Ab initio study of magnetic interactions in KCuF 3 and K 2 CuF 4 low-dimensional systems

Theab initio cluster model approach has been used to study the electronic structure and magnetic coupling of KCuF3 and K2CuF4 in their various ordered polytype crystal forms. Due to a cooperative Jahn-Teller distortion these systems exhibit strong anisotropies. In particular, the magnetic properties strongly differ from those of isomorphic compounds. Hence, KCuF 3 is a quasi-one-dimensional ~1D! nearest neighbor Heisenberg antiferromagnet whereas K 2CuF4 is the only ferromagnet among the K 2MF4 series of compounds ~M5Mn, Fe, Co, Ni, and Cu! behaving all as quasi-2D nearest neighbor Heisenberg systems. Different ab ini io techniques are used to explore the magnetic coupling in these systems. All methods, including unrestricted Hartree-Fock, are able to explain the magnetic ordering. However, quantitative agreement with experiment is reached only when using a state-of-the-art configuration interaction approach. Finally, an analysis of the dependence of the magnetic coupling constant with respect to distortion parameters is presented. @S0163-1829 ~99!09831-8#


I. INTRODUCTION
The study of magnetic interactions in strongly correlated insulators has recently attracted interest due to three main motives.The most important one, no doubt, concerns high-T c superconductivity because most of the cuprate superconductors exhibit strong antiferromagnetic interactions in the normal phase and this may play a role in fundamental theories of superconductivity. 1,2The second reason arises from the difficulty of conventional solid state band structure calculations to properly describe the electronic structure, magnetic interactions, and magnetic structures of these systems. 2 The third likewise important reason arises from the need to understand the strong anisotropies in crystal structures and magnetic ordering in low-dimensional systems and focus the interest of the present work.The low-dimensional term is used to describe crystal structures where some relevant properties can be fully explained without invoking the threedimensional ͑3D͒ periodic structure of the solid.The magnetic order and most of the properties of the normal state of high-T c cuprates can be described by invoking 2D CuO 2 planes only.This dimensionality is further reduced to 1D in ladder compounds 3,4 where magnetic order within nearest neighbors in the Cu-O ladders does imply just one direction.
In the case of magnetic interactions and magnetic order, the appearance of low dimensionality has important consequences.For instance, the mean field approach to the Heisenberg Hamiltonian is not suitable and quantum fluctuations in the Heisenberg nearest neighbor model in one dimension have to be accounted for. 5Therefore, theoretical description of magnetic interactions in these low-dimensional systems faces the problems originated by the strong correlated nature of these wide gap ionic insulators.Hence, instantaneous electron-electron interactions must be explicitly accounted for in any ab initio theoretical approach aimed at properly describing these systems as insulators and at quantitatively predicting the extent of the magnetic coupling. 2Similarly, quantum fluctuations have to be included to achieve a satisfactory agreement between theory and experiment when us-ing the Heisenberg model Hamiltonian.In fact, quantum fluctuations and electronic correlation are two different ways to term the same effect, namely, the need to go beyond a mean field description of the exact or of the model Hamiltonian used in these theoretical approaches.In the case of ab initio approaches to magnetic coupling, inclusion of the subtle electronic correlation effects governing the magnitude of the magnetic coupling constant requires the use of sophisticated state-of-the-art methods.Much of the commonly used spin polarized band structure calculations based either on density functional theory ͑DFT͒ or in the Hartree-Fock approach fail to accurately describe the electronic and magnetic properties of this kind of insulators including those with 3D character. 2On the other hand, a local or cluster approach enables the use of pure spin eigenfunctions and of highly correlated wave functions.][8][9][10][11][12][13][14][15][16] This agreement is complete when specially designed configuration interaction expansions are variationally solved. 13,14,17The fact that the experimentally determined magnetic coupling constant and the ab initio cluster model calculated value quantitatively agree is the fingerprint of the local two-body character of the magnetic coupling constant. 17,18n this work we extend the ab initio cluster model approach [7][8][9][10][11][12][13][14][15][16][17][18] to the study of the electronic structure and magnetic coupling of KCuF 3 and K 2 CuF 4 low-dimensional systems in their various ordered polytype crystal forms.Earlier neutron diffraction studies on KCuF 3 ͑Ref.19͒ suggested that the KCuF 3 structure was slightly distorted from the cubic prototypical KNiF 3 one.This interpretation is in agreement with recent x-ray diffraction measurements 20 and with a particular orbital ordering of the magnetic orbitals claimed by Khomskii and Kugel' 21,22 based on the cooperative Jahn-Teller effect.This fact explained the actual distorted structure of KCuF 3 and also properly predicted that K 2 CuF 4 will exhibit an analogous Jahn-Teller distorted structure with feeble ferromagnetic interactions in the magnetic planes. 23,24t is believed that the distortion from the ideal cubic or tetragonal structures, respectively, induces a strong anisotropy in their mechanical and magnetic properties.Hence, KCuF 3 behaves as a quasi-1D nearest neighbor antiferromagnet Heisenberg system whereas K 2 CuF 4 is the only ferromagnet representative of 2D Heisenberg systems in the K 2 M F 4 ͑M ϭMn, Fe, Co, Ni, and Cu͒ family. 25Therefore, the interest in these systems arises from their electronic and crystal structures, related to the high-T c superconductors and with different low-dimensional magnetic character.This is a challenge for a theoretical study and further motivation for a comparative analysis of its electronic structure and of its influence on the magnetic coupling constant.

II. EXTRACTION OF THE MAGNETIC COUPLING CONSTANTS
Previous cluster model studies using sophisticated configuration interaction techniques [6][7][8][9][10][11][12][13][14][15][16][17][18] have shown that for wide gap ionic insulators, the magnetic coupling constant J is a local property that may be described correctly by means of a properly embedded finite model containing only two interacting magnetic centers.The Heisenberg Hamiltonian for such a cluster model reduces to and, since for two interacting centers with Sϭ 1 2 , as the Cu 2ϩ cations in KCuF 3 and K 2 CuF 4 , the triplet ͉T͘ and singlet ͉S͘ states are the only eigenstates of Eq. ͑1͒, it follows that which allows us to obtain J from ab initio electronic structure calculations.
One may also apply unrestricted Hartree-Fock ͑UHF͒ and DFT based spin polarized solutions to the problem of magnetic coupling in ionic solids 26,27 following a method similar to those earlier used by Noodleman and Davidson 28,29 or by Bagus and Bennett. 30In this approach the ferromagnetic, ͉F͘, state is rather well represented but one is forced to use a broken symmetry, ͉BS͘, solution for the antiferromagnetic, ͉AF͘, state.In a recent work 31 it has been shown that for two Sϭ 1 2 localized magnetic moments J may be obtained as [34][35][36]

III. SUMMARY OF CRYSTAL STRUCTURES AND MAGNETIC PROPERTIES FOR KCuF 3 AND K 2 CuF 4
Both KCuF 3 and K 2 CuF 4 have perovskitelike structures isomorphous to the cubic KNiF 3 and the tetragonal K 2 NiF 4 , respectively.In these structures, Cu 2ϩ is a d 9 cation in an octahedral crystal field, hence e g orbitals have different occupation and a Jahn-Teller ͑JT͒ distortion stabilizes the structure.The distortion on each copper center is equivalent and a few possible cooperative distortions of the solid structure exist depending on the cation coordination and the rest of the ions in the crystal.The appearance of elongated distorted octahedra has been theoretically explained by Khom-skii and Kugel' 21,22 from simple orbital arguments which consists in alternating d z 2 Ϫx 2-and d z 2 Ϫy 2-type orbitals arising from the two possible combinations of the d x 2 Ϫy 2 and d z 2 pure atomic orbitals ͑cf.Fig. 1͒.For both KCuF 3 and K 2 CuF 4 , this orbital alternation on neighbor copper centers in the ab planes results always in a feeble direct exchange interaction leading to a ferromagnetic interaction in the ab plane, J ab Ͼ0.In the case of KCuF 3 for each magnetic center in the ab plane there is always a magnetic center neighbor in the c direction which can interact by a strong superexchange mechanism, J c Ͻ0, and this system can be regarded as formed by linear antiferromagnetic chains in the c direction but with a feeble ferromagnetic interchain interaction.For K 2 CuF 4 the additional nonmagnetic KF planes along the c direction block any magnetic interaction and the only magnetic interaction left is the weak ferromagnetic coupling between neighbor magnetic centers lying in the ab planes.

A. KCuF 3
In the KCuF 3 tetragonal perovskite, there are three Cu-F distances and two nonequivalent bridging F because the JT distortion on the Cu 2ϩ centers slightly displaces the F ions from the midpoint of adjacent Cu sites in the ab planes ͑Fig.1͒.Depending on the stacking of the ab planes along the c axis, KCuF 3 has two ordered polytype structures: twisted, A ͑Fig. 2͒, and untwisted, D. 19,20 In a given sample these two polytype forms usually coexist and preparation of pure phases requires careful synthesis.
Susceptibility measurements and heat capacity curves 37 denoted a 1D antiferromagnetic behavior with a J value of Ϫ380 K. Neutron diffraction experiments 19 found a magnetic order consisting in linear antiferromagnetic chains along the c axis with J c again of ϳϪ380 K with a feeble interchain J ab ferromagnetic coupling of ϳϩ0.4 K and with a Cu 2ϩ magnetic moment of 0.5 B confined in the ab planes.][40][41] Periodic UHF calculations 34 predict that KCuF 3 is a wide gap insulator with a KK ground state orbital ordering and that ordered, A and D, and disordered stacking polytype forms are almost isoenergetic.This study predicts that KCuF 3 has effectively 1D magnetic structure.[34][35] B. K 2 CuF 4 K 2 CuF 4 is a ferromagnetic ionic insulator with crystal structure derived from that of the ideal K 2 NiF 4 crystal.In K 2 CuF 4 the JT distortion displaces the F anions from the middle of the adjacent Cu centers, as in the KCuF 3 ab planes ͑Fig.1͒.The K 2 CuF 4 crystal structure has been unequivocally resolved and an ordered phase similar to that of ideal K 2 NiF 4 was found. 23,24This is precisely the one chosen in the present study; the orthorhombic unit cell, containing four formula units, is represented in Fig. 3, with the basal plane of the K 2 NiF 4 -type pseudocell indicated in the center of the figure for comparative purposes.K 2 CuF 4 exhibits a 1.0 B localized magnetic moment on each Cu atom ͑at Tϭ0 K͒ lying on ab planes [42][43][44] forming a quasi-2D Heisenberg ferromagnetic system at low temperature with a small XY anisotropy ͑ϳ1%͒.The Curie temperature of this compound is 6.25 K, [42][43][44] the magnetic coupling constant has been determined from thermal analysis, 44 and by neutron diffraction 43 and is in the range 15-23 K.The KK orbital ordering of the magnetic planes was also invoked to explain the observed magnetic order. 42ecent theoretical works on electronic and magnetic properties of K 2 CuF 4 are from local density approximation ͑LDA͒, 45 and from ab initio derived Hubbard Hamiltonian 46 studies.The LDA band structure calculations 45 were even unable to predict the distorted K 2 CuF 4 structure and the undistorted structure was predicted to be metallic.Extended Hubbard Hamiltonian studies 46 were able to nicely reproduce the magnetic order of this compound.However, the accuracy of these model Hamiltonians strongly depends on the parameters used and predictions of physical magnitudes, so elusive as the magnetic coupling constants, from model Hamiltonians is of limited value.

IV. COMPUTATIONAL APPROACH
In this work we use a cluster model configuration interaction approach to obtain the relevant magnetic coupling constants for KCuF 3 and K 2 CuF 4 .For KCuF 3 we consider J c in A and D polytypes and J ab in the D polytype.For K 2 CuF 4 we consider J ab in ordered K 2 NiF 4 -type structures.For both compounds we have also investigated the effect of the JT distortion in the ab planes on J c or J ab , respectively, by using idealized, fictitious, structures, I, in which the structural parameters are those of the A polytype for KCuF 3 and those of the K 2 NiF 4 -type structure for K 2 CuF 4 but where distortion has been removed.
Local Cu 2 F 11 cluster model representations of the compounds have been used.These models contain two magnetic centers lying on the c axis or on the ab planes and are embedded in an appropriate crystal environment consisting of one shell of total ion potentials ͑TIPs͒, 47 and an array of point charges to adequately account for the Madelung potential ͑see Refs.48 and 49͒.Further, nonempirical pseudopotentials are used to represent the He and Ne inner cores of outer F ͑Ref. 50͒ and Cu, 51 respectively.For the bridging F Ϫ anion all electrons are explicitly considered.The number of electrons corresponds to that of an ionic system although the complete final clusters are electrically neutral.The wave functions used in this work are flexible enough so as to include any possible covalent character albeit this has been shown to be very small. 34,45Pure spin wave functions of increasing complexity were obtained to describe the electronic structure of the central Cu 2 F 11 unit of the KCuF 3 and K 2 CuF 4 cluster models.The wave functions are constructed from atomic orbitals expressed in a contracted Gaussian-type orbital ͑CGTO͒ basis set.For Cu we use a (10s5 p5d) primitive set contracted to ͓4s3 p3d͔ basis, for the bridge F anion the basis set is (11s5 p2d) contracted to ͓4s3 p2d͔, while for the remaining ligands we use a ͓2s2 p͔ contraction of the (5s5 p) primitive set.0][12][13][14][15][16][17][18] A preliminary restricted open-shell Hartree-Fock calculation for the triplet state is carried out to generate a set of molecular spin orbitals.
First, we consider a complete active space configuration interaction ͑CASCI͒ wave function containing all Slater determinants that can be built by distributing the active electrons in the active orbitals.This CASCI is equivalent to the superexchange Anderson model. 52,53The CAS is defined by the two half filled orbitals arising from the JT splitting of the e g manifold and contains two active electrons, therefore, giving rise to either triplet or singlet states.This model is limited because it neglects terms that involve excitations out of the CAS; these excitations contribute to the external correlation and correspond to important physical mechanisms necessary to correctly describe magnetic interactions. 54ollowing previous works, the CASCI description is improved by making use of the difference dedicated configuration interaction ͑DDCI͒ method 55 ͑see also Ref. 17 for recent applications related to this work͒.Two different DDCI spaces have been considered.The first one, DDCI2, includes excitations, either holes and/or particles, with at most two inactive orbitals.When covalent effects are important the M L ϩ M Ϫ configurations ͑where L is the bridging atom between the metal atoms͒ play an important role in enhancing the superexchange mechanism. 17These charge transfer configurations are included in the DDCI2 space as one-hole configurations, but their effective energy is too high unless instantaneous repolarization of these instantaneous physical situations is explicitly accounted for.This effect is included when considering the full DDCI list which includes configurations having up to two holes, one particle and one hole, two particles in the inactive orbitals and, hence, is often denoted as DDCI3.Both DDCI2 and DDCI methods have been used to compute the J values in various structures of KCuF 3 and K 2 CuF 4 .
In order to provide an adequate comparison to available 34 and forthcoming solid state studies where configuration interaction techniques cannot be applied, we have computed the J values in KCuF 3 and K 2 CuF 4 by using a spin polarized broken symmetry approach in Hartree-Fock, UHF, or DFT methods following the previous works by Martin and Illas. 26,27These BS calculations are carried out on the same cluster models but using all electron basis sets as in Refs.26 and 27.The exchange-correlation functionals used include the well-known B3:LYP hybrid three-parameter functional which mixes the single determinant exchange ͑as in Hartree-Fock͒ with the gradient corrected Becke88 exchange functional, and uses the Lee-Yang-Parr gradient corrected correlation functional, 56 and a set of functionals which include an equal mixture of Fock and DFT ͓local or gradient-corrected, Becke88 ͑Ref.57͔͒ exchange and the correlation functional is either ignored or included in the LYP form; 58 for a detailed description see Ref. 27.
All CI calculations were carried out using the PSHF-CIPSI- CASDI chain of programs. 59UHF and DFT calculations were performed using the GAUSSIAN94 program. 60

V. RESULTS
The theoretical values of magnetic coupling constants for KCuF 3 , J c for A, D, and I structures and J ab for the D-type one are given in Table I whereas the J ab for the I and experimental ordered structures of K 2 CuF 4 are collected in Table II.An important general feature of all theoretical approaches, including the simplest UHF method, is the correct qualitative description of the magnetic ordering.All results show the effective quasi-1D magnetic structure of KCuF 3 (J c ӷJ ab ) and the 2D feeble ferromagnetic character of K 2 CuF 4 .Another feature is the close similarity between the calculated J c ͑at all levels of theory͒ for the A and D structures of KCuF 3 , in agreement with experiment.
The CI calculations show a systematic improvement of the theoretical results which parallels the degree of complexity of the particular CI expansion used.The CASCI calculations provide a qualitative description only.The DDCI2 results lead to a semiquantitative agreement with the experimental values recovering roughly 50-70 % of the experimental J value.This level of theory permits a fair description of magnetic coupling using rather small CI spaces; for the clusters used in this work the dimension of the DDCI2 space is of ϳ10 4 -10 5 determinants.A further improvement is achieved when considering the full DDCI list ͑ϳ3ϫ10 5 -2ϫ10 6 determinants͒.The results are now in very good agreement with experimental results for KCuF 3 and K 2 CuF 4 , in accordance with results for the more symmetric KNiF 3 , K 2 NiF 4 , and La 2 CuO 4 systems. 17This is be- cause the DDCI space includes almost all important physical mechanisms determining the magnitude of the J in this kind of ionic solids.The improvement of the numerical results achieved when the DDCI2 space is enlarged to form the full DDCI space is due to the inclusion of the single excitations of the Cu-F charge transfer determinants which account for the orbital relaxation of these instantaneous ionic forms.Hence, DDCI is able to mimic the orbital relaxation effects accounted for in the nonorthogonal configuration interaction ͑NOCI͒ approach used by Van Oosten, Broer, and Nieuwpoort 14 in their study of superconductor parent compounds.The use of the full DDCI list includes the same effects and can be applied to cases such as KNiF 3 where NOCI becomes prohibitive.We must point out that the use of the full DDCI list is not necessary.In fact, it will be enough to extend the DDCI2 space by including the rather small list of the single excitations of the Cu-F charge transfer determinants.The use of the full DDCI space is just convenient for technical reasons.Now, we turn our attention to the BS approaches which can also be employed in solid state periodic calculations.The crude UHF method has been included to properly compare the numerical results from a local cluster model approach to those that correspond to a fully periodic description as reported in previous works. 10For KCuF 3 the cluster results nicely reproduce those arising from the periodic calculations by Towler, Dovesi, and Saunders. 34Hence, for either J c or J ab we found Ϫ86 and ϩ4 K which have to be compared with Ϫ92 and ϩ2 K reported in Ref. 34.This fact is another clear fingerprint of the local, two-body, character of the magnetic coupling constant in the Heisenberg Hamiltonian. 10,17,18or K 2 CuF 4 the BS UHF also correctly describes the magnetic ordering.However, one must be aware of the fact that these UHF, cluster or periodic, calculations lead to values of J that are too small compared to experiment because of the lack of external correlation.The UHF provides a crude approximation to the CASCI and, hence, cannot go beyond the Anderson model.One must recognize that the magnitudes of the ferromagnetic coupling constants of K 2 CuF 4 and J ab of KCuF 3 are very small and results from this BS UHF method are surprisingly close to the experimental value.The DFT results have been included because they currently provide the only way to study magnetic coupling fully exploiting the 3D character of these systems.We must remark that all methods correctly predict the KK orbital ordering.The fact that all methods predict this KK orbital ordering is a consequence of the electronic structure which is the driving force for the crystal structure.Therefore, the magnetic order is a consequence of the electronic structure and not vice versa.
With respect to the numerical values predicted by the different DFT approaches we find the same trend as in the works by Martin and Illas. 26,27The effect of the correlation functional is minor and all the action is taken by the exchange part.As in previous works, 26,27 a reasonable numerical description is only found when Hartree-Fock and DFT, local or gradient-corrected, exchange are mixed on an equal footing.We must stress that the semiempirical B3:LYP functional which is very popular in chemical applications and performs very well in thermochemistry 61,62 gives values for J that are systematically 80% or more in excess with respect to the experimental ones for ferromagnetic or antiferromagnetic couplings.The rest of the hybrid functionals obtain numerical values close to the DDCI2 ones for all the structures.In order to bring the B3:LYP close to the experimental values one must assume that, in spite of having an expectation value of ϳ1.000 for the total square spin operator, the energy of the BS B3:LYP solution is that of the pure singlet state. 63,64his assumption will also violate the mapping procedures discussed above and will lead to the absurd consequence that two equivalent BS approaches such as UHF or DFT have to use two different mappings ͑see Ref. 10͒ to compute the magnetic coupling constant and to the absurd conclusion that the antiferromagnetic state of a periodic system leads to the energy of the pure singlet state.
To provide further arguments about the influence of the electronic structure on the magnetic order let us have a rapid insight into the ideal, I, structures, in which the distortion parameters in the ab planes have been set equal to zero.The consequence is that the magnetic orbitals become almost pure d z 2 atomic orbitals and the KK ordering disappears.For KCuF 3 this effect enhances the superexchange mechanism along the c axis because the overlap between the magnetic orbitals and the F bridge p z orbitals increases.Therefore, J c becomes more negative and the effective 1D character of the magnetic structure is maintained.For K 2 CuF 4 the removal of the structural distortion causes a different effect on J ab .In terms of the simple Anderson model where one considers that the magnetic coupling constant has simply a weak direct exchange contribution plus a stronger antiferromagnetic superexchange contribution to J, the removal of the distortion maximizes the overlap between the magnetic orbitals and the F bridge p (x,y) orbitals.Hence, the superexchange mechanism contribution dominates over the direct exchange contribution and the sign of the magnetic coupling constant is reversed while maintaining the 2D magnetic structure.A similar line of reasoning holds for J ab in KCuF 3 .

VI. CONCLUSIONS
In this work difference dedicated configuration interaction methods have been applied to properly embedded cluster models of the A-and D-type structures of KCuF 3 , ordered K 2 CuF 4 structure and the respective ideal, I, undistorted, structures to investigate the physical contributions to the magnetic coupling constants and to provide accurate theoretical values of this elusive physical property.To provide adequate comparison to forthcoming periodic calculations, several polarized UHF and DFT formalisms using a broken symmetry approach have also been used.In spite of using cluster models which explicitly consider two magnetic centers only, the DDCI approach reaches a quantitative description of the magnetic coupling constant for all the experimental structures.The success of the DDCI method lies in the inclusion of the instantaneous relaxation of the Cu-F charge transfer forms.This effect is crucial to correctly describe the magnetic coupling constant in this class of wide gap ionic insulators.The DFT results provide a qualitative agreement with experiment for KCuF 3 and even quantitative agreement for K 2 CuF 4 .However, the widely used B3:LYP functional gives J values exceeding the experimental ones by a factor of 1.5 to 4 for ferromagnetic or antiferromagnetic interactions.The rest of the hybrid functionals explored in this work re-cover roughly 60% of the experimental value for antiferromagnetic interactions and match the experimental one for ferromagnetic interactions.The orbital ordering described by Khomskii and Kugel' is predicted at all levels of theory used in this work provided the experimental structures of both solids is used as external input.We must stress the fact that crystal structure results from the electronic structure, and magnetic order is a consequence of this electronic structure.This assertion is supported by computational experiments where the distortions in the ab planes are eliminated, resulting in ideal I-type structures.Calculations on these I structures reveal that the directionality of the magnetic orbitals on each Cu 2ϩ center is enhanced and has two different effects in the two compounds.In KCuF 3 this larger directionality increases the magnitude of the magnetic coupling constant along the c axis because superexchange mechanism is favored due to an increasing in overlap between open shells of neighbor magnetic centers and the bridging ligand.In K 2 CuF 4 the overlap between magnetic orbitals of neighbor Cu 2ϩ centers and F Ϫ orbitals is maximized and the weak enhancement of the antiferromagnetic contribution to J is slightly augmented reversing the sign of J for the interaction in the ab planes.

FIG. 1 .
FIG. 1. Schematic representation of the ordering of the elongated axis of the distorted octahedra on ab planes.The magnetic orbital of each Cu 2ϩ center, d z 2 Ϫx 2 and d z 2 Ϫy 2, are perpendicular to the elongated axis and are also shown.

FIG. 2 .
FIG. 2. The unit cell of the twisted, A, structure for KCuF 3 .Small ͑medium͒ dark spheres represent Cu 2ϩ (K ϩ ) cations and light large spheres represent F Ϫ anions.The cluster model used to calculate J c is also shown, thick lines link the atoms in the cluster model used in the calculations.

TABLE I .
Magnetic coupling constants of KCuF 3 in A, D, and I structures ͑in K͒ obtained at different levels of theory using embedded Cu 2 F 11 cluster models.

TABLE II .
Magnetic coupling constants of K 2 CuF 4 in experimental and I structures ͑in K͒ obtained at different levels of theory using embedded Cu 2 F 11 cluster models.