol −1 K −1 for Ba 4 NbMn 3 O 12 .Both values are smaller than the expected entropy for the S = 2 magnet (dashed horizontal lines in Fig.5); ∆S mag = Rln(5) ∼ 13.38 J mol −1 K −1 , where R is the gas constant.However, these values are nearly the same as that of the unexpected S = 3/2 an-

FIG. 9 .
FIG. 9. (a) Primitive rhombohedral unit cell of Ba4NbMn3O12 and Ba4TaMn3O12, where the two simplest magnetic configurations at Mn trimer are shown on the right.Note that Mn atoms at the center and two sides of the Mn trimer are denoted as Mn1 and Mn2, respectively (consistent with Tables I and II).S = 3/2 moments at Mn1 sites are depicted as dotted arrows.(b) Projected density of states (PDOS) for Ba4NbMn3O12 (a top panel) and Ba4TaMn3O12 (a bottom panel), where the energy splitting by hybridization and exchange energy is shown.(c) Band structure of Ba4TaMn3O12 with the ferrimagnetic trimer in the primitive unit cell shown in (a).Red and blue lines depict spin-up and -down bands, respectively.Note that the band of Ba4NbMn3O12 is qualitatively the same as that of Ba4TaMn3O12 (not shown).Conventions for high-symmetry points follow the ones described in Ref.[26].
FIG. 10. Results of band structure calculations of the two simplest antiferromagnetic magnetic configurations.(a) AF1 and (b) AF2, which employ the Mn-trimer as a magnetic unit.(c) PDOS for Ba4NbMn3O12 (upper panels) and Ba4TaMn3O12 (lower panels), where the top and bottom ones correspond to PDOS of the AF1 and AF2 configuration, respectively.

FIG. 11 .
FIG. 11.HOMO (highest energy occupied molecular orbital) and LUMO (lowest energy unoccupied molecular orbital) wave functions of Ba4TaMn3O12 at the Γ point in Fig. 9(b).Violet and red spheres are Mn and and O ions, respectively.Ta ions are not shown as there is practically no contribution in the wave functions.
) and 3(d)], high-energy magnetic signals excited within the trimer are expected in INS measurements; for example, strong antiferromagnetic interactions between the middle and end Mn ions are anticipated in the AF2 model [Fig.10(b)].Thus INS could be suitable to probe the S = 3/2 trimer based on the energy and intensity of the trimer excitation.
visualizes the wave functions of the partial molecular orbital state at Γ point of Ba 4 TaMn 3 O 12 [Fig.9(b)],demonstrating the absence of the e g electron in the central Mn ion.Also, it confirms that the electrons are not delocalized across the trimers via Ta/Nb d orbitals.It is microscopically because the Ta/Nb orbitals are located at much higher energies, which makes the overlap ineffective [Fig.9(b)].The crucial difference for the different electronic structures of the 3d and 4d/5d compounds comes from the distinct hierarchy of the microscopic energy scales.In the 3d case for Ba 4 MMn 3 O 12 , the Coulomb interaction (U ) is dominant.It is stronger than the intersite hybridization energy within the Mn trimer and the exchange splitting (Hund's coupling) [Fig.9(b)], which stabilizes the dominant local moment.This is a strongly localized scheme.The signs of p-orbital lobes surrounding the central Mn (in the overlap of ligand p orbitals) are inversion antisymmetric for all four partial molecular orbital wave functions, effectively canceling inversion-symmetric d-orbital components at the central Mn ion.On the other hand, in the 4d/5d counterparts Ba 4 NbM 3 O 12 (M = Ru, Ir), the intersite hybridization energy is dominant, stabilizing the molecular orbital state[21].

FIG. 12 .
FIG. 12. Projected densities of states for central and side Mn atoms within the Mn trimer from nonmagnetic DFT calculations without U eff .Note that t2g orbital states are close to the Fermi level (E = 0), and eg orbital states are located between 1.5 and 3 eV.
Appendix A: Nonmagnetic DFT calculations

shows the x-ray diffraction patterns of Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 samples collected at room temperature. Rietveld refinements
confirmed that both Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 crystallized in a trigonal crystal structure (R 3 m, No. 166), consistent with the previous research

TABLE IV .
Parameters obtained from CW fits using the inverse magnetic susceptibility χ −1 (T ) of Ba4NbMn3O12.
and Table IV for Ba 4 TaMn 3 O 12 and Ba 4 NbMn 3 O 12 , respectively.The effective magnetic moments are comparable and θ values fluctuate in all trials in Ba 4 TaMn 3 O 12