Pressure Effects on 3dn (n=4, 9) Insulating Compounds: Long Axis Switch in Na3 MnF6 not Due to the Jahn-Teller Effect.

The pressure-induced switch of the long axis of MnF6 3- units in the monoclinic Na3 MnF6 compound and Mn3+ -doped Na3 FeF6 is explored with the help of first principles calculations. Although the switch phenomenon is usually related to the Jahn-Teller effect, we show that, due to symmetry reasons, it cannot take place in 3dn (n=4, 9) systems displaying a static Jahn-Teller effect. By contrast, we prove that in Na3 MnF6 the switch arises from the anisotropic response of the low symmetry lattice to hydrostatic pressure. Indeed, while the long axis of a MnF6 3- unit at ambient pressure corresponds to the Mn3+ -F3 - direction, close to the crystal c axis, at 2.79 GPa the c axis is reduced by 0.29 Å while b is unmodified. This fact is shown to force a change of the HOMO wavefunction favoring that the long axis becomes the Mn3+ -F2 - direction, not far from crystal b axis, after the subsequent relaxation process. The origin of the different d-d transitions observed for Na3 MnF6 and CrF2 at ambient pressure is also discussed together with changes induced by pressure in Na3 MnF6 . The present work opens a window for understanding the pressure effects upon low symmetry insulating compounds containing d4 or d9 ions.

Introduction

In insulating compounds containing d4 or d9 transition metal cations in sixfold coordination, remarkable changes in the geometry of associated complexes are induced by external pressure. This singular behavior has been observed, for instance, in Na3MnF6, (NH4)2Cu(H2O)6(SO4)2, Cs2Zn(ZrF6)2 ⋅ 6H2O : Cu2+, CuF2(H2O)2(pyz) (pyz=pyrazine) or CuWO4, and it has systematically been related[ , , , , , , , ] to both the existence of a static Jahn‐Teller (JT) effect and a pressure‐induced switch of the principal axis of the complex formed by the 3dn (n=4, 9) cation.

Nonetheless, this explanation, though widely followed, is hard to accept first due to the restrictive symmetry conditions necessary for the presence of a JT effect.[ , ] Indeed, the JT effect requires the existence of an initial geometry where the ground state of the complex is actually degenerate. This condition is fulfilled under cubic or trigonal symmetry, and thus a static JT effect has been observed in cases like KZnF3:Cu2+,[ , , ] d9 ions (Ni+, Cu2+, Ag2+) doped NaCl,[ , , , , , , ] A(H2O)6SiF6 : Cu2+ (A=Mg, Zn)[ , , , ] or AgF2[ , ] but not in K2ZnF4:Cu2+[26] where the host lattice is tetragonal. This key argument thus makes difficult to assume the existence of a JT effect in crystals like Na3MnF6, (NH4)2Cu(H2O)6(SO4)2, CuF2(H2O)2(pyz), or CuWO4, whose structures are either monoclinic or triclinic.[ , , , , , , , ]

Furthermore, in systems displaying a static JT effect, a hydrostatic pressure can hardly switch the principal axis direction. Indeed, for a Cu2+ ion in a cubic crystal like the KZnF3 perovskite,[ , , ] the principal axis of the CuF6 4− unit can be one of the three C4 axes (along X, Y, Z directions) of the lattice as a result of a static JT effect that leads to a local tetragonal symmetry. This symmetry breaking ultimately arises from the unavoidable random strains present in any real crystal, a key point firstly clarified by Ham.[ , ] Accordingly, if in a given region of the KZnF3 cubic crystal the principal axis of the CuF6 4− complex is forced to be Z by random strains, the X and Y axes continue to be equivalent under hydrostatic pressure, which is an isotropic stress. Thus, an applied pressure can modify the value of the tetragonal distortion and the JT stabilization energy, EJT, but not the principal direction of the CuF6 4− unit at a given point of the crystal. In contrast, systems that do not present a JT effect display preferential directions that, as we will show, are associated with the elastic properties of the lattice. Armed with this knowledge, gained for example using closed‐shell systems that are clearly not suspect of displaying the JT effect, one may predict the possible change in direction of the long axis of these transition‐metal complexes. This makes JT and non‐JT systems quite different since in the first class there cannot be a systematic long‐axis switch while in the second class this can be allowed.

Bearing these facts in mind, it has previously been shown that in low symmetry compounds like CuF2, CrF2, or K2CuF4 there is no JT effect.[ , ] Therefore, the local distortion and the ground state of involved MF6 4− (M=Cu, Cr) units obey to different causes that can be uncovered with the help of first principles calculations. Such an analysis stresses the importance of considering the effects of the anisotropy of the low symmetry lattice on the complex. In fact, even if the MF6 4− (M=Cu, Cr) complex is, in principle, octahedral the anisotropic electric field due to the rest of lattice ions generally leads to a non‐degenerate ground state.[ , , , ] Moreover, that anisotropy makes that the six F− ligands are no longer elastically equivalent, and thus the change of the force constant due to the rest of the lattice can be different for each of the involved ligands. Both facts have been proved to play a key role in explaining the different distortion and ground state observed[ , ] for Cu2+ or Ni+ in the layered lattices K2MgF4 and K2ZnF4.[ , ]

According to the present arguments, it is now necessary to understand why in low symmetry compounds involving 3dn (n=4, 9) cations the nature of the longest local axis of the involved complex can be switched by an applied pressure if that phenomenon is not due to a JT effect. Seeking to gain a better insight on this relevant issue, this work focuses on the monoclinic Na3MnF6 compound[ , , ] where pressure effects have been investigated experimentally. This compound is taken as a guide for the present study, carried out with the help of first principles calculations and an analysis of available experimental results.

The singularity of 3dn (n=4, 9) compounds displaying a sixfold coordination just reflects that the ground state electronic density of a MX6 unit (M=3d4 or 3d9 ion), has not a cubic symmetry and thus, if all ligands are at the same distance, not all of them experience the same force.[ , ] However, this general behavior does not appear in high spin MX6 complexes when M is a cation like Fe3+, Mn2+ (d5) or Cr3+ (d3) whose ground state electronic density is close to spherical. For this reason, in the study of Na3MnF6 it is also useful to explore how the structure evolves when all Mn3+ cations are replaced by a cation with a similar ionic radius but a more spherical electronic density, keeping the same space group. This is just the so‐called parent phase previously used[ , , ] as a good starting point for understanding the structure and ground state of compounds like AgF2, CuF2, CrF2, or K2CuF4.

This work is organized as follows. In section 2 we describe the crystal structure of Na3MnF6 as a function of pressure observed experimentally. For the sake of clarity, it is compared to that of cryolite (Na3AlF6) at ambient pressure. The computational tools used in the present study are explained in section 3, while main results are given in section 4. Some final remarks are provided in the last section.

Crystal structure of Na3MnF6 under pressure: Experimental results

According to Englich et al. the structure of Na3MnF6 at ambient pressure can be described by the monoclinic space group P21/n, a non‐standard setting more visual than the standard one, P21/c. A view of the corresponding lattice cell is displayed in Figure 1, whereas the ensemble of lattice parameters (a, b, c, and monoclinic angle, β) and Mn3+−F− distances is reported in Table 1. When pressure increases up to 2.79 GPa, the space group of Na3MnF6 is unmodified, being again P21/n. The structure observed experimentally at P=2.79 GPa is also shown in Figure 1. The crystal data derived by Carlson et al. at P=0.12 GPa and P=2.79 GPa are collected in Table 1, together with those corresponding to the cryolite compound (Na3AlF6) at zero pressure, also displaying a P21/n space group.

Figure 1

Experimental unit cells of Na3MnF6 compound at pressures P=0.12 and 2.79 GPa in the non‐standard monoclinic P21/n setting. {x,y,z} are the local axes of the MnF6 4− complexes, with Ci point symmetry. The 3 Mn−Fi distances (in Angstroms) are also shown.

Table 1

Description of the crystal structure of Na3MnF6 reported at P=0 GPa by Englich et al. and at P=0.12 and 2.79 GPa by Carlson et al. In addition to lattice parameters (a, b, c and monoclinic angle, β), the values of three Mn3+−F− distances, R1, R2 and R3, are also given. In a first approximation, it can be considered that the directions of R1, R2 and R3 are parallel to a, b and c axis, respectively. The value of the longest metal‐ligand distance is written in bold type for each structure. For comparison, the values corresponding to Na3AlF6 are also shown. All distances are in Å and the β angle in degrees.

Compound	Pressure	a	b	c	β	R1	R2	R3
Na3MnF6	0	5.471	5.683	8.073	88.96	1.862	1.897	2.018
	0.12	5.470	5.690	8.066	89.08	1.899	1.886	2.038
	2.79	5.386	5.690	7.783	90.76	1.790	1.971	1.891
Na3AlF6	0	5.402	5.596	7.756	90.27	1.804	1.813	1.805

As shown in Table 1, the crystallographic data obtained at ambient pressure and at 0.12 GPa are practically identical. They reveal that the lattice cell is nearly orthorhombic with a monoclinic angle β=89°. The MnF6 3− complexes formed in Na3MnF6 are practically isolated, i. e., they do not share F− ligands and the distance between two neighbors Mn3+ cations is equal to 5.40 Å. As the F−−Mn3+−F− angle for two F− ions in cis position[ , ] differs by about 2° from 90°, the local symmetry of a MnF6 3− complex is triclinic Ci. The three Mn3+−Fi − lines (i=1, 2, 3) in Figure 1 are not strictly parallel to the crystallographic a, b, and c axes, respectively, due to a tilting of the complex. Still, the Mn−F3 line is mainly oriented along the c axis and the Mn−F2 line along the b axis. For instance, the angle between c axis and the Mn−F3 direction is equal to 22°. The three Mn3+−F− distances are not equal; the largest one, R3, corresponds to the Mn−F3 bond and differs from R2 by 0.152 Å, while R2 and R1 are identical within 1.5 % (Table 1).

The results for Na3MnF6 at zero pressure are somewhat similar to those for Na3AlF6, also displayed in Table 1. Indeed, despite Al3+ being a closed shell cation, the three Al3+−F− distances are not equal, although the differences among them are smaller than 0.01 Å. On the other hand, in the cryolite the largest Al3+−F− distance corresponds to R2 and not to R3, at variance with Na3MnF6 at ambient pressure (Table 1). Interestingly, when we compare the lattice parameters of Na3MnF6 at ambient pressure with those for Na3AlF6 the reduction is not isotropic. Indeed, c reduces by 4 % while b and a decrease by less than 1.5 %.

An increase of pressure reduces the value of lattice parameters, specially c. So, on passing from P=0 to P=0.91 GPa the value of c decreases by 0.081 Å according to experimental data by Carlson et al. At the same time, the value of R3‐R2 at P=0.91 GPa is equal only to 0.081 Å in comparison to R3‐R2=0.121 Å at ambient pressure (Table 1). As a salient feature, Na3MnF6 undergoes a phase transition at about 2.2 GPa, where the lattice parameter c abruptly decreases by about 2 %. Moreover, it is a first‐order phase transition because is isosymmetric (the space group is always P21/n) and involves a significant hysteresis of ∼0.8 GPa. Despite this fact, the phase transition is characterized by a switch of the longest axis of all MnF6 3− complexes. Indeed, while at ambient pressure that axis corresponds to the Mn3+−F3 − direction, it is the Mn3+−F2 − direction that becomes the longest axis after the phase transition, such as it is shown in Table 1. Furthermore, comparing the experimental results at different pressures, we can notice that, on passing from P=0 GPa to P=2.79 GPa the three lattice parameters evolve in a very different way: c is reduced by 0.29 Å while b is unmodified and a decreases only by 0.08 Å (Table 1).

The origin of the switch of the longest axis under pressure is discussed in Section 4 helped by the results of first principles calculations. In that analysis, particular attention is paid to exploring the electronic density changes due to pressure.

Computational tools

First principles periodic calculations have been performed on pure Na3MnF6, its parent phase Na3FeF6, as well as on doped Na3FeF6 : Mn3+ by means of the CRYSTAL17 code. This software works with linear combinations of Gaussian type functions to represent the Bloch orbitals.

All electron triple‐zeta polarized basis set of high quality developed by Peitinger et al.[ , ] were employed, together with the one‐parameter hybrid functionals B1WC and PW1PW, which include 16 % and 20 % of Hartree‐Fock exchange, respectively, and have provided accurate results for crystalline structures and properties of insulating systems containing transition metal ions.[ , ]

First, we have optimized the geometry of the two phases that exist in Na3MnF6, at ambient pressure, and at P=2.79 GPa, at the monoclinic P21/n space group, obtaining that both lattice parameters and bond Mn−F distances are similar to the experimental data, with discrepancies of less than 3.5 %. The sampling of reciprocal space for the numerical integration within the Brillouin zone was 8x8x8 Monkhorst‐Pack grid, which represents a distance between two consecutive k points of 0.144, 0.138 and 0.072 Å−1 conforming to the three reciprocal space directions. For geometry optimizations the tolerance for energy change was 10−8 Hartree and for gradient and nuclei displacement were 0.0002 a.u.

The next step was the optimization of the parent phase structure. Regarding this calculation, all open shell Mn3+ ions, with d4 electronic configuration and S=2, were replaced 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. The symmetry of the optimized parent phase Na3FeF6 remains in the same P21/n space group but the complexes are less distorted, reflecting a more spherical symmetry for Fe3+ in Na3FeF6 than for Mn3+ in Na3MnF6. Further discussion on this issue is provided in Section 4.2.

In addition, cluster simulations on MnF6 3− complexes have been performed in order to explore the relationship between the structural changes due to pressure and the variations undergone by the HOMO as well as the influence of pressure on the spin allowed d‐d transitions. These calculations have been carried out by means of the Amsterdam density functional (ADF) code, which works with DFT techniques in the Kohn‐Sham framework. Hybrid B3LYP functional (25 % of exact exchange) combined with triple zeta polarized basis set have been employed in the simulations. The core electrons (1s‐3p for Mn3+, 1 s for F−) were kept frozen since they do not play a relevant role in the effects we focus on. In these calculations, the MnF6 3− clusters were embedded in the electrostatic potential of the rest of lattice ions,[ , ] which was previously calculated through Ewald‐Evjen summations.[ , ] Although the effects of this internal potential are important in systems like ruby,[ , , ] the Egyptian‐blue pigment or K2ZnF4 : Cu2+,[ , , ] it is not the case of Na3MnF6 where only induces energy variations in the optical d‐d transitions equal or smaller than 0.1 eV. Such transitions are discussed in Section 4.4. Both complex and environment were rotated in such a way that the directions R1, R2 and R3 lies, approximately, along the local axes x, y and z respectively.

Understanding the long axis switch induced by pressure in Na3MnF6 entails the analysis of the wavefunction in the ground state at different stages since its evolution leads to the distortions observed in this system. Therefore, the HOMO wavefunction has been determined for Na3MnF6 at different pressures and also for Na3FeF6:Mn3+. In the latter case, the HOMO corresponding to the MnF6 3− unit has been derived for both the undistorted parent phase and the final equilibrium geometry. In these calculations, the wavefunction coefficients have been obtained following Mulliken population criterion.

Results and Discussion: Na3MnF6

Results of calculations at P=0 and P=2.79 GPa for Na3MnF6

The optimized geometries for Na3MnF6 obtained under periodic boundary conditions working in the P21/n space group are collected in Table 2 for P=0 and 2.79 GPa. Such results are compared to available experimental results.[ , ] The calculated values reasonably reproduce the experimental results. Indeed, calculated lattice parameters deviate from experimental ones less than 1 % while the error on the value of the longest Mn3+−F− distance is smaller than 3.5 %. It is worth noting that the present calculations also lead to the long axis along the Mn3+−F3 − direction at zero pressure while it is along the Mn3+−F2 − direction for P=2.79 GPa. Moreover, they also shed light on the nature of the ground state of the MnF6 3− complex at both pressures.

Table 2

Calculated values of lattice parameters (a, b, c and β) and the three Mn3+−F− distances, R1, R2, R3, for Na3MnF6 in the P21/n space group at both zero pressure and P=2.79 GPa. The results are compared to available experimental results.[ , ] The value of the longest Mn3+−F− distance is written in bold type for both pressures. All distances are in Å and the β angle in degrees.

P		a	b	c	β	R1	R2	R3
0	Experimental	5.471	5.683	8.073	88.96	1.862	1.897	2.018
	Calculated	5.460	5.646	8.137	88.47	1.864	1.880	2.069
2.79	Experimental	5.386	5.690	7.783	89.24	1.790	1.971	1.891
	Calculated	5.381	5.720	7.713	88.37	1.861	2.041	1.866

The electronic ground state of a hypothetically octahedral (Oh point group) MnF6 3− unit with S=2 corresponds to the configuration t2g 3eg 1. At zero pressure, the present periodic and cluster calculations on Na3MnF6 reveal that the unpaired electron coming from eg has a dominant 3z2‐r2 character where z essentially reflects the Mn3+−F3 − direction (Figure 1). This is consistent with R3 being the long axis while R1 and R2 are equal within 1.5 %. Indeed, if the octahedron is elongated along the z axis and R1=R2 the HOMO corresponds to a molecular orbital |3z2‐r2⟩ (transforming like 3z2‐r2) while the LUMO corresponds to |x2‐y2⟩ . This situation is thus quite similar to that found for the low spin (S=1/2) RhCl6 4− complex generated by the d7 Rh2+ cation formed in NaCl.[ , ]

At P=2.79 GPa the present calculations reveal that the unpaired electron coming from eg has a dominant 3y2‐r2 character where y essentially reflects the Mn3+−F2 − direction (Figure 1). Thus, the switch of the longest axis also involves a change of the nature of the HOMO orbital helped by the anisotropic reduction of lattice parameters under pressure (Table 2). As the biggest reduction occurs for the c axis while the b axis is practically unmodified one can already envisage that the electronic density in the HOMO prefers to be oriented along the Mn3+−F2 − direction and then push F2 ligands away.

The change of the calculated electronic spin density (difference between spin up and spin down densities) when pressure moves from P=0 GPa to 2.79 GPa is depicted in Figure 2. These spin density contours corresponds to a good extent to the four electrons placed in the |3z2‐r2⟩, |xy⟩, |xz⟩, |yz⟩ orbitals all of them with spin up. At P=0 GPa, Figure 2 shows the existence of bonding with 2pσ and 2 s orbitals of F3 − ligands placed on the z axis of the MnF6 3− complexes, consistent with an unpaired electron from eg with a dominant 3z2‐r2 character. By contrast, at P=2.79 GPa that bonding involves F2 − ligands placed on the y local axis of the complexes. A further discussion on this issue is given in Section 4.3.

Figure 2

Calculated electronic spin density (difference between spin up and spin down densities) of the ground state of a MnF6 3− unit in Na3MnF6 when pressure moves from P=0 GPa to 2.79 GPa. At P=0 GPa results reflect an antibonding orbital involving the 3z2‐r2(Mn) orbital mixed with with 2pσ(F) and 2 s(F) orbitals of F3 − ligands. By contrast, at P=2.79 GPa that orbital involves orbitals of F2 − ligands. Note that as calculations are spin unrestricted there are small regions with negative spin density (blue colour) showing the polarization of fully occupied bonding orbitals by the four unpaired electrons with spin up.

Results on the parent phase Na3FeF6 pure and doped with Mn3+ at zero pressure

In the study of a dn (n=4, 9) compounds it has proved to be useful to determine the actual structure once each cation is replaced by another one with a similar ionic radius but a more spherical electronic density.[ , , , , ] As during this process, giving rise to the so‐called parent phase, the space group does not change, the symmetry of the parent phase is necessarily equal or higher than that of the initial dn (n=4, 9) compound. For example, for CuF2 (monoclinic P21/n space group) the Cu2+→Zn2+ substitution leads to a ZnF2 parent phase belonging to the tetragonal P42/mnm space group. Accordingly, the rutile structure of ZnF2 becomes unstable as far as Zn2+ ions are progressively substituted by Cu2+.

In the present case of Na3MnF6, the parent phase is obtained by substituting all Mn3+ ions by Fe3+ ones (both with the same ionic radius, 0.785 Å ). A geometry optimization calculation performed on the Na3FeF6 parent phase, starting from the experimental P21/n geometry of Na3MnF6, led to a structure displaying the same P21/n space group (Table 3). Moreover, that phase actually corresponds to the experimental Na3FeF6 compound, whose structure at ambient pressure was firstly reported by Croft and Kestigian from X‐ray diffraction in powder. The values of lattice parameters obtained by these authors are gathered on Table 3 and compared to results of present geometry optimization. Both sets of data are essentially coincident. However, Croft and Kestigian in their powder study did not report the values of metal‐ligand distances and thus we cannot compare them to those derived from present calculations (Table 3).

Table 3

Experimental values of lattice parameters (a, b, c and β) derived from X‐ray diffraction power data for Na3FeF6 at ambient pressure displaying the P21/n space group. Such values are compared to the structural information obtained from present geometry optimizations for the parent phase of Na3MnF6 where the three Fe3+−F− distances have been determined. All distances are in Å and the β angle in degrees.

	a	b	c	β	R1	R2	R3
Experimental	5.506	5.719	7.925	90.5	–	–	–
Calculated	5.483	5.720	7.936	89.5	1.931	1.945	1.943

–

–

–

The local symmetry around a Fe3+ cation in Na3FeF6 is given by the same triclinic Ci point group as that for Mn3+ in Na3MnF6. Despite this fact, it should be noticed that R2‐R3=0.002 Å derived for Na3FeF6 is clearly smaller than R2‐R3=0.180 Å determined for Na3MnF6 at 2.79 GPa (Table 2). This situation is thus rather similar to that found for cryolite (Table 1) where the longest Al3+−F− distance also corresponds to R2 and R2‐R3=0.008 Å.

Particular attention has been paid to study the behavior of a single substitutional Mn3+ impurity in Na3FeF6. Results of the present calculations are gathered in Table 4. Although the Fe3+→Mn3+ substitution does not modify the Ci local symmetry, it enhances the octahedron distortion leading to a local geometry that resembles that derived for pure Na3MnF6 at 2.79 GPa (Table 2) with R2‐R3=0.15 Å. Therefore, the Fe3+→ Mn3+ substitution does not give rise to any symmetry breaking, such as it happens in a static JT effect. Nevertheless, the increase of R2‐R3 on passing from pure Na3FeF6 (R2‐R3=0.002 Å) to Na3FeF6:Mn3+ (R2‐R3=0.15 Å) has to be related to the different ground state electronic density in the corresponding FeF6 3− and MnF6 3− units.

Table 4

Calculated values of three metal‐ligand distances, R1, R2 and R3, obtained for a Mn3+ impurity in the Na3FeF6 parent phase using two different sets of lattice parameters, those optimized for Na3FeF6 (second row) and other set where the value of c increases only by 1.1 % (third row). It can be noticed that in the latter case, the longest axis is along the Mn3+−F3 − direction and not along the Mn3+−F2 − direction. The results for pure Na3FeF6 (first row) are also included for comparison. In every case the longest metal‐ligand distance is written in bold. All distances are in Å and the β angle in degrees.

	a	b	c	β	R1	R2	R3
Na3FeF6	5.483	5.720	7.936	89.5	1.931	1.945	1.943
Na3FeF6:Mn3+	5.483	5.720	7.936	89.5	1.856	2.045	1.895
Na3FeF6:Mn3+	5.490	5.700	8.020	89.5	1.866	1.892	2.049

Let us consider an initial step where Mn3+ replaces Fe3+ in the undistorted Na3FeF6 lattice. We have verified that in that situation the unpaired electron coming from eg has a dominant 3y2‐r2 character such as it is shown in Table 5. Indeed, the calculated wavefunction of the HOMO, |ψH⟩, can shortly be written as

Table 5

Relevant data corresponding to a Mn3+ impurity in Na3FeF6. In the initial step, a Mn3+ ion just substitutes a Fe3+ one in the undistorted Na3FeF6 parent lattice, while the final step describes the equilibrium geometry of a single Mn3+ impurity in the parent phase. In both situations, the calculated wavefunction of the HOMO and the value of the HOMO‐LUMO gap, Δ, are given. The relative energy of the electronic ground state (GS) derived for both situations is also reported. All distances are in Å and energies in eV.

Step	R1, R2, R3	HOMO	Δ	GS energy
Initial	1.931, 1.945, 1.943	0.972|3y2‐r2>+0.234|x2‐z2>	0.06	0
Final	1.856, 2.045, 1.895	0.996|3y2‐r2>+0.078|x2‐z2>	0.76	−0.18

Here |3y2‐r2⟩ just means a molecular orbital of the MnF6 3− unit transforming like 3y2‐r2. Accordingly, the LUMO is mainly made of |x2‐z2⟩, and it is placed only at 0.06 eV above the HOMO just reflecting that in this initial step the local symmetry of the MnF6 3− unit is Ci despite the three Mn3+−F− distances are equal within 0.01 Å. These facts thus stress that in this initial stage the electronic ground state of the MnF6 3− unit is not degenerate, thus excluding the existence of a JT effect. Despite this fact, as the electron placed in the HOMO is described by Equation (1), it exerts a different force on the three kinds of ligands initially placed at the same distance from Mn3+ within 0.01 Å. A different situation holds, however, when we consider the average (3y2‐r2)0.5(x2‐z2)0.5 configuration, whose associated electronic density leads to a similar force on all ligands.[ , , ] Accordingly, in the actual (3y2‐r2)1(x2‐z2)0 configuration, the difference of electronic density pushes the F2 ligands away while it tends to approach F1 and F3 ligands, such as it is shown in Tables 4 and 5. This relaxation process, leading to the final equilibrium geometry of a single MnF6 3− unit in the parent phase, involves an energy gain, ER, calculated to be equal to 0.18 eV (Table 5).

In addition to study the behavior of a Mn3+ impurity in Na3FeF6, we have also investigated the influence of a small increase (1.1 %) on the c lattice parameter upon both the equilibrium geometry and the electronic ground state of the MnF6 3− unit. The results, also displayed in Table 4, show that under that 1.1 % increase on the c parameter the longest distance of the MnF6 3− unit changes from being R2 to R3 and in the final geometry the unpaired electron coming from eg has then a dominant 3z2‐r2 character.

Understanding the long axis switch induced by pressure in Na3MnF6

Bearing the analysis carried out in sections 4.1 and 4.2 in mind, the long axis switch in Na3MnF6 appears to arise from the significant anisotropic response of this compound to hydrostatic pressure. Thus, although the pressure on any crystal face is always the same, the reduction of crystal parameters is far from being isotropic. Indeed, in the pressure range 0–2 GPa the lattice parameters a and, specially, c, experience a progressive decrease, while b is nearly unmodified. This pressure reduction of c and a tends, in turn, to reduce the value of R3 and R1, as it is experimentally shown. For instance, at P=0.91 GPa, R3=1.978 Å and R1=1.810 Å, which are clearly smaller than those reported at 0.12 GPa (Table 1). By contrast, the value R2=1.897 Å measured at P=0.91 GPa is only 0.01 Å higher than the value obtained at 0.12 GPa (Table 1).

A progressive reduction of R3 approaches the two F3 − ligands to Mn3+, thus increasing the energy of the |3z2‐r2⟩ level, which works against having the HOMO of the MnF6 3− unit in Na3MnF6 with a dominant 3z2‐r2 character. For this reason, when pressure increases, it appears more favorable a change of the HOMO wavefunction in order to reach a dominant 3y2‐r2 character and placing the associated density along the y direction where R2 is not reduced by pressure. Moreover, although this switch of the HOMO wavefunction involves, in principle, an energy increase there is an energy reduction in the subsequent relaxation process leading to an increase of R2 and a decrease of R3 (Table 2).

The main steps of this process induced by pressure in Na3MnF6 can quantitatively be clarified looking at Figure 3. Particular attention is paid to analyze, in each step, the shape of the HOMO wavefunction of the MnF6 3− unit, |ψH⟩. In general, according to calculations, |ψH⟩ can simply be written as follows

Figure 3

Relative variation of both energy and wavefunction of the HOMO for MnF6 3− complexes in Na3MnF6 calculated at different values of metal‐ligand distances, whose directions in the Na3MnF6 unit cell are also shown. Step I corresponds to Ri (i=1, 2, 3) values for zero pressure while step II reflects the experimental geometry at about 1.8 GPa and step IV that for a pressure of 2.79 GPa where the highest metal‐ligand distance is already R2 instead of R3. Step III describes a transition state where the geometry is that of step II while the wavefunction is that of the final Step IV. The increase in the HOMO energy by changing the wavefunction at the geometry corresponding to ∼1.8 GPa is only 0.06 eV. By contrast, the energy decrease due to the ligand relaxation process on passing from step III to the final step is of 0.21 eV.

with α2 + β2=1. Four steps are considered in Figure 3. The initial step I describes the situation of a MnF6 3− unit in Na3MnF6 at ambient pressure, while step II reflects the experimental geometry and HOMO wavefunction at about P=1.8 GPa where the longest axis is still R3, as described by our first principles simulations. Similarly, step IV corresponds with data at P=2.79 GPa pressure, where the longest axis is already R2. Step III is a virtual process where the geometry is that of step II (P=1.8 GPa) while the HOMO wavefunction has been switched to that of the final step (P=2.79 GPa). We describe below the process to obtain its energy.

At ambient pressure, the MnF6 3− unit is elongated along the z axis while R1 and R2 are identical within 0.03 Å. Accordingly, the HOMO wavefunction is essentially |3z2‐r2⟩ (β ≈ 0). On passing from step I to step II, R3 decreases while R2‐R1 increases. This favors an increase of the |x2‐y2⟩ component although |ψH(II)⟩ still exhibits a dominant |3z2‐r2⟩ contribution (84.2 %) and is described by

At the same time, on passing from step I to step II the energy of the HOMO level experiences an increase of 0.63 eV (Figure 3).

The calculated HOMO wavefunction for step IV (Figure 3)

involves changes with respect to that of step II which are better seen writing |ψH(IV)⟩ in the {|3y2‐r2⟩, |x2‐z2⟩} basis

Accordingly, in |ψH(IV)⟩ there is a 90 % contribution of |3y2‐r2⟩ and thus the associated density lies mainly along the R2 direction.

To calculate the energy of the HOMO level in the virtual step III we have expressed the orbital wavefunction |ψH(IV)⟩ (Eq. (4)) in terms of that of the HOMO at step II, |ψH(II)⟩, and its LUMO, |ψL(II)⟩. As < ψL(II) |ψH(II)⟩ =0 and the HOMO‐LUMO separation in step II is equal to 0.54 eV, this means that, on passing from step II to step III (Figure 3) which represents just a change in basis, the increase in HOMO energy can be estimated to be only 0.06 eV. This slight increase reflects the relative closeness of |ψH(IV)⟩ and |ψH(II)⟩ as <ψH(II)|ψH(IV)⟩ is equal to 0.944 from Eqs. (3) and (4). Interestingly, the 0.06 eV energy increase obtained on passing from step II to step III is widely compensated by the decrease of 0.21 eV due to the subsequent ligand relaxation involved in the transition from step III to step IV such as it is shown on Figure 3.

If the long axis switch in Na3MnF6 is strongly influenced by the electronic ground state of the involved MnF6 3− unit a different situation would happen in Na3FeF6, where a more spherical electronic density around Fe3+ is expected to take place. Calculations carried out for Na3FeF6 at a pressure of 2.79 GPa support this idea. Indeed, the obtained pattern of metal‐ligand distances (R1=1.922 Å, R2=1.935 Å and R3=1.932 Å) is similar to that derived at zero pressure (Table 4) although all of them are reduced by ∼0.01 Å.

It is worth noting that the switch phenomenon analyzed in Na3MnF6 is different from that observed[ , ] in Tutton salts like (ND4)2Cu(D2O)6(SO4)2 or (NH4)2Cu(H2O)6(SO4)2. As it has recently been shown,[ , ] the structure of Cu(D2O)6 2+ or Cu(H2O)6 2+ units in those salts at ambient pressure arises from a compressed geometry, with Z as principal axis and an unpaired electron in a |3z2‐r2⟩ orbital, followed by an orthorhombic instability in the XY plane. Accordingly, the three metal‐ligand distances are ordered as follows: RY>RX>RZ where RY and RX stand for the Cu−O8 and Cu−O7 bonds respectively while RZ corresponds to the shortest Cu−O9 bond. Interestingly, despite the orthorhombic distortion, the wavefunction of the unpaired electron at ambient pressure keeps an 82 % of 3z2‐r2 character. Upon raising the applied pressure at 0.15 GPa the long axis is no longer the Cu−O8 but the Cu‐O7 bond[ , ] thus implying RX>RY>RZ with RZ being unmodified within 1 %. For this reason, in the switch process corresponding to (ND4)2Cu(D2O)6(SO4)2 or (NH4)2Cu(H2O)6(SO4)2 the unpaired electron wavefunction maintains a dominant 3z2‐r2 character a situation distinct to that encountered in Na3MnF6.

Optical transitions of Na3MnF6 at ambient and high pressure

The ground state, Ψ0, with S=2 and MS=2, of a high spin d4 complex can shortly be written by the following Slater determinant

Accordingly, the spin allowed excitations involve jumps from the four orbitals appearing in Ψ0 to the empty |x2‐y2⟩ orbital and thus they are the most intense among the d‐d transitions coming from Cr2+ or Mn3+ complexes.[ , , ]

In the case of Na3MnF6 at ambient pressure the values of such spin allowed transitions have been reported by Palacio and Moron and Carlson et al. although the last authors have not detected the lowest transition 3z2‐r2↑→x2‐y2↑. Experimental transition energies are collected in Table 6 and compared to the values obtained in the present calculations. The calculated d‐d transitions are reasonably close to those observed experimentally at ambient pressure.

Table 6

Calculated optical transitions for Na3MnF6 at ambient pressure. They are compared to the experimental values reported by Palacio and Moron (I) and Carlson et al. (II). Energies are all given in eV units.

Optical Transition	Calculated	Experim. (I)	Experim. (II)
|3z2‐r2⟩→|x2‐y2⟩	0.71	1.04	–
|xy⟩→|x2‐y2⟩	2.11	2.18	2.17
|yz⟩→|x2‐y2⟩	2.27	2.38	2.38
|xz⟩→ |x2‐y2⟩	2.32	2.38	2.58

–

It is worth noting now that in K3MnF6 there are two different Mn3+ sites and the geometry of the so‐called Mn3+(2) site is very close to that found in Na3MnF6 at ambient pressure (Table 1). Indeed, the values of R1, R2, and R3 differ by less than 3 %, and the value of the average metal‐ligand distance, ⟨R⟩, is equal in both cases to 1.93 Å.[ , ] The experimental d‐d excitations measured in a powder K3MnF6 for the Mn3+(2) site are close to those reported for Na3MnF6 in Table 6, the differences being smaller than 12 %.

For the sake of completeness, we have compared the d‐d transitions for Na3MnF6[ , ] and CrF2 (involving 3d4 cations, Mn3+ and Cr2+) measured at ambient pressure. The three transitions coming from {xy, xz, yz} levels are clearly higher for Na3MnF6[ , ] than for CrF2. For instance, the highest d‐d transition for Na3MnF6 is found at ∼2.45 eV and only at 1.75 eV for CrF2 as shown in Table 7. This fact partially reflects the higher value of ⟨R⟩ for CrF2 than for Na3MnF6. For this reason, it may be surprising that the opposite happens for the lowest d‐d transition (Table 7) as it is equal to 1.04 eV for Na3MnF6 while equal to 1.22 eV for CrF2.[ , ] Nevertheless, the CrF6 4− unit in CrF2 is, in principle, compressed along the z direction but followed by an orthorhombic instability leading to a significant difference of final Cr2+−F− distances along x (R1) and y (R2) directions equal to 0.416 Å (Table 7). As shown in Ref. [28] the existence of this huge orthorhombic distortion increases the value of the x2‐y2↑→3z2‐r2↑ transition in CrF2 by 0.60 eV.

Table 7

Experimental values (in eV) of highest, EH, and lowest, EL, d‐d transitions for CrF2 and Na3MnF6[ , ] measured at ambient pressure. The values of the average metal‐ligand distance, ⟨R⟩, and |R2‐R1| (in Å units) are also given.

	EH	EL	<R>	|R2‐R1|
CrF2	1.75	1.22	2.14	0.42
Na3MnF6	∼2.45	1.01	1.94	0.01

Concerning the influence of pressure upon d‐d excitations we have also calculated the values of four spin allowed transitions of Na3MnF6 at P=2.79 GPa. Results are depicted in Figure 4 and compared to those obtained for zero pressure. In accord with results of section 4.3, the nature of HOMO and LUMO orbitals has been modified as well as the order of three levels coming from t2g in Oh symmetry (Figure 4). The present results shed light on the three optical transitions measured by Carlson et al. in the high‐pressure phase of Na3MnF6. Indeed, according to Figure 4, the peaks observed at 2.19 eV, 2.39 eV and 2.59 eV by Carlson et al. can reasonably be assigned to the xz↑→x2‐z2↑, xy↑→x2‐z2↑ and yz↑→x2‐z2↑ excitations, respectively. As shown in Figure 4, the HOMO‐LUMO gap is calculated to increase by 0.14 eV on going from P=0 to P=2.79 GPa, a fact that is partially helped by the increase of orthorhombicity. Indeed, while at P=2.79 GPa R3‐R1=0.168 Å, at ambient pressure R1‐R2 is equal only to 0.035 Å (Table 1). Unfortunately, in the experimental set‐up by Carlson et al. the absorption measurements under pressure were carried out for photon energies above 1.5 eV and thus the lowest d‐d transition has not been detected.

Figure 4

Energies (in eV) of spin allowed d‐d transitions of Na3MnF6 calculated for a MnF6 3− complex with the experimental geometries at P=0 and P=2.79 GPa. The dominant contribution for each orbital is shown, and the spin distribution corresponds to the electronic ground state (S=2, MS=2). Note that on passing from ambient pressure to P=2.79 GPa, the nature of HOMO and LUMO orbitals has been modified as well as the order of three levels coming from t2g in Oh symmetry.

Conclusions

The present results show the usefulness of the parent phase as a good starting point for understanding the behavior of low symmetry compounds containing dn (n=4, 9) cations. In the present case, when we introduce a substitutional Mn3+ impurity in the parent Na3FeF6 compound, the HOMO wavefunction is already determined by the low local symmetry, which in turn, fixes the subsequent distortion due to an electron density that exerts a different force on the three kinds of ligands. Moreover, the force constant for moving a ligand is different in each type of ligands as a result of the low symmetry of the lattice. This behavior has been well studied in layered lattices like K2MF4 (M=Mg, Zn) doped with d9 ions like Cu2+ or Ni+.[ , ]

Bearing these facts in mind, we can now envisage the differences and similarities between low symmetry and JT systems. If we place a d4 cation in a cubic material with sixfold coordination, the ground state is degenerate, and thus the eg electron is, in general, described by a linear combination of |3z2‐r2⟩ and |x2‐y2⟩ molecular orbitals of the associated complex. If the unpaired electron is located in the |3z2‐r2⟩ orbital the corresponding adiabatic minimum corresponds to a tetragonal elongated complex while it is compressed if placed in |x2‐y2⟩. Usually for systems displaying a static JT effect the elongated conformation is preferred,[ , ] with the known exception of CaO : Ni+ which is compressed, a matter discussed in Ref. [60]. Nevertheless, if the complex is tetragonal elongated, there are still three different possibilities for the electronic ground state of the complex because of the cubic symmetry. Indeed, the electron wavefunction can be either |3z2‐r2⟩, |3x2‐r2⟩ or |3y2‐r2⟩. In real crystals, this equivalence is, however, often destroyed by small random strains, establishing one of the three initially degenerate options at a given point of the lattice. Once the unpaired electron is then fixed in the |3z2‐r2⟩ orbital, the corresponding electron density does not exhibit cubic symmetry and thus, induces a tetragonal distortion. This behavior is usually called a static JT effect[ , ] well observed in cases like KZnF3:Cu2+[11,12] or d9 ions (Ni+, Cu2+, Ag2+) in alkali halides.[ , , , , , , , ] Nevertheless, when random strains play a minor role, there is no symmetry breaking, and EPR spectra of JT systems exhibit a cubic and not a tetragonal symmetry.[ , ] This uncommon phenomenon has been observed for Cu2+ or Ag2+‐doped MgO where EPR spectra[ , , ] and first principle calculations prove the existence of coherent tunneling[ , ] among equivalent adiabatic minima thus leading to a global cubic symmetry.

According to this reasoning, if we deal with a static JT effect with z as main axis, an applied hydrostatic pressure, which is an isotropic stress, cannot destroy the equivalence between x and y axes of a tetragonal complex and thus the nature of the principal axis is unmodified. By contrast, a different situation occurs when the dn (n=4, 9) cation is placed in a low symmetry lattice where the corresponding complex shows at most an orthorhombic geometry such as it is found for Na3MnF6 and K3MnF6. Accordingly, in this circumstance, as the geometry does not arise from a JT effect, one can expect, after careful consideration of the elastic properties of the solid, possible pressure‐induced switches of the longest axis, such as those observed for Na3MnF6.

In accord to present results, the properties of d4 and d9 low symmetry compounds with sixfold coordination cannot be explained transferring the E⊗e JT framework, valid for a parent cubic symmetry, to an initial tetragonal or orthorhombic symmetry. For an initial cubic symmetry, the adiabatic minima have tetragonal symmetry well observed experimentally in cases displaying a static JT effect.[ , , , , , , , , , , , , , ] Thus, when the observed local symmetry of the involved complex is orthorhombic or lower, such as it happens for Na3MnF6, explanations based on the JT effect[ , , , , , , ] are likely to be meaningless. Along this line, the existence of a switch of the JT axis recently put forward for explaining the behavior of [(CH3)2NH2]Cu(HCOO)3 under pressure is certainly doubtful due to the monoclinic symmetry of the lattice at ambient pressure. A similar situation holds for recent results on the rhombohedral CuPyr‐I compound where the six ligands around Cu2+ are not identical (two N and four O) a fact that is already against the existence of a JT effect.

The present work stresses the importance of considering the anisotropic elastic response to pressure in low symmetry systems containing d4 or d9 ions. It opens a window for properly understanding the pressure effects upon insulating compounds such as CuF2(H2O)2(pyz) (pyz=pyrazine). Further work along this line is now in progress.

Conflict of interest

The authors declare no conflict of interest.