Understanding the Structure and Ground State of the Prototype CuF2

Article Cite This: Inorg. Chem. XXXX, XXX, XXX−XXX

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Understanding the Structure and Ground State of the Prototype CuF2 Compound Not Due to the Jahn−Teller Effect Jose ́ Antonio Aramburu* and Miguel Moreno

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Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain ABSTRACT: Insulating CuF2 is considered a prototype compound displaying a Jahn−Teller effect (JTE) which gives rise to elongated CuF64− units. By means of first-principles calculations together with an analysis of experimental data of both CuF2 and Cu2+-doped ZnF2, we demonstrate that such an idea is not correct. For ZnF2:Cu2+, we find that CuF64− units are compressed always along the Z local axis with a hole essentially in a 3z2−r2 antibonding orbital, in agreement with experimental EPR data that already underline the absence of a JTE. The structure of the monoclinic CuF2 crystal also comes from compressed CuF64− complexes, although hidden by an additional orthorhombic instability due to a negative force constant of b2g and b3g local modes. The associated distortion, similar to that involved in K2CuF4 and other layered Cu2+ compounds, is also shown to be developed for ZnF2:Cu2+ upon increasing the copper concentration. The origin of this cooperative effect is discussed together with the differences between non-Jahn−Teller systems like ZnF2:Cu2+ and CuF2 and true Jahn−Teller systems like KZnF3:Cu2+. From present results and those on layered compounds, the usual assumption of a JTE for explaining the properties of d9 ions in low-symmetry lattices can hardly be right.

1. INTRODUCTION In the last decades transition metal compounds containing d9 or d4 ions have attracted a great deal of interest helped by the appearance of phenomena like colossal magnetoresistance or superconductivity. Such systems exhibit a particular interplay between geometrical and electronic structures which needs to be unveiled for a deep understanding of their associated magnetic, optical, and electrical properties. The present work is focused on the actual origin of the structure displayed by CuF2, a compound that due to its relative simplicity is considered as a model system1−4 in the realm of insulating compounds containing Cu2+ or Ag2+ ions. In addition, special attention is currently paid to CuF2 as it is considered a promising cathode for lithium and fluoride ion batteries.5−7 At ambient pressure, the CuF2 lattice, involving 3D linked CuF64− units, is monoclinic associated with the standard P21/c space group8 (Figure 1). Alternatively, that structure is also described by the nonstandard P21/n group (Figure 1) which shows that CuF2 can also be viewed as a rutile structure (Figure 2) although distorted9. The {X, Y, Z} local axes considered for each CuF64− unit in CuF2 in Figure 1 are the natural ones in the tetragonal rutile structure of ZnF2 (space group P42/mnm) displayed Figure 2. The experimental values of three Cu2+−F− distances (Figure 1), equal to RZ = 1.917 Å, RX = 1.932 Å, and RY = 2.298 Å, imply that in a first view each CuF64− unit in CuF2 can be described as an octahedron elongated along the local Y axis, displaying a nearly tetragonal symmetry, as RX and RZ differ only by 0.015 Å. Accordingly, the ground state of CuF64− © XXXX American Chemical Society

complexes in CuF2 would involve a hole in an orbital with strong x2−z2 character. The driving force responsible for such a local geometry in CuF2 has systematically been attributed by different authors1−3,8−16 to the static Jahn−Teller effect (JTE). Even more, CuF2 is currently taken as a prototype1−13 among compounds displaying a JTE. This idea is likely based on the equilibrium geometry usually observed for d9 ions in cubic lattices, where a static JTE actually takes place.17−19 Indeed, in cases like KZnF3:Cu2+ or NaCl:M (M = Ni+, Ag2+, Cu2+) the local equilibrium geometry of MX6 units (X = F−, Cl−) corresponds to a clearly elongated octahedron involving a local tetragonal symmetry.20−27 Although the structural data on CuF2 seem, in a first view, to support the existence of a JTE the main goal of this work is just to prove that such an idea is not correct. On the contrary, we want to show that other mechanisms different from the JTE are responsible for both the local structure and ground state of CuF64− units in the CuF2 compound. Such mechanisms have common points with those previously discussed for layered systems 2 8 , 2 9 like K 2 CuF 4 , (C n H 2 n +1 NH 3 ) 2 CuCl 4 or K2ZnF4:Cu2+. Seeking to clarify this relevant matter the present study is based on three cornerstones: (i) The results of first-principles calculations that thus exclude the use of any model involving fitting parameters.11,12 (ii) The analysis of available experimental data. (iii) Previous studies28,29 on systems displaying a Received: January 18, 2019

A

DOI: 10.1021/acs.inorgchem.9b00178 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry

Figure 1. Structure of CuF2 in (a) the standard monoclinic P21/c space group and the (b) nonstandard P21/n group. The local axes {X, Y, Z} considered for each CuF64− unit are the natural ones in the tetragonal rutile structure of ZnF2 (space group P42/mnm) displayed Figure 2. The values of metal−ligand distances (in Å) are included.

the present calculations on Cu2+-doped ZnF2 show that this system is close to a mechanical instability, which is however developed upon increasing the copper content. This relevant fact helps to understand the actual origin of the structure displayed by CuF2. The present work is organized as follows. Section 2 provides information on the employed computational tools. For the sake of clarity, the structures of both ZnF2 and CuF2 compounds are described in section 3, together with a critical assessment on the assumption of a JTE in CuF2. Experimental and theoretical results on ZnF2:Cu2+ are shown and discussed in section 4, while section 5 is focused on the origin of the CuF2 structure. Some final remarks are provided in the last section.

2. COMPUTATIONAL DETAILS First-principles periodic geometry optimizations on CuF2 and ZnF2 compounds have been performed under the framework of the Density Functional Theory (DFT). Calculations were carried out by means of the CRYSTAL17 code,32 where the Bloch wave functions are represented by a linear combination of Gaussian basis functions centered at the atomic positions. All ions were described by means of basis-sets taken directly from CRYSTAL’s webpage.32 In particular, we have used the all-electron triple-ζ plus polarization (TZP) basis recently developed33 for Peitinger et al. We have also employed the B1WC hybrid exchange-correlation functional (including 16% of Hartree−Fock exchange) that has shown to be able to reproduce with great accuracy the geometry and properties of a large number of both pure and doped crystals.34 Similar results have been found using other basis sets and the PW1PW hybrid functional (including 20% of Hartree−Fock exchange).35 In order to consider the antiferromagnetic structure8 of CuF2, calculations were performed on a 2 × 1 × 1 conventional supercell containing 4 unit formulas and all optimized geometries agree with the experimental values within 3% error. Additional calculations considering ferromagnetic order have also been carried out, but the changes in the optimized

Figure 2. Structure of ZnF2, with tetragonal P42/mnm space group. {X, Y, Z} are the local axes of each tetragonal ZnF64− unit and {X′, Y′, Z′ = Z} the principal axes corresponding to a D2h local symmetry. The values of metal−ligand distances (in Å) are included.

true JTE as well as on pure and doped compounds whose structure and ground state is not the result of a JTE. For clearing up the present questions on CuF2, a parallel study on Cu2+-doped ZnF2 is also performed as this lattice exhibits a rutile structure30 (Figure 2), which is the parent structure of CuF2. In this sense, a detailed electron paramagnetic resonance (EPR) investigation31 on ZnF2:Cu2+, carried out by Swalen et al., clearly reveals that the actual nature of the ground state of CuF64− units in ZnF2 also sheds light on the corresponding ground state in CuF2. Moreover, B


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Inorganic Chemistry

Interestingly, the results of present calculations on ZnF2 reproduce well the experimental values of lattice parameters, Zn2+−F− distances, and the ϕ angle (Table 1). Indeed, the largest deviation with respect to the experimental value is obtained for the c axis being equal only to 1.25%. As shown in Figure 1b, CuF2 can be described as a distorted rutile structure.9 In comparison with ZnF2 the distortion taking place in CuF2 makes that RY is clearly higher than RX; thus, X and Y directions are no longer equivalent. Additionally, this local distortion makes that the lattice is no longer tetragonal and becomes monoclinic. The results of present calculations on CuF2 considering the standard P21/c space group (Figure 1a) are reported in Table 2 and compared to experimental findings.8 The present calculated values are close to experimental ones. The largest deviation is found for RY though it is equal only to 1.4%. Therefore, when comparing CuF64− units in CuF2 (Table 2) with ZnF64− ones in ZnF2 (Table 1) there is a distortion in the former case making that RY > RX, while RX and RZ become near identical as RX − RZ = 0.015 Å. For this reason, CuF64− units in CuF2 appear to display an elongated octahedron with an unpaired electron residing essentially in a x2−z2 orbital. The distortion around Cu2+ ions in CuF2 has widely been ascribed1−3,8−16 to the JTE based on results for d9 ions in cubic lattices under octahedral coordination.17−19 Indeed, in these cases a static JTE usually leads to an elongated octahedron displaying a strict tetragonal geometry. The systems KZnF3:Cu2+, KCl:Ag2+, or CsCaF3:Ni+ are good examples of that behavior.20−22,25,26,42 By contrast, up to now the only known exception to that pattern corresponds to CaO:Ni+ where the octahedron surrounding Ni+ is actually compressed.43 The origin of this surprising fact has recently been explained.44 Although the experimental Cu2+−F− distances in CuF2 seem to correspond to an elongated octahedron driven by a static JTE, that interpretation raises however several doubts. In fact, a distortion driven by a JTE actually requires the existence of an initial electronic and vibrational degeneracy such as that which happens for d9 ions in cubic lattices with octahedral coordination.17,18 However, that initial degeneracy is, in principle, broken if the starting point does not involve a cubic but tetragonal symmetry. Moreover, the experimental Cu2+−F− distances and ϕ angle in CuF2 (Table 2) do not describe a strict tetragonal symmetry around Cu2+, expected for systems where the E⊗e JTE takes place.17,18 It has commonly been argued11,45,46 that in noncubic host lattices involving practically octahedral MX64− units (M = closed shell dication, like Mg2+, Zn2+ ,or Cd2+; X = F−, Cl−) there is always a JTE when M2+ is replaced by Cu2+. However, this assertion only makes sense if the effects of the noncubic lattice are a perturbation of the JTE. This means that the energy splitting induced by the noncubic host lattice on valence levels has to be smaller than the energy barrier, B, separating two equivalent minima in a JTE.29 Although no

geometry are only less than 0.3%. Along this line, the energy per unit formula is found to increase only by about 20 meV with respect to the antiferromagnetic structure. Calculations on Cu2+-doped ZnF2, where a Cu2+ impurity enters replacing a Zn2+ ion of the lattice, were performed with CRYSTAL17 code in the P42/mnm space group using periodic 2 × 2 × 2 and 2 × 2 × 3 conventional supercells containing, respectively, 48 and 72 atoms. The results were very similar to both supercells. For each crystal structure considered in this work, we have calculated the electrostatic potential VR(r) = Σj qj/|Rj − r| felt by the electrons localized in a MF64− (M = Cu2+, Zn2+) complex due to all lattice ions lying outside the complex, with charges qj and positions Rj taken from our first-principles optimizations. Although often ignored, the VR(r) potential is the actual responsible for the ground state36 of K2ZnF4:Cu2+ or the different color displayed by ruby, emerald, or alexandrite gemstones37 due to Cr3+ impurities. Calculation of VR(r) potentials has been performed following the procedure developed by van Gool and Piken38 based on a modified version of the Ewald’s method39 in order to accelerate the convergence. It is worth noting however that the shape and anisotropy displayed by VR(R) is greatly due to few ions lying closer to the complex40,41,36 whose contribution to the VR(R) potential can be calculated by hand.

3. STRUCTURES OF ZnF2 AND CuF2: USUAL INTERPRETATION OF THE CUF2 STRUCTURE The pure compound ZnF2 has a rutile structure (Figure 2) belonging to the tetragonal P42/mnm space group.30 As shown in Figure 2, there are two Zn2+ ions in the unit cell. Nonetheless, due to the existence of a 42 helicoidal axis the two ZnF64− units are physically equivalent although rotated by 90° around the crystal c axis. The experimental values of lattice parameters and Zn2+−F− distances are reported in Figure 2 and Table 1 and compared to results of present calculations. Table 1. Calculated a and c Lattice Parameters, Zn2+−F− Distances, and ϕ Angle for Pure ZnF2a experimental calculated

a

c

RX = RY

RZ

ϕ

4.707 4.668

3.128 3.167

2.040 2.043

2.019 2.009

79.7 78.8

a P42/mnm space group. Compared to experimental findings of ref 30. All distances are given in Å, and the ϕ angles are given in degrees.

Although RX = RY, the value of RZ is slightly smaller. Moreover, the local X and Y directions in Figure 2 are not perpendicular, as they form an angle ϕ = 79.7°. As this angle differs from 90°, the local symmetry around Zn2+ does not exhibit D4h but D2h group whose principal directions are described by {X′, Y′, Z′ = Z} in Figure 2. Accordingly, while the directions X and Y are physically equivalent, the Z direction is inequivalent to the previous ones.

Table 2. Calculated a, b, c, and β Lattice Parameters, and Cu2+−F− Distances for Pure CuF2 in the Monoclinic Standard P21/c Space Groupa experimental calculated

a

b

c

β

RY

RX

RZ

ϕ

3.294 3.284

4.568 4.552

5.358 5.367

121.2 121.8

2.298 2.265

1.932 1.936

1.917 1.918

78.1 77.6

Compared to experimental findings of ref 8. All distances are given in Å, and β and ϕ angles are given in degrees.

a

C


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Inorganic Chemistry

As shown in Tables 1 and 3 the substitution of a Zn2+ ion of the ZnF2 lattice by a Cu2+ impurity leads to an equilibrium situation that preserves the local D2h symmetry although the values of metal−ligand distances are somewhat modified. Indeed, the Zn2+ → Cu2+ substitution makes RZ decrease from 2.02 to 1.92 Å, while RX = RY distances grow slightly from 2.04 to 2.07 Å. However, these changes do not imply a breaking of local symmetry as it remains D2h in the whole vibronic relaxation process, involving an energy gain of 140 meV. We have verified that the relaxation obtained for ligands along the Z axis being bigger than that for ligands in the perpendicular plane is helped by different force constants. Indeed, the motion of F− ligands lying along X or Y axes is found to involve a force constant 40% higher than that of ligands in the Z axis. This situation is thus somewhat akin to that derived36 for divalent impurities in the layered perovskite K2ZnF4. Thus, the equilibrium geometry derived from present calculations for a Cu2+ impurity in ZnF2 corresponds to a CuF64− octahedron compressed along the Z axis. Concerning the associated ground state, it involves a hole with a strong 3z2−r2 character (98%) although not equal to 100%, because ϕ ≈ 78°, thus involving a small admixture of the x2−y2 orbital. This fact underlines the existence of a small orthorhombicity despite the fact that RX = RY. As a salient feature, no other equivalent distortions have been found, thus implying that all Cu2+ impurities should give rise to compressed octahedra whose short axis is always the Z axis of the complex (Figure 2). The present calculations are consistent with the analysis of EPR spectra of ZnF2:Cu2+ by Swalen et al. demonstrating the formation of compressed CuF64− octahedra with a short axis which is always the local Z axis.31 This idea is underpinned by the experimental g tensor31 (gZ′ = 2.06, gX́ = 2.43, and gÝ = 2.35) consistent with a compressed complex along the Z′ = Z axis and a small rhombic distortion in accord with ϕ ≠ 90°. Moreover, the highest superhyperfine constant, corresponding to 19F nuclei of ligand ions, is obtained for the two ligands along the Z axis when the magnetic field, H, is parallel to that axis. This constant, which is more than three times higher than that due to the four ligands along X and Y axes,31 is thus consistent with an unpaired electron with a strong 3z2−r2 character that is around 95% from the analysis by Swalen et al. This view is also supported looking at the experimental hyperfine tensor of ZnF2:Cu2+ reflecting the interaction of the unpaired electron with copper nucleus. Indeed, under D4h symmetry with Z as principal axis and low covalency AZ should be negative49,50 if the hole lies in the antibonding b1g(x2−y2) orbital whereas it becomes positive49,51,52 with a hole in a1g(3z2−r2). The experimental value31 AZ = 51 × 10−4 cm−1 obtained for ZnF2:Cu2+ is thus consistent with a hole in an orbital with a dominant 3z2−r2 character. A positive AZ value has been determined for the tetragonal CuCl4(NH3)22− unit in NH4Cl51−53 where the unpaired electron lies in the 3z2−r2 orbital. In the same vein, it has recently been proved54 that the hole for Cu2+-doped Tutton salts has a strong 3z2−r2 character consistent with AZ = 60 × 10−4 cm−1 measured by Hoffmann et al.55 in K2Zn(H2O)6(SO4)2:Cu2+. Therefore, the EPR data on ZnF2:Cu2+ show the absence of three equivalent distortions that are the footprint of a static JTE. Such three distortions are however well observed for d9 ions (Cu2+, Ag2+, Ni+) in perfect cubic crystals.17−21,23−25,27,42−44 It is worth noting now that Swalen et al. in 1970 already pointed out that the properties of the Cu2+-doped ZnF2 system can hardly be explained invoking the JTE due to the tetragonal P42/mnm symmetry of the host

direct measurements on the barrier B characteristic of Jahn− Teller systems have been reported, valuable information on this crucial matter is however provided by results of first principle calculations.18,24,26,29,44,47 Such results give B values in the range of 1−10 meV for Cu2+ and Ag2+ impurities in hard lattices18,29,47 like MgO, CaO, or KZnF3, while values up to 100 meV are found for softer lattices like KCl.26,47 In particular, the very low value B = 8 meV found29 for the true Jahn−Teller system KZnF3:Cu2+ explains why in tetragonal lattices, like K2ZnF4 or Ba2ZnF6 doped with Cu2+ as well as in the pure compound K2CuF4, the JTE is actually absent.28,29,36,48 Indeed the internal electric field, ER(r), created by ions lying outside the CuF64− unit and displaying a tetragonal symmetry, breaks the degeneracy of the eg level producing a gap between the x2−y2 and 3z2−r2 antibonding orbitals around 350 meV even if the initial geometry of the complex is octahedral.29,36,22 This gap forces the hole of all CuF64− units in doped K2ZnF4 or Ba2ZnF6 lattices to be in the same 3z2−r2 orbital, thus destroying the existence of three equivalent distortions, characteristic of a JTE. Bearing these facts in mind, it is necessary to explore the behavior of the Cu2+ impurity in the tetragonal ZnF2 lattice as a first step to understand both the actual origin of the CuF2 structure and the nature of the ground state of involved CuF64− units. That study is carried out in the next section.

4. LOCAL STRUCTURE AND GROUND STATE FOR Cu2+-DOPED ZnF2 4.1. Results for the Isolated Impurity. Particular attention is paid in this section to compare the ground state and local structure of CuF64− units derived from first-principles calculations with the conclusions reached by Swalen et al. from the analysis of EPR spectra31 obtained for ZnF2:Cu2+. The behavior of a Cu2+ impurity in ZnF2 has been simulated by means of 2 × 2 × 2 and 2 × 2 × 3 periodic supercells where only one Zn2+ ion has been substituted by Cu2+. Thus, in a 2 × 2 × 2 supercell, involving 16 unit formulas, the smallest Cu2+− Cu2+ distance amounts to 6.26 Å, while in a 2 × 2 × 3 supercell, with 24 unit formulas, that distance becomes equal to 9.40 Å. The values of equilibrium lattice parameters, Cu2+− F− distances, and ϕ angle for a Cu2+ impurity in ZnF2 using both supercells are collected in Table 3, where the nature of the ground state of CuF64− units is also reported. Table 3. Calculated Values of a and c Lattice Parameters, Cu2+−F− Distances, and ϕ Angle for a Cu2+ Impurity in ZnF2 Together with the Dominant Character of the Hole in the Ground Statea supercell

a

c

RX = RY

RZ

ϕ

hole nature

2×2×2 2×2×3

4.656 4.659

3.169 3.165

2.074 2.074

1.921 1.922

77.8 77.8

∼3z2−r2 ∼3z2−r2

a

Results correspond to periodic calculations performed by means of 2 × 2 × 2 and 2 × 2 × 3 supercells. All distances are given in Å and angles in degrees.

It can first be noted that the results derived through both periodic supercell calculations are practically identical, a fact which supports the validity of the present approach for describing the isolated impurity. Along this line, the calculated lattice parameters differ from those given in Table 1 for pure ZnF2 only by less than 0.1%. D


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Inorganic Chemistry lattice.31 However, little attention has been paid to this remark in subsequent works1−3,8−16 on CuF2. Searching to understand the present results on ZnF2:Cu2+, we have first explored the dependence of the ground state when we move a little from the equilibrium geometry, RZ = 1.92 Å, RX = RY = 2.07 Å, calculated for CuF64− (Table 3) to that for ZnF64− units in ZnF2 (Table 1). Writing RZ = 2.02 − 0.1λRX = RY = 2.04 + 0.03λ

antibonding orbital, then the associated density does not belong to the A1g irrep of the cubic group; thus, it induces a symmetry reduction well seen in a static JTE. At this stage, we can already remark other differences between the present case and d9 systems displaying a true JTE, where, at the beginning, the antibonding x2−y2 and 3z2−r2 orbitals are degenerate. Accordingly, the equilibrium geometry can, in principle, be either elongated with the hole in a x2−y2 type orbital or compressed involving the hole in a 3z2−r2 type orbital; thus, only by studying the relaxation path of both options can we actually know the stable conformation. If it is the elongated one, such as usually happens for d9 ions in cubic crystals,17−19 then there are three equivalent distortions keeping the hole in a x2−y2 type orbital. By contrast, for ZnF2:Cu2+, the rest of the lattice, displaying a tetragonal symmetry, imposes already at the starting point that the hole of all CuF64− units has to be in the same 3z2−r2 local orbital. A situation somewhat similar to the present one has been found for Cu2+-doped K2ZnF4 and Ba2ZnF6 tetragonal lattices where Zn2+ ions form layers perpendicular to the crystal c axis.29,36 In these lattices the tetragonal internal electric field forces the hole of all CuF64− units to be in the axial 3z2−r2 orbital with the Z direction is parallel to c. This hole density leads in turn to tetragonal CuF64− complexes compressed along Z, although this vibronic relaxation process does not imply any reduction of the initial D4h local symmetry. The anisotropy of the (−e)VR(r) potencial energy in those layered lattices is substantially higher than that found for ZnF2. Indeed, if we define the anisotropy of the potential energy as

(1)

and varying λ from 1 to 0 we have verified that in all cases the hole in the ground state of CuF64− remains in the same orbital with a strong 3z2−r2 character. Moreover, the nature of the ground state remains even when the three Cu2+−F− distances are equal. This key fact strongly suggests that when RZ = RX = RY the ground state of CuF64− complexes in ZnF2:Cu2+ is no longer degenerate. Seeking to clarify this issue, we have explored the shape of the internal electric field, ER(r), that feel the localized electrons of CuF64− units in the ZnF2 lattice and reflect the tetragonal symmetry of the host lattice. Calling VR(r) to the potential generating this internal field, through ER(r) = −∇VR(r), Figure 3 displays the potential energy (−e)VR(r) of an electron

ΔU (s) = ( −e){VR (0, 0, s) − VR (s , 0, 0)}

(2)

then we obtain from Figure 3 a small value ΔU = 0.2 eV for s = 1.90 Å. By contrast, ΔU ≈ 2.5 eV for K2ZnF4 and Ba2ZnF6 tetragonal lattices at the same s value.29,36 For the sake of completeness, we have also explored the characteristics of the metastable state involving a hole in a dominant x2−y2 orbital of the Cu2+ impurity in ZnF2. Particular attention has been paid to determine the value of the energy barrier B defined as follows

Figure 3. Potential energy (−e)VR(r) corresponding to the internal electric field created by the rest of lattice ions of ZnF2 on a CuF64− complex depicted along the local X, Y, and Z directions of the complex.

B = E0(x 2−y 2 ) − E0(3z 2−r 2)

(3)

where E0(j) (j = x −y , 3z −r ) means the energy at equilibrium of the state characterized by a hole in the j orbital. In the case of a true JTE the B quantity is equal to the barrier among two equivalent distortions.17,18,47 The local equilibrium geometry found for ZnF2:Cu2+ with a hole in the ∼x2−y2 orbital and using a 2 × 2 × 3 periodic supercell is given by RZ = 2.089 Å and RX = RY = 1.979 Å, describing an octahedron with Z as the longest axis. Concerning the calculated B quantity, it is compared in Table 4 to values previously derived29 for Ba2ZnF6:Cu2+ and KZnF3:Cu2+. It can be noted that |B| is at least 1 order of magnitude higher for the non-Jahn−Teller systems Ba2ZnF6:Cu2+ and ZnF2:Cu2+ than for KZnF3:Cu2+ where there is a true JTE. This fact reflects that in tetragonal host lattices the barrier B is influenced by the internal VR(r) potential but not in a cubic lattice where it depends on other factors and in particular the anharmonicity of vibrations.56,57,24 In contrast, the values |B| = 550 meV for Ba2ZnF6:Cu2+ and |B| = 120 meV for ZnF2:Cu2+ again support that effects from the rest of the lattice are stronger for the former system. The relaxation described for the present case is also the result of the electron−vibration (vibronic) interaction 2

(charge, −e) under VR(r), when r moves along either the Z or the X, Y local axes. It can be noticed that VR(r) is slightly anisotropic, making (−e)VR(0, 0, s)> (−e)VR(s, 0, 0) when s < 2 Å. We have verified that such an anisotropy is mainly due to the two nearest Zn2+ ions lying at 3.128 Å along the c axis depicted in Figure 2. As the 3z2−r2 molecular orbital is mainly lying along the Z axis, the shape of VR(r) in Figure 3 supports the idea that such an orbital should be the HOMO where the hole resides. Once the rest of the lattice forces the positive hole of CuF64− complexes to be in the 3z2−r2 orbital, the vibronic coupling tends to reduce the distance, RZ, between Cu2+ and axial ligands while increasing the value of RX = RY.36 This explains, albeit qualitatively, the equilibrium geometry reached when a Zn2+ cation is replaced by a Cu2+ impurity in ZnF2. As the ground state electronic density (involving a hole in a dominant 3z2−r2 antibonding orbital) is invariant under the D2h group, then the Zn2+ → Cu2+ substitution does not give rise to any change of local symmetry (Table 3). By contrast, if the host lattice is cubic and the hole is placed in a 3z2−r2 or x2−y2 E

2

2

2


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Inorganic Chemistry Table 4. Calculated Values of the Energy Barrier B for the Cu2+ Impurity in Three Fluoride Latticesa system character B (meV)

KZnF3:Cu2+ Jahn−Teller 8

Ba2ZnF6:Cu2+ No Jahn−Teller −550

Bearing these facts in mind, we have explored the evolution of the frequency of the b3g mode, ω(b3g), upon increasing the Cu2+ concentration. For achieving this goal, we have replaced two Zn2+ ions by Cu2+ ions in 2 × 2 × 2 and 2 × 2 × 3 periodic supercells in such a way that the x and y coordinates of both ions are the same (Figure 5). The calculated value of ω(b3g) for both supercells with two Cu2+ ions is compared in Table 5 to that derived for a 2 × 2 × 3 supercell with a single copper ion. As shown in Table 5, when we have only one copper ion in a 2 × 2 × 3 supercell we obtain a value ℏω(b3g) = 67 cm−1 that is small for a stretching mode but positive. This result is thus fully consistent with a D2h local equilibrium geometry reported in Table 3. However, when we add a second Cu2+ to the supercell, in the way depicted in Figure 5, ω(b3g) becomes imaginary. This result means that a D2h geometry is no longer stable and the equilibrium geometry of CuF64− units in ZnF2 when Cu2+ concentration increases becomes C2h. At the same time, it strongly suggests that the existence of Cu2+ complexes sharing some common ligands tends to soften the ω(b3g) frequency. This idea is supported by the increase of |ω(b3g)| when we compare the value ω(b3g) = 137i obtained for a 2 × 2 × 3 supercell with ω(b3g) = 231i derived for a 2 × 2 × 2 supercell (Table 5) involving a higher Cu2+ concentration. Indeed, in the 2 × 2 × 2 supercell with two copper ions (Figure 5) a Cu2+ ion shares two ligands with nearest neighbor CuF64− complexes along the c axis. It is worthwhile to remark that the present results are somehow comparable to those corresponding to K2CuxZn1−xF4 crystals.28 Indeed, for an isolated Cu2+ impurity in the K2ZnF4 lattice all CuF64− complexes display a D4h compressed geometry with a principal axis parallel to the crystal c axis and the hole placed in a pure 3z2−r2 orbital.59,36,28 However, the value of ω(b1g) is reduced upon increasing the Cu2+ concentration becoming imaginary (negative force constant) for x ≥ 0.4. Accordingly, in the pure compound K2CuF4 there is an additional orthorhombic distortion leading to a local D2h symmetry,28,48 which nevertheless does not modify substantially the nature of the electronic ground state.

ZnF2:Cu2+ No Jahn−Teller −122

a

In addition to ZnF2:Cu2+ results are shown for Cu2+-doped KZnF3, where there is a JTE in accord with the cubic host lattice,20−22,29 and Ba2ZnF6:Cu2+, where the host lattice is tetragonal and there is no JTE.29 The positive sign of B for KZnF3:Cu2+ reflects a hole in the ground sate placed in a x2−y2 type orbital.

described by the potential energy V(r, Q). If V0(r, 0) corresponds to the initial frozen situation, then when the nuclei are slightly moved through the QΓ,γ normal coordinates the new situation can be described58 by V (r, Q ) = V0(r, 0) + Σ Γ, γ (∂V/∂Q Γ, γ )Q Γ, γ + ...

(4)

In the expansion of the V(r, Q) operator the second term on the right-hand depicts the so-called linear vibronic operator, which includes all distortion modes of the system with irreps Γ (γ runs on the rows of the f-degenerate Γ irrep). Thus, if Ψ0(0,r) describes the electronic ground state wave function at the initial situation and if the associated density ρ0(r) = |Ψ0(r, 0)|2 is invariant in the symmetry group of the system, then ⟨Ψ0(r,0)|(∂V/∂QΓ,γ)|Ψ0(r,0)⟩ ≠ 0 only if Γ = A1. Thus, the force on ligands due to the electronic density, related to ⟨Ψ0(r,0)|(∂V/∂QΓ,γ)|Ψ0(r,0)⟩ in first-order perturbation, induces a ligand relaxation that does not break the initial symmetry such as it happens for ZnF2:Cu2+. Nevertheless, nonsymmetric distortions can also take place when we consider the effects of vibronic interactions in second-order perturbation, thus involving changes of the electronic density. Indeed, the linear vibronic interaction of eq 4 induces a softening of the force constants corresponding to the electronic ground state58 and plays a key role for understanding the structure of CuF2 discussed in section 5. 4.2. Instability Induced by Increasing the Cu2+ Concentration. As we have seen, a diluted Cu2+ impurity in ZnF2 displays a D2h local equilibrium geometry where RX = RY (Table 3). Nevertheless, for CuF2 RY > RX (Table 2) making that the local equilibrium geometry is C2h. It is worth noting that a b3g distortion mode of the D2h symmetry group leads to a C2h geometry such as it is shown in Figure 4.

5. COMPRESSED GEOMETRY AND ADDITIONAL C2h INSTABILITY IN CuF2 The results of the preceding section on Cu2+-doped ZnF2 strongly suggest that the structure of CuF2 can be understood through two main steps. If, in a first step, we assume a rutile structure, then the CuF64− units should be compressed along the Z axis thus displaying a local D2h geometry because ϕ ≠ 90°. However, that geometry would be unstable making RX ≠ RY and leading finally to a local C2h geometry and a lattice that is no longer tetragonal but monoclinic (Figure 1). That deformation is displayed by b2g and b3g distortion modes associated with the two complexes involved in the unit cell of the rutile structure (Figure 6). As the two complexes are physically equivalent but with nonparallel Z axes, the b2g and b3g distortions are degenerate. Seeking to assess the reliability of this proposed explanation, we have calculated the equilibrium compressed and elongated geometries for CuF2 under a hypothetical rutile structure (P42/ mnm space group) and compared the results with those obtained for the monoclinic P21/c group. In this comparison, particular attention is paid to the energy difference between both structures. Main results are collected in Table 6. It can first be noted that CuF2 under an imposed P42/mnm space group would give rise to CuF64− units displaying a D2h

Figure 4. Cu2+-doped ZnF2 system, with b3g mode shown as red arrows. F


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Figure 5. Periodic supercells, 2 × 2 × 2 and 2 × 2 × 3, used for the calculation of 2 Cu2+ impurities (in blue color) in ZnF2.

Table 5. Influence of the Cu2+ Concentration on the Calculated b3g Frequency (in cm−1) for Cu2+-Doped ZnF2a supercell

no. Cu ions in supercell

ℏω(b3g)

2×2×3 2×2×3 2×2×2

1 2 2

67 137i 231i

compressed geometry with RY = RX = Req = 2.090 Å and RZ = 1.888 Å. These values are thus close to those found for a Cu2+ impurity in ZnF2 (Table 3). However, as it is shown in Table 6, the P42/mnm structure with local compressed geometry has two instable and degenerate modes (Figure 6) leading to the monoclinic P21/c symmetry. In this change, RZ increases only by 1.5%, while RY = Req + u and RX = Req − u, with u ≈ 0.16 Å, describing a local C2h geometry. This change of structure is driven by an energy gain that amounts only to 15 meV (Table 6). In order to quantify the local deformation associated with the shift from P42/mnm to P21/c we can use the quantity defined by

The ω(b3g) value obtained for a single Cu2+ ion in a 2 × 2 × 3 supercell is compared to results derived when two Cu2+ ions are incorporated in the same supercell as well as in a smaller 2 × 2 × 2 supercell. a

η = 2(RY − RX )/(RY + RX )

(5)

As a figure, η = 16% is derived through the experimental RY and RX values for CuF2 (Table 2) this simple argument underpins that the nature of the electronic ground state is not deeply altered by the local C2h distortion. A similar situation holds for K2CuF4, where η = 14% and the hole still has a 83% 3z2−r2 character.28,48 Therefore, the structure of CuF2 can be understood through CuF64− units compressed along the Z axis followed by an additional vibronic instability making RY > RX. As due to this final distortion RX becomes nearly equal to RZ the CuF2 structure has previously been described through elongated CuF64− units with Y as the principal axis being the result of JTE.1−3,8−16 Nevertheless, the present analysis proves that it actually involves compressed CuF64− units although partially hidden by the subsequent instability leading to a D2h → C2h local symmetry change.

Figure 6. Degenerate b3g (red arrows) and b2g (light blue arrows) modes driven the distortion of CuF2 from the metastable P42/mnm tetragonal structure to the experimental P21/n (standard P21/c) monoclinic structure.

Table 6. Optimized Equilibrium Cu2+−F− Distances and Frequencies of b2g and b3g Modes for CuF2 Imposing the Tetragonal Structure P42/mnm Displayed by ZnF2 and Compressed or Elongated Geometries of CuF64− Complexesa space group

CuF64− geometry

RY

RX

RZ

ℏω(b3g, b2)

relative energy

P42/mnm P42/mnm P21/c

compressed elongated orthorhombic

2.090 1.943 2.265

2.090 1.943 1.936

1.888 2.210 1.918

85i 64

0 +60 −15

a

The results are compared to those derived for the right CuF2 structure (P21/c space group). The relative energy corresponding to a CuF64− complex in both situations is also reported. All distances are given in Å; frequencies are given in cm−1. Relative energies are given in meV. G


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Table 7. Values of the Longest, RL, and Shortest, RS, Cu2+− F− Distances Obtained for Pure and Doped Systems Containing CuF64− Complexesa

According to quantum mechanics, an instability involving a nonsymmetric and nondegenerate distortion mode, Γ, is greatly driven by the linear vibronic interaction as it tends to soften the force constant, K, for the electronic ground state.58 Indeed, the force constant can be written as K = K0 + K v

(6)

where K0 = ⟨Ψ0(r,0)|(∂2V/∂QΓ2)|Ψ0(r,0)⟩ > 0 is the contribution from the QΓ displacement of the nuclei in a fixed (frozen) electronic distribution,58 while Kv arises from the change of electronic density along the distortion path due to the vibronic interaction in eq 4 that in second-order perturbation is given by58

K v = −2 ∑ n≠0

⟨Ψ0(0;

∂V r)| ∂Q |Ψn(0; Γ

En − E0

a

system

RL

RS

RL − RS

CuF2 K2CuF4 KCuF3 ZnF2:Cu2+ K2ZnF4:Cu2+ KZnF3:Cu2+ Ba2ZnF6:Cu2+

2.298 2.234 2.256 2.074 2.043 2.102 2.070

1.917 1.939 1.885 1.921 1.931 1.967 1.885

0.381 0.295 0.371 0.153 0.112 0.135 0.185

All distances are given in Å.

6. FINAL REMARKS The conclusions reached in this work are greatly underpinned by two main factors. On one one, the use of first-principles calculations that allow one to simulate a metastable configuration precursor of the final equilibrium geometry. On the other hand, the valuable information extracted from experiments31 and first-principles calculations carried out on ZnF2:Cu2+. Indeed, the study of impurities sheds light on the behavior of pure insulating compounds because in both cases active electrons are actually localized.61,62 From the present analysis, extreme care has to be taken with the use of parametrized Jahn−Teller models46,59 for every system involving a d9 ion. Indeed, the results on CuF2, K2CuF4,28,48 and K2ZnF4:Cu2+36 underline that the application of such parametrized models to systems where there is no JTE is simply meaningless. According to the present study, the local structure on the model CuF2 compound cannot be understood as an elongated octahedron arising from a JTE. By contrast, we have shown that it actually comes from a compressed octahedron although hidden by an additional D2h → C2h instability. This kind of instability, associated with a negative force constant, is thus similar to that involved in K2CuF4,28,48 the hybrid perovskites28 (CnH2n+1NH3)2CuCl4 (n = 1−3), Cu2+ in Tutton salts,54 or CuCl4(H2O)22−-doped NH4Cl.60 It should be remarked that in the case of fluorides the existence of a cooperative mechanism28 makes possible the development of an instability in pure compounds like CuF2 or K2CuF4 not observed for a Cu2+ impurity in ZnF2 or K2ZnF4.59,36 Accordingly, the present results stress that the force constants of a given complex embedded in a lattice cannot, in principle, be transferred to another compound as they depend on the existence of cooperative effects as well as on the ground state nature. The present results stress that in both ZnF2:Cu2+ and CuF2 systems the hole in the ground state is lying mainly along the Z direction (Figures 1 and 2) which thus exhibits a singular character. This situation is somewhat similar to that observed in layered compounds where the direction perpendicular to the layer planes plays a fundamental role.28 Preliminary calculations on CuF2 under pressures up to 9 GPa indicate that the electronic and geometrical structures are only slightly modified with respect to that at zero pressure. Accordingly, a right understanding of magnetic properties has to consider the actual ground state of CuF64− units in CuF2, thus discarding the usual JTE assumption.63,8,14,15 Although the present work is focused on the ground state of CuF64− complexes in CuF2, the conclusions reached here should also be helpful to understand the experimental d−d

2

r)⟩ (7)

Here E0 and En are, respectively, the energies of ground and the n-excited state computed at the initial situation, while Ψn(r, 0) is the n-excited state wave function. It should be remarked that Kv is negative for the electronic ground state, and accordingly, an instability takes place when |Kv| > K0, that is, K = K0 + Kv < 0. As a greater value of |Kv| is helped by the presence of lowlying excited states, this contribution is, in principle, more relevant for complexes like CuF64− than for MgF64− or ZnF64− involving cations with a closed shell structure.28 For this reason, |Kv| is enhanced when a F− ligand is shared by two adjacent Cu2+ ions in comparison with the situation for an isolated Cu2+ impurity found for Cu2+-doped ZnF2. Indeed, in the last case, a F− ligand is shared by a Cu2+ and a closed shell structure cation like Zn2+. This argument, developed in more detail in ref 28, explains albeit qualitatively why an instability takes place in pure compounds like CuF2 or K2CuF4 but not in diluted doped systems like K2ZnF4:Cu2+ or ZnF2:Cu2+. However, in chloride complexes, involving smaller force constants than in fluoride ones, a D4h → D2h symmetry reduction associated with K < 0 has been observed28 for (CH3NH3)2CdCl4:Cu2+ and also for CuCl4(H2O)22− units in NH4Cl.53,60 The present analysis thus stresses that the local structure in CuF2 or K2CuF4 reflects an instability associated with a force constant which becomes negative. By contrast, this relaxation mechanism is not present for Cu2+ impurities in ZnF2 or K2ZnF4 where the involved force constants are positive. A positive force constant is also involved in Jahn−Teller systems29,24,26 like KZnF3:Cu2+, NaCl:Ni+, or KCl:Ag2+. By virtue of this simple reasoning, one would expect, in principle, a bigger ligand relaxation for systems where a force constant becomes negative. For exploring this issue, we take as a measure of the relaxation involved in a CuF64− unit the quantity RL − RS, where RL and RS stand for the longest and shortest Cu2+−F− distance, respectively. Experimental RL and RS values for pure compounds are gathered in Table 7 together with those calculated for Cu2+ impurities in some fluoride compounds. While RL − RS lies in the range 0.29−0.38 Å for pure compounds like CuF2 or K2CuF4, such a quantity is calculated29,36,22 to be only in the range 0.11−0.18 Å for Cu2+ impurities in ZnF2, K2ZnF4, KZnF3, or Ba2ZnF6 host lattices. H


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reaction voltage and the reversibility of the CuF2 electrode in Li-ion batteries. Nano Res. 2017, 10, 4232−4244. (7) Thieu, D. T.; Fawey, M. H.; Bhatia, H.; Diemant, T.; Chakravadhanula, V. S. K.; Behm, R. J.; Kübel, C.; Fichtner, M. CuF2 as reversible cathode for fluoride ion batteries. Adv. Funct. Mater. 2017, 27, 1701051−1701060. (8) Fischer, P.; Haelg, W.; Schwarzenbach, D.; Gamsjaeger, H. Magnetic and crystal structure of copper(II) fluoride. J. Phys. Chem. Solids 1974, 35, 1683−1689. (9) Burns, P. C.; Hawthorne, F. C. Rietveld refinement of the crystal structure of CuF2. Powder Diffr. 1991, 6, 156−158. (10) Wells, A. F. Structural Inorganic Chemistry; Oxford, 1975. (11) Reinen, D.; Krause, S. Local and cooperative Jahn-Teller interactions of copper(2+) in host lattices with tetragonally compressed octahedra. Spectroscopic and structural investigation of the mixed crystals K(Rb)2Zn1‑xCux F4. Inorg. Chem. 1981, 20, 2750− 2759. (12) Reinen, D. Cu2+a chameleon in coordination chemistry. Comments Inorg. Chem. 1983, 2, 227−246. (13) Burdett, J. K. Structural-electronic relationships in rutile. Acta Crystallogr., Sect. B: Struct. Sci. 1995, 51, 547−558. (14) Reinhardt, P.; Habas, M.-P.; Dovesi, R.; de P. R. Moreira, I.; Illas, F. Magnetic coupling in the weak ferromagnet CuF2. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1016−1023. (15) Reinhardt, P.; Moreira, I. P. R.; de Graaf, C.; Dovesi, R.; Illas, F. Detailed ab-initio analysis of the magnetic coupling in CuF2. Chem. Phys. Lett. 2000, 319, 625−630. (16) Grzelak, A.; Gawraczyński, J.; Jaroń, T.; Kurzydłowski, D.; Budzianowski, A.; Mazej, Z.; Leszczyński, P. J.; Prakapenka, V. B.; Derzsi, M.; Struzhkin, V. V.; Grochala, W. High-pressure behavior of silver fluorides up to 40 GPa. Inorg. Chem. 2017, 56, 14651−14661. (17) Ham, F. S. Jahn-Teller effects in EPR spectra. In Electron Paramagnetic Resonance; Geschwind, S., Ed.; Plenum: New York, 1972. (18) Garcia-Fernandez, P.; Trueba, A.; Barriuso, M. T.; Aramburu, J. A.; Moreno, M. Dynamic and static Jahn-Teller effect in impurities: determination of the tunneling splitting. Prog. Theor. Chem. Phys. 2011, 23, 105−142. (19) Bill, H. Observation of the Jahn-Teller effect with electron paramagnetic resonance. In The Dynamical Jahn-Teller Effect in Localized Systems; Perlin, Y. E., Wagner, M., Eds.; Elsevier: Amsterdam, 1984. (20) Minner, E. M. C. Etude spectroscopique des ions Jahn-Teller cuivre et argent bivalents dans des monocristaux de fluoroperovskites de composition chimique AMF3. Ph.D. Thesis. University of Geneva, Switzerland, 1993 (21) Dubicki, L.; Riley, M. J.; Krausz, E. R. Electronic-structure of the copper(ii) ion doped in cubic KZnF3. J. Chem. Phys. 1994, 101, 1930−1938. (22) Garcia-Fernandez, P.; Barriuso, M. T.; Garcia-Lastra, J. M.; Moreno, M.; Aramburu, J. A. Compounds containing tetragonal Cu2+Complexes: Is the dx2−y2-d3z2−r2 gap a direct reflection of the distortioń? J. Phys. Chem. Lett. 2013, 4, 2385−2390. (23) Shengelaya, A.; Drulis, H.; Macalik, B.; Suszyńska, M. Low temperature ESR spectra of nickel doped NaCl. Z. Phys. B: Condens. Matter 1997, 101, 373−376. (24) Barriuso, M. T.; Ortiz-Sevilla, B.; Aramburu, J. A.; GarcíaFernández, P.; García-Lastra, J. M.; Moreno, M. Origin of small barriers in Jahn−Teller systems: quantifying the role of 3d−4s hybridization in the model system NaCl:Ni+. Inorg. Chem. 2013, 52, 9338−9348. (25) Sierro, J. Paramagnetic resonance of the Ag2+ Ion in irradiated alkali chlorides. J. Phys. Chem. Solids 1967, 28, 417−422. (26) Trueba, A.; Garcia-Lastra, J. M.; de Graaf, C.; GarciaFernandez, P.; Barriuso, M. T.; Aramburu, J. A.; Moreno, M. JahnTeller effect in Ag2+ doped KCl and NaCl: Is there any influence of the host latticé. Chem. Phys. Lett. 2006, 430, 51−55. (27) Borcherts, R. H.; Kanzaki, H.; Abe, H. EPR spectrum of a JahnTeller system, NaCl-Cu2+. Phys. Rev. B 1970, 2, 23−27.

transitions. In this sense, under the JTE assumption the hole would be in a x2−z2 orbital, and the energy, E1, of the lowest excitation x2−z2 → 3y2−r2 seen in optical absorption should be equal to 4EJT, where EJT means the Jahn−Teller stabilization energy.17,18,22 However, when we compare the experimental value E1 = 0.93 eV for CuF211,64 with the value EJT = 0.091 eV for KZnF3:Cu2+,22,54,29 where there is a true JTE, we see that 4EJT is actually smaller than E1/2. Preliminary first-principles calculations considering the actual ground state of CuF2 reproduce the value of the experimental E1 excitation. Additional work on this issue is under way. However, the properties of a transition metal complex in an insulating lattice cannot be understood considering only the isolated complex at the right equilibrium geometry.65 In fact, the effects from the rest of the lattice on localized electrons65 play an important role such as we have seen for ZnF2:Cu2+. In the same vein, the different optical spectra of Mn2+ in normal and inverted perovskites40 and the color of the Egyptian Blue pigment66 and that of ruby and emerald37,41,67 have been shown to be greatly dependent on the shape of the internal electric field, ER(r). Moreover the shape of ER(r) plays an important role for explaining68,36 why the equilibrium conformation is elongated in K2MgF4:Ni+ but compressed for K2ZnF4:Cu2+. The present results strongly suggest that the distortions taking place in low symmetry compounds involving d9 or d4 ions can hardly be ascribed to the static JTE. By contrast, the mechanisms analyzed in this work can play an important role. Further work along this line is now in progress.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

José Antonio Aramburu: 0000-0002-5030-725X Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The support by the Spanish Ministerio de Ciencia y Tecnologiá under Project FIS2015-64886-C5-2-P is acknowledged. We appreciate helpful discussions and suggestions of ́ Dr. P. Garcia-Ferná ndez.

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REFERENCES

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