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

Abstract

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 MnF₆^3– 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 MnF₆^3– units. This study underlines the usefulness of first-principles calculations for a right interpretation of experimental findings.

Short abstract

Accurate ab initio calculations on CsMnF₄ layered material show that the phase transition detected at 37.5 GPa by optical absorption is not the first known case of fluoride complexes with spin crossover from S = 2 to S = 1. Moreover, contrary to what is assumed in all the literature on this system, our calculations rule out the existence of a Jahn−Teller effect. The origin of the orthorhombic distortion of the complexes is also analyzed.

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. These compounds are involved in a number of technological applications such as solid-state lasers,¹⁻³ devices with colossal magnetoresistance,⁴ or solar cells⁵ and in their understanding appear concepts like orbital ordering, superexchange, or vibronic instabilities.⁶⁻⁹

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. In this realm the application of high pressures on a TM compound opens a new window for detecting attractive phenomena such as phase transitions or changes in electronic states of TM complexes where active electrons are localized.¹⁰⁻¹⁵ In particular, pressure can change the ground-state spin of a complex shifting from a high- to a low-spin configuration¹⁰ such as happens for a variety of Fe²⁺ complexes.¹⁶

To 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 CsMnF₄.¹⁷ This compound belongs to the interesting family of fluoromanganates¹⁸⁻²⁰ with formula AMnF₄ (A = alkali monocation or NH₄⁺) involving the 3d⁴ cation Mn³⁺. 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 CsMnF₄ 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, CsMnF₄ at ambient pressure belongs to the P4/n space group.¹⁸ The structure is depicted in Figure 1 showing that Mn³⁺ ions are disposed in layers in the ab plane although the F^– ions of MnF₆^3– units are not strictly in that plane just reflecting the existence of buckling. In the involved MnF₆^3– units the shortest Mn–F distance corresponds to the z direction, perpendicular to the ab layer plane (R_z = 1.817 Å), while the longest one (R_y = 2.168 Å) is nearly perpendicular to the z direction. The last Mn–F distance (R_x = 1.854 Å) differs from that of R_z by only 0.037 Å.

Figure 1.

Experimental unit cell of the CsMnF₄ structure corresponding to the P4/n phase at P = 0. Red arrows indicate the local {X, Y, and Z} axes of a MnF₆^3– complex. Blue numbers indicate the Mn–F distances (in Å). The lattice parameters are a = b = 7.944 Å and c = 6.338 Å.

Interestingly, the F–Mn–F angle of MnF₆^3– units in CsMnF₄ 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 Mn³⁺ ion that is not strictly orthorhombic (D_2h) but C_i. CsMnF₄ exhibits an antiferrodistortive arrangement and thus in two adjacent MnF₆^3– units that share a common ligand the longest axes are essentially perpendicular. The RbMnF₄ and KMnF₄ compounds of the AMnF₄ 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.^18,19 Both compounds are antiferromagnetic but only below 5 K.^19,20

In the interpretation of structural and optical data (Figure 2) of CsMnF₄ there are two relevant questions that need to be clarified: (1) The local distortion of MnF₆^3– units in CsMnF₄ has usually been ascribed to the Jahn–Teller effect¹⁷⁻²¹ 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 MnF₆^3– units in CsMnF₄ 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 CsMnF₄ hampers the interpretation of optical absorption results, a circumstance that can, however, be overcome by means of theoretical calculations.

Figure 2.

Experimental optical spectra of CsMnF₄ 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.

Finally, the present work also pays attention to the ferromagnetism displayed by CsMnF₄ 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 K₂CuF₄ or Cs₂AgF₄ shed light on that issue.^9,22,23

This 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 CsMnF₄ 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 CsMnF₄. 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.

On the one hand, the Crystal17 code²⁴ has two relevant implementations, significantly speeding up the calculations: first, it makes full use of the symmetries of the material’s space group, and second, Bloch orbitals are represented through linear combinations of Gaussian type functions.²⁵⁻²⁷ On the other hand, Vienna Ab initio Simulation Package (VASP)^28,29 employs a set of plane waves to describe the Bloch orbitals, enabling highly accurate results in reproducing experimental geometries in all kinds of compounds, including organic and hybrid organic–inorganic materials.³⁰⁻³³

First, we optimized with the Crystal17 code the geometry of CsMnF₄ at the experimental P4/n phase¹⁸ 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 CsMnF₄P4/n structure. For this goal we started from the experimental CsMnF₄ structure substituting all open shell Mn³⁺ ions, with d⁴ electronic configuration and S = 2, by Fe³⁺ ions (d⁵ electronic configuration and S = 5/2) with spherical density (in vacuo) and equal ionic radius, r(Mn³⁺) ≈ r(Fe³⁺) ≈ 0.785 Å.³⁴ The symmetry of the optimized CsFeF₄ parent phase was P4/nmm, precisely the experimental phase of this compound.³⁵ Moreover, in order to elucidate the nature of the phase transition experimentally observed in CsMnF₄ 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 MnF₆^3– complexes by means of the Amsterdam density functional (ADF) code.^36,37 In these pseudomolecular calculations, the MnF₆^3– clusters were embedded in the electrostatic potential of the rest of lattice ions,³⁸ which was previously calculated through Ewald–Evjen summations.^39,40

In 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 Mn³⁺ atoms in the unit cell with Ga³⁺, atoms with a d¹⁰ 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.^41,42 The 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 CsMnF₄ 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 CsMnF₄

The optical properties in the V–UV domain of an insulating compounds like CsMnF₄ greatly depend on the involved MnF₆^3– units, whose ground state in O_h symmetry can, in principle, be either ⁵E_g(t³e¹) or ³T_1g(t⁴e⁰).⁴³ Nevertheless, experimental data, at ambient pressure, on compounds containing MnF₆^3– support a high-spin configuration (S = 2) as ground state.^12,19,20 Octahedral complexes with inorganic ligands like F^– or Cl^– display a high-spin configuration for 3dⁿ ions (n = 5, 6) like Fe³⁺ or Fe²⁺ while a low-spin configuration is found for complexes of 4d and 5d ions, like Ru³⁺ or Ir⁴⁺, involving higher 10Dq values.^10,44 A low-spin (S = 1/2) ground state has also been reported^45,46 for NiF₆^3– complexes in A₃NiF₆, Cs₂ANiF₆ (A = Na, K), and Rb₂NaNiF₆ fluorides and also in oxides doped with Ni³⁺.⁴⁷ However, electron paramagnetic resonance (EPR) measurements carried out in Ni³⁺-doped KMgF₃ and CsCaF₃ perovskites^48,49 lead to a high-spin (S = 3/2) ground state. These results show that, in NiF₆^3– complexes, the high-spin/low-spin energy separation must be very small.^45,46 Moreover, magnetic measurements performed on Li₃NiF₆ characterized at low temperature a low-spin configuration (S = 1/2), which tends to a high-spin one with increasing temperature.⁵⁰ To the best of our knowledge, a spin crossover under pressure in fluorides containing NiF₆^3– complexes has not been found.

As in 6-fold coordinated compounds of d⁴ ions (Mn³⁺, Cr²⁺), the local symmetry of a MnF₆^3– complex corresponds to a distorted octahedron, and the ground state wave function, Ψ_g, can briefly be written as

1

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(3z²–r², x²–y²) orbitals in O_h symmetry, and thus, the e₂ orbital is empty in the ground state.¹³ Accordingly, the spin-allowed transitions (ΔS = 0) are simply described by t_i↑ → e₂↑ (i = 1, 2, 3) and e₁↑ → e₂↑. As the e₁↑ → e₂↑ transition is usually the lowest, its energy is termed E₀ while those associated with the t_i↑ → e₂↑ transitions are simply called E_i (i = 1, 2, and 3). In addition to the spin-allowed transitions, sharp peaks with smaller oscillator strengths associated with forbidden transitions (ΔS = −1) are sometimes observed in optical spectra of d⁴ ions (Mn³⁺, Cr²⁺). They correspond to excited states with S = 1 and are described by determinants like |t₁↑ t₂↓ t₃↑ e₁↑| so the total spin of the t-subshell is only 1/2. Such forbidden transitions have well been detected for CrF₂.⁵¹ Similar transitions with ΔS = −1 are also observed in the absorption spectra of CrO₆^9– or CrF₆^3– units and are little sensitive to pressure.^52,53,14,15

Figure 2 reproduces the experimental optical absorption spectra at room temperature of CsMnF₄ 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 37.5–46.0 GPa only one broad band is observed in the optical spectrum of CsMnF₄ 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 CsMnF₄.

Seeking to understand the optical spectra of CsMnF₄ at ambient pressure it has been proposed that the experimental geometry of MnF₆^3– units is the result of a static Jahn–Teller (JT) effect.¹⁷ Under that assumption, also followed in other works,¹⁸⁻²¹ the JT effect would be responsible for having tetragonal MnF₆^3– units with Y as the local principal axis of the complex (Figure 1). Such units are elongated in accord with the geometry widely observed for systems actually displaying a static JT effect.⁵⁴⁻⁵⁶ Consequently, e₁ would be a molecular orbital transforming like 3y²–r² while the unoccupied e₂ orbital corresponds to z²–x². According to this hypothesis, the optical absorption spectrum of CsMnF₄ at ambient pressure would involve three spin-allowed transitions in the MnF₆^3– unit whose energies have been proposed¹⁷ to be E₀ = 1.80 eV, E₁ = 2.26 eV, and E₂ = E₃ = 2.80 eV from Figure 2. Such values are close to E₀ = 1.92 eV, E₁ = 2.23 eV, and E₂ = E₃ = 2.73 eV reported by Morón and Palacio.⁵⁷

Nonetheless, doubts are raised by the JT assumption due to the following reasons:^54−57,9

• (i)The existence of a JT effect requires a degenerate electronic state in the initial geometry. Therefore, as CsMnF₄ is a layered compound, even if the geometry of the initial parent phase is tetragonal the electronic ground state of an MnF₆^3– unit should not be degenerate according to symmetry.
• (ii)CsMnF₄ is a layered compound where layers are perpendicular to the crystal c axis (Figure 1). Accordingly, one would expect that the axis of the MnF₆^3– 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 MnF₆^3– in CsMnF₄ 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 CsMnF₄.
• (iv)Although most of the d⁹ systems which exhibit a static JT effect are elongated with a hole in an x²–y² type orbital⁵⁴⁻⁵⁷ this is not a general rule. For instance, in the cubic CaO lattice doped with Ni⁺, the hole is in a 3z²–r² orbital and the octahedron compressed.^58,59 In non-JT systems like K₂ZnF₄:Cu²⁺, where the host lattice is tetragonal, the hole is also in 3z²–r².^60,61

Looking at higher pressures, the optical spectrum of CsMnF₄ 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 ³T_1g(t⁴e⁰) when P > 37 GPa with an O_h local geometry for MnF₆^3– units.¹⁷

No further arguments are given for underpinning that assumption that casts in principle some doubts. Indeed, just looking at Tanabe–Sugano diagrams⁴³ one would expect that in octahedral symmetry the transition of a ⁵E_g(t³e¹) ground state to ³T_1g(t⁴e⁰) in MnF₆^3– 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 Na₃MnF₆ (10Dq = 1.88 eV) is close to 10Dq measured for octahedral CrF₆^3– units in cubic elpasolites involving also a trivalent TM cation.^14,62,63 For instance, in the case of Rb₂KCrF₆, 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.⁶³ Accepting this value for a hypothetical O_h MnF₆^3– 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 CsMnF₄: (1) the origin of the local geometry of MnF₆^3– units in CsMnF₄ 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 CsMnF₄ 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.^18,19 As 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 MnF₆^3– unit is found to come from the ⁵E_g(t³e¹) configuration in cubic symmetry, thus implying a high-spin configuration with S = 2.

Table 1. Calculated Lattice Parameters and Mn–F Distances^a for CsMnF₄ at Zero Pressure in the P4/n Space Group Using VASP (First Row) and CRYSTAL (Second Row) Codes and Compared to Experimental Values^18,19.

	a = b	c	RZ	RX	RY
Experimental	7.944	6.338	1.817	1.854	2.168
Calculated	8.029	6.401	1.829	1.871	2.189
	7.961	6.347	1.818	1.877	2.161

^aAll distances are given in Å.

Seeking to understand the origin of the local structure in CsMnF₄, it is useful to analyze the evolution of the crystal structure when Mn³⁺ is replaced by a cation like Fe³⁺ with a similar ionic radius, giving rise to the so-called high-symmetry parent phase.^6,13,22,23 Under strict octahedral symmetry, Fe³⁺ in high-spin configuration (S = 5/2) exhibits a nearly spherical electronic density which is however never found for a 3d⁴ ion like Mn³⁺ or a 3d⁹ ion like Cu²⁺ in the same situation. After the Mn³⁺ → Fe³⁺ substitution, we carried out a geometry optimization of the CsFeF₄ structure maintaining fixed the experimental P4/n space group of CsMnF₄. 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 FeF₆^3– unit in CsFeF₄ 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 CsFeF₄ compound at normal conditions.^64,65 The calculated lattice parameters and Fe–F distances correspond to experimental ones within 1.5%.

Table 2. Calculated Lattice Parameters and Fe–F Distances for CsFeF₄ at Zero Pressure Assuming, in Principle, the P4/n Space Group^a.

	a = b	c	RZ	RX	RY
Calculated	7.926	6.631	1.863	1.989	1.989
Experimental	7.787	6.540	1.884	1.959	1.959

^aThe results, derived through the CRYSTAL code, show that the system evolves until reaching the P4/nmm space group, where R_X = R_Y due to the higher symmetry. Experimental results⁶⁵ are given for comparison. All distances are given in Å.

Bearing the preceding considerations in mind, we first consider CsMnF₄ in the tetragonal P4/nmm geometry of the parent phase. In that case, the MnF₆^3– 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 MnF₆^3– unit transforming like 3z²–r² and is simply designated by |3z²–r²⟩ while the highest occupied molecular orbital, HOMO, is |x²–y²⟩. Accordingly, the ground state belongs to ⁵A_1g in tetragonal symmetry and the |x²–y²⟩ → |3z²–r²⟩ excited state to ⁵B_1g.

Figure 3.

Qualitative scheme of the energy levels of the 5 antibonding orbitals with mainly d character of the MnF₆^3– complex in CsMnF₄ 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.

At this point it is important to highlight that the gap, E₀, associated with the |x²–y²⟩ → |3z²–r²⟩ excitation in MnF₆^3– does not only reflect that in the parent phase R_Z < R_X = R_Y. Indeed, although active electrons are essentially localized in the MnF₆^3– unit they also feel the internal electric field, E_R(r), due to ions of CsMnF₄ lying outside the complex, that gives rise to an extrinsic contribution to E₀ such as has been proved for layered compounds like K₂CuF₄ or La₂CuO₄^6,9,23,61,66

Figure 4 depicts the electrostatic potential V_R(r) associated with the internal field through . According to the shape of (−e)V_R(r) it favors to increase the energy of |3z²–r²⟩ and reduce that of |x²–y²⟩, thus enhancing the value of E₀. We have derived a value E₀ = 0.7 eV considering only the isolated MnF₆^3– unit while E₀ = 1.2 eV is obtained once V_R(r) is incorporated into the calculation. Therefore, V_R(r) not only helps to stabilize |x²–y²⟩ as HOMO but likely has a significant influence on optical transitions, a matter that will be discussed later.

Figure 4.

Potential energy (−e)V_R(r) corresponding to the internal electric field created by the rest of the lattice ions of CsMnF₄ in the metastable P4/nmm phase of CsFeF₄ on a MnF₆^3– complex. Energies are depicted along the local X, Y, and Z directions of the complex.

Interestingly, if CsMnF₄ is in the P4/nmm phase the electronic density in the MnF₆^3– unit with an unpaired electron in |x²–y²⟩ 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 K₂ZnF₄:Cu²⁺.^61,76 Nevertheless, the equilibrium geometry of CsMnF₄ exhibits a lower P4/n symmetry with R_Y > R_X for the MnF₆^3– complex. This instability of the P4/nmm structure for CsMnF₄ implies the existence of at least one vibration mode with imaginary frequency.²³ Accordingly, we have calculated the vibrational frequencies of CsMnF₄ 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 MnF₆^3– 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 CsMnF₄ 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 K₂CuF₄ or Cs₂AgF₄,^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.^8,67

Figure 5.

Picture of the local b_1g vibrational mode, which is unstable in CsMnF₄ in the parent phase P4/nmm at P = 0.

If H₀ 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

2

It should be noted that H_vib exhibits the same symmetry as H₀ 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

3

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

4

Here, K₀ stands for the positive contribution associated with the frozen electronic density of H₀ while K_ν reflects the electronic density changes due to H_vib and is given by

5

Thus, the instability responsible for the equilibrium structure of CsMnF₄ appears because the K₀ < K_ν condition is fulfilled in this case. Interestingly, this implies the admixture of the ⁵A_1g ground state of MnF₆^3– with the excited ⁵B_1g through a b_1g local mode, thus modifying the electronic density of the complex. It is worth noting that, in CsMnF₄, two adjacent MnF₆^3– units share a common ligand, a fact that is behind the instability developed in K₂CuF₄ or Cs₂AgF₄^23,9 but not in K₂ZnF₄:Cu^2+23,9,60,61 or even in KAlCuF₆⁶⁸⁻⁷⁰ where the CuF₆^4– units are well-separated. In addition, the parent phase of CsMnF₄ involves compressed MnF₆^3– 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.^23,9,22

In the equilibrium geometry of CsMnF₄ at zero pressure the highest occupied molecular orbital (HOMO) wave function of the MnF₆^3– unit, |φ_H⟩, is not purely |x²–y²⟩ but involves an admixture of |3z²–r²⟩ as a result of the symmetry reduction due to the instability

6

The present calculations yield α² = 85%, stressing that the HOMO wave function keeps a dominant |x²–y²⟩ character once the distortion takes place and R_Y > R_X. A similar situation has been found in other layered systems like K₂CuF₄ or Cs₂AgF₄.^23,9 Interestingly, if we write eq 6 using the {|x²–z²⟩, |3y²–r²⟩} basis, then

7

If α² = 85%, it is simple to find β′² = 98% demonstrating that the HOMO wave function is essentially |3y²–r²⟩ and thus greatly localized along the longest Y axis. In the same way, the LUMO wave function, |φ_L⟩, is basically equal to |x²–z²⟩ despite the electronic structure of CsMnF₄ not being due to the JT effect.

For the sake of completeness, a qualitative picture of the electronic ground state of MnF₆^3– at the equilibrium geometry of CsMnF₄ at zero pressure is shown on Figure 3. Accordingly, the lowest d–d excitation is simply described by |3y²–r²⟩ → |x²–z²⟩. This matter will be discussed later.

Structural Changes Induced by Pressure

Results of calculations on CsMnF₄ 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 CsMnF₄ 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.

Table 3. Evolution of Lattice Parameters and Mn–F Distances for CsMnF₄ with Pressure Calculated with VASP (First Row) and CRYSTAL (Second Row) Codes^a.

P (GPa)	Symmetry	a (Å)	c (Å)	RZ (Å)	RX (Å)	RY (Å)
0	P4/n	8.029	6.401	1.829	1.871	2.189
		7.961	6.347	1.818	1.877	2.161
10	P4/n	7.649	5.970	1.802	1.858	2.058
		7.632	5.949	1.794	1.868	2.042
20	P4/n	7.442	5.763	1.784	1.839	1.997
		7.437	5.759	1.779	1.852	1.985
30	P4/n	7.282	5.632	1.771	1.821	1.957
		7.285	5.637	1.766	1.835	1.947
40	P4/n	7.146	5.545	1.760	1.806	1.925
		7.154	5.557	1.757	1.819	1.918
40	P4	7.172	5.430	1.755	1.833	1.939
		7.181	5.444	1.751	1.844	1.924
50	P4	6.988	5.532	1.745	1.853	1.851
		7.106	5.283	1.733	1.869	1.885

^aBelow P = 40 GPa the equilibrium structure is that observed at ambient pressure (space group P4/n) with MnF₆^3– units in the high-spin 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.

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 MnF₆^3– units is found to correspond to the high-spin configuration involving S = 2.

Figure 6.

Qualitative picture of the small distortions produced by the unstable a_2g modes on the P4/n structure of CsMnF₄ at P = 37.5 GPa producing the P4 phase, where the MnF₆^3– units have triclinic C₁ symmetry.

It should be noted that in the P4 phase at P = 40 GPa the MnF₆^3– complexes have local triclinic symmetry (point group C₁), 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.

Table 5. Calculated Energies (in eV) of Four Spin-Allowed d–d Transitions of MnF₆^3– Units in CsMnF₄ for Different Pressures, P (in GPa)^a.

P	Symmetry	xy → x2–z2	yz → x2–z2	xz → x2–z2	3y2–r2 → x2–z2
0	P4/n	2.83	2.48	2.18	1.84
		2.81	2.60	2.26	1.92
10	P4/n	2.94	2.53	2.33	1.55
		2.84	2.48	2.28	1.54
20	P4/n	3.02	2.63	2.45	1.44
		2.93	2.54	2.38	1.48
30	P4/n	3.10	2.73	2.57	1.38
			2.61	2.47	1.43
40	P4	2.99	2.72	2.56	1.47
		3.02	2.56	2.47	1.43

^aFirst 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.

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 MnF₆^3– 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 MnF₆^3– 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 (C₂H₅NH₃)₂CdCl₄ doped with Cu²⁺ and plays a key role for 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 MnF₆^3– 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 MnF₆^3– units coming from the ⁵E_g(t³e¹) high-spin configuration in O_h symmetry, we have also paid attention to determine the enthalpy of the lowest state emerging from the ³T_1g(t⁴e⁰) low-spin configuration in O_h symmetry. The difference of the enthalpy per molecule, ΔH, between low-sping (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 0.45–0.68 eV. Therefore, the assumption of a transition from S = 2 to S = 1 induced by pressure at about 40 GPa¹⁷ is highly unlikely.

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.

P	Phase	ΔH
40	P4/n	0.63
		0.53
40	P4	0.65
		0.56
50	P4/n	0.48
		0.39
50	P4	0.45
		0.50

^aResults obtained using VASP (first line) and CRYSTAL (second line) codes are both displayed.

Spin-Allowed d–d Transitions in CsMnF₄ under Pressure

Considering the equilibrium geometries derived for CsMnF₄ 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 MnF₆^3– units. As expected from Figure 3 the lowest d–d excitation actually corresponds to |3y²–r²⟩ → |x²–z²⟩ for all pressures.

The experimental spectrum of CsMnF₄ obtained at room temperature and ambient pressure (Figure 2) was assumed to involve only three spin-allowed d–d transition with energies¹⁷E₀ = 1.80 eV, E₁ = 2.26 eV, and E₂ = E₃ = 2.80 eV. According to Table 5, such transitions can now reasonably be assigned as |3y²–r²⟩ → |x²–z²⟩, |xz⟩ → |x²–z²⟩, and |xy⟩ → |x²–z²⟩, respectively. In the poorly resolved experimental spectrum (Figure 2), the |yz⟩ → |x²–z²⟩ transition, calculated at about 2.55 eV, is not well seen likely due to the width of the |xz⟩ → |x²–z²⟩ and |xy⟩ → |x²–z²⟩ 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. It should be noted that the first |3y²–r²⟩ → |x²–z²⟩ transition experiences a drastic red shift with pressure as it moves from 1.84 eV at zero pressure to 1.47 eV at P = 40 GPa. By contrast, the three transitions associated with t_2g orbitals in O_h symmetry undergo a moderate blue-shift under pressure. Interestingly, the energy difference, Δ(xy,xz), between |xy⟩ → |x²–z²⟩ and |xz⟩ → |x²–z²⟩ transitions is reduced significantly on passing from zero (Δ(xy,xz) = 0.65 eV) to 40 GPa (Δ(xy,xz) = 0.43 eV). At the same time, the energies of the two transitions associated with |xz⟩ and |yz⟩ orbitals become equal within ∼0.1 eV at P = 40 GPa such as it is shown in Table 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.

Figure 7.

Variation of the four d–d transition energies of CsMnF₄ in the 0–60 GPa pressure range. Values were calculated with the VASP code, but similar variations were obtained with CRYSTAL + ADF codes.

The evolution of four spin-allowed d–d transitions of CsMnF₄ under pressure (Table 5 and Figure 7) is consistent with the calculated variations undergone by the Mn–F distances (Table 3). Indeed, we have seen that pressure favors a geometry of the MnF₆^3– complex progressively closer to the octahedral one thus reducing Δ(xy,xz) as well as the energy of the |3y²–r²⟩ → |x²–z²⟩ 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 perovskites⁶⁹ like (CH₃NH₃)₂CuCl₄, (C₂H₅NH₃)₂CuCl₄, or (C₂H₅NH₃)₂CdCl₄:Cu²⁺.

In a further step, it is necessary to disclose the influence of the internal field on the optical transitions of CsMnF₄ as it plays an important role in the case of inorganic layered perovskites like K₂CuF₄ or Cs₂AgF₄.^6,8,23 For clarifying this issue, we have calculated in a first step the energies of four d–d transitions considering only the isolated MnF₆^3– 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 |3y²–r²⟩, lying in the layer plane thus enhancing the |3y²–r²⟩ → |x²–z²⟩ transition energy. A similar situation is encountered in K₂CuF₄ or Cs₂AgF₄.²³

Table 6. Influence of the Internal Electric Field, E_R(r), on the Energy (in eV) of Four d–d Transitions Corresponding to MnF₆^3– Units in CsMnF₄ for Two Pressures, P = 0 and 20 GPa^a.

Pressure	xy → x2–z2	yz → x2–z2	xz → x2–z2	3y2–r2 → x2–z2
0	2.62	2.51	2.23	1.44
	2.81	2.60	2.26	1.92
20	2.69	2.54	2.38	0.95
	2.93	2.54	2.38	1.48

^aThe first row corresponds to calculated values with CRYSTAL + ADF codes on an isolated MnF₆^3– 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.

Finally, for the sake of clarity, we have performed an analysis of three contributions responsible for the energy E₀ of the first spin-allowed transition |3y²–r²⟩ → |x²–z²⟩ of CsMnF₄ at zero pressure. According to the present discussion, we divide the calculation in 3 steps: (1) We consider an isolated MnF₆^3– unit in the tetragonal geometry of the parent phase, obtaining a value E₀ = 0.89 eV. (2) In a second step, we still keep the isolated MnF₆^3– unit but in the final P4/n geometry where MnF₆^3– exhibits a practical orthorhombic symmetry with R_Y – R_X = 0.31 Å, obtaining E₀ = 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 E₀ = 1.92 eV. Therefore, the contributions of both the orthorhombic distortion and V_R(r) enhance the E₀ value by ∼1 eV.

Survey of Other Compounds Containing MnF₆^3– Units

Thanks to the analysis carried out in preceding sections on CsMnF₄ we can now gain better insight into the different optical properties at zero pressure displayed by other fluorides involving Mn³⁺. In a first step we pay attention to the Na₃MnF₆ compound^12,13 where the metal–ligand distances are R_z = 2.018 Å, R_x = 1.862 Å, and R_y = 1.897 Å. 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 |3z²–r²⟩. The energies of allowed d–d transitions measured experimentally¹² and derived by means of first-principles calculations¹³ are reported in Table 7. It can be seen that the transitions |t_i⟩ → |3z²–r²⟩ (t = xy, xz, yz) all are in the 2–3 eV range, as has been found for CsMnF₄. However, the energy of the first |3z²–r²⟩ → |x²–y²⟩ transition is practically half the value E₀ = 1.9 eV obtained for CsMnF₄, thus involving a remarkable difference. A similar situation is encountered when looking at the first transition of K₃MnF₆ or Cs₃MnF₆ where E₀ = 1.1 eV.^72,57

Table 7. Energies of Spin-Allowed d–d Transitions for MnF₆^3– Units in Na₃MnF₆ Derived at Ambient Pressure^a.

	3z2–r2 → x2–y2	xy → x2–y2	xz → x2–y2	yz → x2–y2	ref.
Experimental	1.04	2.18	2.38	2.38	(54)
		2.17	2.38	2.58	(12)
Calculated	0.71	2.11	2.27	2.32	(13)

^aIn this compound the metal–ligand distances¹² are R_z = 2.018 Å, R_x = 1.862 Å, and R_y = 1.897 Å giving rise to a small orthorhombicity (R_y – R_x = 0.035 Å) and a HOMO practically equal to |3z²–r²⟩. Transition energies (in eV) have been obtained from both experiments and calculations.

There are two main reasons behind a E₀ value around 1 eV for Na₃MnF₆. On one hand, the orthorhombicity of MnF₆^3– units in this compound is 1 order of magnitude smaller than that found for CsMnF₄. On the other hand, Na₃MnF₆ is not a typical layered compound like K₂CuF₄ or CsMnF₄ 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 Cu²⁺, such as KZnF₃:Cu²⁺, K₂ZnF₄:Cu²⁺, and K₂CuF₄. As KZnF₃ is cubic there is a static JT effect in KZnF₃:Cu²⁺, with an unpaired electron in |x²–y²⟩.⁷³⁻⁷⁵ Due to the cubic symmetry of the host lattice, E_R(r) has no effect on the first d–d transition, |3z²–r²⟩ → |x²–y²⟩, whose energy is E₀ = 0.40 eV.^74,75 As K₂ZnF₄ 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 |3z²–r²⟩ orbital by the action of V_R(r), and consequently the |x²–y²⟩ → |3z²–r²⟩ transition energy is enhanced having a value E₀ = 0.70 eV.^76,61,74 Finally, as in K₂CuF₄ two adjacent CuF₆^4– units share a common ligand, this favors an orthorhombic instability which still increases the E₀ value up to 1.03 eV.^77,78,74,9

Magnetic Structure of CsMnF₄: Influence of Pressure

As shown in Figure 8, the present calculations support that in CsMnF₄ at ambient pressure layers are ferromagnetically ordered. This behavior, consistent with experimental data, is the same found for layered compounds like K₂CuF₄ or Cs₂AgF₄ where the M–F–M angle, θ, (M = Cu, Ag) is 180° due to the absence of buckling at the Cmca equilibrium structure.^9,22 The ferromagnetism displayed by K₂CuF₄ or Cs₂AgF₄ is surprising as tetragonal K₂MnF₄ and K₂NiF₄ compounds, where the θ angle is also equal to 180°, exhibit an AFM ordering the same found for KMnF₃ and KNiF₃ perovskites.⁷⁹ Very recently, the ferromagnetism in K₂CuF₄ and Cs₂AgF₄ at the Cmca equilibrium structure has proved to come from the orthorhombic distortion undergone by MF₆^4– units (M = Cu, Ag), which in turn is actually responsible for the orbital ordering in these compounds.²² Indeed, in the I4/mmm parent phase of K₂CuF₄ and Cs₂AgF₄, involving tetragonal MF₆^4– units (M = Cu, Ag), the calculations lead to an AFM ordering similar to that observed for K₂MnF₄ or K₂NiF₄ at ambient pressure.

Figure 8.

Evolution of magnetic ordering in CsMnF₄ 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 CsMnF₄ is always in the P4/n structure.

We verified that a similar situation holds for CsMnF₄. Indeed, in the P4/nmm parent phase, where R_y = R_x we also find that the layers of CsMnF₄ 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, CsMnF₄ easily becomes ferromagnetic following the increase of the orthorhombic distortion on the MnF₆^3– units, as shown in Figure 9.

Figure 9.

Energy (given per formula unit) of CsMnF₄ 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 CsMnF₄ are both considered.

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 MnF₆^3– units at equilibrium, play an important role for explaining the magnetic structure in CsMnF₄.

In K₂CuF₄ and Cs₂AgF₄, 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 MF₆^4– 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.²² As the AFM contribution to the exchange constant depends on q(R_X) × q(R_Y), this fact favors the shift to a FM phase.²²

When pressure increases, the results of the present calculations (Figure 8) reveal that above a pressure of 15 GPa the layers of CsMnF₄ 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 CsMnF₄ 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 CsMnF₄ changes from θ = 162.6° at zero pressure to θ = 153.4° at P = 15 GPa. Although in this process θ changes only by 5.7%, we have to recall that RbMnF₄ at zero pressure (space group P2₁/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 CsMnF₄ or K₂CuF₄. However, that instability is surprisingly not developed in CsFeF₄ whose structure belongs to the P4/nmm space group and the FeF₆^3– 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 FeF₆^3– complexes in CsFeF₄ 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 d⁵ configuration of free Fe³⁺ ion, there are a total of 246 states.⁴³ However, in such excited states the spin is at most S_n = 3/2 and thus none of them can be coupled to the ⁶A_1g state of the FeF₆^3– unit. This spin barrier thus hampers the existence of orthorhombic instability in high-spin complexes of Fe³⁺ or Mn²⁺ ions.

That barrier does not exist for MnF₆^3– complexes in CsMnF₄ as the ⁵A_1g ground state can be mixed with excited ⁵B_1g through the orthorhombic b_1g mode. A similar situation holds for CuF₆^4– units in layered lattices like K₂CuF₄, where the orthorhombic distortion has been shown to be directly responsible for its surprising FM behavior.^9,22

By contrast, for tetragonal CrF₆^3– complexes, there is also a hindrance against orthorhombic instability. Under O_h symmetry the ground state of CrF₆^3– is ⁴A_2g which becomes ⁴B_1g in D_4h. Among the 120 multiplets arising from the d³ configuration of free Cr³⁺ ion, only the states ⁴T_1g and ⁴T_2g have the same spin S = 3/2 as the ground state ⁴A_2g in O_h symmetry.⁴³ Thus, a ⁴B_1g ground state in D_4h symmetry requires ⁴A_1g excited states for having a nonzero vibronic coupling associated with the b_1g mode. However, neither ⁴T_1g nor ⁴T_2g give rise to ⁴A_1g states under the O_h → D_4h symmetry reduction. No orthorhombic distortion is observed for the tetragonal K₂MgF₄ compound doped with Cr³⁺.^82,83 A similar lack of excited states for tetragonal NiF₆^4– units hampers the orthorhombic instability in K₂NiF₄.²²

These reasons thus underline the origin of low-symmetry complexes widely observed for Cu²⁺ or Mn³⁺ systems and are not due to the Jahn–Teller effect. They are also behind the so-called plasticity property of compounds of Cu²⁺ and Mn³⁺ ions.⁸⁴

We have seen that in CsMnF₄ 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 K₂CuF₄.²³ It 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 CaCuSi₄O₁₀, the basis for the historical Egyptian blue pigment,⁸⁵ involving square-planar CuO₄^6– 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.⁸⁶

According to the present discussion, the behavior of d⁴ and d⁹ 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.⁸⁷ For instance, in the Cu²⁺-doped cubic SrCl₂ compound, the Cu²⁺ ion, initially replacing Sr²⁺, experiences a big off-center motion along ⟨001⟩ type directions driven by a force constant that becomes negative.^88,89 A similar situation is found for Ag²⁺ and Ni⁺ in SrCl₂ and also for SrF₂:Cu²⁺ and CaF₂:Ni⁺.⁸⁷

Further work on the optical and magnetic properties of insulating transition metal compounds is now underway.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.inorgchem.4c00599.

• Additional computational details for both CRYSTAL17 (S1)25-27, and VASP codes (S2) and equilibrium geometries (S3); equilibrium geometries for P4/n phase at P = 0 GPa for S = 2, P4/n phase at P = 40 GPa for S = 2, and P4 phase at P = 40 GPa for S = 2 (PDF)

Author Contributions

^† G.S. and T.F.-R. contributed equally to this work.

The authors declare no competing financial interest.