Understanding Pressure Effects on Structural, Optical, and Magnetic Properties of CsMnF4 and Other 3dn Compounds

The pressure dependence of structural, optical, and magnetic properties of the layered compound CsMnF4 are explored through first-principles calculations. The structure at ambient pressure does not arise from a Jahn–Teller effect but from an orthorhombic instability on MnF63– units in the tetragonal parent phase, while there is a P4/n → P4 structural phase transition at P = 40 GPa discarding a spin crossover transition from S = 2 to S = 1. The present results reasonably explain the evolution of spin-allowed d–d transitions under pressure, showing that the first transition undergoes a red-shift under pressure following the orthorhombic distortion in the layer plane. The energy of such a transition at zero pressure is nearly twice that observed in Na3MnF6 due to the internal electric field and the orthorhombic distortion also involved in K2CuF4. The reasons for the lack of orthorhombic distortion in K2MF4 (M = Ni, Mn) or CsFeF4 are also discussed in detail. The present calculations confirm the ferromagnetic ordering of layers in CsMnF4 at zero pressure and predict a shift to an antiferromagnetic phase for pressures above 15 GPa consistent with the reduction of the orthorhombicity of the MnF63– units. This study underlines the usefulness of first-principles calculations for a right interpretation of experimental findings.


■ INTRODUCTION
Insulating transition metal (TM) compounds are an important family of materials characterized by the presence of localized d electrons with strong correlation, giving rise to the interplay of electronic, charge, spin, and orbital degrees of freedom.−9 Due to the open shell structure of the cation, TM compounds usually exhibit optical response in the V−UV domain and magnetic ordering, and thus, optical and magnetic tools are widely used in their characterization.−15 In particular, pressure can change the ground-state spin of a complex shifting from a high-to a low-spin configuration 10 such as happens for a variety of Fe 2+ complexes. 16o understand optical data under pressure, it is crucial to know the evolution of interatomic distances and the nature of involved optical transitions.This requirement is however more difficult to fulfill when the TM complex is distorted from octahedral symmetry.As both conditions are often not fulfilled, first-principles calculations can be of help for gaining the right insight into this matter.Furthermore, in high-pressure experiments optical spectra are sometimes only recorded at room temperature 12,17 where the resolution is certainly poorer than at 4.2 K, and thus, theoretical calculations can aid to overcome that hindrance as well.
This work is devoted, in a first step, to understand the optical absorption spectra under pressure of CsMnF 4 . 17This compound belongs to the interesting family of fluoromanganates 18−20 with formula AMnF 4 (A = alkali monocation or NH 4 + ) involving the 3d 4 cation Mn 3+ .This family has deserved much attention as a model to correlate the various structural, magnetic, and optical properties, both at ambient pressure (study of the chemical pressure effects linked to the change of monocation A) and under high hydrostatic pressure.Among these layered compounds the most studied system is just CsMnF 4 as it is the only one with a tetragonal structure and ferromagnetic (FM) order in the layer planes at ambient pressure when T < T c = 23 K. 19 According to X-ray diffraction data by Molinier and Massa, CsMnF 4 at ambient pressure belongs to the P4/n space group. 18The structure is depicted in Figure 1 showing that Mn 3+ ions are disposed in layers in the ab plane although the F − ions of MnF 6 3− units are not strictly in that plane just reflecting the existence of buckling.In the involved MnF 6 3− units the shortest Mn−F distance corresponds to the z direction, perpendicular to the ab layer plane (R z = 1.817Å), while the longest one (R y = 2.168 Å) is nearly perpendicular to the z direction.The last Mn−F distance (R x = 1.854Å) differs from that of R z by only 0.037 Å.
Interestingly, the F−Mn−F angle of MnF 6 3− units in CsMnF 4 differs by less than 3°from 90°.This fact and a Mn−F−Mn angle of 162°, a consequence of buckling, leads to a local symmetry around a Mn 3+ ion that is not strictly orthorhombic (D 2h ) but C i .CsMnF 4 exhibits an antiferrodistortive arrangement and thus in two adjacent MnF 6  3− units that share a common ligand the longest axes are essentially perpendicular.The RbMnF 4 and KMnF 4 compounds of the AMnF 4 family also involve puckered layers but do not belong to the P4/n space group and have a Mn−F−Mn angle equal to 148°and 140°, respectively. 18,19Both compounds are antiferromagnetic but only below 5 K. 19,20 In the interpretation of structural and optical data (Figure 2) of CsMnF 4 there are two relevant questions that need to be clarified: (1) The local distortion of MnF 6 3− units in CsMnF 4 has usually been ascribed to the Jahn−Teller effect 17−21 despite the low symmetry of the compound.(2) From the optical absorption data under pressure (Figure 2) it has been proposed that the ground state of MnF 6 3− units in CsMnF 4 undergoes a spin crossover transition from S = 2 to S = 1 for a pressure around 38 GPa. 17 As these kinds of transitions have been observed for complexes with ligands such as CN − or SCN −10,16 its possible existence for a fluoride complex certainly deserves a further investigation.For achieving this goal, the lack of data on the evolution of Mn−F distances under pressure for CsMnF 4 hampers the interpretation of optical absorption results, a circumstance that can, however, be overcome by means of theoretical calculations.
Finally, the present work also pays attention to the ferromagnetism displayed by CsMnF 4 at zero pressure, as well as the evolution of the magnetic order under an applied pressure.At this point, the recent results derived for insulating layered compounds like K 2 CuF 4 or Cs 2 AgF 4 shed light on that issue. 9,22,23his work is organized as follows.The computational tools employed in this work are briefly described in the next section, and we then discuss the previous interpretations of experimental data on CsMnF 4 together with the main results obtained in this work.Some final remarks are reported in the last section.

■ COMPUTATIONAL TOOLS
First-principles density functional theory (DFT) calculations (performed using hybrid exchange−correlation functionals) were conducted to analyze the influence of the pressure on the geometry, magnetic order, and optical transitions of the layered compound CsMnF 4 .In order to be sure of the calculated properties, we have used two codes, Crystal17 and VASP, with different implementations of the DFT for periodic crystals, obtaining very similar results.
−33 First, we optimized with the Crystal17 code the geometry of CsMnF 4 at the experimental P4/n phase 18 using different magnetic phases, obtaining the minimum energy for the experimental FM order with values of both lattice parameters  and bond Mn−F distances similar to the experimental data, with discrepancies of less than 3.5%.The next step was the acquisition of the parent phase of the experimental CsMnF 4 P4/n structure.For this goal we started from the experimental CsMnF 4 structure substituting all open shell Mn 3+ ions, with d 4 electronic configuration and S = 2, by Fe 3+ ions (d 5 electronic configuration and S = 5/2) with spherical density (in vacuo) and equal ionic radius, r(Mn 3+ ) ≈ r(Fe 3+ ) ≈ 0.785 Å. 34 The symmetry of the optimized CsFeF 4 parent phase was P4/nmm, precisely the experimental phase of this compound. 35Moreover, in order to elucidate the nature of the phase transition experimentally observed in CsMnF 4 under a pressure P ≈ 38 GPa, 17 we have also optimized the structure of this compound under hydrostatic pressures in the range of P = 0−60 GPa.Furthermore, for each optimized structure, we have followed the unstable harmonic modes in the supercell and found the corresponding ground state.
Geometry optimizations under pressure from 0 to 60 GPa, in 10 GPa increments, were also performed with the VASP code using initial geometries provided by the Crystal17 optimizations, significantly expediting the VASP convergence process.
Energies of the d−d optical transitions were calculated for each optimized geometry under pressure.For the Crystal17 optimized geometries, DFT calculations have been carried out on MnF 6 3− complexes by means of the Amsterdam density functional (ADF) code. 36,37In these pseudomolecular calculations, the MnF 6 3− clusters were embedded in the electrostatic potential of the rest of lattice ions, 38 which was previously calculated through Ewald−Evjen summations. 39,40n a different strategy, we have also calculated the energy values of these d−d transitions with VASP, using a different embedding by substituting three of the four Mn 3+ atoms in the unit cell with Ga 3+ , atoms with a d 10 configuration, and thus a symmetric electron density.The energies of the optical transitions were determined for each pressure ranging from 0 to 60 GPa, by using the ΔSCF method. 41,42The ΔSCF method consists of a self-consistent calculation of the total energy for the ground state, yielding the corresponding E GS .The geometries used are the ones obtained from the optimization of the periodic CsMnF 4 system.At the same geometry the electronic configuration of an excited state is imposed by manually adjusting the occupations of the d-like Kohn−Sham orbitals, and the SCF procedure is performed again, to obtain the corresponding E exc energy.The energy difference, E exc − E GS , is taken as the electronic transition energy.Results of these calculations are very similar to the ones performed by means of the ADF code.
More details on the calculations with both codes can be found in the Supporting Information.

■ ANALYSIS OF PREVIOUS INTERPRETATIONS OF STRUCTURAL AND OPTICAL DATA FOR CSMNF 4
The optical properties in the V−UV domain of an insulating compounds like CsMnF 4 greatly depend on the involved MnF 6 3− units, whose ground state in O h symmetry can, in principle, be either 5 E g (t 3 e 1 ) or 3 T 1g (t 4 e 0 ). 43Nevertheless, experimental data, at ambient pressure, on compounds containing MnF 6 3− support a high-spin configuration (S = 2) as ground state. 12,19,20Octahedral complexes with inorganic ligands like F − or Cl − display a high-spin configuration for 3d n ions (n = 5, 6) like Fe 3+ or Fe 2+ while a low-spin configuration is found for complexes of 4d and 5d ions, like Ru 3+ or Ir 4+ , involving higher 10Dq values. 10,44A low-spin (S = 1/2) ground state has also been reported 45,46 for NiF 6 3− complexes in A 3 NiF 6 , Cs 2 ANiF 6 (A = Na, K), and Rb 2 NaNiF 6 fluorides and also in oxides doped with Ni 3+ . 47However, electron paramagnetic resonance (EPR) measurements carried out in Ni 3+ -doped KMgF 3 and CsCaF 3 perovskites 48,49 lead to a highspin (S = 3/2) ground state.These results show that, in NiF 6 3− complexes, the high-spin/low-spin energy separation must be very small. 45,46Moreover, magnetic measurements performed on Li 3 NiF 6 characterized at low temperature a low-spin configuration (S = 1/2), which tends to a high-spin one with increasing temperature. 50To the best of our knowledge, a spin crossover under pressure in fluorides containing NiF 6 3− complexes has not been found.
As in 6-fold coordinated compounds of d 4 ions (Mn 3+ , Cr 2+ ), the local symmetry of a MnF 6 3− complex corresponds to a distorted octahedron, and the ground state wave function, Ψ g , can briefly be written as Here the three t i (i = 1, 2, 3) and e j (j = 1, 2) orbitals come from the t 2g (xz, xy, yz) and e g (3z 2 −r 2 , x 2 −y 2 ) orbitals in O h symmetry, and thus, the e 2 orbital is empty in the ground state. 13Accordingly, the spin-allowed transitions (ΔS = 0) are simply described by t i ↑ → e 2 ↑ (i = 1, 2, 3) and e 1 ↑ → e 2 ↑.As the e 1 ↑ → e 2 ↑ transition is usually the lowest, its energy is termed E 0 while those associated with the t i ↑ → e 2 ↑ transitions are simply called E i (i = 1, 2, and 3).In addition to the spinallowed transitions, sharp peaks with smaller oscillator strengths associated with forbidden transitions (ΔS = −1) are sometimes observed in optical spectra of d 4 ions (Mn 3+ , Cr 2+ ).They correspond to excited states with S = 1 and are described by determinants like |t 1 ↑ t 2 ↓ t 3 ↑ e 1 ↑| so the total spin of the t-subshell is only 1/2.Such forbidden transitions have well been detected for CrF 2 . 51Similar transitions with ΔS = −1 are also observed in the absorption spectra of CrO 6 9− or CrF 6 3− units and are little sensitive to pressure. 52,53,14,15igure 2 reproduces the experimental optical absorption spectra at room temperature of CsMnF 4 in the 1.5−3.5 eV range for pressures up to 46 GPa. 17 The spectra for P < 25 GPa involve three poorly resolved broad bands and two sharp peaks that are little sensitive to pressure and progressively disappear at higher pressures.Accordingly, it is not easy to know from experimental results at ambient pressure (Figure 2) the number of spin-allowed transitions and the corresponding energies.In the pressure range 37.5−46.0GPa only one broad band is observed in the optical spectrum of CsMnF 4 whose maximum is around 2.5 eV, and its bandwidth is higher than ∼1 eV.No signs of a structural phase transition around 1.4 GPa, early suggested by Moron et al., 21 have been found in the optical measurements on CsMnF 4 .
Seeking to understand the optical spectra of CsMnF 4 at ambient pressure it has been proposed that the experimental geometry of MnF 6 3− units is the result of a static Jahn−Teller (JT) effect. 17Under that assumption, also followed in other works, 18−21 the JT effect would be responsible for having tetragonal MnF 6 3− units with Y as the local principal axis of the complex (Figure 1).−56 Consequently, e 1 would be a molecular orbital transforming like 3y 2 −r 2 while the unoccupied e 2 orbital corresponds to z 2 −x 2 .According to this hypothesis, the optical absorption spectrum of CsMnF 4 at ambient pressure would Inorganic Chemistry involve three spin-allowed transitions in the MnF 6 3− unit whose energies have been proposed 17 to be E 0 = 1.80 eV, E 1 = 2.26 eV, and E 2 = E 3 = 2.80 eV from Figure 2.Such values are close to E 0 = 1.92 eV, E 1 = 2.23 eV, and E 2 = E 3 = 2.73 eV reported by Moroń and Palacio. 57onetheless, doubts are raised by the JT assumption due to the following reasons: [54][55][56][57]9 (i) The existence of a JT effect requires a degenerate electronic state in the initial geometry.Therefore, as CsMnF 4 is a layered compound, even if the geometry of the initial parent phase is tetragonal the electronic ground state of an MnF 6 3− unit should not be degenerate according to symmetry. (ii)MnF 4 is a layered compound where layers are perpendicular to the crystal c axis (Figure 1).Accordingly, one would expect that the axis of the MnF 6 3− unit perpendicular to the layer plane plays a singular role, a fact seemingly not consistent with the JT assumption.
(iii) The local equilibrium geometry for MnF 6 3− in CsMnF 4 is not tetragonal.Indeed, even assuming Y as the main axis (Figure 1) the symmetry would be at most orthorhombic because R X − R Z = 0.037 Å. Accordingly, one should expect four and not only three d−d transitions with ΔS = 0 for CsMnF 4 .(iv) Although most of the d 9 systems which exhibit a static JT effect are elongated with a hole in an x 2 −y 2 type orbital 54−57 this is not a general rule.For instance, in the cubic CaO lattice doped with Ni + , the hole is in a 3z 2 −r 2 orbital and the octahedron compressed. 58,59In non-JT systems like K 2 ZnF 4 :Cu 2+ , where the host lattice is tetragonal, the hole is also in 3z 2 −r 2 . 60,61oking at higher pressures, the optical spectrum of CsMnF 4 seems to undergo some change around 37 GPa as above this pressure the optical spectrum at room temperature involves only one very broad band (Figure 2).Such effect has been assumed to arise from a change of the ground-state configuration which would be 3 T 1g (t 4 e 0 ) when P > 37 GPa with an O h local geometry for MnF 6 3− units. 17o further arguments are given for underpinning that assumption that casts in principle some doubts.Indeed, just looking at Tanabe−Sugano diagrams 43 one would expect that in octahedral symmetry the transition of a 5 E g (t 3 e 1 ) ground state to 3 T 1g (t 4 e 0 ) in MnF 63− takes place for 10Dq ≥ 3 eV.For estimating whether this condition is fulfilled, it is useful to consider that the 10Dq value derived from the four allowed transitions in Na 3 MnF 6 (10Dq = 1.88 eV) is close to 10Dq measured for octahedral CrF 6 3− units in cubic elpasolites involving also a trivalent TM cation. 14,62,63For instance, in the case of Rb 2 KCrF 6 , 10Dq = 1.97 eV at ambient pressure while a value d(10Dq)/dP = 0.014 eV/GPa has been measured in the range 0−10 GPa. 63Accepting this value for a hypothetical O h MnF 6 3− unit, one would expect 10Dq = 2.41 eV for P = 38 GPa.This figure is thus below 10Dq = 3 eV, which is the estimated value required for the spin crossover.
Given these facts, this work addresses the following questions centered on the interpretation of optical data under pressure in CsMnF 4 : (1) the origin of the local geometry of MnF 6 3− units in CsMnF 4 and the nature of the electronic ground state at zero pressure; (2) the number and energies of the spin-allowed transitions at zero pressure and its evolution as a function of the applied pressure; and (3) the nature of the phase transition observed around 37 GPa, paying particular attention to a possible shift of the ground state spin from S = 2 to S = 1.

■ RESULTS AND DISCUSSION
Structure and Electronic Ground State at Zero Pressure.The calculated values of the lattice parameters and Mn−F distances of CsMnF 4 at zero pressure are collected in Table 1.Such values, derived using both CRYSTAL and VASP codes in a P4/n space group, are compared to experimental findings. 18,19As can be seen in Table 1 the differences between calculated and experimental values are always smaller than 1.5%.The electronic ground state of the MnF 6 3− unit is found to come from the 5 E g (t 3 e 1 ) configuration in cubic symmetry, thus implying a high-spin configuration with S = 2.
Seeking to understand the origin of the local structure in CsMnF 4 , it is useful to analyze the evolution of the crystal structure when Mn 3+ is replaced by a cation like Fe 3+ with a similar ionic radius, giving rise to the so-called high-symmetry parent phase. 6,13,22,23Under strict octahedral symmetry, Fe 3+ in high-spin configuration (S = 5/2) exhibits a nearly spherical electronic density which is however never found for a 3d 4 ion like Mn 3+ or a 3d 9 ion like Cu 2+ in the same situation.After the Mn 3+ → Fe 3+ substitution, we carried out a geometry optimization of the CsFeF 4 structure maintaining fixed the experimental P4/n space group of CsMnF 4 .The obtained final structure has a higher symmetry belonging to the tetragonal P4/nmm space group, although it still involves buckled layers (Table 2).It should be noted that the obtained local geometry of each FeF 6 3− unit in CsFeF 4 corresponds to a compressed octahedron with R X = R Y and R Z < R X (Table 2) and a symmetry practically tetragonal with Z as the main axis.Interestingly in the present case the calculated parent phase coincides with the experimental structure of the CsFeF 4 compound at normal conditions. 64,65The calculated lattice parameters and Fe−F distances correspond to experimental ones within 1.5%.Bearing the preceding considerations in mind, we first consider CsMnF 4 in the tetragonal P4/nmm geometry of the parent phase.In that case, the MnF 6 3− units are essentially tetragonal with R Z = 1.80 Å and R X = R Y = 2.011 Å.The main axis is Z, perpendicular to the layer plane, and the four antibonding valence orbitals are ordered as shown in Figure 3, giving rise to an orbitally singlet ground state.Following the compressed geometry, the LUMO corresponds to the molecular orbital of the MnF 6 3− unit transforming like 3z 2 − r 2 and is simply designated by |3z 2 −r 2 ⟩ while the highest occupied molecular orbital, HOMO, is |x 2 −y 2 ⟩.Accordingly, the ground state belongs to 5  .According to the shape of (−e)V R (r) it favors to increase the energy of | 3z 2 −r 2 ⟩ and reduce that of |x 2 −y 2 ⟩, thus enhancing the value of E 0 .We have derived a value E 0 = 0.7 eV considering only the isolated MnF 6 3− unit while E 0 = 1.2 eV is obtained once V R (r) is incorporated into the calculation.Therefore, V R (r) not only helps to stabilize |x 2 −y 2 ⟩ as HOMO but likely has a significant influence on optical transitions, a matter that will be discussed later.
Interestingly, if CsMnF 4 is in the P4/nmm phase the electronic density in the MnF 6 3− unit with an unpaired electron in |x 2 −y 2 ⟩ is compatible with a tetragonal symmetry (R Z < R X = R Y ) of the complex.This fact is thus consistent with the lack of JT effect under an initial tetragonal symmetry such as happens for K 2 ZnF 4 :Cu 2+ . 61,76Nevertheless, the equilibrium geometry of CsMnF 4 exhibits a lower P4/n symmetry with R Y > R X for the MnF 6 3− complex.This instability of the P4/nmm structure for CsMnF 4 implies the existence of at least one vibration mode with imaginary frequency. 23Accordingly, we have calculated the vibrational frequencies of CsMnF 4 in the optimized P4/nmm geometry finding an a 2g lattice mode with a frequency equal to 367i cm −1 .The effect of that mode on the MnF 6 3− unit is described by the orthorhombic b 1g local mode (Figure 5) which is thus responsible for having R Y − R X = 0.31 Å in the final equilibrium geometry of CsMnF 4 at P = 0 (Table 1).The difference, ΔU, between the calculated energy per molecule for the equilibrium P4/n and the parent P4/nmm phase amounts to 71 meV and is thus the source for the orthorhombic instability shown in Figure 5.That instability, similar to that found for K 2 CuF 4 or Cs 2 AgF 4 , 9,23 is driven by the electron− vibration coupling, H vib , and involves changes in the ground state wave function and the associated electronic density. 8,67f H 0 denotes the Hamiltonian where all nuclei are frozen at a given position, the Hamiltonian H describing the small motions around it following the distortion coordinate Q of a nondegenerate mode can simply be written as It should be noted that H vib exhibits the same symmetry as H 0 provided that symmetry operations are carried out on both electron and nuclei coordinates.Accordingly, in eq 2, V(r) transforms like the coordinate Q, and thus both operators belong to the same Γ irrep.If Ψ g (r) is the wave function of the ground-state orbital singlet then ⟨Ψ g (r)| V(r)| Ψ g (r)⟩ = 0 unless Q refers to the totally symmetric vibration belonging to a 1g .However, in second-order perturbations H vib can couple Ψ g (r) with excited states, Ψ n (r), belonging to Γ n , giving rise to a decrement, ΔE g , of the ground-state energy Thus, the excited states verifying Γ g × Γ n ⊃ Γ can be coupled to the ground state.This fact modifies the electronic density and also yields a negative contribution, −K ν ,to the total force constant, K, which can be written as Here, K 0 stands for the positive contribution associated with the frozen electronic density of H 0 while K ν reflects the electronic density changes due to H vib and is given by Thus, the instability responsible for the equilibrium structure of CsMnF 4 appears because the K 0 < K ν condition is fulfilled in this case.Interestingly, this implies the admixture of the 5 A 1g ground state of MnF 6 3− with the excited 5 B 1g through a b 1g local mode, thus modifying the electronic density of the complex.It is worth noting that, in CsMnF 4 , two adjacent MnF 6 3− units share a common ligand, a fact that is behind the instability developed in K 2 CuF 4 or Cs 2 AgF 4 23,9 but not in K 2 ZnF 4 :Cu 2+23,9,60,61 or even in KAlCuF 6 68−70 where the CuF 6 4− units are well-separated.In addition, the parent phase of CsMnF 4 involves compressed MnF 6 3− units giving rise to softer bonds in the layer plane, a fact that also helps to develop the orthorhombic instability such as has previously been discussed. 23,9,22n the equilibrium geometry of CsMnF 4 at zero pressure the highest occupied molecular orbital (HOMO) wave function the MnF 6 3− unit, |φ H ⟩, is not purely |x 2 −y 2 ⟩ but involves an admixture of |3z 2 −r 2 ⟩ as a result of the symmetry reduction due to the instability The present calculations yield α 2 = 85%, stressing that the HOMO wave function keeps a dominant |x 2 −y 2 ⟩ character once the distortion takes place and R Y > R X .A similar situation has been found in other layered systems like K 2 CuF 4 or Cs 2 AgF 4 . 23,9Interestingly, if we write eq 6 using the {|x 2 −z 2 ⟩, | 3y 2 −r 2 ⟩} basis, then If α 2 = 85%, it is simple to find β′ 2 = 98% demonstrating that the HOMO wave function is essentially |3y 2 −r 2 ⟩ and thus greatly localized along the longest Y axis.In the same way, the LUMO wave function, |φ L ⟩, is basically equal to |x 2 −z 2 ⟩ despite the electronic structure of CsMnF 4 not being due to the JT effect.
For the sake of completeness, a qualitative picture of the electronic ground state of MnF 6 3− at the equilibrium geometry of CsMnF 4 at zero pressure is shown on Figure 3. Accordingly, the lowest d−d excitation is simply described by |3y 2 −r 2 ⟩ → | x 2 −z 2 ⟩.This matter will be discussed later.
Structural Changes Induced by Pressure.Results of calculations on CsMnF 4 under applied pressure were performed using both CRYSTAL and VASP codes.By means of them, we can derive the enthalpy per molecule, H = U + PV, responsible for the equilibrium structure at T = 0 K. Below P = 40 GPa the equilibrium structure of CsMnF 4 is always found to be described by the space group at ambient pressure (P4/n).The variations of lattice parameters and Mn− F distances induced by applied pressure are displayed in Table 3.Both codes lead to very similar results.Nevertheless, at P = 40 GPa the present calculations indicate that the P4/n structure becomes unstable, being slightly distorted to another one with a P4 space group (Figure 6).In both phases, the ground state of MnF 6 3− units is found to correspond to the high-spin configuration involving S = 2.
It should be noted that in the P4 phase at P = 40 GPa the MnF 6 3− complexes have local triclinic symmetry (point group C 1 ), with small distortions in the Mn−F distances and in the F−Mn−F angles with respect to the P4/n structure at the same  pressure (see Figure S1 of the Supporting Information).For simplicity, Tables 3 and 5 show the average values of the Mn− F distances in the 3 local directions X, Y, and Z of the complexes, although the calculations of the d−d transitions have been carried out with the optimized geometries.
It should be also noted that within the P4/n phase, the length reduction due to pressure is much bigger for the long bond of MnF 6 3− than for the two others.For instance, in the range 0−40 GPa, R Y decreases by 0.25 Å while R X and R Z are reduced only by 0.03 and 0.07 Å, respectively (Table 3).In other words, pressure helps the geometry of the MnF 6 3− complex to become closer to the octahedral one as R Y − R Z goes from 0.36 Å at ambient pressure to only 0.16 Å when P = 40 GPa.This behavior is very similar to that found for the hybrid layered perovskite (C 2 H 5 NH 3 ) 2 CdCl 4 doped with Cu 2+ and plays a key role explaining the shifts undergone by d−d transitions under pressure, 71 a question analyzed in the next subsection.
Concerning the P4/n → P4 phase transition at P = 40 GPa, calculations lead to an enthalpy difference ΔH = H(P4) − H(P4/n) equal only to −0.023 eV and −0.030 eV from VASP and CRYSTAL codes, respectively.It is worth noting that according to calculations, the three Mn−F distances of MnF 6 3− are only slightly influenced by the phase transition.Indeed, the variations undergone by R X , R Y , and R Z at P = 40 GPa on changing from P4/n to P4 are smaller than 1.4% (Table 3).This fact thus suggests that the P4/n → P4 phase transition does not produce significant jumps in optical transitions, a matter treated in the next subsection.
Although the present calculations lead to an electronic ground state of MnF 6 3− units coming from the 5 E g (t 3 e 1 ) highspin configuration in O h symmetry, we have also paid attention to determine the enthalpy of the lowest state emerging from the 3 T 1g (t 4 e 0 ) low-spin configuration in O h symmetry.The difference of the enthalpy per molecule, ΔH, between lowsping (S = 1) and high-spin (S = 2) configurations derived for both the P4/n and P4 structures at 40 and 50 GPa is displayed in Table 4.The calculated ΔH values by means of VASP and CRYSTAL codes are all in the range 0.45−0.68eV.Therefore, the assumption of a transition from S = 2 to S = 1 induced by pressure at about 40 GPa 17 is highly unlikely.
Spin-Allowed d−d Transitions in CsMnF 4 under Pressure.Considering the equilibrium geometries derived for CsMnF 4 at different pressures in Table 3, we have calculated in a further step the evolution of spin-allowed d−d transitions for pressures up to 40 GPa using both VASP and CRYSTAL codes.Results displayed in Table 5 correspond to the P4 phase for P = 40 GPa and to the P4/n phase for the rest of the pressures.Both codes lead to similar results.
The results presented in Table 5 show the existence of four allowed d−d transitions, in accord with the orthorhombic symmetry of MnF 6 3− units.As expected from Figure 3  The experimental spectrum of CsMnF 4 obtained at room temperature and ambient pressure (Figure 2) was assumed to involve only three spin-allowed d−d transition with energies 17 E 0 = 1.80 eV, E 1 = 2.26 eV, and E 2 = E 3 = 2.80 eV.According to Table 5, such transitions can now reasonably be assigned as | 3y 2 −r 2 ⟩ → |x 2 −z 2 ⟩, |xz⟩ → |x 2 −z 2 ⟩, and |xy⟩ → |x 2 −z 2 ⟩, respectively.In the poorly resolved experimental spectrum (Figure 2), the |yz⟩ → |x 2 −z 2 ⟩ transition, calculated at about 2.55 eV, is not well seen likely due to the width of the |xz⟩ → | x 2 −z 2 ⟩ and |xy⟩ → |x 2 −z 2 ⟩ transitions as well as to the presence of spin-forbidden transitions in the spectrum.
The calculated evolution of four spin-allowed d−d transitions with pressure (Table 5) is depicted in Figure 7 5.Therefore, the very broad band observed at around 40 GPa that peaked at 2.5 eV likely involves the unresolved contributions of three transitions associated with the three t 2g orbitals in O h symmetry.
The evolution of four spin-allowed d−d transitions of CsMnF 4 under pressure (Table 5 and Figure 7) is consistent   a First and second lines show the results obtained through VASP and CRYSTAL + ADF codes, respectively.The energies of all transitions are in eV.Note that data for P = 40 GPa correspond to the stable P4 phase, while for lower pressures the equilibrium structure corresponds to P4/n.Transitions are described though the dominant character of the involved orbitals.The influence of the internal electric field, E R (r), on the calculated d−d transitions is systematically taken into account.
with the calculated variations undergone by the Mn−F distances (Table 3).Indeed, we have seen that pressure favors a geometry of the MnF 6 3− complex progressively closer to the octahedral one thus reducing Δ(xy,xz) as well as the energy of the |3y 2 −r 2 ⟩ → |x 2 −z 2 ⟩ transition.Along this line, we have found that R Y related to the softest Mn−F bond is much more reduced by pressure than R X or R Z (Table 3), and thus, we can reasonably expect a red-shift for that transition under pressure.This behavior is fully similar to that found in hybrid layered perovskites 69  In a further step, it is necessary to disclose the influence of the internal field E (r) R on the optical transitions of CsMnF 4 as it plays an important role in the case of inorganic layered perovskites like K 2 CuF 4 or Cs 2 AgF 4 . 6,8,23For clarifying this issue, we have calculated in a first step the energies of four d−d transitions considering only the isolated MnF 6 3− unit at the equilibrium geometry while in a second step we have added the influence of the electrostatic potential, V R (r).Results have been derived for P = 0 and 20 GPa and are shown in Table 6.It can be noticed that the addition of E R (r) produces an important shift of about 0.5 eV on the energy of the lowest d− d transition, a result qualitatively consistent with the shape of V R (r) (Figure 4).Indeed (−e)V R (r) tends to decrease the energy of orbitals, such as |3y 2 −r 2 ⟩, lying in the layer plane thus enhancing the |3y 2 −r 2 ⟩ → |x 2 −z 2 ⟩ transition energy.A similar situation is encountered in K 2 CuF 4 or Cs 2 AgF 4 . 23inally, for the sake of clarity, we have performed an analysis of three contributions responsible for the energy E 0 of the first spin-allowed transition |3y 2 −r 2 ⟩ → |x 2 −z 2 ⟩ of CsMnF 4 at zero pressure.According to the present discussion, we divide the calculation in 3 steps: (1) We consider an isolated MnF 6 3− unit in the tetragonal geometry of the parent phase, obtaining a value E 0 = 0.89 eV.(2) In a second step, we still keep the isolated MnF 6 3− unit but in the final P4/n geometry where MnF 6 3− exhibits a practical orthorhombic symmetry with R Y − R X = 0.31 Å, obtaining E 0 = 1.44 eV.(3) In the final step, we include the shift due to the internal V R (r) potential on the energy of the transition, obtaining a value E 0 = 1.92 eV.Therefore, the contributions of both the orthorhombic distortion and V R (r) enhance the E 0 value by ∼1 eV.

Survey of Other Compounds Containing MnF 6 3−
Units.Thanks to the analysis carried out in preceding sections on CsMnF 4 we can now gain better insight into the different optical properties at zero pressure displayed by other fluorides involving Mn 3+ .In a first step we pay attention to the Na 3 MnF 6 compound 12,13 where the metal−ligand distances are R z = 2.018 Å, R x = 1.862Å, and R y = 1.897Å. Accordingly, in this case the longest metal−ligand distance is along the Z axis, the orthorhombicity is small (R y − R x = 0.035 Å), and the HOMO practically equal to |3z 2 −r 2 ⟩.The energies of allowed d−d transitions measured experimentally 12 and derived by means of first-principles calculations 13 are reported in Table 7.It can be seen that the transitions |t i ⟩ → |3z 2 −r 2 ⟩ (t = xy, xz, yz) all are in the 2−3 eV range, as has been found for CsMnF 4 .However, the energy of the first |3z 2 −r 2 ⟩ → |x 2 −y 2 ⟩ transition is practically half the value E 0 = 1.9 eV obtained for CsMnF 4 , thus involving a remarkable difference.A similar situation is encountered when looking at the first transition of K 3 MnF 6 or Cs 3 MnF 6 where E 0 = 1.1 eV. 72,57here are two main reasons behind a E 0 value around 1 eV for Na 3 MnF 6 .On one hand, the orthorhombicity of MnF 6 3− units in this compound is 1 order of magnitude smaller than that found for CsMnF 4 .On the other hand, Na 3 MnF 6 is not a typical layered compound like K 2 CuF 4 or CsMnF 4 and the internal electric field, E R (r), has proven to induce shifts on d− d transitions not higher than 0.1 eV. 13 It is worth noting now that a similar situation has been found when comparing fluoride compounds containing Cu 2+ , such as KZnF 3 :Cu 2+ , K 2 ZnF 4 :Cu 2+ , and K 2 CuF 4 .As KZnF 3 is cubic there is a static JT effect in KZnF 3 :Cu 2+ , with an unpaired electron in |x 2 −y 2 ⟩. 73−75 Due to the cubic symmetry  The first row corresponds to calculated values with CRYSTAL + ADF codes on an isolated MnF 6 3− unit at the equilibrium geometry corresponding to the applied pressure, while in the second row are shown the values derived including the electrostatic potential V R (r) in the calculation.Transitions are described though the dominant character of involved orbitals. of the host lattice, E R (r) has no effect on the first d−d transition, |3z 2 −r 2 ⟩ → |x 2 −y 2 ⟩, whose energy is E 0 = 0.40 eV. 74,75As K 2 ZnF 4 is a layered compound where the shape of V R (r) is similar to that of Figure 4, the unpaired electron is forced to be in a |3z 2 −r 2 ⟩ orbital by the action of V R (r), and consequently the |x 2 −y 2 ⟩ → |3z 2 −r 2 ⟩ transition energy is enhanced having a value E 0 = 0.70 eV. 76,61,74Finally, as in K 2 CuF 4 two adjacent CuF 6 4− units share a common ligand, this favors an orthorhombic instability which still increases the E 0 value up to 1.03 eV. 77,78,74,9agnetic Structure of CsMnF 4 : Influence of Pressure.As shown in Figure 8, the present calculations support that in CsMnF 4 at ambient pressure layers are ferromagnetically ordered.This behavior, consistent with experimental data, is the same found for layered compounds like K 2 CuF 4 or Cs 2 AgF 4 where the M−F−M angle, θ, (M = Cu, Ag) is 180°d ue to the absence of buckling at the Cmca equilibrium structure. 9,22The ferromagnetism displayed by K 2 CuF 4 or Cs 2 AgF 4 is surprising as tetragonal K 2 MnF 4 and K 2 NiF 4 compounds, where the θ angle is also equal to 180°, exhibit an AFM ordering the same found for KMnF 3 and KNiF 3 perovskites. 79Very recently, the ferromagnetism in K 2 CuF 4 and Cs 2 AgF 4 at the Cmca equilibrium structure has proved to come from the orthorhombic distortion undergone by MF 6 4− units (M = Cu, Ag), which in turn is actually responsible for the orbital ordering in these compounds. 22Indeed, in the I4/ mmm parent phase of K 2 CuF 4 and Cs 2 AgF 4 , involving tetragonal MF 6 4− units (M = Cu, Ag), the calculations lead to an AFM ordering similar to that observed for K 2 MnF 4 or K 2 NiF 4 at ambient pressure.
We verified that a similar situation holds for CsMnF 4 .Indeed, in the P4/nmm parent phase, where R y = R x we also find that the layers of CsMnF 4 are AFM ordered although the FM ordering has an energy that is only 20.6 meV above (Figure 9).Moreover, on passing progressively at zero pressure from the P4/nmm parent phase to the equilibrium P4/n structure, CsMnF 4 easily becomes ferromagnetic following the increase of the orthorhombic distortion on the MnF 6 3− units, as shown in Figure 9.
It is worth noting that, on passing from R Y − R X = 0 to R Y − R X = 0.32 Å, the orthorhombic distortion implies an energy gain of 206.7 and 166.7 meV for the FM and AFM ordering, respectively (Figure 9).These values are thus 1 order of magnitude higher than the energy difference (20.6 meV) between both magnetic structures at the P4/nmm parent phase.This simple result just stresses that vibronic interactions, which are behind the orthorhombic MnF 6 3− units at equilibrium, play an important role for explaining the magnetic structure in CsMnF 4 .
In K 2 CuF 4 and Cs 2 AgF 4 , the shift from AFM to FM ordering when R y − R x increases has proved to arise from deep changes in chemical bonding in the MF 6 4− units (M = Cu, Ag).Indeed, an increase of the orthorhombicity enhances the charge, q(R X ), transferred to the closest ligands, placed at R X from the cation, at the expense of two ligands at R Y whose charge, q(R Y ), is drastically reduced, being null at equilibrium. 22As the AFM contribution to the exchange constant depends on q(R X ) × q(R Y ), this fact favors the shift to a FM phase. 22hen pressure increases, the results of the present calculations (Figure 8) reveal that above a pressure of 15 GPa the layers of CsMnF 4 should be AFM ordered.Experimental data with pressures up to 4 GPa obtained by Ishizuka et al. 80 show that the critical temperature, T c , decreases with pressure, a fact qualitatively consistent with results gathered in Figure 8.Along this line, recent GGA+U calculations by Behatha et al. 81 also find a transition from the ferromagnetic to the AFM ordering although at a lower pressure of 2.4 GPa.
Two relevant facts are behind the pressure-induced shift from FM to AFM ordering in CsMnF 4 shown in Figure 8.On one hand, there is a significant reduction of the orthorhombicity (Table 3).Indeed, while R y − R x = 0.32 Å at zero pressure, it becomes 44% smaller under a pressure of 15 GPa.On the other hand, the Mn−F−Mn angle in CsMnF 4 changes from θ = 162.6°atzero pressure to θ = 153.4°atP = 15 GPa.Although in this process θ changes only by 5.7%, we have to recall that RbMnF 4 at zero pressure (space group P2 1 /a) is AFM with a very low transition temperature (4 K) and an angle θ equal to 148°. 19

■ CONCLUSIONS
The existence of an orthorhombic instability plays a central role in understanding the optical and magnetic properties of layered compounds like CsMnF 4 or K 2 CuF 4 .However, that instability is surprisingly not developed in CsFeF 4 whose structure belongs to the P4/nmm space group and the FeF 6 3− units are essentially tetragonal with R x = R y .
According to eq 3, a negative force constant requires the admixture of the electronic ground state, Ψ g (r), with an excited state, Ψ n (r), via operator V(r) reflecting the electron− vibration coupling.As V(r) is a purely orbital operator, a necessary condition for having such an admixture and a force constant K < 0 is a matrix element ⟨Ψ g (r), S g | V(r) | Ψ n (r), S n ⟩ different from zero, where S g and S n stand for the spin of ground and excited states, respectively.As FeF 6 3− complexes in CsFeF 4 are in a high-spin configuration this means a ground state with S g = 5/2.If we now consider all excited states emerging from the d 5 configuration of free Fe 3+ ion, there are a total of 246 states. 43However, in such excited states the spin is at most S n = 3/2 and thus none of them can be coupled to the 6 A 1g state of the FeF 6 3− unit.This spin barrier thus hampers the existence of orthorhombic instability in high-spin complexes of Fe 3+ or Mn 2+ ions.
That barrier does not exist for MnF 6 3− complexes in CsMnF 4 as the 5 A 1g ground state can be mixed with excited 5 B 1g through the orthorhombic b 1g mode.A similar situation holds for CuF 6 4− units in layered lattices like K 2 CuF 4 , where the orthorhombic distortion has been shown to be directly responsible for its surprising FM behavior. 9,22y contrast, for tetragonal CrF 6 3− complexes, there is also a hindrance against orthorhombic instability.Under O h symmetry the ground state of CrF 6 3− is 4 A 2g which becomes 4 B 1g in D 4h .Among the 120 multiplets arising from the d 3 configuration of free Cr 3+ ion, only the states 4 T 1g and 4 T 2g have the same spin S = 3/2 as the ground state 4 A 2g in O h symmetry. 43Thus, a 4 B 1g ground state in D 4h symmetry requires 4 A 1g excited states for having a nonzero vibronic coupling associated with the b 1g mode.However, neither 4 T 1g nor 4 T 2g give rise to 4 A 1g states under the O h → D 4h symmetry reduction.No orthorhombic distortion is observed for the tetragonal K 2 MgF 4 compound doped with Cr 3+ . 82,83A similar lack of excited states for tetragonal NiF 6 4− units hampers the orthorhombic instability in K 2 NiF 4 . 22hese reasons thus underline the origin of low-symmetry complexes widely observed for Cu 2+ or Mn 3+ systems and are not due to the Jahn−Teller effect.They are also behind the socalled plasticity property of compounds of Cu 2+ and Mn 3+ ions. 84e have seen that in CsMnF 4 the internal electric field, E R (r), increases the value of the first d−d excitation by 0.5 eV, and a similar effect takes place in other layered compounds like K 2 CuF 4 . 23It is worth noting now that this behavior is not necessarily general, as in other systems E R (r) can give rise to an energy reduction.This is just what happens in the singular compound CaCuSi 4 O 10 , the basis for the historical Egyptian blue pigment, 85 involving square-planar CuO 4 6− complexes.In that compound the internal electric field produces a reduction of 0.9 eV in the highest d−d transition, which is thus directly responsible for its blue color. 86ccording to the present discussion, the behavior of d 4 and d 9 ions in tetragonal insulating lattices can hardly be ascribed to the Jahn−Teller effect.Along this line it is worth noting that even when such ions are initially located in a cubic symmetry there is not necessarily a static Jahn−Teller effect. 87For instance, in the Cu 2+ -doped cubic SrCl 2 compound, the Cu 2+ ion, initially replacing Sr 2+ , experiences a big off-center motion along ⟨001⟩ type directions driven by a force constant that becomes negative. 88,89A similar situation is found for Ag 2+ and Ni + in SrCl 2 and also for SrF 2 :Cu 2+ and CaF 2 :Ni + . 87urther work on the optical and magnetic properties of insulating transition metal compounds is now underway.

Figure 1 .
Figure 1.Experimental unit cell of the CsMnF 4 structure corresponding to the P4/n phase at P = 0. Red arrows indicate the local {X, Y, and Z} axes of a MnF 6 3− complex.Blue numbers indicate the Mn−F distances (in Å).The lattice parameters are a = b = 7.944 Å and c = 6.338Å.

Figure 2 .
Figure 2. Experimental optical spectra of CsMnF 4 in the 1.5−3.5 eV range measured at room temperature for zero pressure and also P = 10.0, 25.5, 36.0,37.5, and 46.0 GPa, (adapted from ref 17).Dotted lines are the approximate variations in the energies of the band maxima as proposed in ref 17, although this proposal is discussed in the text.
Figure 4 depicts the electrostatic potential V R (r) associated with the internal field through = r r E ( ) V ( ) R R

Figure 3 .
Figure 3. Qualitative scheme of the energy levels of the 5 antibonding orbitals with mainly d character of the MnF 6 3− complex in CsMnF 4 at zero pressure depicted in 3 steps: (1) complex in vacuo with D 4h geometry, (2) adding the internal V R (r) potential, and (3) in D 2h geometry and including the V R (r) potential.

Figure 4 .
Figure 4. Potential energy (−e)V R (r) corresponding to the internal electric field created by the rest of the lattice ions of CsMnF 4 in the metastable P4/nmm phase of CsFeF 4 on a MnF 6 3− complex.Energies are depicted along the local X, Y, and Z directions of the complex.

Figure 5 .
Figure 5. Picture of the local b 1g vibrational mode, which is unstable in CsMnF 4 in the parent phase P4/nmm at P = 0.

Figure 6 .
Figure 6.Qualitative picture of the small distortions produced by the unstable a 2g modes on the P4/n structure of CsMnF 4 at P = 37.5 GPa producing the P4 phase, where the MnF 6 3− units have triclinic C 1 symmetry.

Figure 7 .
Figure 7. Variation of the four d−d transition energies of CsMnF 4 in the 0−60 GPa pressure range.Values were calculated with the VASP code, but similar variations were obtained with CRYSTAL + ADF codes.

Figure 8 .
Figure 8. Evolution of magnetic ordering in CsMnF 4 as a function of pressure.In the figure is depicted the enthalpy difference (given per formula unit, in meV) between ferromagnetic and antiferromagnetic ordering calculated for pressures up to 30 GPa where CsMnF 4 is always in the P4/n structure.

Figure 9 .
Figure 9. Energy (given per formula unit) of CsMnF 4 obtained by single-point calculations throughout the antiferrodistortive distortion from the parent tetragonal P4/nmm phase (R Y − R X = 0) to the P4/n structure where R Y − R X = 0.32 Å at equilibrium.In this process, the AFM and FM ordering in the layers of CsMnF 4 are both considered.

Table 1 .
18,19lated Lattice Parameters and Mn−F Distances a for CsMnF 4 at Zero Pressure in the P4/n Space Group Using VASP (First Row) and CRYSTAL (Second Row) Codes and Compared to Experimental Values18,19 a All distances are given in Å.

Table 2 .
Calculated Lattice Parameters and Fe−F Distances for CsFeF 4 at Zero Pressure Assuming, in Principle, the P4/ n Space Group a

Table 3 .
Evolution of Lattice Parameters and Mn−F Distances for CsMnF 4 with Pressure Calculated with VASP (First Row) and CRYSTAL (Second Row) Codes a a Below P = 40 GPa the equilibrium structure is that observed at ambient pressure (space group P4/n) with MnF63− units in the highspin configuration (S = 2).At P = 40 GPa the P4/n structure becomes unstable giving rise to a new equilibrium structure described by the P4 space group.

Table 4 .
Difference of the Enthalpy Per Molecule, ΔH (in eV), between Low-Spin (S = 1) and High-Spin (S = 2) Configurations Calculated for Both P4/n and P4 Phases at Pressures of P = 40 and 50 GPa a

Table 5 .
Calculated Energies (in eV) of Four Spin-Allowed d−d Transitions of MnF 6 3− Units in CsMnF 4 for Different Pressures, P (in GPa) a

Table 6 .
Influence of the Internal Electric Field, E R (r), on the Energy (in eV) of Four d−d Transitions Corresponding to MnF 6 3− Units in CsMnF 4 for Two Pressures, P = 0 and 20 GPa a Pressure xy→ x 2 −z 2 yz → x 2 −z 2 xz → x 2 −z 2 3y 2 −r 2 → x 2 −z 2

Table 7 .
Energies of Spin-Allowed d−d Transitions for MnF 6 3− Units in Na 3 MnF 6 Derived at Ambient Pressure a In this compound the metal−ligand distances 12 are R z = 2.018 Å, R x = 1.862Å, and R y = 1.897Å giving rise to a small orthorhombicity (R y − R x = 0.035 Å) and a HOMO practically equal to |3z 2 −r 2 ⟩. a