Unusual Spin Exchanges Mediated by the Molecular Anion P2S64−: Theoretical Analyses of the Magnetic Ground States, Magnetic Anisotropy and Spin Exchanges of MPS3 (M = Mn, Fe, Co, Ni)

We examined the magnetic ground states, the preferred spin orientations and the spin exchanges of four layered phases MPS3 (M = Mn, Fe, Co, Ni) by first principles density functional theory plus onsite repulsion (DFT + U) calculations. The magnetic ground states predicted for MPS3 by DFT + U calculations using their optimized crystal structures are in agreement with experiment for M = Mn, Co and Ni, but not for FePS3. DFT + U calculations including spin-orbit coupling correctly predict the observed spin orientations for FePS3, CoPS3 and NiPS3, but not for MnPS3. Further analyses suggest that the ||z spin direction observed for the Mn2+ ions of MnPS3 is caused by the magnetic dipole–dipole interaction in its magnetic ground state. Noting that the spin exchanges are determined by the ligand p-orbital tails of magnetic orbitals, we formulated qualitative rules governing spin exchanges as the guidelines for discussing and estimating the spin exchanges of magnetic solids. Use of these rules allowed us to recognize several unusual exchanges of MPS3, which are mediated by the symmetry-adapted group orbitals of P2S64− and exhibit unusual features unknown from other types of spin exchanges.


Introduction
In an extended solid, transition-metal magnetic cations M are surrounded by maingroup ligands L to form ML n (typically, n = 3-6) polyhedra, and the unpaired spins of M are accommodated in the singly occupied d-states (i.e., the magnetic orbitals) of ML n . Each dstate has the metal d-orbital combined out-of-phase with the p-orbitals of the surrounding ligands L. The tendency for two adjacent magnetic ions to have a ferromagnetic (FM) or an antiferromagnetic (AFM) spin alignment is determined by the spin exchange between them, which takes place through the M-L-M or M-L . . . L-M exchange path [1][2][3][4]. Whereas the characteristics (e.g., the angular and distance dependence) of the M-L-M exchanges is conceptually well understood [5][6][7][8], the properties of the M-L . . . L-M exchanges involving several main-group ligands have only come into focus in the last two decades [1][2][3][4].   To a first approximation, it may be assumed that each MPS 3 layer has a trigonal symmetry (see below for further discussion), so there are three types of spin exchanges to consider, i.e., the first nearest-neighbor (NN) spin exchange J 12 , the second NN spin exchange J 13 , and the third NN exchange J 14 (Figure 1d). J 12 is a spin exchange of the M-L-M type, in which the two metal ions share a common ligand, while J 13 and J 14 are nominally spin exchanges of the M-L . . . L-M type, in which the two metal ions do not share a common ligand. In describing the magnetic properties of MPS 3 in terms of the spin exchanges J 12 , J 13 and J 14 , an interesting conceptual problem arises. Each P 2 S 6 4− anion is coordinated to the six surrounding M 2+ cations simultaneously (Figure 1c,d), so one P 2 S 6 4− anion participates in all three different types of spin exchanges simultaneously with the surrounding six M 2+ ions. Furthermore, the lone-pair orbitals of the S atoms of P 2 S 6 4− , responsible for the coordination with M 2+ ions, form symmetry-adapted group orbitals, in which all six S atoms participate (for example, see Figure 1e). Consequently, there is no qualitative argument with which to even guess the possible differences in J 12 , J 13 , and J 14 . Over the past two decades, it became almost routine to quantitatively determine any spin exchanges of a magnetic solid by performing an energy-mapping analysis based on first principles DFT calculations. From a conceptual point of view, it would be very useful to have qualitative rules with which to judge whether the spin exchange paths involving complex intermediates are usual or unusual.
A number of experimental studies examined the magnetic properties of MPS 3 (M = Mn [9,[11][12][13][14], Fe [9,11,[15][16][17][18], Co [11,19], Ni [11,20]). The magnetic properties of MPS 3 (M = Mn, Fe, Co, Ni) monolayers were examined by DFT calculations to find their potential use as single-layer materials possessing magnetic order [21]. The present work is focused on the magnetic properties of bulk MPS 3 . For the ordered AFM states of MPS 3 , the neutron diffraction studies reported that the layers of MnPS 3 exhibits a honeycomb-type AFM spin arrangement, AF1 (Figure 2a), but those of FePS 3 , CoPS 3 and NiPS 3 a zigzag-chain spin array, AF2 (Figure 2b), in which the FM chains running along the a-direction are antiferromagnetically coupled (hereafter, the ||a-chain arrangement). An alternative AFM arrangement, AF3 (Figure 2c), in which the FM zigzag chains running along the (a + b)-direction are antiferromagnetically coupled (hereafter, the ||(a + b)-chain arrangement), is quite similar in nature to the ||a-chain arrangement. A number of experimental studies examined the magnetic properties of MPS3 (M = Mn [9,[11][12][13][14], Fe [9,11,[15][16][17][18], Co [11,19], Ni [11,20]). The magnetic properties of MPS3 (M = Mn, Fe, Co, Ni) monolayers were examined by DFT calculations to find their potential use as single-layer materials possessing magnetic order [21]. The present work is focused on the magnetic properties of bulk MPS3. For the ordered AFM states of MPS3, the neutron diffraction studies reported that the layers of MnPS3 exhibits a honeycomb-type AFM spin arrangement, AF1 (Figure 2a), but those of FePS3, CoPS3 and NiPS3 a zigzag-chain spin array, AF2 (Figure 2b), in which the FM chains running along the a-direction are antiferromagnetically coupled (hereafter, the ||a-chain arrangement). An alternative AFM arrangement, AF3 (Figure 2c), in which the FM zigzag chains running along the (a + b)-direction are antiferromagnetically coupled (hereafter, the ||(a + b)-chain arrangement), is quite similar in nature to the ||a-chain arrangement. At present, it is unclear why the spin arrangement of MnPS3 differs from those of FePS3, CoPS3 and NiPS3 and why FePS3, CoPS3 and NiPS3 all adopt the ||a-chain arrangement rather than the ||(a + b)-chain arrangement. To explore these questions, it is necessary to examine the relative stabilities of a number of possible ordered spin arrangements of MPS3 (M = Mn, Fe, Co, Ni) by electronic structure calculations and analyze the spin exchanges of their spin lattices.
Other quantities of importance for the magnetic ions M of an extended solid are the preferred orientations of their magnetic moments with respect to the local coordinates of the MLn polyhedra. These quantities, i.e., the magnetic anisotropy energies, are also readily determined by DFT calculations including spin orbit coupling (SOC). For the purpose of interpreting the results of these calculations, the selection rules for the preferred spin orientation of MLn were formulated [2,3,[22][23][24] based on the SOC-induced interactions between the highest-occupied molecular orbital (HOMO) and lowest-unoccupied molecular orbital (LUMO) of MLn. With the local z-axis of MLn taken along its n-fold rotational axis (n = 3, 4), the quantity needed for the selection rules is the minimum difference, |ΔLz|, in the magnetic quantum numbers Lz of the d-states describing the angular behaviors of the HOMO and LUMO. It is of interest to analyze the preferred spin orientations of the M 2+ ions in MPS3 (M = Mn, Fe, Co, Ni) from the viewpoint of the selection rules.
Our work is organized as follows: Section 2 describes simple qualitative rules governing spin exchanges. The details of our DFT calculations are presented in Section 3.1. The magnetic ground states of MPS3 (M = Mn, Fe, Co, Ni) are discussed in Section 3.2, the preferred spin orientations of M 2+ ions of MPS3 in Section 3.3, and the quantitative values of the spin exchanges determined for MPS3 in Section 3.4. We analyze the unusual features of the calculated spin exchanges via the P2S6 4− anion in Section 3.5, and investigate in Section 3.6 the consequences of the simplifying assumption that the honeycomb spin lattice has a trigonal symmetry rather than a slight monoclinic distortion found experimentally. Our concluding remarks are summarized in Section 4. At present, it is unclear why the spin arrangement of MnPS 3 differs from those of FePS 3 , CoPS 3 and NiPS 3 and why FePS 3 , CoPS 3 and NiPS 3 all adopt the ||a-chain arrangement rather than the ||(a + b)-chain arrangement. To explore these questions, it is necessary to examine the relative stabilities of a number of possible ordered spin arrangements of MPS 3 (M = Mn, Fe, Co, Ni) by electronic structure calculations and analyze the spin exchanges of their spin lattices.
Other quantities of importance for the magnetic ions M of an extended solid are the preferred orientations of their magnetic moments with respect to the local coordinates of the ML n polyhedra. These quantities, i.e., the magnetic anisotropy energies, are also readily determined by DFT calculations including spin orbit coupling (SOC). For the purpose of interpreting the results of these calculations, the selection rules for the preferred spin orientation of ML n were formulated [2,3,[22][23][24] based on the SOC-induced interactions between the highest-occupied molecular orbital (HOMO) and lowest-unoccupied molecular orbital (LUMO) of ML n . With the local z-axis of ML n taken along its n-fold rotational axis (n = 3, 4), the quantity needed for the selection rules is the minimum difference, |∆L z |, in the magnetic quantum numbers L z of the d-states describing the angular behaviors of the HOMO and LUMO. It is of interest to analyze the preferred spin orientations of the M 2+ ions in MPS 3 (M = Mn, Fe, Co, Ni) from the viewpoint of the selection rules.
Our work is organized as follows: Section 2 describes simple qualitative rules governing spin exchanges. The details of our DFT calculations are presented in Section 3.1. The magnetic ground states of MPS 3 (M = Mn, Fe, Co, Ni) are discussed in Section 3.2, the preferred spin orientations of M 2+ ions of MPS 3 in Section 3.3, and the quantitative values of the spin exchanges determined for MPS 3 in Section 3.4. We analyze the unusual features of the calculated spin exchanges via the P 2 S 6 4− anion in Section 3.5, and investigate in Section 3.6 the consequences of the simplifying assumption that the honeycomb spin lattice has a trigonal symmetry rather than a slight monoclinic distortion found experimentally. Our concluding remarks are summarized in Section 4.

Spin Exchange between Magnetic Orbitals
For clarity, we use the notation (ϕ i , ϕ j ) to represent the spin exchange arising from the magnetic orbitals ϕ i and ϕ j at the magnetic ion sites A and B, respectively. It is well known that (ϕ i , ϕ j ) consists of two competing terms [1][2][3][4]25] (ϕ i , ϕ j ) = J F + J AF (1)  The FM component J F (>0) is proportional to the exchange repulsion, which increases with increasing the overlap electron density ρ ij = ϕ i ϕ j . In case when the magnetic orbitals ϕ i and ϕ j are degenerate (e.g., between the t 2g states or between e g states of the magnetic ions at octahedral sites), the AFM component J AF (<0) is proportional to the square of the energy split ∆e ij between ϕ i and ϕ j induced by the interaction between them, The energy split ∆e ij is proportional to the overlap integral S ij = ϕ i |ϕ j , so that the magnitude of the AFM component J AF increases with increasing that of (S ij ) 2 . If ϕ i and ϕ j are not degenerate (e.g., between the t 2g and e g states of the magnetic ions), the magnitude of J AF is approximately proportional to −(S ij ) 2 .

p-Orbital Tails of Magnetic Orbitals
The spin exchanges between adjacent transition-metal cations M are determined by the interactions between their magnetic orbitals, which in turn are governed largely by the overlap and the overlap electron density that are generated by the p-orbitals of the ligands present in the magnetic orbitals (the p-orbital tails, for short) [1][2][3][4]. Suppose that the metal ions M are surrounded by main-group ligands L to form ML 6 octahedra. In the t 2g and e g states of an ML 6 octahedron (Figure 3a,b), the d-orbitals of M make σ and π antibonding combinations with the p-orbitals of the ligands L. Thus, the p-orbital tails of the t 2g and e g states are represented as in Figure 4a,b, respectively, so that each M-L bond has the p π and p σ tails in the t 2g and e g states, respectively, as depicted in Figure 4c. The triple-degeneracy of the t 2g and the double-degeneracy of the e g states are lifted in a ML 5 square pyramid and a ML 4 square plane, both of which have a four-fold rotational symmetry; the t 2g states (xz, yz, xy) are split into (xz, yz) and xy, and the e g states (3z 2 − r 2 , x 2 − y 2 ) into 3z 2 − r 2 and x 2 − y 2 . Nevertheless, the description of the ligand p-orbital tails of the d-states depicted in Figure 4c The energy split ij e Δ is proportional to the overlap integral ij

p-Orbital Tails of Magnetic Orbitals.
The spin exchanges between adjacent transition-metal cations M are determined by the interactions between their magnetic orbitals, which in turn are governed largely by the overlap and the overlap electron density that are generated by the p-orbitals of the ligands present in the magnetic orbitals (the p-orbital tails, for short) [1][2][3][4]. Suppose that the metal ions M are surrounded by main-group ligands L to form ML6 octahedra. In the t2g and eg states of an ML6 octahedron (Figure 3a,b), the d-orbitals of M make σ and π antibonding combinations with the p-orbitals of the ligands L. Thus, the p-orbital tails of the t2g and eg states are represented as in Figure 4a,b, respectively, so that each M-L bond has the pπ and pσ tails in the t2g and eg states, respectively, as depicted in Figure 4c. The triple-degeneracy of the t2g and the double-degeneracy of the eg states are lifted in a ML5 square pyramid and a ML4 square plane, both of which have a four-fold rotational symmetry; the t2g states (xz, yz, xy) are split into (xz, yz) and xy, and the eg states (3z 2 − r 2 , x 2 − y 2 ) into 3z 2 − r 2 and x 2 − y 2 . Nevertheless, the description of the ligand p-orbital tails of the d-states depicted in Figure 4c

Spin Exchanges in Terms of the p-Orbital Tails
In this section, we generalize the qualitative rules of spin exchanges formulated for the magnetic solids of Cu 2+ ions [4]. Each Cu 2+ ion has only one magnetic orbital, i.e., the

Spin Exchanges in Terms of the p-Orbital Tails
In this section, we generalize the qualitative rules of spin exchanges formulated for the magnetic solids of Cu 2+ ions [4]. Each Cu 2+ ion has only one magnetic orbital, i.e., the x 2 −y 2 state in which each Cu-L bond has a p σ tail. The d-electron configuration of the magnetic ion is (t 2g ↑) 3 (e g ↑) 2 (t 2g ↓) 0 (e g ↓) 0 in MnPS 3 , (t 2g ↑) 3 (e g ↑) 2 (t 2g ↓) 1 (e g ↓) 0 in FePS 3 , (t 2g ↑) 3 (e g ↑) 2 (t 2g ↓) 2 (e g ↓) 0 in CoPS 3 , and (t 2g ↑) 3 (e g ↑) 2 (t 2g ↓) 3 (e g ↓) 0 in NiPS 3 . Thus, the Mn 2+ , Fe 2+ , Co 2+ , and Ni 2+ ions possess 5, 4, 3, and 2 magnetic orbitals, respectively. For magnetic ions with several magnetic orbitals, the spin exchange J AB between two such ions located at sites A and B is given by the sum of all possible individual exchanges (ϕ i , ϕ j ), where n A and n B are the number of magnetic orbitals at the sites A and B, respectively. Each individual exchange (ϕ i , ϕ j ) can be FM or AFM depending on which term, J F or J AF , dominates. Whether J AB is FM or AFM depends on the sum of all individual (ϕ i , ϕ j ) contributions.

M-L-M Exchange
As shown in Figure 5, there occur three types of M-L-M exchanges between the magnetic orbitals of t 2g and e g states.

Spin Exchanges in Terms of the p-Orbital Tails
In this section, we generalize the qualitative rules of spin exchanges formulated for the magnetic solids of Cu 2+ ions [4]. Each Cu 2+ ion has only one magnetic orbital, i.e., the x 2 −y 2 state in which each Cu-L bond has a pσ tail. The d-electron configuration of the magnetic ion is (t2g↑) 3 (eg↑) 2 (t2g↓) 0 (eg↓) 0 in MnPS3, (t2g↑) 3 (eg↑) 2 (t2g↓) 1 (eg↓) 0 in FePS3, (t2g↑) 3 (eg↑) 2 (t2g↓) 2 (eg↓) 0 in CoPS3, and (t2g↑) 3 (eg↑) 2 (t2g↓) 3 (eg↓) 0 in NiPS3. Thus, the Mn 2+ , Fe 2+ , Co 2+ , and Ni 2+ ions possess 5, 4, 3, and 2 magnetic orbitals, respectively. For magnetic ions with several magnetic orbitals, the spin exchange JAB between two such ions located at sites A and B is given by the sum of all possible individual exchanges where A n and B n are the number of magnetic orbitals at the sites A and B, respectively.  If the M-L-M bond angle θ is 90° for the (eg, eg) and (t2g, t2g) exchanges, and also when θ is 180° for the (eg, t2g) exchange, the two p-orbital tails have an orthogonal arrangement so that 〈 i φ | j φ 〉 = 0 (i.e., AF J = 0). However, the overlap electron density j i φ φ is nonzero (i.e., F J ≠ 0), hence predicting these spin exchanges to be FM. When the θ angles of the (eg, eg) and (t2g, t2g) exchanges increase from 90° toward 180°, and also when the angle θ of the (eg, t2g) exchange decreases from 180° toward 90°, both AF J and F J are nonzero If the M-L-M bond angle θ is 90 • for the (e g , e g ) and (t 2g , t 2g ) exchanges, and also when θ is 180 • for the (e g , t 2g ) exchange, the two p-orbital tails have an orthogonal arrangement so that ϕ i |ϕ j = 0 (i.e., J AF = 0). However, the overlap electron density ϕ i ϕ j is nonzero (i.e., J F = 0), hence predicting these spin exchanges to be FM. When the θ angles of the (e g , e g ) and (t 2g , t 2g ) exchanges increase from 90 • toward 180 • , and also when the angle θ of the (e g , t 2g ) exchange decreases from 180 • toward 90 • , both J AF and J F are nonzero so that the balance between the two determines if the overall exchange (ϕ i , ϕ j ) becomes FM or AFM. These trends are what the Goodenough-Kanamori rules [5][6][7][8] predict.

M-L . . . L-M Exchange
There are two extreme cases of M-L . . . L-M exchange. When the p σ -orbital tails are pointing toward each other (Figure 6a), the overlap integral, ϕ i |ϕ j , can be substantial if the contact distance L . . . L lies in the vicinity of the van der Waals distance. However, the overlap electron density ρ ij = ϕ i ϕ j is practically zero because ϕ i and ϕ j do not have an overlapping region. Consequently, the in-phase and out-of-phase states Ψ + and Ψare split in energy with a large separation ∆e ij . Thus, it is predicted that the M-L . . . L-M type exchange can only be AFM [1][2][3][4]. When the L . . . L linkage is bridged by a d 0 cation such as V 5+ or W 6+ , for example, only the out-of-phase state Ψis lowered in energy by the d π orbital of the cation A, reducing the ∆e ij so that the M-L . . M type exchange can only be AFM [1][2][3][4]. When the L…L linkage is bridged by a d 0 cation such as V 5+ or W 6+ , for example, only the out-of-phase state Ψ-is lowered in energy by the dπ orbital of the cation A, reducing the    M type exchange can only be AFM [1][2][3][4]. When the L…L linkage is bridged by a d cation such as V 5+ or W 6+ , for example, only the out-of-phase state Ψ-is lowered in energy by the dπ orbital of the cation A, reducing the    When a magnetic ion has several unpaired spins, the spin exchange between two magnetic ions is given by the sum of all possible individual (ϕ i , ϕ j ) exchanges. These qualitative rules governing spin exchanges can serve as guidelines for exploring how the calculated spin exchanges are related to the structures of the exchange paths and also for ensuring that important exchange paths are included the set of spin exchanges to evaluate by the energy-mapping analysis.

Details of Calculations
We performed spin-polarized DFT calculations using the Vienna ab initio Simulation Package (VASP) [26,27], the projector augmented wave (PAW) method, and the PBE exchange-correlation functionals [28]. The electron correlation associated with the 3d states of M (M = Mn, Fe, Co, Ni) was taken into consideration by performing the DFT+U calculations [29] with the effective on-site repulsion U eff = U − J = 4 eV on the magnetic ions. Our DFT + U calculations carried out for numerous magnetic solids of transitionmetal ions showed that use of the U eff values in the range of 3 − 5 eV correctly reproduce their magnetic properties (see the original papers cited in the review articles [1][2][3]22,24]). The primary purpose of using DFT + U calculations is to produce magnetic insulating states for magnetic solids. Use of U eff = 3 − 5 eV in DFT + U calculations leads to magnetic insulating states for magnetic solids of Mn 2+ , Fe 2+ , Co 2+ , and Ni 2+ ions. The present work employed the representative U eff value of 4 eV. We carried out DFT + U calculations (with U eff = 4 eV) to optimize the structures of MPS 3 (M = Mn, Fe, Co, Ni) in their FM states by relaxing only the ion positions while keeping the cell parameters fixed and using a set of (4 × 2 × 6) k-points and the criterion of 5 × 10 −3 eV/Å for the ionic relaxation. All our DFT + U calculations for extracting the spin-exchange parameters employed a (2a, 2b, c) supercell, the plane wave cutoff energy of 450 eV, the threshold of 10 −6 eV for self-consistent-field energy convergence, and a set of (4 × 2 × 6) k-points. The preferred spin direction of the cation M 2+ (M = Mn, Fe, Co, Ni) cation was determined by DFT + U + SOC calculations [30], employing a set of (4 × 2 × 6) k-points and the threshold of 10 −6 eV for self-consistent-field energy convergence.

Magnetic Ground States of MPS 3
We probed the magnetic ground states of the MPS 3 phases by evaluating the relative energies, on the basis of DFT + U calculations, of the AF1, AF2 and AF3 spin configurations shown in Figure 2 as well as the FM, AF4, AF5, and AF6 states depicted in Supplementary Materials, Figure S1. As summarized in Table 1, our calculations using the experimental structures of MPS 3 show that the magnetic ground states of MnPS 3 and NiPS 3 adopt the honeycomb state AF1 and the ||a-chain state AF2, respectively, in agreement with experiment. In disagreement with experiment, however, the magnetic ground state is predicted to be the ||(a + b)-chain state AF3 for FePS 3 , and the honeycomb state AF1 for CoPS 3 . Since the energy differences between different spin ordered states are small, it is reasonable to speculate if they may be affected by small structural (monoclinic) distortion. Thus, we optimize the crystal structures of MPS 3 (M = Mn, Fe, Co, Ni) by performing DFT + U calculations to obtain the structures presented in the supporting material. Then, we redetermined the relative stabilities of the FM and AF1-AF6 states using these optimized structures. Results of these calculations are also summarized in Table 1. The optimized structures predict that the magnetic ground states of MnPS 3 , CoPS 3 and NiPS 3 are the same as those observed experimentally, but that of FePS 3 is still the ||(a+b)-chain state AF3 rather than the ||a-chain state AF2 reported experimentally. This result is not a consequence of using the specific value of U eff = 4 eV, because our DFT + U calculations for FePS 3 with U eff = 3.5 and 4.5 eV lead to the same conclusion. To resolve the discrepancy between theory and experiment on the magnetic ground state of FePS 3 , we note that the magnetic peak positions in the neutron diffraction profiles are determined by the repeat distances of the rectangular magnetic structures, namely, a and b for the AF2 state (Figure 2b), and a' and b' for the AF3 state (Figure 2c). In both the experimental and the optimized structures of FePS 3 , it was found that a = a' = 5.947 Å and b = b' = 10.300 Å. Thus, for the neutron diffraction refinement of the magnetic structure for FePS 3 , the AF2 and AF3 states provide an equally good model. In view of our computational results, we conclude that the AF3 state is the correct magnetic ground state for FePS 3 .
The experimental and optimized structures of MPS 3 (M = Mn, Fe, Co, Ni) are very similar, as expected. The important differences between them affecting the magnetic ground state would be the M-S distances of the MS 6 octahedra, because the d-state splitting of the MS 6 octahedra is sensitively affected by them. The M-S distances of the MS 6 octahedra taken from the experimental and optimized crystal structures of MPS 3 are summarized in Table 2, and their arrangements in the honeycomb layer are schematically presented in Figure 8. All Mn-S bonds of MnS 6 in MnPS 3 are nearly equal in length, as expected for a high-spin d 5 ion (Mn 2+ ) environment. The Fe-S bonds of FeS 6 in the optimized structure of FePS 3 are grouped into two short and four long Fe-S bonds. This distinction is less clear in the experimental structure. The Co-S bonds of CoS 6 in the experimental and optimized structures of CoPS 3 are grouped into two short, two medium and two long Co-S bonds. However, the sequence of the medium and long Co-S bonds is switched between the two structures. In the experimental and optimized structures of NiPS 3 , the Ni-S bonds of NiS 6 are grouped into two short, two medium and two long Ni-S bonds. This distinction is less clear in the experimental structure. Thus, between the experimental and optimized structures of MPS 3 , the sequence of the two short, two medium and two long M-S bonds do not switch for M = Fe and Ni whereas it does for M = Co. The latter might be the cause for why the relative stabilities of the AF1 and AF2 states in CoPS 3 switches between the experimental and optimized structures. is less clear in the experimental structure. The Co-S bonds of CoS6 in the experimental and optimized structures of CoPS3 are grouped into two short, two medium and two long Co-S bonds. However, the sequence of the medium and long Co-S bonds is switched between the two structures. In the experimental and optimized structures of NiPS3, the Ni-S bonds of NiS6 are grouped into two short, two medium and two long Ni-S bonds. This distinction is less clear in the experimental structure. Thus, between the experimental and optimized structures of MPS3, the sequence of the two short, two medium and two long M-S bonds do not switch for M = Fe and Ni whereas it does for M = Co. The latter might be the cause for why the relative stabilities of the AF1 and AF2 states in CoPS3 switches between the experimental and optimized structures.

Quantitative Evaluation
We determine the preferred spin orientations of the M 2+ ions in MPS3 (M = Mn, Fe, Co, Ni) phases by performing DFT + U + SOC calculations using their FM states with the ||z and ⊥z spin orientations. For the ⊥z direction we selected the ||a-direction. As summarized in Table 3, these calculations predict the preferred spin orientation to be the ||z direction for FePS3, and the ||x direction for MnPS3, CoPS3 and NiPS3. These predictions are in agreement with experiment for FePS3 [9,18], CoPS3 [19], and NiPS3 [20], while this is not the case for MnPS3 [9,12,14,31]. Our DFT + U + SOC calculations for the AF1 state of MnPS3 show that the ||x spin orientation is still favored over the ||z orientation just as found from the calculations using the FM state of MnPS3. The Mn 2+ spins of MnPS3 were reported to have the ||z orientation in the early studies [9,12], but were found to be

Preferred Spin Orientation of MPS 3 3.3.1. Quantitative Evaluation
We determine the preferred spin orientations of the M 2+ ions in MPS 3 (M = Mn, Fe, Co, Ni) phases by performing DFT + U + SOC calculations using their FM states with the ||z and ⊥z spin orientations. For the ⊥z direction we selected the ||a-direction. As summarized in Table 3, these calculations predict the preferred spin orientation to be the ||z direction for FePS 3 , and the ||x direction for MnPS 3 , CoPS 3 and NiPS 3 . These predictions are in agreement with experiment for FePS 3 [9,18], CoPS 3 [19], and NiPS 3 [20], while this is not the case for MnPS 3 [9,12,14,31]. Our DFT + U + SOC calculations for the AF1 state of MnPS 3 show that the ||x spin orientation is still favored over the ||z orientation just as found from the calculations using the FM state of MnPS 3 . The Mn 2+ spins of MnPS 3 were reported to have the ||z orientation in the early studies [9,12], but were found to be slightly tilted away from the z-axis (by 8 • ) [14,31]. In our further discussion (see below), this small deviation is neglected.  a The same result is obtained by using the AF1 state, which is the magnetic ground state of MnPS 3 . b The same results are obtained from our DFT+U calculations with U eff = 3.5 and 4.5 eV.

Qualitative Picture Selection Rules of Spin Orientation and Implications
With the local z-axis of a ML 6 octahedron along its three-fold rotational axis (Figure 1a), the t 2g set is described by {1a, 1e'}, and the e g set by {2e'} [22][23][24], where Using these d-states, the electron configurations expected for the M 2+ ions of MPS 3 (M = Mn, Fe, Co, Ni) are presented in Figure 9. In the spin polarized description of a magnetic ion, the up-spin d-states lie lower in energy than the down-spin states so that the HOMO and LUMO occur in the down-spin d-states for the M 2+ ions with more than the d 5 electron count, so only the down-spin states are shown for FePS 3 , CoPS 3 , and NiPS 3 in  a The same result is obtained by using the AF1 state, which is the magnetic ground state of MnPS3. b The same results are obtained from our DFT+U calculations with Ueff = 3.5 and 4.5 eV.

Qualitative Picture Selection Rules of Spin Orientation and Implications
With the local z-axis of a ML6 octahedron along its three-fold rotational axis (Figure  1a), the t2g set is described by {1a, 1e'}, and the eg set by {2e'} [22][23][24], where Using these d-states, the electron configurations expected for the M 2+ ions of MPS3 (M = Mn, Fe, Co, Ni) are presented in Figure 9. In the spin polarized description of a magnetic ion, the up-spin d-states lie lower in energy than the down-spin states so that the HOMO and LUMO occur in the down-spin d-states for the M 2+ ions with more than the d 5 electron count, so only the down-spin states are shown for FePS3, CoPS3, and NiPS3 in  In terms of the d-orbital angular states |L, Lz〉 (L = 2, Lz = −2, −1, 0, 1, 2), the 1e' state consists of the |2, ±2〉 and |2, ±1〉 sets in the weight ratio of 2:1, and the 2e' state in the weight ratio of 1:2 ratio. Consequently, the major component of the 1e' set is the |2, ±2〉 set, while that of the 2e' set is the |2, ±1〉 set. In terms of the d-orbital angular states |L, L z (L = 2, L z = −2, −1, 0, 1, 2), the 1e' state consists of the |2, ±2 and |2, ±1 sets in the weight ratio of 2:1, and the 2e' state in the weight ratio of 1:2 ratio. Consequently, the major component of the 1e' set is the |2, ±2 set, while that of the 2e' set is the |2, ±1 set.
The selection rules of the spin orientation are based on the |∆L z | value between the HOMO and LUMO of ML n . If the HOMO and LUMO both occur in the up-spin state or in down-spin states (Figure 9a-c), the ||z spin orientation is predicted if |∆L z | = 0, and the ⊥z spin orientation if |∆L z | = 1. When |∆L z | > 1, the HOMO and LUMO do not interact under SOC and hence do not affect the spin orientation. Between the 1a, 1e' and 2e' states, we note the following cases of values: between the major components of the 1e set between the major components of the 2e set (6) |∆L z | = 1    between 1a and the minor component of 1e between 1a and the major component of 2e between the major components of 1e and 2e (7) We now examine the preferred spin orientations of MPS 3 from the viewpoint of the selection rules and their electron configurations (Figure 9). The d-electron configuration of FePS 3 can be either (d↑) 5 (1e'↓) 1 or (d↑) 5 (1a↓) 1 (Figure 9a), where the notation (d↑) 5 indicates that all up-spin d-states are occupied. The (d↑) 5 (1e'↓) 1 configuration, for which |∆L z | = 0, predicts the ||z spin orientation, while the (d↑) 5 (1a↓) 1 configuration, for which |∆L z | = 1, predicts the ⊥z spin orientation. Thus, the (d↑) 5 (1a↓) 1 configuration is correct for the Fe 2+ ion of FePS 3 . Since this configuration has the degenerate level 1e' unevenly occupied, it should possess uniaxial magnetism [2,3,[22][23][24] and hence a large magnetic anisotropy energy. This is in support of the experimental finding of the Ising character of the spin lattice of FePS 3 [16] or the single-ion anisotropic character of the Fe 2+ ion [17,18]. The d-electron configuration of CoPS 3 can be either (d↑) 5 (1e'↓) 2 or (d↑) 5 (1a↓) 1 (1e'↓) 1 (Figure 9b). The (d↑) 5 (1e'↓) 2 configuration, for which |∆L z | = 1, predicts the ⊥z spin orientation, while the (d↑) 5 (1a↓) 1 (1e'↓) 1 configuration, for which |∆L z | = 0, predicts the ||z spin orientation. Thus, the (d↑) 5 (1e'↓) 2 configuration is correct for the Co 2+ ion of CoPS 3 . Since this configuration has the degenerate level 1e' evenly occupied, it does not possess uniaxial magnetism [2,3,[22][23][24] and hence a small magnetic anisotropy energy. The d-electron configuration of NiPS 3 is given by (d↑) 5 (1a) 1 (1e'↓) 2 (Figure 9c), for which |∆L z | = 1, so the ⊥z spin orientation is predicted in agreement with experiment.
Let us now consider the spin orientation of the Mn 2+ ion of MnPS 3 . First, it should be noted that, if the HOMO and LUMO occur in different spin states as in MnPS 3 (Figure 9d), the selection rules predict the opposite to those found for the case when the HOMO and LUMO occur all in up-spin states or all in down-spin states [2,3,[22][23][24]. Namely, the preferred spin orientation is the ||z spin orientation if |∆L z | = 1, but the ⊥z spin orientation if |∆L z | = 0 [2,3,[22][23][24]. According to Equation (7), |∆L z | = 1 for the Mn 2+ ion of MnPS 3 , which predicts the ⊥z orientation as the preferred spin direction in agreement with the quantitative estimate of the magnetic anisotropy energy obtained from the DFT + U + SOC calculations, although this is in disagreement with experiment [5,[8][9][10]. It has been suggested that the ||z spin orientation is caused by the magnetic dipole-dipole (MDD) interactions [13]. This subject will be probed in the following.

Magnetic Dipole-Dipole Interactions
Being of the order of 0.01 meV for two spin-1/2 ions separated by 2 Å, the MDD interaction is generally weak. For two spins located at sites i and j with the distance r ij and the unit vector e ij along the distance, the MDD interaction is defined as [32] where a 0 is the Bohr radius (0.529177 Å), and (gµ B ) 2 /(a 0 ) 3 = 0.725 meV. The MDD effect on the preferred spin orientation of a given magnetic solid can be examined by comparing the MDD interaction energies calculated for a number of ordered spin arrangements. In summing the MDD interactions between various pairs of spin sites, it is necessary to employ the Ewald summation method [33][34][35]. Table 4 summarizes the MDD interaction energies calculated, by using the optimized structures of MPS 3 (M = Mn, Fe, Co, Ni), for the ||z and ||x spin directions in the AF1, AF2 and AF3 states. The corresponding results obtained by using the experimental structures of MPS 3 are summarized in Table S1. These results can be summarized as follows: for the ||z spin orientation, the AF1 state is more stable than the AF2 and AF3 states. For the ||x spin orientation, the AF2 state is more stable than the AF1 and AF3 states. The ||x spin direction of the AF2 state is more stable than the ||z spin direction of the AF1 state. However, none of these results can reverse the relative stabilities of the ||z and ||x spin directions determined for FePS 3 , CoPS 3 , and NiPS 3 from the DFT + U + SOC calculations ( Table 3). The situation is slightly different for MnPS 3 , which adopts the AF1 state as the magnetic ground state. For MnPS 3 in this state, the MDD calculations predict that the ||z spin orientation is more stable than the ||x spin orientation by 0.3 K per formula unit ( Table 4). Note that this prediction is the exact opposite to what the DFT + U + SOC calculations predict for MnPS 3 in the AF1 state (Table 3). Thus, the balance between these two opposing energy contributions will determine whether the ||z spin orientation is more stable than the ⊥z spin orientation in agreement with the experimental observation. Consequently, for MnPS 3 the MDD interaction dominates over the SOC effect which is plausible because of the half-filled shell electronic configuration. This is because the AF1 magnetic structure is forced on MnPS 3 ; in terms of purely MDD interactions alone, the ⊥z spin orientation in the AF2 state is most stable.

Quantitative Evaluations of Spin Exchanges
Due to the monoclinic crystal structure that MPS 3 adopts, each of the exchanges J 12 , J 13 and J 14 (Figure 10a) are expected to split into two slightly different spin exchanges ( Figure  10b) so that there are six spin exchanges J 1 -J 6 to consider. To extract the values of the six spin exchanges J 1 -J 6 ( Figure 3), we employ the spin Hamiltonian expressed as: Molecules 2021, 26, x 12 of 18 the MDD interaction energies calculated for a number of ordered spin arrangements. In summing the MDD interactions between various pairs of spin sites, it is necessary to employ the Ewald summation method [33][34][35]. Table 4 summarizes the MDD interaction energies calculated, by using the optimized structures of MPS3 (M = Mn, Fe, Co, Ni), for the ||z and ||x spin directions in the AF1, AF2 and AF3 states. The corresponding results obtained by using the experimental structures of MPS3 are summarized in Table S1. These results can be summarized as follows: for the ||z spin orientation, the AF1 state is more stable than the AF2 and AF3 states. For the ||x spin orientation, the AF2 state is more stable than the AF1 and AF3 states. The ||x spin direction of the AF2 state is more stable than the ||z spin direction of the AF1 state. However, none of these results can reverse the relative stabilities of the ||z and ||x spin directions determined for FePS3, CoPS3, and NiPS3 from the DFT + U + SOC calculations ( Table 3). The situation is slightly different for MnPS3, which adopts the AF1 state as the magnetic ground state. For MnPS3 in this state, the MDD calculations predict that the ||z spin orientation is more stable than the ||x spin orientation by 0.3 K per formula unit ( Table 4). Note that this prediction is the exact opposite to what the DFT + U + SOC calculations predict for MnPS3 in the AF1 state (Table 3). Thus, the balance between these two opposing energy contributions will determine whether the ||z spin orientation is more stable than the ⊥z spin orientation in agreement with the experimental observation. Consequently, for MnPS3 the MDD interaction dominates over the SOC effect which is plausible because of the half-filled shell electronic configuration. This is because the AF1 magnetic structure is forced on MnPS3; in terms of purely MDD interactions alone, the ⊥z spin orientation in the AF2 state is most stable.

Quantitative Evaluations of Spin Exchanges
Due to the monoclinic crystal structure that MPS3 adopts, each of the exchanges J12, J13 and J14 (Figure 10a) are expected to split into two slightly different spin exchanges (Figure 10b) so that there are six spin exchanges J1-J6 to consider. To extract the values of the six spin exchanges J1-J6 (Figure 3), we employ the spin Hamiltonian expressed as:  Then, the energies of the FM and AF1-AF6 states of MPS 3 (M = Mn, Fe, Co, Ni) per 2 × 2 × 1 supercell are written as: where S is the spin on each M 2+ ion (i.e., S = 5/2, 2, 3/2 and 1 for M = Mn, Fe, Co, and Ni, respectively). By mapping the relative energies of the FM and AF1-AF6 states determined in terms of the spin exchange J 1 -J 6 onto the corresponding relative energies obtained from the DFT + U calculations (Table 1), we find the values of J 1 -J 6 listed in Table 5. (The spin exchanges of MPS 3 determined by using their experimental crystal structures are summarized in Table S2) With the sign convention adopted in Eq. 1, AFM exchanges are represented by J ij < 0, and FM exchanges by J ij > 0. From Table 5, the following can be observed: In all MPS 3 (M = Mn, Fe, Co, Ni), J 1 = J 2 , J 3 = J 4 , and J 5 = J 6 , reflecting that the exchange paths are different between J 1 and J 2 , between J 3 and J 4 , and between J 5 and J 6 ( Figure 10). b. J 1 ≈ J 2 < 0, J 3 ≈ J 4 ≈ 0, and J 5 ≈ J 6 < 0 for MnPS 3 while J 1 ≈ J 2 > 0, J 3 ≈ J 4 ≈ 0, and J 5 ≈ J 6 < 0 NiPS 3 . To a first approximation, the electron configurations of MnPS 3 and NiPS 3 can be described by (t 2g ) 3 (e g ) 2 and (t 2g ) 6 (e g ) 2 , respectively. That is, they do not possess an unevenly occupied degenerate state t 2g . c.
In FePS 3 and CoPS 3 , J 1 and J 2 are quite different, and so are J 3 and J 4 . While J 5 and J 6 are comparable in FePS 3 , they are quite different in CoPS 3 . The electron configurations of FePS 3 and NiPS 3 can be approximated by (t 2g ) 4 (e g ) 2 and (t 2g ) 5 (e g ) 2 , respectively. Namely, they possess an unevenly occupied degenerate state t 2g . d.
The strongest exchange is J 1 in MnPS 3 , but J 6 in other MPS 3 (M = Fe, Co, Ni). e.
The second NN exchange J 3 is strongly FM in CoPS 3 , while the third NN exchange J 6 is very strongly AFM in CoPS 3 and NiPS 3 .
From the viewpoints of the expected trends in spin exchanges, the observation (e) is quite unusual. This will be discussed in the next section. As pointed out in the previous section, the second NN exchange J 3 of CoPS 3 is strongly FM despite that it is a M-L . . . L-M exchange to a first approximation. This implies that the J F component in some (ϕ i , ϕ j ) exchanges is nonzero, namely, the overlap electron density associated with those exchanges is nonzero. This implies that the p-orbital tails of the two magnetic orbitals are hybridized with the group orbitals of the P 2 S 6 4− anion, i.e., they become delocalized into the whole P 2 S 6 4− anion. Each MS 6 octahedron has three mutually orthogonal "MS 4 square planes" containing the yz, xz and xy states (Figure 11a). At the four corners of these three square planes, the p-orbital tails of the d-states are present (Figure 3a).
octahedron. Due to the bonding requirement of the P2S6 anion, such lone pair orbitals become symmetry-adapted. An example in which the p-orbitals of all the S atoms are present is shown in Figure 1e.
With the (t2g) 5 (eg) 2 configuration, the Co 2+ ion of CoPS3 has five electrons in the t2g level, namely, it has only one t2g magnetic orbital. This magnetic orbital is contained in one of the three CoS4 square planes presented in Figure 11b-d. When the S p-orbital at one corner of the P2S6 4− anion interacts with a d-orbital of M, the S p-orbitals at the remaining corners are also mixed in. Thus, when P2S6 4− anion shares corners with both MS4 square planes of the J3 exchange path, a nonzero overlap electron density is generated, thereby making the spin exchange FM. For convenience, we assume that the magnetic t2g orbital of the Co 2+ ion is the xy state. Then, there will be not only the (xy, xy) exchange, but also the (xy, x 2 −y 2 ) and (x 2 −y 2 , xy) exchanges between the two Co 2+ ions of the J3 path. All these individual exchanges lead to nonzero overlap electron densities by the delocalization of the p-orbital tails with the group orbitals of the molecular anion P2S6 4− . In other words, the spin exchange J3 in CoPS3 is nominally a M-L…L-M, which is expected to be AFM by the qualitative rule, but it is strongly FM. It is clear that, if the L…L linkage is a part of the covalent framework of a molecular anion such as P2S6 4

Third Nearest-Neighbor Exchange
Unlike in MnPS3 and FePS3, the M-S…S-M exchange J6 is unusually strong in CoPS3 and NiPS3 (Section 3.3). This is so despite that the S…S contact distances are longer in CoPS3 and NiPS3 than in MnPS3 and FePS3 (i.e., the S…S contact distance of the J6 path in MPS3 is 3.409, 3.416, 3.421 and 3.450 Å for M = Mn, Fe, Co and Ni, respectively). We note that a strong M-L…L-M exchange (i.e., a spin exchange leading to a large energy split ij e Δ ) becomes weak, when the L…L contact is bridged by a d 0 cation like, e. g., V 5+ and The lone-pair orbitals of the S atoms are important for the formation of each MS 6 octahedron. Due to the bonding requirement of the P 2 S 6 4− anion, such lone pair orbitals become symmetry-adapted. An example in which the p-orbitals of all the S atoms are present is shown in Figure 1e.
With the (t 2g ) 5 (e g ) 2 configuration, the Co 2+ ion of CoPS 3 has five electrons in the t 2g level, namely, it has only one t 2g magnetic orbital. This magnetic orbital is contained in one of the three CoS 4 square planes presented in Figure 11b-d. When the S p-orbital at one corner of the P 2 S 6 4− anion interacts with a d-orbital of M, the S p-orbitals at the remaining corners are also mixed in. Thus, when P 2 S 6 4− anion shares corners with both MS 4 square planes of the J 3 exchange path, a nonzero overlap electron density is generated, thereby making the spin exchange FM. For convenience, we assume that the magnetic t 2g orbital of the Co 2+ ion is the xy state. Then, there will be not only the (xy, xy) exchange, but also the (xy, x 2 −y 2 ) and (x 2 −y 2 , xy) exchanges between the two Co 2+ ions of the J 3 path. All these individual exchanges lead to nonzero overlap electron densities by the delocalization of the p-orbital tails with the group orbitals of the molecular anion P 2 S 6 4− . In other words, the spin exchange J 3 in CoPS 3 is nominally a M-L . . . L-M, which is expected to be AFM by the qualitative rule, but it is strongly FM. It is clear that, if the L . . . L linkage is a part of the covalent framework of a molecular anion such as P 2 (Figure 6b), because the out-of-phase combination ψ − is lowered in energy by interacting with the unoccupied d π orbital of the cation A. Conversely, then, one may ask if the strength of a M-L . . . L-M spin exchange can be enhanced by raising the ψ − level. The latter can be achieved if the L . . . L path provides an occupied level of π-symmetry that can interact with ψ − . As depicted in Figure 12a, the J 6 path has the two MS 4 square planes containing the x 2 -y 2 magnetic orbitals (Figure 12b). The lone-pair group orbital of the S 4 rectangular plane (Figure 12c) of the P 2 S 6 4− anion has the correct symmetry to interact with ψ − , so that the ψ − level is raised in energy thereby enlarging the energy split between ψ + and ψ − and strengthening the J 6 exchange (Figure 12d). Although this reasoning applies equally to MnPS 3 and FePS 3 , the latter do not have a strong J 6 exchange. This can be understood by considering Equation (1), which shows that a magnetic ion with several magnetic orbitals leads to several individual spin exchanges that can lead to FM contributions. an occupied level of π-symmetry that can interact with − ψ . As depicted in Figure 12a, the J6 path has the two MS4 square planes containing the x 2 -y 2 magnetic orbitals ( Figure  12b). The lone-pair group orbital of the S4 rectangular plane (Figure 12c) of the P2S6 4− anion has the correct symmetry to interact with − ψ , so that the − ψ level is raised in energy thereby enlarging the energy split between + ψ and − ψ and strengthening the J6 exchange (Figure 12d). Although this reasoning applies equally to MnPS3 and FePS3, the latter do not have a strong J6 exchange. This can be understood by considering Eq. 1, which shows that a magnetic ion with several magnetic orbitals leads to several individual spin exchanges that can lead to FM contributions. In view of the above discussion, which highlights the unusual nature of the second and third NN spin exchanges mediated by a molecular anion such as P2S6 4− , we propose to use the notation M-(L-L)-M to distinguish it from M-L-M. M-L…L-M and M-L…A…L-M type exchanges. The notation (L-L) indicates two different ligand sites of a multidentate molecular anion, each with lone pairs for the coordination with a cation M. Such M-(L-L)-M exchanges can be strongly FM or strongly AFM, as discussed above. Currently, there are no qualitative rules with which to predict whether they will be FM or AFM. A similar situation was found, for example, for the mineral Azurite Cu3(CO3)2(OH)2, in which every molecular anion CO3 2− participates in three different Cu-(O-O)-Cu exchanges. DFT + U calculations show that one of these three is substantially AFM, but the remaining two are negligible. So far, this observation has not been understood in terms of qualitative reasoning.

Description Using Three Exchanges
Experimentally, the magnetic properties of MPS3 have been interpreted in terms of three exchange parameters, namely, by assuming that J1 = J2 (≡ J12), J3 = J4 (≡ J13), and J5 = J6 (≡ J14). To investigate whether this simplified description is justified, we simulate the relative energies of the seven ordered spin states of MPS3 by using the three exchanges J12, J13 and J14 as parameters in terms of the least-square fitting analysis. Our results, summarized in Table 6, show that the standard deviations of J12, J13 and J14 are small for MnPS3 and NiPS3, In view of the above discussion, which highlights the unusual nature of the second and third NN spin exchanges mediated by a molecular anion such as P 2 S 6 4− , we propose to use the notation M-(L-L)-M to distinguish it from M-L-M. M-L . . . L-M and M-L . . . A . . . L-M type exchanges. The notation (L-L) indicates two different ligand sites of a multidentate molecular anion, each with lone pairs for the coordination with a cation M. Such M-(L-L)-M exchanges can be strongly FM or strongly AFM, as discussed above. Currently, there are no qualitative rules with which to predict whether they will be FM or AFM. A similar situation was found, for example, for the mineral Azurite Cu 3 (CO 3 ) 2 (OH) 2 , in which every molecular anion CO 3 2− participates in three different Cu-(O-O)-Cu exchanges. DFT + U calculations show that one of these three is substantially AFM, but the remaining two are negligible. So far, this observation has not been understood in terms of qualitative reasoning.

Description Using Three Exchanges
Experimentally, the magnetic properties of MPS 3 have been interpreted in terms of three exchange parameters, namely, by assuming that J 1 = J 2 (≡ J 12 ), J 3 = J 4 (≡ J 13 ), and J 5 = J 6 (≡ J 14 ). To investigate whether this simplified description is justified, we simulate the relative energies of the seven ordered spin states of MPS 3 by using the three exchanges J 12 , J 13 and J 14 as parameters in terms of the least-square fitting analysis. Our results, summarized in Table 6, show that the standard deviations of J 12 , J 13 and J 14 are small for MnPS 3 and NiPS 3 , moderate in FePS 3 , but extremely large in CoPS 3 (for details, see Figures S2-S5). The exchanges experimentally deduced for FePS 3 are J 12 = −17 K, J 13 = −0.5 K, and J 14 = 7 K from neutron inelastic scattering measurements [17], −17 K ≤ J 12 ≤ −5.6 K, −7.2 K ≤ J 13 ≤ 2.8 K, and 0 ≤ J 14 ≤ 10 K from powder susceptibility measurements [9], and J 12 = −19.6 K, J 13 = 10.3 K, and J 14 = −2.2 K from high field measurements [17]. These experimental estimates are dominated by J 12 , but the theoretical estimates of Table 6 by J 14 . One might note from Table 6 that the magnetic properties of MnPS 3 , FePS 3 and NiPS 3 can be reasonably well approximated by two exchanges, that is, by J 12 and J 14 for MnPS 3 , by J 14 and J 12 for NiPS 3 , and by J 14 and J 13 for FePS 3 . However, this three-parameter description leads to erroneous predictions for the magnetic ground states of MPS 3 ; it predicts the AF1 state to be the ground state for both MnPS 3 and CoPS 3 . This prediction is correct for MnPS 3 , but incorrect for CoPS 3 . In addition, it predicts that the AF2 and AF3 states possess the same stability for all MPS 3 (M = Mn, Fe, Co, Ni), and are the ground states for FePS 3 and NiPS 3 . These two predictions are both incorrect.