Successive Short- and Long-Range Magnetic Ordering in Ba2Mn3(SeO3)6 with Honeycomb Layers of Mn3+ Ions Alternating with Triangular Layers of Mn2+ Ions

Mixed-valent Ba2Mn2+Mn23+(SeO3)6 crystallizes in a monoclinic P21/c structure and has honeycomb layers of Mn3+ ions alternating with triangular layers of Mn2+ ions. We established the key parameters governing its magnetic structure by magnetization M and specific heat Cp measurements. The title compound exhibits a close succession of a short-range correlation order at Tcorr = 10.1 ± 0.1 K and a long-range Néel order at TN = 5.7 ± 0.1 K, and exhibits a metamagnetic phase transition at T < TN with hysteresis most pronounced at low temperatures. The causes for these observations were found using the spin exchange parameters evaluated by density functional theory calculations. The title compound represents a unique case in which uniform chains of integer spin Mn3+ (S = 2) ions interact with those of half-integer spin Mn2+ (S = 5/2) ions.


Introduction
Compounds of transition metal magnetic ions exhibit a wide range of phenomena, which are commonly grouped in terms of their spin values: quantum magnetism for systems of small spins and classical magnetism for systems of large spins. In both groups, the ground state that a magnetic compound reaches at low temperatures is governed largely by the dimensionality of its spin exchange interactions and also by the degree of geometrical spin frustration (hereafter, spin frustration) [1]. It quickly becomes complicated to analyze the properties of a magnetic system with magnetic ions of two (or more) different oxidation states and two (or more) different values of spins in terms of these two factors unless the relative values of its various spin exchanges are known. These days, it has become almost routine to determine the values of spin exchanges for any complex magnetic system by performing energy mapping analysis [2][3][4]. This method employs a number of brokensymmetry spin states of a given magnetic system, then evaluates their energies using a spin Hamiltonian made up of the spin exchanges, in addition using density functional theory (DFT) calculations and finally maps the relative energies of the broken-symmetry states from the spin Hamiltonian to those of the DFT calculations. In other words, this method relates the energy spectrum of a model Hamiltonian to that of an electronic Hamiltonian using a set of broken-symmetry states. Over the years, the energy mapping analysis has led to correct spin lattice models with which to understand the properties of numerous magnetic materials.

Materials and Methods
Mixed-valent Ba 2 Mn 2+ Mn 2 3+ (SeO 3 ) 6 was synthesized by a hydrothermal reaction of BaCO 3 (2 mmol), MnCl 2 ·4H 2 O (1 mmol) and H 2 SeO 3 (3 mmol) as precursors with 1.5 mL of 65% HNO 3 and 3 mL of water added. The mixture was placed into a Teflon chamber of a steel autoclave (10 mL) after the degassing was finished. The autoclave then was placed into the furnace, where the temperature was raised to 200 • C for 1 week. After this, the brown powder of Ba 2 Mn 3 (SeO 3 ) 6 was rinsed with water to wash out the contaminants. The obtained powder sample was found to crystallize in the monoclinic P2 1 /c space group with a = 5.4717(3) Å, b = 9.0636(4) Å, c = 17.6586(9) Å, β = 94.519(1), V = 873.03(8) Å 3 , Z = 2 in agreement with the original solution [10] and its powder XRD pattern (BRUKER D8 Advance diffractometer Cu Kα, λ = 1.54056, 1.54439 Å, LYNXEYE detector) is shown in Figure 1.  Physical properties of Ba2Mn3(SeO3)6 were characterized by measuring the magnetization M and the specific heat Cp on ceramic samples (well-pressed pellets of 3   Physical properties of Ba2Mn3(SeO3)6 were characterized by measuring the magnetization M and the specific heat Cp on ceramic samples (well-pressed pellets of 3 Physical properties of Ba 2 Mn 3 (SeO 3 ) 6 were characterized by measuring the magnetization M and the specific heat C p on ceramic samples (well-pressed pellets of 3 mm in diameter and 1 mm in thickness) using various options of "Quantum Design" Physical Properties Measurements System PPMS-9 T taken in the range 2-300 K under magnetic field µ 0 H up to 9 T.

Magnetic Susceptibility
The magnetic susceptibility χ = M/H of Ba 2 Mn 3 (SeO 3 ) 6 taken at µ 0 H = 0.1 T in the field-cooled regime is shown in Figure 3. In the high-temperature region, it follows the Curie-Weiss law: with the temperature-independent term χ 0 = −7.6 × 10 −4 emu/mol, the Curie constant C = 10.98 emu K/mol and the Weiss temperature Θ = −27.8 K. The value of χ 0 exceeds the sum of the Pascal constants of ions and groups constituting the title compound χ 0,calc = −3.5 × 10 −4 emu/mol [13]. This should be attributed to the effect of sample holder. The value of C somewhat exceeds the value C calc = 10.375 emu K/mol expected under the assumption of g-factor, g = 2, for both the Mn 2+ and Mn 3+ ions. Use of g = 2 is reasonable for Mn 2+ (S = 5/2) ions with no orbital-moment contribution, but it underestimates the g-factor for Mn 3+ ions (S = 2). The negative value of the Weiss temperature Θ points to the predominance of antiferromagnetic exchange interactions at elevated temperatures. Its absolute value can be influenced by the competition of ferromagnetic and antiferromagnetic exchange interactions.
magnetic field µ0H up to 9 T.

Magnetic Susceptibility
The magnetic susceptibility χ = M/H of Ba2Mn3(SeO3)6 taken at µ0H = 0.1 T in the fiel cooled regime is shown in Figure 3. In the high-temperature region, it follows the Curi Weiss law: with the temperature-independent term 0 = −7.6 × 10 −4 emu/mol, the Curie constant C 10.98 emu K/mol and the Weiss temperature Θ = −27.8 K. The value of 0 exceeds the su of the Pascal constants of ions and groups constituting the title compound 0, = −3.5 10 −4 emu/mol [13]. This should be attributed to the effect of sample holder. The value of somewhat exceeds the value Ccalc = 10.375 emu K/mol expected under the assumption g-factor, g = 2, for both the Mn 2+ and Mn 3+ ions. Use of g = 2 is reasonable for Mn 2+ (S = 5/ ions with no orbital-moment contribution, but it underestimates the g-factor for Mn 3+ io (S = 2). The negative value of the Weiss temperature Θ points to the predominance antiferromagnetic exchange interactions at elevated temperatures. Its absolute value c be influenced by the competition of ferromagnetic and antiferromagnetic exchan interactions. On lowering the temperature, the χ(T) curve passes through a broad maximum Tcorr = 10.2 K and shows a kink at TN = 5.6 K, which is more pronounced in the Fishe specific heat d(χT)/dT (not shown). This broad maximum is typically found for a qua one-dimensional (1D) antiferromagnetic chain system; hence, suggesting th Ba2Mn3(SeO3)6 has a 1D-like antiferromagnetic subsystem. The kink at a low temperature shows that Ba2Mn3(SeO3)6 undergoes a long-range antiferromagnetic orde The drop of magnetic susceptibility χ below its largest value at Tcorr is less than one-thi of that expected for a three-dimensional easy-axis antiferromagnet [14]. The absence o so-called Curie tail at lowest temperatures signals the high chemical purity of the samp On lowering the temperature, the χ(T) curve passes through a broad maximum at T corr = 10.2 K and shows a kink at T N = 5.6 K, which is more pronounced in the Fisher's specific heat d(χT)/dT (not shown). This broad maximum is typically found for a quasi-onedimensional (1D) antiferromagnetic chain system; hence, suggesting that Ba 2 Mn 3 (SeO 3 ) 6 has a 1D-like antiferromagnetic subsystem. The kink at a lower temperature shows that Ba 2 Mn 3 (SeO 3 ) 6 undergoes a long-range antiferromagnetic order. The drop of magnetic susceptibility χ below its largest value at T corr is less than one-third of that expected for a three-dimensional easy-axis antiferromagnet [14]. The absence of a so-called Curie tail at lowest temperatures signals the high chemical purity of the sample.

Field Dependence of Magnetization
The field dependencies of the magnetization M taken at selected temperatures in the T < T N and T N < T < T corr regions are shown in Figure 4. At the highest temperature of our measurement, the M(H) curve is linear indicating that the system is in the paramagnetic state, but starts to deviate from linearity as the temperature is lowered toward T N . Below T N , the M(H) curves exhibit hysteresis, which becomes most pronounced at the lowest temperature of our measurement. In general, the Heisenberg magnets of quasi-isotropic magnetic moment experience a spin-flop transition prior to the full saturation at the spinflip transition. This is not the case for Ba 2 Mn 3 (SeO 3 ) 6 , although it has isotropic Mn 2+ ions. Instead, it exhibits a metamagnetic transition inherent to the Ising magnets. Such behavior should be associated with the presence of highly anisotropic Mn 3+ ions in the system. The well-pronounced hysteresis underlines the first-order nature of the metamagnetic transition [15].

Field Dependence of Magnetization
The field dependencies of the magnetization M taken at selected temperature T < TN and TN < T < Tcorr regions are shown in Figure 4. At the highest temperature measurement, the M(H) curve is linear indicating that the system is in the param state, but starts to deviate from linearity as the temperature is lowered toward TN TN, the M(H) curves exhibit hysteresis, which becomes most pronounced at the temperature of our measurement. In general, the Heisenberg magnets of quasi-is magnetic moment experience a spin-flop transition prior to the full saturation at th flip transition. This is not the case for Ba2Mn3(SeO3)6, although it has isotropic Mn Instead, it exhibits a metamagnetic transition inherent to the Ising magnets. Such be should be associated with the presence of highly anisotropic Mn 3+ ions in the syste well-pronounced hysteresis underlines the first-order nature of the metam transition [15].

Heat Capacity
The magnetization data are fully consistent with the specific heat data, sh Figure 5. In a wide temperature range, the Cp(T) curve can be described by the sum Debye function with ΘD = 223 K and two Einstein functions with ΘE1 = 556 K an 1449 K. The first of the Einstein functions can be ascribed to the oscillation mode MnO6 octahedra and the second one to that of the SeO3 pyramids. These paramete obtained by fitting the data in the 70-290 K region with the fixed sum of the Deb Einstein functions. The remaining data were considered as a purely m contribution. Indeed, the magnetic entropy is nearly equal to the theoretical v R(2ln(5) + ln(6)) = 41.6 J/mol K, confirming the accuracy of the fit. Neverthe nonmagnetic analogue is still needed to obtain more accurate values of Deb Einstein temperatures.

Heat Capacity
The magnetization data are fully consistent with the specific heat data, shown in Figure 5. In a wide temperature range, the C p (T) curve can be described by the sum of the Debye function with Θ D = 223 K and two Einstein functions with Θ E1 = 556 K and Θ E2 = 1449 K. The first of the Einstein functions can be ascribed to the oscillation mode of the MnO 6 octahedra and the second one to that of the SeO 3 pyramids. These parameters were obtained by fitting the data in the 70-290 K region with the fixed sum of the Debye and Einstein functions. The remaining data were considered as a purely magnetic contribution. Indeed, the magnetic entropy is nearly equal to the theoretical value of R(2ln(5) + ln(6)) = 41.6 J/mol K, confirming the accuracy of the fit. Nevertheless, a nonmagnetic analogue is still needed to obtain more accurate values of Debye and Einstein temperatures.
On lowering the temperature, the specific heat C p passes through a broad maximum at T corr = 10 K and shows a peak at T N = 5.8 K. Under external magnetic field, the broad maximum retains its position, but the sharp anomaly shifts to lower temperatures. Such behavior is typical of low-dimensional antiferromagnets experiencing successive shortrange and long-range orders. On lowering the temperature, the specific heat Cp passes through a broad ma at Tcorr = 10 K and shows a peak at TN = 5.8 K. Under external magnetic field, th maximum retains its position, but the sharp anomaly shifts to lower temperature behavior is typical of low-dimensional antiferromagnets experiencing successive range and long-range orders.

Spin Exchanges and Interpretation
The two important issues concerning the observed magnetic proper Ba2Mn3(SeO3)6 are the cause for the broad maximum of the magnetic susceptibility = 10.1 ± 0.1 K, suggesting a short-range correlation as found for a 1D antiferrom chain and a sharp kink at 5.7 ± 0.1 K, suggesting a long-range antiferromagnetic or With Θ = −27.8 K and TN = 5.7 K (the index of spin frustration f = 5.0), the spin fru in Ba2Mn3(SeO3)6 is not strong enough to prevent it from adopting a lon antiferromagnetic ordering. This is somewhat surprising because one might e strong spin frustration in Ba2Mn3(SeO3)6. Figure 2c shows that the interaction bet chain of Mn 2+ ions with the Mn 3+ ions in the surrounding hexagonal prism ge numerous spin exchange triangles, which is a common arrangement leading t frustration. Furthermore, these interactions must give rise to a 1D antiferromagnet behavior to explain the 1D-like short range correlation at 10.1 K. To explore these we first evaluate the spin exchanges of various exchange paths J in Ba2Mn3(SeO3) 6 All adjacent magnetic ions of Ba2Mn3(SeO3)6 are interconnected by the SeO3 py except for the Mn 3+ ions encircled by dashed ellipses in Figure 2a a-direction  6b). For convenience, these chains will be referred to as J2-and J1-chains, respe Note that each J2-chain is coupled to two adjacent J2-chains and also to two adja chains. Between adjacent J1-and J2-chains four different spin exchange paths (i.e.,

Spin Exchanges and Interpretation
The two important issues concerning the observed magnetic properties of Ba 2 Mn 3 (SeO 3 ) 6 are the cause for the broad maximum of the magnetic susceptibility at T corr = 10.1 ± 0.1 K, suggesting a short-range correlation as found for a 1D antiferromagnetic chain and a sharp kink at 5.7 ± 0.1 K, suggesting a long-range antiferromagnetic ordering. With Θ = −27.8 K and T N = 5.7 K (the index of spin frustration f = 5.0), the spin frustration in Ba 2 Mn 3 (SeO 3 ) 6 is not strong enough to prevent it from adopting a long-range antiferromagnetic ordering. This is somewhat surprising because one might expect a strong spin frustration in Ba 2 Mn 3 (SeO 3 ) 6. Figure 2c shows that the interaction between a chain of Mn 2+ ions with the Mn 3+ ions in the surrounding hexagonal prism generates numerous spin exchange triangles, which is a common arrangement leading to spin-frustration. Furthermore, these interactions must give rise to a 1D antiferromagnetic chain behavior to explain the 1Dlike short range correlation at 10.1 K. To explore these issues, we first evaluate the spin exchanges of various exchange paths J in Ba 2 Mn 3 (SeO 3 ) 6 .
All adjacent magnetic ions of Ba 2 Mn 3 (SeO 3 ) 6 are interconnected by the SeO 3 pyramids except for the Mn 3+ ions encircled by dashed ellipses in Figure 2a Figure 6a. The spin exchange paths J 2 (J 1 ) form chains of Mn 3+ (Mn 2+ ) ions along the a-direction (Figure 6b). For convenience, these chains will be referred to as J 2 -and J 1 -chains, respectively. Note that each J 2 -chain is coupled to two adjacent J 2 -chains and also to two adjacent J 1 -chains. Between adjacent J 1 -and J 2 -chains four different spin exchange paths (i.e., J 4 , J 5 , J 6 and J 7 ) occur (Figure 6c), leading to (J 1 , J 4 , J 5 ), (J 2 , J 4 , J 5 ), (J 1 , J 6 , J 7 ) and (J 2 , J 6 , J 7 ) exchange triangles. A more extended view of Figure 6c is presented in Figure S1 in the Supporting Information (SI). To determine the values of these exchanges, we employ the spin Hamiltonian defined as where the spin exchange Jij between two spin sites can be any one of J1-J7. To evaluate J1-J7, we carry out the energy-mapping analysis [2][3][4] using the eight ordered spin states, i.e., AFi, where i = 1-8, depicted in Figure S2 of the SI. First, we express the energies of the eight ordered states in terms of the spin exchanges J1-J7 using the spin Hamiltonian of Equation (2) and then determine the relative energies of these states (Table 1) by DFT calculations using the frozen core projector augmented plane wave [16,17] encoded in the Vienna ab Initio Simulation Package [18] and the exchange-correlation functional of Perdew, Burke and Ernzerhof [19]. The electron correlations associated with the 3d states of Mn were taken into consideration by DFT + U calculations with effective on-site repulsion Ueff = U − J = 3 eV and 4 eV [20]. All our DFT + U calculations used the plane wave cutoff energy of 450 eV, a set of (6 × 4 × 4) k-points, and the threshold of 10 −6 eV for self-consistent-field energy convergence. Finally, the numerical values of J1-J7 were obtained by mapping the relative energies of the eight ordered spin states onto the corresponding energies determined by DFT + U calculations. The results of these energy-mapping analyses are summarized in Table 2. To determine the values of these exchanges, we employ the spin Hamiltonian defined as where the spin exchange J ij between two spin sites can be any one of J 1 -J 7 . To evaluate J 1 -J 7 , we carry out the energy-mapping analysis [2][3][4] using the eight ordered spin states, i.e., AFi, where i = 1-8, depicted in Figure S2 of the SI. First, we express the energies of the eight ordered states in terms of the spin exchanges J 1 -J 7 using the spin Hamiltonian of Equation (2) and then determine the relative energies of these states (Table 1) by DFT calculations using the frozen core projector augmented plane wave [16,17] encoded in the Vienna ab Initio Simulation Package [18] and the exchange-correlation functional of Perdew, Burke and Ernzerhof [19]. The electron correlations associated with the 3d states of Mn were taken into consideration by DFT + U calculations with effective on-site repulsion U eff = U − J = 3 eV and 4 eV [20]. All our DFT + U calculations used the plane wave cutoff energy of 450 eV, a set of (6 × 4 × 4) k-points, and the threshold of 10 −6 eV for self-consistent-field energy convergence. Finally, the numerical values of J 1 -J 7 were obtained by mapping the relative energies of the eight ordered spin states onto the corresponding energies determined by DFT + U calculations. The results of these energy-mapping analyses are summarized in Table 2. Table 2. Geometrical parameters of the exchange paths and the values of the spin exchanges in Ba 2 Mn 3 (SeO 3 ) 6 . The plus and minus signs of the spin exchanges represent ferromagnetic and antiferromagnetic couplings, respectively.

Geometrical Parameters
Spin Exchanges (in K) As already pointed out, each J 2 -chain interacts with two adjacent J 2 -chains and with two adjacent J 1 -chains (Figure 6b). In terms of the spin exchanges of Table 2, the nature of these interchain interactions can be stated as follows:

Ions
(1) Each J 1 -chain is an antiferromagnetic chain, and so is each J 2 -chain.
(2) Each J 2 -chain is ferromagnetically coupled to two adjacent J 2 -chains via the exchange J 3 (Figure 7a).  As already pointed out, each J2-chain interacts with two adjacent J2-chains and with two adjacent J1-chains (Figure 6b). In terms of the spin exchanges of Table 2, the nature of these interchain interactions can be stated as follows: (1) Each J1-chain is an antiferromagnetic chain, and so is each J2-chain.
(4) Each J2-chain is coupled to another J1-chain via the antiferromagnetic exchanges J4 and J5 (Figure 7c), forming the (J1, J4, J5) and (J2, J4, J5) exchange triangles. With all three antiferromagnetic spin exchanges, each exchange triangle is spin frustrated. Thus, as depicted in Figure 7c,d, one can have two different spin arrangements between these J1and J2-chains. This explains the presence of spin frustration in Ba2Mn3(SeO3)6 as indicated by its index of spin frustration of f = 5. Since J4 is more strongly antiferromagnetic than J5 (by a factor of approximately 3), the spin configuration of Figure 7c is energetically more stable than that of Figure 7d. The antiferromagnetic ordering at TN = 5.7 K means that the spin configuration of Figure 7c dominates over that of Figure 7d in the population. (3) Each J 2 -chain is coupled to one J 1 -chain via the antiferromagnetic exchange J 6 and the ferromagnetic exchange J 7 (Figure 7b), forming the (J 1 , J 6 , J 7 ) and (J 2 , J 6 , J 7 ) exchange triangles. With one ferromagnetic and two antiferromagnetic exchanges, each exchange triangle is not spin-frustrated, so the coupling between these J 2 -and J 1 -chains is ferromagnetic.
(4) Each J 2 -chain is coupled to another J 1 -chain via the antiferromagnetic exchanges J 4 and J 5 (Figure 7c), forming the (J 1 , J 4 , J 5 ) and (J 2 , J 4 , J 5 ) exchange triangles. With all three antiferromagnetic spin exchanges, each exchange triangle is spin frustrated. Thus, as depicted in Figure 7c,d, one can have two different spin arrangements between these J 1and J 2 -chains. This explains the presence of spin frustration in Ba 2 Mn 3 (SeO 3 ) 6 as indicated by its index of spin frustration of f = 5. Since J 4 is more strongly antiferromagnetic than J 5 (by a factor of approximately 3), the spin configuration of Figure 7c is energetically more stable than that of Figure 7d. The antiferromagnetic ordering at T N = 5.7 K means that the spin configuration of Figure 7c dominates over that of Figure 7d in the population.
(5) As already noted, each J 2 -chain is an antiferromagnetic chain and interacts with two adjacent J 2 -chains and two adjacent J 1 -chains. These interchain interactions are all ferromagnetic except for the one with one of the two J 1 -chains. The latter is spin-frustrated as described above. Above T N = 5.7 K, where the latter spin frustration is not settled, the magnetic behavior of Ba 2 Mn 3 (SeO 3 ) 6 should have a strong 1D antiferromagnetic chain character because the antiferromagnetic J 1 -and J 2 -chains are ferromagnetically coupled (via ferromagnetic J 7 and antiferromagnetic J 6 , Figure 6b). This explains the occurrence of the broad maximum in the magnetic susceptibility and the specific heat of Ba 2 Mn 3 (SeO 3 ) 6 . However, interactions between the J 1 -and J 2 -chains via J 4 and J 5 are spin-frustrated, because the (J 1 , J 4 , J 5 ) and (J 2 , J 4 , J 5 ) exchange triangles are spin-frustrated so two different arrangements between the J 1 -and J 2 -chains are possible (Figure 6c,d).
The two sets of the spin exchanges obtained with U eff = 3 and 4 eV are similar in trend. To see which set is better, one might estimate the Weiss temperature Θ using the mean field theory [21,22], to see which set leads to a value closer to the experimental value of Θ = −27.8 K observed for Ba 2 Mn 3 (SeO 3 ) 6 . According to Figure 6a,c, the Θ calc value for Mn 2+ (S = 5/2) ions is predicted to be which is -21.9 K for U eff = 3 eV and -9.9 K for U eff = 4 eV. Similarly, the Θ calc value for Mn 3+ (S = 2) ions is predicted to be which is −10.2 K for U eff = 3 eV and −1.2 K for U eff = 4 eV. Thus, the spin exchanges obtained from of U eff = 3 eV better describes the observed Weiss temperature.

Discussion and Conclusions
In summary, our magnetization and specific heat measurements of Ba 2 Mn 3 (SeO 3 ) 6 reveal that it is a low-dimensional antiferromagnet with a short-range one-dimensional antiferromagnetic chain behavior followed by a long-range antiferromagnetic order as marked by a succession of a broad maximum at T corr = 10.1 ± 0.1 K and a sharper anomaly at T N = 5.7 ± 0.1 K in both magnetic susceptibility (Fisher's specific heat) and specific heat. These observations are well-explained in terms of the spin exchanges of Ba 2 Mn 3 (SeO 3 ) 6 , both antiferromagnetic and ferromagnetic, evaluated by the energy-mapping analysis. In the ordered state, Ba 2 Mn 3 (SeO 3 ) 6 exhibits a metamagnetic phase transition inherent for the Ising magnets with magnetization hysteresis most pronounced at low temperatures. Notably, no hysteresis is observed at µ 0 H = 0 which points to the absence of spontaneous magnetization in Ba 2 Mn 3 (SeO 3 ) 6 .
Structurally, the title compound is organized by the honeycomb layers of Mn 3+ ions alternating with the triangular layers of Mn 2+ ions. Magnetically, it consists of uniform chains of integer spins S = 2 of Mn 3+ ions and half-integer spins S = 5/2 of Mn 2+ ions running along the a axis. Qualitatively different quantum ground states, i.e., gapped and gapless spin liquid, correspondingly, can be expected for these entities [23]. However, the interaction of these chains leads to the formation of a long-range antiferromagnetic order. It would be interesting to synthesize and study isostructural phases of Ba 2 Mn 3 (SeO 3 ) 6 where either the divalent or the trivalent magnetic ions is replaced with nonmagnetic counterparts. The exchange interactions through a chalcogenide anion such as SeO 3 2makes the scales of magnetic fields and temperatures quite convenient for experiments with equipment readily available.
While preparing this article, we became aware of an independent unpublished study on Ba 2 Mn 3 (SeO 3 ) 6 [24], which reported experimental data similar to ours but did not provide any theoretical analysis.

Supplementary Materials:
The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/ma16072685/s1, Figure S1: an extended view of the spin exchange paths in Ba 2 Mn 3 (SeO 3 ) 6 . Figure S2: Eight ordered spin states used for the energy-mapping analysis.