Structural, electronic, and magnetic properties of Fe2SiO4 fayalite

PHYSICAL REVIEW B 67, 094106 共2003兲

Structural, electronic, and magnetic properties of Fe2 SiO4 fayalite: Comparison of LDA and GGA results Matteo Cococcioni, Andrea Dal Corso, and Stefano de Gironcoli SISSA–Scuola Internazionale Superiore di Studi Avanzati and INFM-DEMOCRITOS National Simulation Center, via Beirut 2-4, I-34014 Trieste, Italy 共Received 9 August 2002; revised manuscript received 2 January 2003; published 7 March 2003兲 We present a first principle investigation of the structural, electronic and magnetic properties of Fe2 SiO4 Fayalite, the iron-rich end member of the (Mg,Fe) 2 SiO4 olivine solid solution, naturally occurring in the Earth’s upper mantle. Local spin-density approximation and spin-polarized generalized gradient approximation ( ␴ -GGA) results are compared; ␴ -GGA appears to provide an overall better description of the structural properties. The ground-state spin configuration is investigated and the antiferromagnetic spin arrangement consistent with a superexchange mechanism through oxygen orbitals is found to be preferred. Electronic structure calculations using both exchange and correlation functionals predict a metallic ground state, contrary to experimental evidence that indicates a insulating, possibly Mott-Hubbard, behavior. In fact, by comparison of our DFT results with the RPA solution of a simple ad hoc Hubbard model, we were able to estimate the average short-range electron-electron repulsion parameter U. This quantity turns out to be larger than the relevant band width, and therefore, we support the Mott-Hubbard nature of the experimentally observed insulating behavior. DOI: 10.1103/PhysRevB.67.094106

PACS number共s兲: 71.15.Mb, 75.30.Et, 91.60.Ed

I. INTRODUCTION

Fayalite is the iron-rich end member of (Mg,Fe) 2 SiO4 olivine 共orthorhombic structure兲, one of the most abundant minerals in Earth’s upper mantle. The main discontinuity observed at 410 Km depth inside Earth in seismic wave velocities, density and electrical conductivity is commonly attributed to the olivine-to-spinel transition that these solid solutions undergo at upper-mantle pressure and temperature conditions.1,2 Occurrence of phase-transition ranges from 6 ⫼8 GPa 共Fayalite兲 to 12⫼14 GPa 共Forsterite兲 for pure compounds.3,4 At room temperature and pressure Fayalite is insulating and experimental work indicates a Mott-Hubbard type mechanism.5,6 The magnetic structure of Fe2 SiO4 Fayalite has been studied with Mo¨ssbauer spectroscopy and neutron diffraction and it is reported to be a 共noncollinear兲 antiferromagnetic compound below a Ne´el temperature of ⬃65 K. 7 The strong anisotropy in the measured magnetic susceptibility 共and also the noncollinearity of the spin arrangement兲 supports the idea of strong correlations among the crystal structure and the electronic or magnetic properties. In fact, residual 共nonperfectly quenched兲 orbital moments on iron ions, whose magnetic moments are reported to be larger than their spin only value, allows the crystal field to act on the magnetic moments via the spin-orbit coupling and produces the observed magnetocrystalline anisotropy.7 In recent years the application of ab initio techniques to the study of systems of geophysical interest has expanded considerably and even detailed studies of the thermodynamical properties of some systems are available.8 –12. However, the treatment of iron containing minerals bears additional difficulties due to the possible presence of strong electroncorrelation effects. In this work the static, low pressure, equilibrium properties of Fe2 SiO4 Fayalite are studied from first 0163-1829/2003/67共9兲/094106共7兲/$20.00

principles. To our knowledge, no previous theoretical study of the structural and electronic properties of Fayalite has been presented to date and the nature of the observed insulating behavior and even the precise ground-state magnetic configuration need to be established. We use-density functional theory 共DFT兲 in a plane-wave pseudopotential framework comparing results obtained with local spin-density 共LSDA兲 and spin-polarized generalized gradient ( ␴ -GGA兲 approximations. This allows to establish the relative merit of the two schemes and to identify those properties of Fayalite that are reasonably described by these electron-gas based schemes and which ones are instead poorly described and point toward strong correlation effects. The paper is organized as follows. In the following section we study the ground-state equilibrium geometry and the magnetic properties of Fayalite by comparing the 共relaxed兲 crystal structures and the total energies obtained with both LSDA and ␴ -GGA functionals. Comparison of the energetics of the two spin configurations that are compatible with the crystallographic evidence allows us to identify the groundstate magnetic configuration. The ␴ -GGA approximation is found to provide an overall more accurate description of the structural properties of this compound thus we adopt this functional in the study of the band structure in the following section 共Sec. III兲. In spite of the fact that Fayalite is experimentally observed to be insulating at ambient pressure and temperature we obtain, within ␴ -GGA, a metallic behavior. Studying the ␴ -GGA Fermi surface and the charge distribution of the electronic states close the Fermi level, strong indication is found of the importance of the electronic correlations along iron-ion chains extending in the 关010兴 direction. In Sec. IV, we estimate the average on-site repulsion associated to local charge fluctuation on iron ions comparing constrained electron-number DFT calculations with the RPA electronic compressibility. The obtained, Hubbard U, on-site

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COCOCCIONI, DAL CORSO, AND DE GIRONCOLI

FIG. 1. The unit cell of Fayalite. Large dark ions are Fe, small dark ions are O, light ions are Si.

interaction turns out to be larger than the relevant electronic band width and shows that Mott-Hubbard charge localization is the driving mechanism of the observed insulating behavior. II. CRYSTAL AND MAGNETIC STRUCTURE

In this section we study the structural and magnetic ground-state properties of Fayalite. The main purpose of this investigation is to find the magnetic ground state of the system and to understand the interplay among the crystal structure and the magnetic configuration. From x-rays diffraction studies it is known that Fayalite has an orthorhombic cell, whose experimental lattice parameters are 共in atomic units兲 a⫽19.79, b⫽11.50, c⫽9.11. The unit cell 共depicted in Fig. 1兲 contains four formula units, 28 atoms: 8 iron, 4 silicon, and 16 oxygen atoms. Silicon ions are tetrahedrally coordinated to oxygens, whereas iron-ions occupy the centers of distorted oxygen octahedra. The point group symmetry of the nonmagnetic crystal is mmm (D2h in the Schoenflies notation兲 and the space group is Pnma. The magnetization of iron reduces the original symmetry and only half of the symmetry operations survive. The general expression for the internal structural degrees of freedom is given in Table I in the Wyckoff notation.13 Iron sites can be divided in two classes 共see Fig. 1 and Table I兲: Fe1 centers which are structured in chains running parallel to the b, 关010兴, side of the orthorhombic cell, and Fe2 sites which belong to mirror planes for the nonmagnetic crystal structure perpendicular to the b side and cutting it at 1/4 and 3/4 of its length. The main structural units are the iron centered oxygen octahedra, which are distorted from the cubic symmetry and tilted with respect to each other both

FIG. 2. The two possible spin configurations. 共1兲 AF interaction between edge-sharing octahedra; 共2兲 AF interaction between cornersharing octahedra. The zig-zag lines connect first neighbor Fe1-Fe2 iron sites 共almost on the same 001 plane兲.

along the chains and on nearest Fe2 sites. As for the magnetic structure, Fayalite is known to be an antiferromagnetic 共AF兲 compound with slightly noncollinear arrangement of spin on Fe1 iron site. This noncollinearity 共expected to be due to the coupling of spin orbit with the crystal field of the tilted octahedra7兲 will not be addressed here. Magnetic moments along the central and the edge Fe1 chains are antiferromagnetically oriented, but no constraint about the relative orientation of moments on iron belonging to different classes comes from experiment. In order to investigate the ground-state spin configuration of Fayalite we performed DFT calculations in a plane-wave pseudopotential framework using the PWSCF open source package14 both in the LSDA and ␴ -GGA approximations 共the PBE functional15 is used in the latter case兲. Ultrasoft

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STRUCTURAL, ELECTRONIC, AND MAGNETIC . . . TABLE I. The Wyckoff parameters for each ionic species. Ion

Class

Coordinates

Fe1

4a

Fe2, Si, O1, O2

4c

O3

8d

共0,0,0兲, 共1/2,0,1/2兲共0,1/2,0兲, 共1/2,1/2,1/2兲 ⫾共u,1/4,v兲, ⫾共u⫹1/2,1/4,1/2-v兲 ⫾共x,y,z兲, ⫾共x,1/2-y,z兲, ⫾共x⫹1/2,1/2-y,1/2-z兲, ⫾共x⫹1/2,y,1/2-z兲

TABLE II. The relaxed lattice parameters and the total energy for spin configuration 1 and 2. a 关a.u.兴

b 关a.u.兴

c 关a.u.兴

19.79

11.50

9.11

⌬E 关ryd/cell兴

Expt.

pseudopotentials16 were used for Fe 共generated in the 3d 7 4s 1 4p 0 configuration兲 and O while a norm-conserving pseudopotential was used for Si atoms. A 31 Ry kineticenergy cutoff was found sufficient for the plane-wave expansion of the electronic states, while a 248 Ry cutoff for the augmented charge density was used. The special point technique was used for efficient sampling of the first Brillouin17 zone 共BZ兲 and 16 points in the irreducible BZ 共IBZ兲 were found to give a good convergence of the total energy and the ionic forces. As it will be discussed in the following section Fayalite electronic structure turns out to be metallic and the smearing technique18 was adopted to smooth the Fermi distribution with a broadening width of 10 mRy. Two spin configurations, shown in Fig. 2, are compatible with experiments: in the first one magnetization of Fe2 ion is opposite to that of the closest Fe1 iron, and one obtains AF order between ions at the center of edge-sharing oxygen octahedra and ferromagnetic order between corner sharing octahedra. In the second spin configuration the opposite is true and AF order occurs between corner sharing octahedra. This second magnetic structure is consistent with an iron-iron magnetic interaction via a superexchange mechanism through oxygen p orbitals. This mechanism provides AF coupling when the Fe-O-Fe angle is close to 180 degrees 共corner sharing octahedra兲 and a weak ferromagnetic interaction when this angle is about 90 degrees 共edge sharing octahedra兲. For each spin configuration 共and both exchangecorrelation functionals兲 we performed a structural relaxation of the internal ionic degrees of freedom 共Table I兲 and of the unit cell lattice parameters. In the final relaxed geometries force components did not exceeded 10⫺3 a.u. and the stress tensor vanished within 4 –5 kBar corresponding to errors in both the atomic positions and the lattice parameters of about 共and usually less than兲 two parts per thousand. The results of our calculations are collected in Tables II and III. For both exchange-correlation functionals the second spin configuration has lower total energy and is, therefore, the magnetic ground state of the system. Superexchange mechanism is thus confirmed to be responsible of magnetic interaction and the difference between the total energies of the spin configurations 共about 10 mRy/cell in ␴ -GGA兲 gives a measure of the exchange parameters between Fe1 and Fe2 iron 共it is about sixteen times the average J Fe1,Fe2 ). This small value is consistent with the low value of Ne´el temperature (⬃65 K at ambient pressure兲. Despite the fact we cannot account for the canting of the magnetic moments on iron sites 共because only collinear magnetism is allowed in the

LSDA 1 2 ␴ -GGA 1 2

19.23 (⫺2.8%) 11.09 (⫺3.6%) 8.99 (⫺1.2%) 0.0598 18.36 (⫺7.2%) 10.77 (⫺6.3%) 9.42 共⫹3.4%兲 19.96 共⫹0.9%兲 11.45 (⫺0.4%) 9.21 共⫹1.1%兲 19.78 (⫺0.1%) 11.29 (⫺1.8%) 9.36 共⫹2.7%兲

0.0113

calculation and spin-orbit coupling is not included兲, their absolute value is found to be in good agreement with experimental results.7 Our calculated value for the magnetic moment is 3.8 ␮ B per magnetic ion and compares quite well with the experimental result of 4.4 ␮ B 共both for Fe1 and Fe2 ions兲 at a temperature of about 10 K if we consider that this latter moment contain a residual orbital contribution 共the spin-only value is 4 ␮ B ). 7 Although both LSDA and ␴ -GGA functionals provide a reasonable description of the structural properties of the system it can be observed that for both spin configurations ␴ -GGA gives lattice parameters that are in better agreement with experimental data than LSDA does. Moreover ␴ -GGA structural properties are not very sensitive to spin ordering while LSDA appears to be more sensitive. A detailed comparison of the internal structural degrees of freedom is given in Table III, where the theoretical values of these quantities obtained for the ground-state spin configuraTABLE III. The relaxed ionic positions in the unit cell described by the Wyckoff parameters. Ion

u

v

Fe2 Si O1 O2 O3

0.7800 0.5975 0.5929 0.9530

0.5147 0.0708 0.7313 0.2923

Fe2 Si O1 O2 O3

0.7591 0.5905 0.5886 0.9353

Fe2 Si O1 O2 O3

0.7766 0.5955 0.5964 0.9467

x

y

z

0.1637

0.0384

0.2885

0.1663

0.0224

0.2822

0.1650

0.0292

0.2782

Expt.

LSDA 0.5080 0.0716 0.7471 0.2878

␴ -GGA

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0.5170 0.0710 0.7376 0.2979



FIG. 4. The electronic density of states. The Fermi level is set to 0. FIG. 3. The electronic band structure originated from the iron d bands.

tion using LSDA and ␴ -GGA are compared with the experimental data. We can observe an overall good agreement between theory and experiment but again ␴ -GGA results are in closer agreement with experimental results. III. ELECTRONIC STRUCTURE

In the preceding section, especially using ␴ -GGA approximation, we have obtained a good description of the ground-state crystal structure and spin configuration of Fayalite. However, the same cannot be said for its electronic band structure. In Fig. 3 we report the ␴ -GGA band structure of Fayalite around Fermi energy that appears to be a band metal, with two iron d-bands crossing the Fermi level, in spite of the fact that experimentally it is found to be insulating. A finite density of state 共DOS兲 is present at the Fermi energy, see Fig. 4. Notwithstanding the obvious failure of ␴ -GGA to describe the nature of the experimental ground state 共that indicates the importance of stronger electron correlation effects兲 the analysis of the theoretical band structure is not without interest as it embodies some information on the nature of the electronic level around the Fermi energy and their lowenergy excitations. All the electronic bands displayed in Fig. 3 originate from iron d level. Due to the AF ordering, spin-up and spin-down levels are degenerate. Fe2⫹ ions occupy the center of distorted oxygen octahedra. The quasioctahedral local symmetry determines a crystal field splitting of the d levels in two groups: a lower-energy t 2g 共three-fold兲 level and an higherenergy e g 共two-fold兲 level. It is convenient to refer to the simple scheme reported in Fig. 5 and compare it to Fig. 4. Every iron ion in Fayalite is in its high-spin configuration because the crystal-field splitting U CF due to the oxygen octahedra, is smaller than the exchange splitting U X due to first Hund’s rule. From an analysis of the band structure it results

that U CF⬃ 21 U X . Therefore, five electrons per iron fill up completely the majority-spin d states 共Hund’s rule is fulfilled兲, whereas one more electron per iron atom partially fills the minority-spin t 2g bands that are crossed by the Fermi level. The distortion of the Fe-O octahedra from cubic symmetry induces a partial mixing of majority-spin t 2g and e g states and a small splitting of the minority-spin t 2g levels in two groups of 12 bands each 共visible in Fig. 4 only as a very deep dip in the DOS at ⬇0.3 eV above the Fermi level due to the finite smearing width used in the plot兲. The t 2g band width around the Fermi level is about 1.5 eV 共or about 1 eV if only the 12 lowest t 2g bands are considered兲. One important feature of the electronic band structure is the marked flatness of the bands around the Fermi surface along 关001兴 (⌫Z line in Fig. 3兲 and 关100兴 (⌫X, SY, ZU, RT兲 directions and the relative large dispersion in the 关010兴 direction (⌫Y, XS, ZT, UR兲. This means that the calculated electronic states in the minority-spin t 2g bands are rather localized in the x an z direction and mainly extend along the zig-zag chains in the y direction. A confirm of the quasi-onedimensional character of these states comes from Fig. 6, where the Fermi surface of Fayalite, as obtained from ␴ -GGA calculation, is shown. Two inter-crossing sheets run almost perpendicular to the ⌫Y direction. This scenario is confirmed by Fig. 7, where the chargedensity corresponding to the electronic states at the Fermi energy is shown by isosurfaces drawn at a value equal to 2% of the maximum charge density. As evident from the plot,

FIG. 5. A simple scheme about the splitting of the iron d levels in the crystal.

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STRUCTURAL, ELECTRONIC, AND MAGNETIC . . .

FIG. 6. The GGA Fermi surface of Fayalite, plotted in 1/8 of the BZ.

electronic states at the Fermi-level actually belong to both Fe1 and Fe2 irons but extreme localization occurs on iron sites so that electrons are very little extended away from the atomic sites. The spatial localization of these states suggests that correlation effect due to on-site Hubbard repulsion, treated only at the mean-field level in the LSDA and ␴ -GGA calculations, may be important and may explain the insulating behavior of Fayalite.19

vary at will the corresponding occupation, it is costumary to remove hybridization of the localized orbitals with the environment by zeroing all their hopping integrals. This is a natural option in the all-electrons 共LMTO or LAPW兲 methods employed in Ref. 20–23, but less so in the plane-wave pseudopotential approach used here, where hopping integrals are not readily available, and therefore some adjustments are needed. In our approach, presented in more detail in the following, a density fluctuation around the GGA ground state is induced by injecting or removing a fraction of electron in the material. The effective U is computed as usual from the quadratic variation of the total energy, obtained via the linear shift in the Fermi energy with respect the occupation number. Owing to the fact that the states at Fermi energy originate from Fe ions, see for instance Fig. 7, it should be at least approximately correct to assign the resulting Coulomb repulsion to these states. The effect of orbital rehybridization upon charge redistribution, avoided in the original approaches via the removal of hopping integrals, is computed here from the noninteracting response of the system and subtracted from the total response. We assume that GGA provides a good mean-field-like solution, of Hartree or Hartree-Fock type, for the electronic states around the Fermi energy and compare its electronic compressibility, that is the curvature of the total energy with respect to long wavelength density fluctuations, with the RPA result obtained for a simplified Hubbard model that describes the physics of the electrons responsible for conduction 共or lack of it兲 in Fayalite. This is given in the following expression:

IV. ON-SITE HUBBARD REPULSION

We want now to estimate the on-site repulsion for electrons on the iron sites and compare the result with the relevant band width around the Fermi level. This problem has been addressed previously in the literature and several related methods have been proposed20–23 to calculate the onsite Hubbard repulsion in systems with localized d or f electrons. The basic idea in all these approaches is to calculate the change in the total energy as a function of the local occupation. The curvature of the energy surface gives the effective U. In order to isolate the orbitals of interest and

H⫽

U

n i,l n i,l ⬘ , 兺兺 t i j c i,l† c j,l ⬘⫹ 2 兺i 兺l l 兺具 i, j 典 l,l (⫽l) ⬘

共1兲

where t i j is the hopping amplitude between two nearest iron on the same zig-zag chain 共the model is therefore effectively unidimensional兲, U is the average on-site repulsion, c †il and c il are creation and annihilation fermionic operators on the lth orbital of the ith site, while n il (⫽c †il c il ) are the corresponding occupation numbers. The spin degrees of freedom never appear in this simple model because we consider only the three t 2g minority spin levels for each iron atom and thus, fixing the site, actually fixes also the spin. This simple model contains the two competing parameters 共the hopping term and the on-site repulsion兲, which describe the physics of the electrons around the Fermi level. In order to establish a connection between the parameter in this model and the DFT calculation we need to treat it at the same level of approximation, i.e., at the mean-field level. One can thus assume that E DFT ⬇ 具 H 典 M F ⫽T 关具 n 典兴 ⫹ ⫹E x 关具 n 典兴 ⫹E bg ,

FIG. 7. The charge-density generated by states around the Fermi level.

⬘

N s U 共 m⫺1 兲具n典2 2 m 共2兲

where T is the average kinetic energy, m is the number of states per site 共3 t 2g levels on each ion in our case兲, 具 n 典 is the average electronic occupation of each iron site, and N s is the number of iron site in the unit cell 共8兲. The exchange term, E x , that would contain information about the exchange pa-

094106-5



rameter J, will be neglected here. One can think of it as adsorbed approximately in the Hartree term so that the resulting U is effectively an U e f f ⬇U Coul ⫺J. The total energy does not allow to extract a value for U due to the presence in the actual system not only of the mentioned states close to the Fermi energy but also of all the other electrons that determine the structural stability of the system and are hidden in Eq. 共2兲 in the background energy E bg . The electronic response to long-wavelength external perturbing potential is however, to a large extent insensitive to the background. And we can therefore compare the electronic compressibility obtained from the DFT calculation 共which is proportional to limq→0 ␹ (q,q), where ␹ is the density response function of the electronic system兲 with the second-order derivative of the total energy in Eq. 共2兲 with respect to 具 n 典 . For a charged Fermi liquid the compressibility only depends on the short-range part of the screened electron-electron interaction,24 which can be identified with the localized U parameter in our model. We obtain

冋

册

⳵␮ 1 m ⫺ , U⫽ m⫺1 ⳵ n ␯ 兲共共0兲具典

共3兲

where ⳵ ␮ / ⳵ 具 n 典 is the linear variation of the Fermi energy, when the number of electron in the unit cell is varied and a uniform compensating background is implicitly added in order to keep the system neutral. As already mentioned this quantity is related to the q→0 limit of the density response function and analysis of this limit shows that, since all macroscopic Coulomb divergences must cancel exactly, the presence of a compensating background in the definition of the compressibility is required.24 This leaves in the q→0 limit only a finite average shift in the macroscopic component of the effective self-consistent potential, that is correctly accounted for in the computed Fermi-energy variation, when the number of electrons is varied in the periodic system 共which corresponds to a q⫽0 calculation兲. The inverse of the electronic DOS at the Fermi-level 1/␯ (0) is nothing but the same quantity for the noninteracting case that must be subtracted from the interacting result, in order to isolate the contribution originating from 共short-range兲 interaction. The relevant Fermi-energy derivative have been computed numerically adding ⫾0.01 electrons per unit cell and monitoring the variation of the Fermi energy and of the iron 3d states occupancy 具 n 典 . This has been done in two ways: 共i兲 assuming that the injected 共or removed兲 fraction of electron goes completely into the iron 3d states how one would naively expect from the analysis of the density of state at the Fermi energy, and 共ii兲 actually computing the variation of

1

Syun-Iti Akimoto and Hideyuki Fujisawa, J. Geophys. Res. 70, 443 共1965兲. 2 Syun-Iti Akimoto, Eiji Komada, and Ikuo Kushiro, J. Geophys. Res. 72, 679 共1967兲. 3 C.R. Bina and B.J. Wood, J. Geophys. Res., 关Solid Earth兴 92, 4853 共1987兲.

具 n 典 defined as the average d-level occupancy per iron sites, as obtained from the projected density of states. This second value gives an enhanced value of the derivative, that takes into account the fact that, due to screening effects, part of the additional electron can redistribute to other orbitals. In fact we found that the actual variation of 具 n 典 in this case is only 65% of the bare variation. From Eq. 共3兲 we obtain U ⫽2.4 eV and U⫽4.5 eV for the two recipes, respectively. Both values, especially the second one, computed taking into account the possibility of charge redistribution to other orbitals, are much larger than the calculated electronic band width (⬇1.5 eV) around the Fermi level. This implies that the electrons in the minority t 2g bands do not have enough kinetic energy to overcome the repulsion they experience on each iron site and thus a Mott localization occurs giving rise to the observed insulating behavior. V. CONCLUSIONS

Using a DFT approach within the LSDA and ␴ -GGA approximations a good description of the structural and magnetic properties of Fayalite at ambient conditions has been achieved. In particular our calculations indicate that the magnetic ground state of Fayalite corresponds to an AF order between corner sharing octahedra and ferromagnetic order between edge sharing octahedra, in agreement with an oxygen-mediated superexchange mechanism for the ironiron magnetic interaction. The ␴ -GGA electronic structure of Fayalite is qualitatively incorrect since this mineral is described as a band metal by ␴ -GGA, whereas it is experimentally an insulator. However, we have been able to show that the average on-site repulsion among electrons around the Fermi level is much larger than the relevant band width thus providing support to the Mott-Hubbard origin for its insulating behavior. This latter result is quite remarkable from a methodological point of view because it allows to extract some 共nonquantitative兲 information about a strongly correlated system only using ingredients from a conventional ␴ -GGA DFT calculation. ACKNOWLEDGMENTS

We gratefully acknowledge illuminating discussions with S. Baroni, M. Fabrizio, and G. Santoro. We thank A. Kokalj for suppling XCrySDen graphical package. This work has been supported by the MIUR under the PRIN program and by the INFM in the framework of the Iniziativa Trasversale Calcolo Parallelo.

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