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

Abstract

A recent theoretical study of CsMnF₄ under pressure presents conclusions on its structural, optical, and magnetic behavior that conflict with established experimental evidence. Crucially, that work omits key prior experimental results on CsMnF₄ and related Mn³⁺ fluorides under pressure. This commentary examines the resulting discrepancies, arguing that omissions of this data undermine the theoretical estimates and methodological validity. This paper provides a critical overview centered on two main points: the contested nature of the pressure-induced high-spin to low-spin transition observed in CsMnF₄ at ~37 GPa, and a detailed discussion of Jahn–Teller physics in this archetypal system. By reconciling the existing literature with the new theoretical claims, this work aims to clarify the high-pressure behavior of CsMnF₄. A thorough analysis on the spectroscopic and magnetic properties of Mn³⁺ fluorides is included to support this commentary.

1. Introduction

CsMnF₄ is a foundational example of a cooperative Jahn–Teller (JT) system, whose structural distortions mediate its distinctive optical and magnetic properties. While the effects of pressure on this material have been extensively studied [1,2,3,4,5,6,7,8,9,10,11,12], its high-pressure structural evolution and electronic ground state remain subjects of active debate [1,7]. A recent publication [1] has intensified this controversy by presenting theoretical calculations that contradict interpretations from prior experimental work. That study, however, omits several key experimental findings on CsMnF₄ and other Mn³⁺ fluorides under pressure [2,3,4,5,6,7,8,9,10,11]. This commentary aims to critically reexamine the structural, optical, and magnetic properties of CsMnF₄ under pressure, specifically to address the interpretation in Ref. [1] that conflicts with the established literature. A primary focus is the disputed assignment of the high-spin to low-spin transition observed in CsMnF₄ near 37 GPa [7].

At ambient pressure, CsMnF₄ crystallizes in a tetragonal structure (space group P4/n), forming a layered perovskite. The MnF₆³⁻ octahedra are subject to a cooperative JT distortion, resulting in locally elongated complexes of D_2h symmetry. The crystal structure is characterized by an antiferrodistortive arrangement where the in-plane equatorial ligands of one MnF₆³⁻ unit act as the axial ligands for two adjacent complexes (Figure 1). The in-plane Mn-F bond distances are 2 × 2.168 Å and 2 × 1.817 Å, while the out-of-plane distances are 2 × 1.854 Å [12]. This specific connectivity, with Mn-F-Mn bond angles close to 180°, favors ferromagnetic superexchange within the layers, successfully explaining the emergence of long-range ferromagnetic order below Tc = 19 K [2,3,4,5].

This paper provides an overview that contrasts the findings of Ref. [1] with the body of existing research. We will first delineate the fundamental JT distortions in this system before critically assessing its high-pressure behavior, with particular emphasis on the debated spin-state transition [1,7]. Furthermore, this paper provides a thorough analysis on the spectroscopic and magnetic properties of Mn³⁺ fluorides, enabling the establishment of correlations between crystal and electronic structures, based on experimental results. This aims to clarify the behavior of these systems as a function of pressure and across the fluoride series. Focusing on CsMnF₄, our goal is to reconcile the theoretical predictions with experimental evidence and clarify the current understanding of this important material.

This commentary is organized in six different subsections within the Results and Discussions section, plus an appendix. The first subsection deals with the structure of CsMnF₄ and its pressure dependence; the second analyzes the electronic structure of Mn³⁺ fluorides by optical absorption spectroscopy; and the third focusses on optical band assignment, which is crucial for understanding magneto-optical correlations in these Mn³⁺ fluorides. Their magnetic properties are analyzed in the fourth subsection, and the high-spin to low-spin transition from optical spectroscopy is analyzed in the fifth. Finally, the sixth subsection is devoted to clarifying concepts from the JT theory that often lead to controversy—specifically, whether the strong low-symmetry local distortions around Mn³⁺ or Cu²⁺ in sixfold coordinated fluorides are associated with the JT effect and are therefore generally referred to as JT distortions. This analysis is supported by the JT theoretical model developed in Appendix A.

2. Results and Discussions

The main scientific concerns related to the theoretical work about the structural, electronic and magnetic properties of CsMnF₄ published elsewhere [1] are summarized in the six following sections.

2.1. Structure of CsMnF₄

The authors theoretically describe the structural evolution of CsMnF₄ in terms of a tetragonal structure P4/n (0–40 GPa), which experiences a structural phase transition at 37.5 GPa to another tetragonal phase (P4) that is stable up to at least 50 GPa. In this structural evolution, the ground state of Mn³⁺ is high spin (S = 2). The point is that there are three publications [2,3,4] dealing with the structural evolution of CsMnF₄ with pressure using X-ray diffraction (XRD), namely, energy-dispersive XRD at the Daresbury Synchrotron, UK [2,3], and angle dispersive XRD at the European Synchrotron Radiation Facility (ESRF), France [4]. These independent works concluded that the tetragonal structure P4/n of CsMnF₄ is unstable above 2 GPa, transitioning to a monoclinic structure stable up to at least 16 GPa [4] (Figure 1). Energy-dispersive XRD experiments were performed using ethanol-methanol-water mixture as pressure-transmitting medium with added Ag or NaCl powder as internal pressure standard [2,3], in contrast with the ESRF data using a fully hydrostatic pressure-transmitting medium like helium [4]. In the former experiment, it was concluded that a tetragonal to orthorhombic phase transition at 1.4 GPa followed by a monoclinic transition above 6 GPa (see Orthorhombic to Monoclinic section of Ref. [3]). In the latter, it is shown that under hydrostatic conditions, CsMnF₄ directly transits to a monoclinic structure at 2 GPa. None of these publications were discussed in [1] in relation to the experimentally observed structural evolution—Refs. [2,4] were not cited– and therefore, this low-pressure structural transition calls into question all results derived from a tetragonal structure above 2 GPa. Although Ref. [3] was cited in the article, it literally states that “No signs of a structural phase transition around 1.4 GPa, early suggested by Moron et al. have been found in the optical measurements on CsMnF₄” (page 13233 in Ref. [1]). Besides the recognition of a structural phase transition around 1.4 GPa earlier reported by Moron et al. [3], the authors ignored two other publications [2,4] not cited in Ref. [1], which confirm the tetragonal to monoclinic structural phase transition (around 2 GPa in Ref. [4]). In addition to the XRD results, Ref. [4] also reports precise optical data in the 0–16 GPa range, which provides clear evidence of the influence of the phase transition at 2 GPa in the optical spectra of CsMnF₄ (Figure 1).

The measured spectra show that the pressure shift rate of the ⁵B₁ ⟶ ⁵A₁ transition energy at 1.89 eV changes from −42 meV/GPa to +2 meV/GPa at 2 GPa, coinciding with the tetragonal to monoclinic phase transition [4] (Figure 1). It must be emphasized that Refs. [2,3,4] are the only studies dealing with the crystal structure of CsMnF₄ under high pressure conditions, yet they were not considered or cited [2,3,4] in the reported work [1], although one of its authors cited Ref. [4] in a previous publication [6]. Therefore, the structural phase transition detected by XRD by two different research groups at two different synchrotron facilities, between 1–2 GPa, was neither considered in the article [1] nor detected by using the authors’ DFT methodology. Importantly, this structural phase transition indicates that the tetragonal structure of CsMnF₄ is no longer stable above 2 GPa, transitioning to a monoclinic structure that persists at least up to the maximum pressure reached in XRD experiments. Furthermore, the structural evolution of CsMnF₄ with increasing pressure reported in Ref. [4] aligns with the structural evolution across the AMnF₄ (A: Cs, Rb, Tl, K, Na) series with decreasing the unit cell volume [2,3,4,5], as shown in Figure 2 and Figure 3. As discussed later, the absence of the CsMnF₄ monoclinic structure for pressures above 2 GPa implies missing relevant structural and electronic variations under pressure.

2.2. Pressure Dependence of the Optical and Electronic Properties

The pressure dependence of the optical and electronic properties derived theoretically from the tetragonal phase [1] are unable to explain the experimental results by optical spectroscopy published previously [4,7,8,9]. CsMnF₄ in the tetragonal phase is a uniaxial crystal instead of a biaxial crystal attained in the monoclinic structure (p > 2 GPa). Furthermore, the structure of the Mn³⁺ one-electron d-orbitals obtained theoretically [1], predicts that the energy of the first absorption band associated with the ⁵B₁ ⟶ ⁵A₁ (denoted by the one-electron orbital transition 3y² − r² ⟶ x² − z² in Figure 4 of Ref. [1]), decreases continuously from 1.92 eV to 1.43 eV in the 0–40 GPa range, while experimentally it changes from 1.89 to 1.81 eV in the 0–2 GPa range (tetragonal phase) but blueshifts by only 0.03 eV in the 2–16 GPa range (monoclinic phase) [4,7] giving a total shift of +0.07 eV at 37 GPa [4,8]. It means that the band shifts from 1.89 eV at ambient pressure to 1.88 eV at 37 GPa, which is two orders of magnitude lower than the shift predicted theoretically of −0.49 eV in the same pressure range (Figure 1). Neither the magnitude of the pressure band shift, nor the two opposite pressure rates below and above 2 GPa were accounted for theoretically [1]. It should be noted that the theoretically predicted pressure shift rate for this band is −38 meV/GPa from in the 0–10 GPa (using a CRYSTAL + ADF code [1]), which aligns well with the experimental shift rate of −42 meV/GPa in the tetragonal phase. However, the transition to the monoclinic structure completely reverses this trend, changing the shift rate from −42 meV/GPa to +0.07 meV/GPa. This discrepancy highlights that failing to account for the monoclinic structure above 2 GPa results in erroneous predictions—both in magnitude and sign—of the pressure-induced shift rate when compared with experimental observations.

Refs. [4,8] were not cited in Ref. [1], and their experimental results could have guided the authors to determine the actual structural evolution of CsMnF₄ and its associated properties. It must be noted that the pressure-induced blueshift of the first absorption band experimentally observed in the CsMnF₄ monoclinic phase above 2 GPa is also observed in NaMnF₄ in the 0–6 GPa range, which has a monoclinic structure at ambient conditions [8,9] (Figure 3). The preservation of the tetragonal structure across the entire explored pressure range [1] appears to be unable to explain the observed optical properties.

2.3. Optical Band Assignment of Mn³⁺ Fluorides

The band assignment of the crystal-field spectra related to d-d transitions in CsMnF₄, as well as the high-spin (HS) to low-spin (LS) transition detected by optical spectroscopy at 37 GPa reported elsewhere [7] (Figure 4)—the spectra of which are reproduced in Figure 4 of Ref. [1]—is questioned in the article [1]. Although no alternative interpretation to the assignment of Ref. [7] is provided in the article [1], it must be noted that the optical absorption band assignment of Mn³⁺ fluorides is based on spectroscopic results reported elsewhere [9]. This work, which is not cited in Ref. [1], involves single-crystal polarized optical absorption spectroscopy and the temperature dependence of the various bands. Based on their transition energy, spectral shape, oscillator strength, and temperature dependence (Figure 5, Figure 6 and Figure 7), it was concluded that two types of bands exist in the optical spectra, namely, three broad bands associated with single-electron transitions within the d-orbitals of Mn³⁺ in an almost D_4h local environment, and two narrow spin–flip peaks arising from the 3d⁴ electronic configuration of Mn³⁺ (Figure 5). It is experimentally shown that the former ones are electric dipole spin-allowed transitions, whose oscillator strength is increased with increasing temperature by coupling to odd parity vibrations within the (MnF₆)³⁻ complex. The spin–flip transitions are shown to be electric dipole spin-forbidden transitions within a single Mn³⁺ ion, but are activated by the spin-effective mechanism in exchange-coupled systems via the Mn-F-Mn superexchange pathway [9] (Figure 7). The temperature dependence of their oscillator strength demonstrates the spin–flip nature of these narrow peaks observed in the optical spectra of exchange-coupled systems [7,8,9]. In addition, these spin–flip transitions can be easily identified in the optical spectra because, unlike the spin-allowed broad bands whose energies are strongly dependent on the (MnF₆)³⁻ octahedral distortion, their energy is not as sensitive to the distortion [7,8,9] (Figure 5). Spin–flip transitions are within 0.05 eV at the same position in all series of manganese (III) fluorides, namely, E_SP1 = 2.380 eV and E_SP2 = 2.873 eV in 2D CsMnF₄ [7], 2.397 and 2.890 eV in 2D NaMnF₄ [8,9], 2.380 and 2.860 eV in 2D TlMnF₄ (2D fluorides) [8], and 2.418 and 2.914 eV in the 1D Tl₂MnF₅.H₂O [9].

In this respect, the d-orbital electronic structure derived theoretically for CsMnF₄ in the article [1] does not agree with the measured spectra [7] (Figure 4), which are reproduced in Figure 4 of Ref. [1]. The spectra show three broad bands associated with single-electron transitions from 3z² -r², xy, and the nearly degenerate (xz, yz) to x²-y², in order of increasing energy (note that the local z-axis (7–9) in Ref. [7] refers to the long Mn-F bond in CsMnF₄ of the nearly D_4h (MnF₆)³⁻ complex). In the article [1], the xy, xz, and yz d-orbitals have different energies with respect to the x²-y² level, namely, 2.2, 2.5 and 2.8 eV (Figure 7 of Ref. [1]). Except for the first absorption band 3z²-r² ⟶ x²-y² at about 1.9 eV, the other three transition energies cannot give rise to a two-broad-band structure as observed experimentally (Figure 5), but rather to three almost equally spaced broad bands. This would likely produce a structureless broad band instead of the two well-resolved bands observed experimentally [7]. Furthermore, the calculations performed in the article deal with one-electron d-orbitals and not with the states arising from the d⁴ electronic configuration; thus, the observed spin–flip transitions are missed in the reported calculations [1]. This is a significant point, as the authors should recognize that the spin–flip transitions observed in the spectra of CsMnF₄ (Refs. [4,7]) are crucial for explaining the spin transition at 37 GPa on the basis of optical spectroscopy [7] (Figure 4 and Figure 5).

$$f\left(T\right)={\displaystyle \frac{2f\left(0\right)}{(2S-1)}}\left[S\left(U^2-1\right)+{\displaystyle \frac{4SU\left(S+1\right)UV}{2V}}\right]$$

(1)

where f(0) is the 0 K oscillator strength, U = cothV − 1/V, V = 2JS(S + 1)/k_BT, S = 2 and J (<0) is the intrachain exchange constant [9] (left side). We also see the temperature dependence of the oscillator strength for the ⁵B_1g ⟶ ⁵A_1g broad band (1.45 eV) in π and σ polarizations. Solid lines are fits to $f\left(T\right)=f\left(0\right) \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\hslash {\omega }_u/{2k}_{B}T\right)$ (vibronic mechanism, parity forbidden electric-dipole transitions activated by odd parity vibrations ${\omega }_u$). The distinct temperature dependences and polarization reveal different activation mechanisms: exchange for spin–flip peaks and vibronic for spin-allowed broad bands (right side). Adapted with permission from Ref. [9]. Copyright 1994 American Chemical Society.

2.4. Magnetic Properties of CsMnF₄

Regarding the magnetic properties, the theoretical work finds that CsMnF₄ undergoes an abrupt change from ferromagnetic to antiferromagnetic at 10 GPa within the tetragonal phase [1]. However, this conflicts with the magnetic measurements under high pressure reported elsewhere [11]. That paper describes singular magnetic behavior above 2 GPa, including a reduction in the Curie temperature and a sharp decrease in the saturation magnetization of CsMnF₄ above 3.3 GPa. The authors explain this in terms of a progressive transition from ferromagnetic to antiferromagnetic with a permanent ferromagnetic component, which the authors attribute to canted antiferromagnetism [11]. This behavior is overlooked in the article [1]; consequently, properties observed in the monoclinic phase are not predicted for the tetragonal phase. In fact, this behavior has been explained across the AMnF₄ series of layered perovskites, where the only tetragonal member is the ferromagnetic CsMnF₄; the remaining compounds with a monoclinic structure are all antiferromagnetic [2,5,11,12]. Moreover, Refs. [2,3] show that the AFM Neel temperature correlates with the Mn-F-Mn tilting angle in the monoclinic phase—a relationship that cannot be reproduced in the same way using a tetragonal phase due to structural constrains.

2.5. High-Spin to Low-Spin Transition at 37 GPa

The change in the optical spectra (Figure 4 of Ref. [1]; or Figure 2 of Ref. [7]) associated with a HS ⟶ LS crossover transition at 37 GPa and room temperature in the original paper [7] (Figure 4) is now questioned and attributed to a tetragonal-to-tetragonal phase transition P4/n ⟶ P4 at 37.5 GPa, with both phases being HS (S = 2). However, this interpretation fails to account for the spectral evolution at the crossover point. There are three main reasons for retaining the HS-to-LS transition assignment, as follows: (1) The three broad bands observed below 30 GPa, associated with the strong octahedral distortion due mainly to the E ⊗ e JT effect, transform into a structureless broad band characteristic of a nearly octahedral coordination. The T ⊗ e JT effect in the LS ³T₁ ground state (O_h) is a factor of 4 weaker than the strongly E ⊗ e JT-distorted HS ⁵E ground state [7,13]; its stabilization energy in the LS state is an order of magnitude smaller. (2) The spin–flip transitions E_SP1 and E_SP2, characteristic of a HS ground state, disappear completely above 37 GPa. According to the Tanabe–Sugano diagrams for d⁴ ions [13,14], these transitions are absent in a LS ground state (³T₁), making them a precise spectroscopic probe for layered AMnF₄ (A: Cs, Na, Tl) perovskites in HS [7,9] (Figure 4, Figure 5 and Figure 6). (3) The LS broad band peaking at 2.5 eV (Figure 4) coincides with the centroid of the spin-allowed crystal-field bands at the spin crossover (S = 2 ⟶ 1), corresponding to a crystal field splitting 10Dq of 27B (C/B = 4.6), where B and C are the Racah parameters for (MnF₆)³⁻ [7,13,14]. On the other hand, the argument given in Ref. [1] to support a structural transition at 37 GPa instead of a HS ⟶ LS crossover is not properly justified. The authors state that the 10Dq required to stabilize the LS state should be higher than 3 eV, without any further justification. The value of 10Dq derived from the Tanabe–Sugano diagram at the transition point requires accurate knowledge of the Racah parameter B, which is known to be B = 0.097 eV (780 cm⁻¹) in (MnF₆)³⁻ at ambient conditions [10]. This value can also be obtained from the spin–flip transition energies, as the first observed spin–flip transition depends on pressure as E_SP2 = 2.397 − 0.0018 P (in eV and GPa, respectively) [7,9]. From the Tanabe–Sugano diagram [13,14] (Figure 4), we see that its energy varies as 24.6 B in the HS state up to the spin crossover point. This indicates that B slightly decreases from 0.097 eV at ambient pressure to about 0.094 eV at 37 GPa, considered the spin crossover pressure [7]. Consequently, 10Dq should be about 2.54 eV (with an estimated accuracy of 0.04 eV) lower than 3 eV. Considering that the single-electron t_2g⁴ e_g⁰ ⟶ t_2g³ e_g¹ transition gives rise to several spin-allowed transitions within the d⁴ electronic configuration—from the LS ³T₁ ground state to the ³Γ_i excited states, whose energies spread over the 0.4 eV range (2.39 eV (³T₁), 2.48 eV (³T₂), 2.65 eV (³A₂), and 2.76 eV (³A₂)) [13,14]—and all these transitions produce a broad band whose centroid peaks at about 2.54 eV. This aligns with the observed spectra of CsMnF₄ above 37 GPa in the LS state [7] (Figure 4). (4) The argument given to rule out a spin transition based on the enthalpy difference between HS and LS states in tetragonal symmetry is not supported by any information or evidence, even in the Supporting Information [1]. It is well known that the process of estimating a spin crossover pressure from ab initio calculations is subtle, requiring accurate knowledge of the actual HS and LS structures at the crossover; such calculations are highly sensitive to the choice of the functional and other parameters. A thorough theoretical study of the HS–LS transition pressure in these systems is needed for a consistent analysis [15,16]. Therefore, it is scientifically unsound to state that the enthalpies for LS and HS differ by a few tenths of eV based on an inadequate HS structure and an unknown LS structure, without providing calculation details. Optical spectroscopy offers a more robust means of identifying spin crossover phenomena, as it effectively distinguishes the distinct spectra of each spin state regardless of the specific crystal structures involved, as well-established for Fe²⁺ compounds [17,18,19].

2.6. The Jahn–Teller Effect: Theorem, Theory, and Molecular Distortion

Finally, some concepts related to the JT effect must be clarified in order to avoid misunderstandings among scientists working with systems exhibiting JT distortions. This arises from sentences (i–iii) on page 13234 of Ref. [1], which state the following: (i) “The existence of a JT effect requires a degenerate electronic state in the initial geometry”, (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 MnF₆³⁻ unit perpendicular to the layer plane plays a singular role, a fact seemingly not consistent with the JT assumption.”, and (iii) “The local equilibrium geometry for MnF₆³⁻ 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.”

Due to these statements, it is necessary to distinguish between the JT theorem, the JT theory, and the JT distortion. The JT theorem states that a nonlinear polyatomic molecule with an orbital degenerate ground state is unstable and must distort to a lower energy configuration, or literally, “All nonlinear configurations are therefore unstable for an orbital degenerate electronic state” [20]. Based on this theorem, JT theory explores molecular distortions under different structural perturbations. For highly distorted E ⊗ e JT systems such as octahedral MX₆ complexes (M: Cr²⁺, Cu²⁺, Mn³⁺, and X: Cl⁻, F⁻), most of which exhibit elongated octahedral distortions [21,22,23,24,25], the JT theory shows that the equilibrium geometry caused by a tetragonal symmetry perturbation of the octahedron corresponds to the JT-distorted geometries (three degenerate minima associated with tetragonally elongated octahedra along the x, y and z axes) slightly modified by the axial perturbation [21,22,23,24,25] (Appendix A). JT theory demonstrates that the equilibrium coordination geometry of the JT ion varies from elongated tetragonal along a tensile perturbation direction to two equivalent minima corresponding to the JT-elongated octahedra with a slight orthorhombic distortion for a compressive D_4h site perturbation, to a tetragonally compressed geometry if the compressive perturbation is sufficiently large (Appendix A). Typically, a compressive D_4h perturbation results in two equivalent elongated octahedra associated with the two JT-distorted geometries whose long axes are perpendicular to the D_4h site perturbation axis. The two short M-X bonds are generally different; their difference, which is zero for an O_h site (three equivalent D_4h elongated geometries), increases with the D_4h compressive perturbation, yielding local JT distortions of D_2h symmetry. Thus, E ⊗ e JT theory shows that slight tetragonal distortions of the initial octahedral symmetry lead to equilibrium geometries corresponding to the JT distortions obtained for O_h slightly perturbed by the D_4h compressive strain. Therefore, JT theory indicates that equilibrium geometries of the MX₆ complex can correspond to JT distortions when the JT ion is placed in a D_4h-compressed, near-O_h symmetry. This results in two out of three JT-equivalent D_2h elongated equilibrium geometries, with their elongated M-X bonds perpendicular to each other and both perpendicular to the D_4h axis of the compressive perturbation, despite an initial non-degenerate ground state [22,23,24,25].

This implies a reduction in local symmetry relative to the D_4h symmetry of the site, although the average of the two orthorhombic local symmetries retains the site’s D_4h symmetry (Appendix A). This result contradicts the authors’ assertion that the local symmetry of Mn³⁺ or Cu²⁺ in a compressed D_4h symmetry must remain unchanged, contrary to JT theory. Other structural perturbations (e.g., non-centrosymmetric strains) may be present but do not affect the JT distortion, as such distortion modes are not coupled to the parent octahedral degenerate electronic e_g levels in first-order perturbation.

3. Conclusions

In conclusion, JT theory shows that JT ions placed in O_h-perturbed low-symmetry sites, where electronic degeneracy is lifted, exhibit JT distortions different from the initial site symmetry. This makes the exclusion of such distortions as non-JT distortions questionable. The authors in Ref. [1] conclude that reasons for not considering low-symmetry complex formation in Cu²⁺ and Mn³⁺ as a JT effect include that “They are also behind the so-called plasticity property of compounds of Cu²⁺ and Mn³⁺ ions”, citing Ref. [26] herein. In this context, the term “plasticity” appears euphemistic, as Ref. [26], Chapter E, is entitled “The Jahn–Teller and pseudo-Jahn–Teller origin of the plasticity of Cu^II coordination sphere and distortion isomerism”.

Finally, it is crucial to emphasize that translating the behavior of a complex like (MnF₆)³⁻ within any fluoride lattice, such as CsMnF₄, is plausible. This is because the phenomenon is inherently local, and the JT model parameters can effectively mimic the real structural conditions of the complex within the lattice—including crystal anisotropy, cooperative effects, and electron-lattice coupling. This approach, often called the cluster model, significantly simplifies the problem and provides a general framework for understanding distortions induced by the JT or pseudo-JT effect [27].