Faraday Discuss., 1997, 106, 173»187 Structural, electronic and magnetic properties of KMF3 (M = Mn, Fe, Co, Ni) Roberto Dovesi,a* Federica Freyria Fava,a Carla Roettia and Victor R. Saundersb a Department of Chemistry IFM, University of T orino, via P. Giuria 5, I-10125 T orino, Italy b Daresbury L aboratory, Daresbury, W arrington, UK W A4 4AD The structural, electronic and magnetic properties of the perovskite systems (M\Mn, Fe, Co, Ni) have been investigated with CRYSTAL95, a KMF3 periodic ab initio Hartree»Fock program.An all-electron Gaussian basis set has been used. The equation of state has been determined –rst for the cubic structure ; then deviations from cubic symmetry have been explored, with the result that the Mn, Fe and Co systems are found to be slightly more stable in a tetragonal geometry.The systems are almost fully ionic, with net charges for K and M of ]1, ca. [0.9 and ca. ]1.8 o e o , respectively. The antiferromagnetic (AFM) is always more stable than the ferromagnetic (FM) phase; the energy diÜerence *E\E(FM)[E(AFM) is shown: (a) to be additive with respect to the number of MwM –rst neighbours; (b) to increase with decreasing lattice parameter according to an inverse power law; and (c) to become zero when the angle approaches 90°.The MwFå wM super-exchange coupling constants, evaluated from *E by using an Ising model hamiltonian, are in qualitative agreement with the experimental data (from 30% to 45% of the latter). Mulliken population data, charge and spin density maps and density of states are used to illustrate the electronic structure. 1 Introduction Substantial progress towards a full ab initio account of the structural and electronic properties of important classes of crystalline compounds such as simple metals, semiconductors, ceramics and silicates has been achieved in recent years. For other classes of compounds and properties ab initio methods have been less successful.A typical area where additional eÜort is required is that of transition metal (TM) ionics (oxides, sul- –des, halides) and their magnetic properties ; the latter are of intrinsic interest and also because of the relationship between magnetic order and superconductivity.1 The perovskites (M\Mn, Fe, Co, Ni) represent ideal prototypes for the investigation of KMF3 the magnetic super-exchange interaction in a large class of ionic compounds, and of the relationship between structural, electronic and magnetic properties.Their highly ionic character and high symmetry, and the low coordination of the anions, are expected to allow the interpretation of their electronic and magnetic properties in terms of simple models. In the past they have been the object of intensive experimental and theoretical investigation.Much experimental data concerning the MwM magnetic interaction have been collected by de Jongh et al.,2,3 and interpreted in terms of simple model spin hamiltonians. Values of J, the super-exchange coupling constant, are proposed for the compounds with Mn, Co and Ni;2 their dependence on the MwM distance is discussed by comparing results obtained from diÜerent compounds (A\K, Rb, Tl).AMF3 173174 Structural, electronic and magnetic properties of KMF3 As regards theoretical ab initio methods, both cluster and periodic approaches have been adopted in the study of these systems. The super-exchange magnetic interaction in has been investigated with the cluster model by Mejiç as and Fernaç ndez Sanz4 KNiF3 and Illas and co-workers in a series of papers.5h10 The main advantage of the cluster scheme is related to the possibility of using sophisticated many-body techniques able to take into account electron correlation eÜects ; pure eigenstates of the spin operators can be obtained.Limitations are related to the –nite size of the cluster (usually only two TM atoms are considered) : the wavefunction is not correctly periodic, and border eÜects are expected to in—uence the obtained energies.Even so, J values close to those obtained with the present periodic method have been obtained by the above authors when the same level of theory (unrestricted Hartree»Fock, UHF) is adopted.4,10 Periodic calculations have been performed on the systems under study within the local density approximation (LDA);11 band structures and density of states are produced, but the total energies of the FM and AFM states were not provided.Density functional theory based computational schemes very often describe as metallic, TM systems which are known to be large-gap insulators.12 Corrections to the LDA formalism, such as the self-interaction correction (SIC)13,14 or the on-site Coulomb repulsion (U)15h17 do not seem to be able to improve the description of the ground-state total energy of the FM and AFM states.The present calculations are based on the periodic ab initio UHF method as implemented in the CRYSTAL code.18 This method has been applied in recent years to many TM oxides (NiO, MnO;19 and —uorides Fe2O3 ;20 Cr2O321) (FeF2 ,22 NiF2 , MnF223).In all cases, the systems are large-gap insulators ; the sign of the AFM»FM energy diÜerence is correctly reproduced and its magnitude is proportional to the values of the Neç el temperature. The UHF solutions are eigenfunctions of the spin operator, not of however, Så z Så 2; owing to the localized nature of the unpaired spins and the distance between the M ions, the eÜect of spin contamination on the energies of the magnetic states is expected to be small.The main limitation of the present scheme is related to the neglect of electron correlation, which may account for about half of the FM»AFM energy diÜerence.4,10 There are, however, many advantages in the present scheme: periodicity is fully exploited and border eÜects are absent ; basis set eÜects, additivity with respect to the number of neighbours, the relative importance of –rst- and second-neighbour interactions, and the eÜect of distance and angles on J can all be investigated.Preliminary results concerning have been presented in a previous paper.24 KNiF3 Here, the study is extended to and The structural, electronic KMnF3 , KFeF3 KCoF3 . and magnetic properties of which exhibits a considerable Jahn»Teller distor- KCuF3 , tion, have been presented elsewhere.25 The present work is organized as follows : in Section 2 computational details concerning the method and the basis set are presented.Section 3.A is devoted to the discussion of the geometry and, in particular, to the cubic]tetragonal distortion of the Mn, Fe and Co compounds.In Section 3.B, the electronic structure is discussed in terms of Mulliken population data, charge and spin density maps, and density of states. Section 3.C is devoted to the discussion of the FM»AFM energy diÜerence, additivity of the super-exchange interactions, and eÜects of geometry on J. 2 Computational details In the present work, we have used the CRYSTAL9518 code, which is based on the ab initio periodic Hartree»Fock method.26,27 All the systems have been investigated within the UHF approximation, to allow a description of the spin polarization due to the unpaired d electrons of the TM ions.Bloch functions are constructed from local functions (ìatomic orbitals œ, AOs), which are linear combinations (ìcontractionsœ) of Gaussian-type functions (GTFs); these areR.Dovesi et al. 175 Table 1 Exponents and coefficients of the contracted Gaussian-type basis functions used for Co2` in the present work coef–cient type exponents s/d p s 341 701 0.000227 48 850.0 0.001929 10 400.9 0.0111 2718.99 0.0501 819.661 0.1705 283.878 0.3692 111.017 0.4033 46.4757 0.1433 sp 855.558 [0.0054 0.0088 206.504 [0.0684 0.062 69.0516 [0.1316 0.2165 27.2653 0.2616 0.4095 11.5384 0.6287 0.3932 4.2017 0.2706 0.225 sp 51.5542 0.0182 [0.0287 18.9092 [0.2432 [0.0937 7.7251 [0.849 0.2036 3.5428 0.8264 1.4188 sp 1.4914 1.0 1.0 sp 0.6031 1.0 1.0 d 29.9009 0.0617 8.1164 0.2835 2.6433 0.529 0.8869 0.4976 d 0.3011 1.0 Coefficients multiply individually normalized basis functions.Fig. 1 Cubic unit cell of systems. The FM (left) and two possible AFM (AFM, centre and KMF3 AFM@, right) structures are shown.Open and –lled small circles indicate spin up and down M ions, respectively. In the central –gure, the dashed lines connect M ions with the same spin in the (111) plane.176 Structural, electronic and magnetic properties of KMF3 the product of a Gaussian and a real solid spherical harmonic. Extended all-electron basis sets are used containing 27, 17 and 13 AOs for the TMs, —uorine and potassium ions, respectively.The —uorine basis set can be denoted as 7-311G (the –rst shell is of s type and is a contraction of 7 GTFs; there are then three sp shells). For the TMs an 8-6- 411-(41 d)G basis is used, with two d shells. The potassium basis set is an 8-6-511G. The basis set is as used in ref. 19 (Mn), 24 (Ni, K and F) and 22 (Fe) ; the Co basis is given in Table 1. The FM unit cell contains one formula unit (5 atoms, 83 AOs in the basis) ; for the AFM phases (see Fig. 1) a double cell (10 atoms) is required. As regards the computational conditions, high numerical accuracy is required to investigate the relative stability of the FM and AFM phases,24 whose energy diÜerence is ca. 0.5]10~3 (cell)~1; for Eh this reason the following values have been used for the truncation tolerances in the evaluation of the Coulomb and exchange series :18,26 7 7 7 7 14. A shrinking factor of 4 has been used to de–ne the reciprocal net, corresponding to diagonalization of the Fock matrix at 10 points belonging to the irreducible Brillouin zone. At this level of accuracy the energy diÜerence between the single and double cells in the FM state is smaller than 10~6 (cell)~1.The use of larger sampling nets con–rms the uncertainty due to this Eh factor in the total energy to be less than 10~6 (cell)~1. Eh 3 Results and Discussion 3. A Equilibrium geometry The four systems under consideration are cubic (space group at room tem- Pm3 6 m) perature.The experimental lattice parameters are given in Table 2; the unit cell is shown in Fig. 1. Each M ion is surrounded by six second-nearest-neighbour M ions with the same (FM phase, Fig. 1, left) or opposite spin (AFM, Fig. 1, middle). At very low temperature, small deviations from the cubic symmetry have been observed by Okazaki and Suemune28 for the –rst three compounds of the series.was found to be mono- KMnF3 clinic (but the three lattice parameters were very close to each other, 4.168, 4.171 and Table 2 Calculated and experimental lattice parameters (a and c, in fractional coordinate (x), ”), bulk moduli (B, in GPa) and cell volumes (V , in for cubic and tetragonal systems, and ”3) KMF3 energy diÜerences between the two structures [*E(c»t), in kcal mol~1] cubic tetragonal system parameter calc.expt. ref. parameter calc. expt. ref. *E(c»t) KMnF3 a 4.28 4.19 35 a 4.267 4.168 0.43 B 65 65 35 c 4.298 4.174 36 V 78.4 73.6 x 0.279 0.273 V 78.3 72.51 KFeF3 a 4.22 4.12 37 a 4.237 0.31 B 69 c 4.139 V 75.2 69.9 V 74.30 KCoF3 a 4.16 4.07 28 a 4.139 4.057 28 0.48 B 74 c 4.225 4.049 V 72.0 67.4 V 72.37 66.64 KNiF3 a 4.10 4.01 28 » » » 0.00 B 79 85 38 » » » V 68.9 64.5 » » »R.Dovesi et al. 177 4.185 and b\89° 51@), rhombohedral (a\4.108, a\89° 51@) and ” KFeF3 KCoF3 tetragonal (a\4.057, c\4.049 Two subsequent studies29,30 indicate, however, that ”). is tetragonal (space group I4/mcm; see Table 2). The small diÜerence between a KMnF3 and b and the small deviation of b from 90° found in ref. 38 for are, thus, KMnF3 probably attributable to experimental error.A similar inaccuracy might aÜect the and structural data. KFeF3 KCoF3 The results of the present study are shown in Table 2. When cubic symmetry is enforced, the lattice parameter is overestimated by ca. 2% for all the systems, in agreement with previous Hartree»Fock results for TM compounds. When a lower symmetry is allowed (for the Co and Fe compounds we considered only the tetragonal subgroup P4/mmm of the ideal cubic perovskite) a small deformation is observed for KMnF3 , and as shown in the Table.In the case, the distortion, which is KFeF3 KCoF3 , KMnF3 attributed to anion»anion repulsion, is a nearly rigid rotation of the octahedra (see sketch in Fig. 2) by an angle d\6° 19@ (or 5° 43@, according to experiment).The total energy of the system as a function of d is reported in Fig. 3; the curve is symmetric with respect to the cubic situation (d\0). The barrier separating the two minima is very low (0.69]10~3 as expected because of the low experimental transition temperature Eh), between the cubic and the tetragonal phases (180»187 K29,30). In the case of the Fe and Co compounds, the deformation involves the octahedra, because of Jahn»Teller eÜects.Fe2` has a d6 high-spin con–guration; and up-spin t2g eg levels are fully occupied; the remaining down-spin electron occupies one of the three t2g levels (say a shortening of the octahedra along the z and elongation along the x and dxy ; y axes is then expected because of non-bonded repulsion between the TM d orbitals and the anion) and the triple degeneracy is lifted.Co2` has a d7 con–guration, so that t2g two down-spin electrons occupy, say, the and orbitals, with a consequent elon- dxz dyz gation of the octahedra along z. This Jahn»Teller eÜect for Co and Fe is small, because the radius of the d orbitals is small compared with the size of the octahedra, the latter being largely determined by anion»anion non-bonded repulsions, and because the lobes of the d orbitals are not directed towards the anions.For all three systems the t2g stabilization energy with respect to the cubic structure is very small (a fraction of 1 kcal mol~1). The tetragonal deformation determined experimentally for is much KCoF3 Fig. 2 Visualization along the z axis of the relative rotation of the octahedra in the tetrago- MnF6 nal structure of KMnF3178 Structural, electronic and magnetic properties of KMF3 Fig. 3 Total energy of tetragonal as a function of d, the angle (see previous KMnF3 MnwMn’ wF –gure) smaller in magnitude and of opposite sign with respect to that determined here (a»c of ]0.01 to be compared with [0.09 and is also contradictory with respect to the ”), explanation given above.More accurate low-temperature X-ray experimental data are required for both and KFeF3 KCoF3 . 3. B Electronic properties The electronic properties (which are similar for the FM and the AFM solutions) were evaluated at the experimental room-temperature geometry (cubic). The four systems are nearly fully ionic insulators ; the net charges, evaluated according to a Mulliken partition of the electron density, are ca.]0.99, ]1.8 and [0.93 o e o for K, M and F, respectively (see Table 3), very close to the formal charges]1,]2 and [1 o e o of completely ionic compounds. The unpaired electrons are almost completely localized on the transition metal 3d orbitals, whose spin populations are 4.93 (Mn), 3.92 (Fe), 2.93 (Co) and 1.93 (Ni).The strongly ionic character of these systems is also con–rmed by the bond population data (see Table 4) : these are null for MwK (no interaction) ; small and nega- Table 3 Electron population data (in o e o units) according to a Mulliken analysis Q Ns q3d ns system M K F M M K F M KMnF3 ]1.77 ]0.99 [0.92 5.19 4.94 [0.00 0.02 4.93 KFeF3 ]1.83 ]1.00 [0.94 6.13 3.93 0.0 0.0 3.93 KCoF3 ]1.82 ]1.00 [0.94 7.15 2.93 0.0 0.0 2.92 KNiF3 ]1.85 ]1.00 [0.95 8.10 1.95 0.0 0.02 1.94 Q and are the net charges and the 3d orbital populations, respectively ; and are q3d Ns ns the corresponding spin quantities.Numbers refer to the FM solutions (AFM data are very similar).R. Dovesi et al. 179 Table 4 Bond population data (in electrons) at the experimental cubic geometry according to a Mulliken analysis system MwFap MwFeq MwK KwF KMnF3 FM [0.010 0.0 [0.001 AFM [0.009 0.0 [0.001 KFeF3 FM [0.021 [0.019 0.0 0.0 AFM 0.001 [0.006 0.0 [0.003 KCoF3 FM [0.016 [0.018 0.0 0.0 AFM [0.007 0.001 0.0 [0.003 KNiF3 FM [0.004 0.0 [0.005 AFM [0.003 0.0 [0.005 and indicate apical and equatorial —uorine ions of the octa- Fap Feq hedra.tive, as a consequence of the short-range repulsion, for MwF and KwF.This picture is supported by the diÜerence density maps, obtained by subtracting the superposition of the isolated spherical ion electron distributions from the bulk electron density (the same basis set as for the bulk has been used), reported in Fig. 4 for Only minor KMnF3 . modi–cations arise in going from the superposition of ions to the bulk; the most total (top) and diÜerence (bottom) electron density maps in a (001) plane (left) Fig. 4 KMnF3 through the Mn and F atoms, and in a (110) plane (right) through the three type of atoms. The diÜerence maps are obtained by subtracting, from the bulk density, the superposition of the isolated spherical ion distributions obtained with the basis set used for the bulk.For the total maps, the separation between two contiguous isodensity lines is 0.01 the innermost curves in o e o … a0~3; the atomic region correspond to 0.08 For the diÜerence maps, the separation between o e o … a0~3. contiguous line is 0.005 the function is truncated in the core regions at ^0.03 o e o … a0~3; o e o … a0~3; continuous, dashed and dot»dashed lines correspond to positive, negative and zero values, respectively.180 Structural, electronic and magnetic properties of KMF3 obvious feature is the usual shrinking of the electron charge density of both cations and anions, which is a consequence of the Madelung –eld and short-range repulsion ; this contraction is spherical for K, whereas it is directed along the MwF bonds for M and F; on the latter a small outward displacement of charge is observed in directions orthogonal to the MwF bond.The picture of the electronic properties of these systems can be completed by analysing the projected density of states (DOS). The four systems are large-gap insulators ; the gap ranges from 0.25 to 0.35 The highest occupied states are predominantly of TM d Eh . character with small contributions from —uorine p orbitals ; for the lowest unoccupied states, the DOS shows small contributions from —uorine p orbitals ; at slightly higher energies a sharp peak due to TM d orbitals appears. In Fig. 5 the valence DOS of is reported. The valence states are largely KMnF3 constructed from —uorine p and TM d orbitals. The potassium 4s states, and the Mn 4s and 4p states are almost completely unoccupied; Mn 3sp states are much lower in energy.The DOS of the AFM structure presents relatively narrow and well separated peaks, that can be interpreted in terms of molecular levels perturbed by the MF6 Fig. 5 Valence bands projected density of states of the AFM (top) and FM (bottom) phases of In the top –gure, only the up-spin Mn atom projection is given.KMnF3 .R. Dovesi et al. 181 environment. Assume that M is up-spin, and consider the interaction of the M d orbitals with the up-spin p orbitals on neighbouring F atoms. The six p —uorine orbitals oriented along the MwF bonds generate hybrids of and symmetry; the other 12 p A1g, Eg T1u orbitals combine to give hybrids of and symmetry; the M atoms T1g, T2g, T1u T2u contribute only with d orbitals (4sp orbitals are empty), with symmetry and The Eg T2g .valence DOS therefore shows states which exhibit MF hybridization with and Eg T2g symmetry. The bonding»antibonding splitting for the former is larger, because the interaction in this case is of p character. The other states generated by F p orbitals do not hybridize with M d orbitals for symmetry reasons, giving rise to a large peak at the centre of Fig. 5. For the FM solution the peaks in the up-spin DOS are wider as a consequence of the greater number of orthogonality constraints. Atomic like peaks then overlap, and the simple interpretation given for the AFM DOS is not appropriate. In Fig. 6 the DOS for in the AFM state is reported ; the most evident feature is the KCoF3 appearance of a down-spin sharp band in the valence DOS.As a consequence dxz]dyz Fig. 6 Valence bands projected density of states of the AFM phase of The d orbital KCoF3 . projection is shown in the bottom –gure. Only one type of Co atom is shown; the projection for the other type is symmetric with respect to the horizontal line.182 Structural, electronic and magnetic properties of KMF3 of the symmetry reduction and degeneracy removal, four d peaks are now easily identi- –ed for up-spin states, corresponding to and from low to high dx2~y2, dz2, dxy dxz]dyz energies.The reason for this ordering is clear : the presence of two down-spin electrons in the and orbitals destabilizes the (with respect to and (with dxz dyz dz2 dx2~y2) dxz]dyz respect to up-spin levels because of on-site Coulomb interactions.dxy) 3. C Magnetic properties Two AFM cells are shown in Fig. 1 (centre and right) ; they correspond to a sequence of (111) and (001) TM planes, alternatively with up- (open circles) and down-spin (–lled circles) ; in the –rst case (AFM) each M ion has the six –rst neighbours with opposite spin ; this structure corresponds to the experimentally observed situation.In the second case (AFM@) only two –rst neighbours have opposite spin ; this structure will be used for discussing the additivity of the super-exchange interactions. The AFM is more stable than the FM phase for all four systems considered, even when the unit cell undergoes relatively large geometrical modi–cations (tetragonal or trigonal deformation, isotropic expansion and compression).The energy diÜerence, *E (per formula unit), between the FM and the AFM structures is reported in Table 5. *E has been obtained at the experimental high-temperature (cubic) geometry; it is always less than 1]10~3 and increases nearly linearly with the atomic number. Eh , The experimental data are usually expressed in terms of magnetic coupling constants, J, of simple spin hamiltonians, such as the Heisenberg or Ising models.According to the Ising model, under the hypothesis of additivity of the MwM interactions and taking into account only –rst-neighbour spin interactions, the energy diÜerence (per formula unit) between the FM and AFM solutions is related to J through the following equation (see ref. 10 for a more explicit discussion of the relationship between UHF and Ising states ; see also ref. 24 for the diÜerence between the present and previous results for KNiF3) : *E\2zJS2 k (1) where z is the number of nearest-neighbour TM ions of opposite spin with respect to the TM selected ion, S is its conventional spin (5/2 for Mn, for example) and k\3.1577]105 is the ratio between the conversion factor from to J (1 Eh Eh\ 4.359 748]10~18 J) and the Boltzmann constant (1.380 655 8]10~23 J K~1).The calculated J values (see Table 5) are always smaller than the experimental ones (they range from 30% to 45%). The discrepancy can, in principle, be attributed to many factors : (i) Table 5 Calculated and experimental coupling constants (in K) at the experimental geometry J 102 J2/J system calc.expt. 102 calc./expt. ref. calc. expt. ref. T KMnF3 1.24 3.65 34.0 2 0.34 1»3 39, 40 295 3.70 33.5 2 200 KFeF3 2.54 6.0 42.3 32 KCoF3 5.48 19.1 28.9 33 0.52 114 19.2 28.5 33 KNiF3 14.91 44.5 33.5 32 0.30 0.5 41 300»700 50.8 29.4 42 4.2 J and are the coupling constants between the nearest-neighbour and next-nearest-neighbour J2 TM ions ; T is the temperature (K) to which the experimental determination refers.R.Dovesi et al. 183 inadequacy of the model (Ising) adopted in converting the calculated *E to J and (ii) the experimental data to J; (iii) the UHF solutions are not eigenfunctions of (iv) corre- Så 2; lation eÜects are disregarded at the HF level. As regards point (iv), according to cluster calculations5h8 performed at the UHF level and including correlation treatments of various degrees of sophistication, interelectronic correlation is expected to increase J by about a factor two.An estimate of correlation energy through a density functional correction scheme,31 based on the HF charge density, increases *E by 10% to 40% (see Table 6, last column). The data of Table 7 exclude large basis set eÜects on *E. Presently, we are unable to estimate the importance of the other factors listed above.The additivity hypothesis, which is implicit in the z factor in eqn. (1), can be checked by using the energy of the AFM@ structure (see Fig. 1). Under the hypothesis that only –rst-neighbour interactions are relevant, *E/*E@ should be equal to 6/2 ; this is the case, with very minor deviations, for Mn and Ni (only 1.7%).For Fe and Co the deviation is larger (17%), because the two interactions along z (which are the only ones considered in AFM@) are diÜerent from the four in the xy plane as a consequence of the anisotropic spin density on the M ions. In these cases (Fe and Co) it is more appropriate to use the following expressions, instead of eqn. (1) : g*E\2S2 k (4Jxy]2Jz) *E@\2S2 k 2Jz (2) The resulting values for Jxy and Jz are 2.78 and 2.16 (Fe) and 5.11 and 6.22 (Co).Comparison with experiment has been performed by using (see Table 5), J1 \(4Jxy]2Jz)/6 because only a single J value is provided.32,33 Table 6 Energy diÜerence (in 10~3 per formula unit) between the FM *EHF Eh , and AFM Hartree»Fock total energies system *EHF *EC *Etot dEHF dX dK dC KMnF3 0.293 0.122 0.419 1 [3.84 14.98 [10.14 KFeF3 0.385 0.088 0.479 1 [3.56 13.05 [8.49 KCoF3 0.468 0.070 0.542 1 [3.39 12.22 [7.84 KNiF3 0.569 0.051 0.619 1 [3.28 13.09 [8.81 is the correlation contribution to this diÜerence (evaluated a posteriori *EC according to Perdewœs formula31,43).In the second half of the *Etot\*EHF]*EC . table the exchange (X), kinetic (K) and Coulomb (C) contributions to are *EHF reported.d quantities have been obtained after the division by so that *EHF dX ]dK]dC\dEHF\1. Table 7 Total energy diÜerence between the FM and AFM phases (*E) of as a function of the basis set KMnF3 case basis set *E a as described in Section 2 0.293 b case (a)]d on F (a\0.7) 0.293 c case (a) but 5-1d on Mn instead of 4-1d 0.292 d case (c)]sp (a\0.25) on Mn 0.334 e case (d)]d (a\0.4) on K 0.325184 Structural, electronic and magnetic properties of KMF3 *E and *E@ can be used to estimate the importance of magnetic interactions between second-nearest neighbours magnetic constants).From inspection of Fig. 1, and (J2 assuming that the interaction is additive, the following equations hold for AFM and AFM@, respectively : g*E\2S2 k 6J *E@\2S2 k (2J]8J2) (3) By solving for J and the latter is found to be 200»300 times smaller than the former, J2 , in qualitative agreement with experimental –ndings (see Table 5).The present model can be used to investigate two other aspects of the superexchange interaction, namely its dependence on R, the MwM distance, and h, the MwFwM angle. When the system is isotropically compressed, *E increases very rapidly ; if it is assumed that *E\cRn, as proposed by de Jongh and Miedema,3 the calculated n values are [12.2 (Ni) and [13.9 (Mn, see Fig. 7), in reasonable agreement with the experimental values (n\12^2) determined by measuring J for various AMF3 systems, with A\K, Rb, Tl.3 The variation in *E as a function of d (h\n[2d) has been explored in the interval 0OdO30° and the results are shown in Fig. 8 for If the data are –tted with a KMnF3 . parabola and extrapolated, we –nd that *E\0 when dB45°; this result supports the simple model of super-exchange proposed, for example, in ref. 34 (Fig. 3.22) and discussed below. The reason for the higher stability of the AFM with respect to the FM phase can be discussed with reference to the spin density maps (see Fig. 9 and 10). These maps are very similar for the two magnetic structures (apart, obviously, from the inversion of the spin density on half of the TMS in the AFM structure). Potassium ions appear to be unpolarized. In the —uorine region, on the contrary, a small but extremely important spin polarization appears, which is responsible for the (very small) energy diÜerence Fig. 7 Energy diÜerence per formula unit between the FM and AFM phases as a func- KMnF3 tion of the lattice parameterR. Dovesi et al. 185 Fig. 8 Energy diÜerence per formula unit between the FM and AFM phases as a function of d (see Fig. 3) between the two phases. In the FM –gure, owing to the Pauli repulsion (which is re—ected in our calculation by the orthogonality constraints on the crystalline orbitals) between up-spin F electrons and the up-spin (unpaired) electrons of the two neighbouring TMS, the up-spin density slightly contracts along the MwFwM direction in the neighbourhood of the F ion ; down-spin F electrons, not directly involved in the MwF spin interactions, are slightly more diÜuse.This gives rise to the spin polarization of the F ion shown in the –gure.In the AFM case the F spin polarization is much smaller (two isodensity lines instead of –ve), because the —uorine ion is surrounded by two Mn2` ions whose d electrons have opposite spin. In this situation a (very small) up-spin and downspin electron shift in opposite directions is sufficient to account for the Pauli repulsion between electrons with the same spin.Thus, the spin polarization on F is antisymmetric with respect to the plane orthogonal to the MwFwM direction, and much smaller than for the FM phase. Maps for the Mn and Fe compounds look very similar, apart from the obvious diÜerence that Mn (d5 con–guration) is spherical [so that the spin density in Fig. 9 Spin density maps for the FM (left) and AFM (right) solutions of in the (001) KMnF3 plane.The separation between contiguous isodensity lines is 0.005 the function is trun- o e o … a0~3; cated in the core region at ^0.03 Continuous, dashed and dot»dashed lines correspond o e o … a0~3. to up-spin, down-spin and zero values of the spin density, respectively.186 Structural, electronic and magnetic properties of KMF3 Fig. 10 Spin density maps for the FM (top) and AFM (bottom) solutions of in the (001) KFeF3 (left) and (110) (right) planes.Scale and symbols as in Fig. 9. the (001) and (110) planes is the same, and only the –rst is given in Fig. 9], whereas Fe is not, as the spin density map in the two sections shows; as a consequence the polarization of the two types of F ions is diÜerent, as is also indicated by the bond population data given in Table 4.There is a –nal point that deserves comment. In Table 6, *E, the HF energy diÜerence E(FM)[E(AFM), is decomposed into three contributions, namely Coulomb (C), exchange (X) and kinetic (K) ; it turns out that, in going from the FM to the AFM solution, a large reduction in kinetic energy takes place (*K is 10»15 times larger than *E) as a consequence of the removal of the symmetry constraint requiring the two M atoms to be equivalent.Coulomb and exchange contributions favour the FM solution, and cancel more than 90% of the kinetic energy gain ; *C is 2 to 3 times larger than *X. As anticipated, correlation eÜects, as estimated by a density functional scheme, increase the AFM stability with respect to the FM solution. 4 Conclusions The UHF method, as implemented in the CRYSTAL95 program, has been shown to be a useful tool for understanding the electronic and magnetic properties of crystalline compounds.Such a delicate quantity as the FM»AFM energy diÜerence is always qualitatively reproduced; its dependence on the MwM distance and MwFwM angle is easily obtained and described. The super-exchange interaction is shown, at least for these simple cases, to be additive with respect to the number of neighbours.The distortion of the octahedra in the and cases is easily explained in terms of KFeF3 KCoF3 Jahn»Teller eÜects. In the case the rotation of the octahedra is well reproduced. 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