First-principles studies of the structural and electronic properties of

PHYSICAL REVIEW B, VOLUME 65, 054107

First-principles studies of the structural and electronic properties of pyrite FeS2

Joseph Muscat,1 Andrew Hung,2 Salvy Russo,2 and Irene Yarovsky21CSIRO Minerals, Bayview Avenue, Clayton South, Victoria, 3169, Australia

2RMIT University, GPO Box 2476V, Melbourne 3001, Victoria, Australia~Received 6 September 2000; revised manuscript received 7 May 2001; published 11 January 2002!

We present a study of the structural and electronic properties of Pyrite FeS2 performed using both localizedbasis set and plane wave first-principles calculations. Calculations performed using either Gaussian or planewave basis sets yield results consistent with each other. Small differences in the computed geometries areshown to be due to the choice of pseudopotential employed in the plane wave calculations. The computeddensities of states are relatively insensitive to the form of basis set and pseudopotential used. We find thatdensity functional and hybrid approaches predict properties such as geometry and densities of states in goodagreement with experiment but that the agreement between the results from Hartree–Fock~HF! calculationsand experiment is poor. The reasons for the poor performance of HF theory in this system are examined and arefound to be due to the neglect of electronic correlation.

DOI: 10.1103/PhysRevB.65.054107 PACS number~s!: 31.15.Ew

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I. INTRODUCTION

Metal sulphide minerals are of vast industrial and enomic importance. This is because they are the main soof metals such as zinc, copper and lead. In addition, precmetals such as gold and platinum group elements are oassociated with the sulphide ores.1 Sulphides exhibit a widerange of physical, electronic and chemical properties whare under investigation for a range of possible applicatiincluding solar cells,2 solid state batteries3 and catalysis.

Of the sulphide minerals, pyrite structured FeS2 is themost common form,4 and serves as a useful model systemunderstanding the chemical and physical properties ofclass of minerals. Its relatively simple structure, illustratedFig. 1, makes it amenable for study using a range ofab initiotechniques. The chemistry of sulphides, and in particuFeS2, is of great importance from an environmental persptive since pyrite is a major component of waste matefrom mining processes. The oxidation and subsequent dilution of FeS2 results in the formation of sulphuric aciwhich causes a significant increase in the acidity of lakrivers and streams.5 This increase of acidity can have aadverse effect on marine wildlife and needs to be carefmonitored and controlled in order to minimize the enviromental impact of any mining operation.

There have been a number of recent experimental stuof the bulk and surfaces of pyrite using a wide rangetechniques including x-ray photoelectron spectrosco~XPS!,6,7 low energy electron diffraction~LEED!, ultravioletphotoelectron spectroscopy~UPS! and scanning-tunnellingmicroscopy ~STM!,8 inverse ultraviolet photoemissiospectroscopy9,10 and x-ray adsorption spectroscopy.11 How-ever, in many cases, properties such as the atomic struof surfaces and interfaces, the nature of defects andchemistry and energetics of molecular adsorption are dcult, if not impossible, to determine using available expemental techniques.

The increased sophistication ofab initio methods coupledwith a growth in computer power means that it is now posible to model rather complex physical and chemical s

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tems. Simulation is becoming established as a complemtary technique to experiment for the study of minerals acan provide invaluable primary information on well charaterized systems and allows for structural, chemical andichiometric effects to be isolated. The additional informatiprovided by simulation can facilitate the interpretationexperimental results. Recent work has demonstratedcomputer simulations based onab initio techniques can provide remarkable insights into a diverse range of industriaimportant systems and processes such as the oxidatiometals12 and the chemistry of water on catalytically impotant metal oxides.13–15

Most ab inito schemes are based on either the HartreFock ~HF! approximation, in which electronic exchange

FIG. 1. A bulk unit cell of Pyrite, FeS2.

©2002 The American Physical Society07-1

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MUSCAT, HUNG, RUSSO, AND YAROVSKY PHYSICAL REVIEW B65 054107

treated exactly but correlation effects are neglecteddensity-functional theory~DFT! in which both exchange ancorrelation are approximated. In DFT, in the form of texchange-correlation functional, EXC@r# is approximated ei-ther using the local density approximation~LDA !29 or thegeneralized-gradient approximation~GGA!.33 A number ofcomputational investigations of bulk iron pyrite FeS2 havebeen performed using various DFT based methods; thesclude Xa with LDA;16 linear muffin-tin orbital atomic sphereaproximation~LMTO-ASA! within the LDA,17 more recentLMTO-ASA calculations using both the LDA and GGA;18

pseudopotential-mixed plane wave Gaussian basis LDAGGA;19 and full-potential nonorthogonal local-orbital minmum basis band-structure method within the LDA.20 Calcu-lations of the band structure by these methods predictFe21 ion to be in the low-spin, diamagnetic configuratio~filled t2g states, empty eg states!, with calculated band gapin reasonable agreement with experiment varying fraround 0.319 to 0.85 eV.20

Calculations of the optimal cell volume are typicalfound to be underestimated by LDA, while applicationgradient corrections in the form of the GGA tends to ovcorrect yielding slightly larger predicted cell volumes18 withrespect to experiment. Studies including geometry optimtions of the pyrite unit cell have tended to result in a pdicted S–S bond length greater than the experimentally msured value. Calculated bulk moduli are usuallyreasonable qualitative agreement with experiment, with vues of about 185 GPa20 and 165 GPa.21 A value of 675 GPa~overestimating the experimental value by roughly fourfo!obtained by Temmermanet al.17 has been attributed to thneglect of sulphur coordinate relaxation.

To establish the reliability of the results produced frosimulation, it is essential that the theoretical frameworkwell as the particular computational parameters ofmethod utilized are tested and validated against experimeresults, and can therefore be shown to reliably modelsystem in question. However, this is often limited by tapproximations which are made in differentab initio tech-niques in order to make a calculation computationally trtable. By evaluating what influence some of the differeapproximations in common use have on the computed perties of the system of interest, it is possible to judgereliability of a given type of theoretical approximation fopredicting a particular set of properties. The level of acracy and applicability of some of the most commonab initiotechniques has yet to be quantified for metal sulphide merals such as FeS2.

Before undertaking the study of the surface propertieschemistry of pyrite, it is essential to have a good understaing of the bulk material. Hence, in the current work, we seto further our understanding of the relationship betweenelectronic structure and the physical and chemical propeof FeS2. In addition, we seek to determine the reliabilitythese theoretical techniques and computational parametea prerequisite for future studies of the surfaces and interfaof pyrite. We present the results from one of the first studusing two differentab inito methodologies on FeS2, namelythe use of HF and DFT in conjunction with a localize

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II. METHOD

All calculations have been performed using the periodlinear combination of atomic orbitals~LCAO! formalism asimplemented in theCRYSTAL9822 package and the plane wavpseudopotential technique using theCASTEP23 software pack-age.

In the LCAO methodology employed within thCRYSTAL98 code, the Bloch orbitals of the crystal are epanded using atom centred Gaussian orbitals with s, psymmetry. The main approximation in the current calcutions is the choice of the local basis set. Initial attemptsperform a single point energy calculation of bulk FeS2 at theexperimental cell parameters with the Fe basis set frTowler et al.24 and a S basis set used in previous studiessulphides25 resulted in convergence problems with hundreof SCF cycles needed to achieve convergence. Tests revethat the cause of these problems was due to the S basiwhich was optimized for the S22 ion in PbS. The diffuseouter exponents required to describe the S22 ion in galenaresult in linear convergence problems when used to descS2 ions in pyrite due, in part to the significantly shorter S–bond length in FeS2 compared to PbS. We have used tbasis set from Mianet al.25 incorporatingd-symmetry polar-ization functions as a starting point and reoptimized the ou4sp and 5sp exponents with respect to the total energy forS2 ion in free space before reoptimizing in bulk FeS2 at theexperimental geometry. Using the resulting S2 basis set, forwhich the reoptimized 4sp and 5sp basis functions aresented in Table I, a single point bulk FeS2 calculation can beconverged in a few tens of SCF cycles. It is more difficultdetermine whether a basis set is converged when usingcalized basis functions as opposed to plane waves. Noneless we perfomed tests using an hierarchy of modified FeS basis sets incorporating different numbers of valenced-symmetry functions. Higher angular momentum f-g-symmetry functions are not available in theCRYSTAL98

code. We found that the Fe basis set from Towler andmodifed S2 basis set described above gave the best commise between computational expense and accuracy.

A variety of self consistent treatments of exchange acorrelation have been used including HF, the LDA to DFthe GGA to DFT and the B3LYP functional based on Beckthree parameter hybrid functional~B3LYP! employing acombination of HF and DFT exchange.26 The DFT-LDA cal-culations were performed using LDA exchange27 and the

TABLE I. The optimized S2 Gaussian basis set.ak representsthe exponents and Cs , Cp and Cd represent the coefficients of th4sp, 5sp and 3d functions.

Shell ak Cs Cp Cd

4sp 0.280 1.0 1.05sp 0.126 1.0 1.03d 0.300 1.0

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FIRST-PRINCIPLES STUDIES OF THE STRUCTURAL . . . PHYSICAL REVIEW B65 054107

correlation functional parametrized from Monte Carcalculations28 by Perdew and Zunger29 whereas the GGAcalculations employed the exchange and correlation futionals by Perdew, Burke and Ernzerhof.30,31The B3LYP hy-brid functional used in this study is a modified versionBecke’s original three-parameter exchange-correlafunctional26 as suggested by Stephenset al.32 where the ex-change and correlation is given by

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change functional,33 and the correlation is given by a combination of the Lee, Yang and Parr~LYP!34 and the Vosko,Wilk and Nussair35 functionals. Becke’s original three empirical parameters (a050.2, ax50.72, andac50.81) areused, which were found to optimize the atomization energionization potentials and proton affinities of a numbersmall molecules.26 Solid state calculations using this funtional have been shown to yield band gaps in excellagreement with experiment.36

Sampling of K-space has been performed using PaMonkhorst grids22,37 of shrinking parameter 8, yielding 4symmetry inequivalent K-points for bulk FeS2. Tests revealthat increasing the number of K-points produces no signcant difference in the computed structure or energy ofcrystals~the total energy of the bulk crystals is convergedwithin about 1E–5 eV!. CRYSTAL98 computes the matrix elements of the Coulomb and exchange terms by direct smation of the infinite periodic lattice. The truncation of thesummations is controlled by five Gaussian overlap critedetails of the control of these parameters is availaelsewhere.22,38 The values of the overlap criteria chosenthe current study were high~ITOLS parameters set to 1027,1027, 1027, 1027 and 10214) in order to converge numericaerrors to 1 meV in the total energy.39 The structural optimi-zations were converged to a tolerance of 0.01 Å in cellrameters and 1024 eV in the total energy using a modifieBroyden–Fletcher–Goldfarb–Shanno~BFGS! minimizationalgorithm.40

CASTEPis based on the DFT framework with the crystaline orbitals expanded in a truncated plane wave basis, inporating all terms with kinetic energy below a prescribenergy cutoff Ecut. The exchange-correlation energy hbeen computed using functionals based on the LDA paretrized by Perdew–Zunger~as used in theCRYSTAL98 calcu-lations! and GGA functionals by Perdew and Wang.41 Testswere performed to ensure that lattice parameters andenergy are converged with respect to plane wave cutofwithin the tolerances described in Sec. III. The basis set cvergence was checked by performing geometry optimtions using a range of increasing plane wave cutoffs usatisfactory convergence in the geometry and total enewas achieved.

Fe and S core orbitals were removed by applyingfrozen-core approximation, whereby a pseudopotentiagenerated which replaces the core states and incorporatenucleus as well as the core states as a single entity. Ra

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nodes present in the original valence orbitals around the cregion, which are required for orthonormalization with thcore orbitals, are removed producing ‘‘smooth,’’ nodelepseudo-valence orbitals, which reproduce the exact valeorbitals outside a prescribed core cutoff radius rc . The use ofa larger rc in generating the pseudopotential~thus generatinga ‘‘softer’’ pseudopotential! allows the use of a lower planwave cutoff and thus enables more efficient computatiThe effects of pseudopotential hardness on FeS2 structuraland electronic properties have been examined by comparesults acquired using Ultrasoft pseudopotentials42 againstthose obtained from using Troullier–Martinpseudopotentials.43 Wave function minimization is achieveby simultaneous all-bands optimization using a conjugagradients algorithm. The optimization of the unit cell wperformed by minimization of the stress tensor on the cusupercell through a BFGS40 scheme. Internal sulphur coordnates were optimized using a conjugate-gradient algoritThe sampling of the Brillouin zone was performed using tMonkhorst–Pack scheme. Tests were performed to enthat a sufficientK-point density was used for convergencethe lattice parameter and total energy. Fast Fourier traforms~FFT! were used to evaluate matrix elements. The Fintegration mesh is related to the number of basis functiused to describe the crystal, and thus varies according toplane wave cutoff energy as well as the supercell volumFor cell volume optimizations, the FFT mesh density pscribed is preferably higher than that required for total eergy convergence of the initial supercell.

We have calculated the bulk modulus, B, and its deritive, B8, of pyrite by fitting the Murnaghan equation ostate44 to the computed energy-volume curve.

III. RESULTS

A. Electronic structure

Experimentally, FeS2 exists as a diamagnetic mineral witthe metal ions in a low-spin Fe21 configuration.45 We haveperformed tests to determine the lowest energy electrostructure by performing single point calculations at the eperimental geometry with the Fe21 ions converged in eithethe high-spin ~paramagnetic! or low-spin ~diamagnetic!states. We find that the LDA, GGA and B3LYP calculatioyield a diamagnetic ground state, lower in energy compato the paramagnetic state by 17.0, 12.3 and 3.90 eV, restively. However, HF theory predicts the paramagnetic statebe the most stable by 2 eV. These differences are ducorrelation effects in Fe21 and will be discussed in Sec. IV

TABLE II. The charge state of the Fe and S ions~in ueu) com-puted using a Mulliken scheme.

Method Fe S

HF 1.66 20.83LDA 1.19 20.59GGA 1.24 20.62B3LYP 1.19 20.59

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We have performed an analysis of the charge on theand S sites using the Mulliken scheme.46 We note that thepartitioning of the charge density among individual atomsarbitrary and, in the case of the Mulliken analysis, depeon the local basis functions used. In the current calculatithe use of a consistent basis set approximation meansvariations in the population analysis with functional astructure do provide a useful guide to changes in the naof the electronic state. The Mulliken charge populationsthe Fe and S sites are given in Table II. HF theory yieldcharge of11.66ueu on the Fe sites whereas DFT and B3LYpredict pyrite to be far more covalent with a charge on thesites of about11.2 ueu. The tendency for HF theory to oveestimate ionicity has been reported in previous studiessemi-ionic systems such as TiO2.47

The densities of states~DOS! for bulk FeS2 computedwith the HF approximation, DFT and hybrid functionals uing CRYSTAL98 are presented in Fig. 2 and the main featuof the DOS are summarized in Table III. The valence banpredominantly composed of S 3p and 3d states hybridiwith Fe 3d states whereas the conduction band is mainlyto the unoccupied S 3d states~see Fig. 3! in agreement withrecent theoretical studies.48 The two peaks at around 13 to 1eV correspond to bonding and antibonding S 3s states~Fig.3! .

The LDA and GGA predict FeS2 to be a conducting material whereas HF and B3LYP calculations predict a bagap of 11 and 2 eV, respectively. The reason behind the oestimation of the band gap by HF theory arises from a lof screening in the exchange term leading to an overstazation of the occupied states. It is also interesting to noteHF theory predicts that the Fe 3d states lie at the bottomthe valence band at around26 eV whereas LDA, GGA andB3LYP place these states at top of the valence band, a21 eV. Furthermore, the gap between the bonding and abonding S 3s states is significantly overestimated bytheory~4.5 eV! compared with experiment~3 eV!; see TableIII. Clearly, HF theory yeilds a significantly different electronic structure to the other treatments of exchange andrelation tested. It is of interest to note that Eyertet al.,48

using the augmented spherical wave~ASW! method withinthe LDA, acquired a computed band gap of 0.95 eV, inmost exact agreement with the current most accurate exmental value~0.95 eV! ~see Table III!, in contrast to ourfindings using the LDA. However, it has been found thatcomputed electronic structure of pyrite~and hence the ban

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gap! can be sensitive to the choice of radii and locationempty muffin-tin spheres within the crystal lattice~see, forexample, Eyertet al.48 and Folkertset al.49!. Their choice ofatom-centered and empty sphere radii location, while givminimum linear overlap between spheres, probably resuin a computed band gap which is in fortuitous agreemwith experiment.

The DOS computed with LDA and GGA usingCASTEP

are presented in Fig. 4. TheCASTEPcalculations also predica metallic ground state within the LDA and GGA treatmenof exchange and correlation. Peak positions and band wiare in excellent agreement with the results computed us

FIG. 2. The computed densities of states of FeS2 computed us-ing CRYSTAL98 with HF, LDA, GGA and B3LYP treatments of exchange and correlation. Solid lines represent states due to Fedashed lines represent states due to S ions. The Fermi level hasset to 0 eV and the y-axis is in arbitary units.

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TABLE III. The positions and widths of the main features~in eV! in the computed densities of states
Method Band gap S 3p width S 3ssb S 3ssa Fe 3d S 2p S 2s

HF 11 7 13.5 18 5.5 - 6.5 171 234LDA Conductor 8 12.5 15 0 - 1.5 152 204GGA Conductor 7.5 12.5 15 0 - 1.5 152 205B3LYP 2 7 12.5 15.5 0 - 1 156 211CASTEPLDA Conductor 8 12.5 15 0 - 1.5CASTEPGGA Conductor 7.5 12 14.5 0 - 1.5LDA-ASW ~Ref. 48! 0.95 7.5 13.3 16.4 0 - 1.5Expt. ~Refs. 6,7,58,65,66! 0.95 6 13 16 0 - 1 162.5

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CRYSTAL98 ~see Table III! and are in good agreement witexperiment.

B. Structural properties

The structure of pyrite FeS2 is defined by the cell lengtha0 and the sulphur internal coordinate, xs ~see Fig. 1!. Thepyrite structure is cubic, similar to NaCl but with the anioreplaced by S2 dimers with their molecular axes orientatealong the four crystallographic111& orientations. Each sulphur atom is coordinated to another sulphur atom~the otherhalf of the S2 dimer! and three Fe atoms. Each Fe atomcoordinated to six sulphurs with equal Fe–S bond distanbut the octahedron is compressed along one axis.

The lattice parameters of the optimized structures of F2as computed using HF, LDA, GGA and B3LYP treatmentsexchange and correlation withCRYSTAL98 and with LDA andGGA usingCASTEP are presented in Table IV. All the optimized structures presented in Table IV are for FeS2 in thelow-spin diamagnetic state although we note that HF theyields a paramagnetic ground state for FeS2. The values of a0computed using theCRYSTAL98 code with LDA GGA, andB3LYP treatments of exchange and correlation are in gagreement with experiment, with LDA underestimating0.37% and GGA and B3LYP overestimating by 1.9% a3.7%, respectively. However, the cell volume compuwithin the HF approximation is in poor agreement with eperiment, with an overestimate of over 10%. The reason

FIG. 3. The projected densities of states for pyrite compuwithin the LDA. Dashed lines represent the total DOS.

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the failure of the HF approximation to predict the structuof FeS2 will be discussed in Sec. IV. Calculations of the cevolumes performed withCASTEP using the LDA yield anunderestimate of the cell volume~by 3.1%! while the GGAcorrects for this overbinding effect leading to an increascell volume~although it is still underestimated by 1.0% witrespect to experiment!. Similar trends were reported in thCRYSTAL98 results.

We have performed tests to check that the cell parameare converged with respect to the basis set and to determthe influence of the pseudopotential on the computed resIn the case of theCRYSTAL98 calculations, we find that thecomputed lattice constant is very sensitive to the presencd-symmetry polarization functions on the sulphur atomsreported for PbS.50 Removal of the d-symmetry functionleads to a systematic increase in the cell volume of betw1% and 2% and an increase in the total energy of the sysirrespective of the treatment of exchange and correlatThis suggests that quadrupolar polarization effects mustaken into account in order to yield reliable structural paraeters.

With regards to the plane waveCASTEPcalculations, testsrevealed that when using Ultrasoft pseudopotentials onand S, a plane wave cutoff energy of 400 eV~correspondingto a 27327327 FFT mesh! and Monkhorst–Pack k-poingrid density of 83838 were sufficient for convergence othe total energy to 0.06 eV/unit cell. These computatio

d FIG. 4. The densities of states of FeS2 computed usingCASTEP

with LDA, GGA treatments of exchange and correlation. Solid linrepresent states due to Fe ions, dashed lines represent statesS ions.

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TABLE IV. The computed cell parameters for pyrite. a0 is the cell parameter in Å and xs is the internalcoordinate in fractional units.

HF LDA GGA B3LYPMethod a0 xs a0 xs a0 xs a0 xs

CRYSTAL98 6.000 ~10.8! 0.397 5.386 ~-0.37! 0.378 5.520 ~1.9! 0.380 5.614 ~3.7! 0.387CASTEP 5.247 ~-3.1! 0.382 5.360 ~-1.0! 0.384VASP ~Ref. 53! 5.299 ~-2.2! 0.383Zeng ~Ref. 19! 5.441 ~0.46! 5.455 ~0.72!FP ~Ref. 20! 5.302 ~-2.1! 0.386Expt. ~Ref. 67! 5.416 0.385

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parameters result in convergence of the lattice parametewithin 0.01 Å for a0 and 1024 for the internal sulphur coordinate. Using the harder, Troullier–Martins pseudopotentfor Fe and S, an Ecut of 890 eV and Monkhorst–Pack grid o83838 were required for convergence of the total enera0 and xs to the same tolerance levels.

As noted above, the lattice parameters calculated withe LDA using Ultrasoft pseudopotentials converge toproximately 5.247 Å and 0.3818 for a0 and xs , respectively,deviating from the experimental values by23.3% and20.83%, respectively. Using the GGA functionals resultslattice parameter values which are in better agreementexperiment (a0 of 5.360 Å and xs of 0.3845!, studies reduc-ing the discrepancy in the cell volume and internal coornate to21.0% and20.13% with respect to experiment.can be seen that Ultrasoft pseudopotentials consistently oestimate the Fe-Fe cohesive energy, even with incorporaof gradient corrections. This suggests somewhat poor trferability of the Fe pseudopotential due to inadequatescription of the outer-core region as a result of an overly hcore cutoff radius. This is further supported by GGA calclations where we performed a cell optimization using tharder Troullier-Martins pseudopotentials. The calculatiopredicted lattice parameters of 5.478 Å and 0.3847, deviafrom experimental values by11.14% and20.08%. Thisresult is consistent with previous findings from GGA calclations on transition metals and transition metal solids, whGGA consistently underestimates the cohesive energyhence overestimates the cell length.51 We have found similartrends in lattice parameter variation with pseudopotenhardness from recent plane wave calculations on pureFe.52

The nearest S–S bond distances predicted by the diffetheories are summarized in Table V. The S–S bond dista

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computed withCRYSTAL98 using HF and B3LYP are in goodagreement with experiment whereas the LDA and GGA bodistances are over 5% too large. Overstimates of the Sbond distances of a similar magnitude were also reporterecent localized basis set DFT calculations.19 The currentCASTEPresults yield S–S bond distances in excellent agrment with experiment and with recent plane wave GGsimulations.53

The bulk moduli and their derivatives of FeS2 computedwith CRYSTAL98 using the LDA, GGA, and B3LYP function-als and withCASTEPusing the LDA and GGA are presentein Table VI. TheCASTEP calculations were performed usinUltrasoft pseudopotentials, other computational paramefor CRYSTAL98 andCASTEPcalculations were as describedSec. II. The bulk moduli predicted by the two programs wealmost identical within the LDA yielding a value of aroun210 GPa and the GGA giving 164 GPa. This level of agrment is probably somewhat fortunate considering theresmall difference in the computed cell volumes from the twcodes. B3LYP yields a lower bulk modulus than DFT, proably due to the overestimate of the cell parameter. The Gand B3LYP bulk moduli are in reasonable agreement wexperimental determinations of around 150 GPa54–56 al-though some experiments have measured the bulk modto be as high as 215 GPa.57 The value of the volume derivative of the bulk modulus is significantly harder to determiexperimentally~and is often just assumed to be 4! althoughone experimental study has measured it to be 5.557 in goodagreement with the current GGA and B3LYP results.

IV. DISCUSSION

We find that plane wave and Gaussian basis set calctions give results consistent with each other with LDA, GG

TABLE V. The computed S–S bond distances in Å.

Method HF LDA GGA B3LYP

CRYSTAL98 2.143 ~-0.9! 2.274 ~5.2! 2.295 ~6.2! 2.197 ~1.6!CASTEP 2.145 ~-0.8! 2.154 ~-0.4!VASP ~Ref. 53! 2.148 ~-0.6!Zeng ~Ref. 19! 2.294 ~6.1! 2.300 ~6.4!FP ~Ref. 20!Expt. ~Ref. 67! 2.162

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TABLE VI. The computed bulk moduli~in GPa! and their derivatives~in parentheses! of pyrite.

Method LDA GGA B3LYP

CASTEP 208 ~4.4! 164 ~5.2!CRYSTAL98 209 ~7.9! 164 ~3.5! 113 ~5.7!Expt. 143~4, assumed! 54 148~5.5! 55 157~-! 56 215~5.5! 57

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and B3LYP treatments of exchange and correlation all yieing structural parameters in reasonable agreement withperiment. TheCRYSTAL98 andCASTEPcalculations performedwithin the LDA overestimate binding energy, leading tounderestimate in the cell volumes. GGA functionals corrfor the overbinding and result in increased cell volumesabout 2%. The differences in the cell parameter compuusing CRYSTAL98 and CASTEP is probably due to the use othe pseudopotential approximation inCASTEP whereas inCRYSTAL98 an all electron basis set was used throughoOther GGA studies using the VASP53 plane wave code withdifferent computational parameters and parametrizationthe Ultrasoft pseudopotential gave slightly different lattiparameters~1.1% for a0, 0.25% for xs) compared to the current CASTEPresults.

The HF approximation fails to predict accurate structuparameters due to the nature of the spin on the Fe ions. Spyrite is diamagnetic, the Fe21 ions are in a low-spin configuration with fully occupied t2g and empty eg states. Elec-tron correlation effects will tend to reduce the electroelectron repulsion in the fully occupied orbitals leading toincrease in the binding of the electrons in these states.results in a decrease in the Fe21 ionic radius and hencesmaller lattice cell length. It seems likely that the compleneglect of correlation effects in HF theory would contributo the large overestimate of the cell parameter with respecexperiment. To test this hypothesis, we have performsupplementary geometry optimizations using HF theory wa posterioricorrelation corrections. We find that the additioof correlation effects, even at such a crude level, leadssignificant reduction in the overestimate of the cell parame~5.89 Å with correlation corrections, 6.00 Å uncorrecte!.Similarly, structural optimizations performed at the LDlevel of theory but without computing the correlation enerresults in an increase in the cell parameter~5.470 Å, com-pared to 5.386 Å with correlation!. Our calculations also revealed that the LDA, GGA and B3LYP methods give lowspin ~diamagnetic! solutions as the ground state of pyritwith computed total energies which are lower than the hispin ~paramagnetic! state by 17, 12.3 and 3.9 eV, respetively, while Hartree–Fock theory predicts the high-spstate to be more stable by 2 eV. This erroneous predictiodue to the lack of correlation in HF theory, as discussedSec. III. The ‘‘pure DFT’’ methods incorporate correlationsome extent, resulting in improved calculated structuralrameters and the correct prediction of spin state. The valuthe energy of the low-spin state relative to the high-spin sacquired using B3LYP is in line with expectations; since tB3LYP functional is a linear combination of HF and GGA,is not unreasonable to expect it to give a value whichbetween those acquired using these two methods, i.

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23.9 eV obtained from B3LYP lies between212.3 eV~GGA! and12 eV ~HF!.

In the current study, B3LYP calculations of the band gof pyrite predict a value of around 2 eV. The experimenvalue, determined using a number of different techniquesbeen measured to be between 0.7 and 2.62 eV, with the mreliable values~0.9 to 0.95 eV! coming from photoconduc-tivity measurements.58 As the B3LYP calculations are of thground state of the system, a disagreement with respecexperiment is not surprising. However, a recent studyshowed that calculations using the B3LYP treatment ofchange and correlation give predictions of the band gapwide range of minerals with an accuracy comparable to tof far more computationally expensive quantum Monte Casimulations.36

Considering the findings reported in Ref. 36, it seems sprising that B3LYP overstimates the band gap of pyrite bfactor of 2. However, the overestimate of the band gap mbe understood by considering the influence that defewould have on the real crystal. Spectroscopic studies hprobed the defect nature of natural and synthetic pyrit59

Luck59 found that pyrite crystals are exclusively sulphur dficient, with the empirical formula better expressedFeS22x with x ranging from 0.05 to 0.25. Birkholzet al.60

used ligand field theory arguments to suggest that the 5-~tetragonal pyramidial! coordinated Fe ions resulting fromsulphur vacancies would have a modified splitting of the3d orbitals compared to the 6-fold~octahedrally! coordi-nated, stoichiometric Fe ions. The crystal field splitting in t5-coordinate Fe ions would result in the formation of nedefect states lying between the valence and conduction band would lead to a reduction in the measured band gap,as such may account for the discrepancy between the calated and experimentally measured values.

Other features of the densities of states computed uCRYSTAL98 are in excellent agreement with experiment. Tsplitting of the S 3s states has been attributed to the fortion of bonding (sb) and antibonding (sa) states arisingfrom overlap between sulphur nearest neighbors. The deof overlap and hence the gap between the states is depenon the S–S bond distance. In Fig. 5, we show the Dcomputed using the LDA for a range of S–S bond distanfrom 2.66 Å (xs50.358) to 1.91 Å (xs 5 0.398! at the op-timized LDA cell volume. The S–S bond distance has bevaried by changing the internal coordinate of the S ions. Teffect of changing this bond distance on the splitting of thestates is dramatic. The gap between thesb and sa variesfrom 5 eV at 1.91 Å to about 0.5 eV at 2.47 Å with the gadisappearing at larger separations. We note that our Lcalculations of the DOS at the xs parameter as compute

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using CRYSTAL98 or CASTEP yields a conducting state foboth methods although the predicted value of xs varies some-what~CRYSTAL, 0.378, CASTEP 0.382!. We have performed testscomputing the DOS usingCRYSTAL98 at xs50.382 and find aconducting state.

The influence of the lattice parameters, a0 and xs on thepyrite band structure were determined by calculation ofdensity of states and band structure at a series of app

FIG. 5. The influence of the S–S bond distance on the Sbonding and antibonding peak positions computed usingCRYSTAL98

within the LDA.

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isotropic pressures ranging from215 GPa to120 GPa~seeFig. 6! using B3LYP. Note that in this case ‘‘negative’’ applied isotropic pressures are fictitious quantities, and simcause the cell volume to increase with respect to the calated cell volume at 0 GPa. Both a0 and xs were found todecrease with increasing pressure~see Fig. 7!. While bothFe–S and S–S bond distances decrease with pressure,bonds were found to compress more readily than S–S boin agreement with the calculations of Opahleet al.20 As aconsequence, the unit cell also decreases in volume mrapidly with pressure than S–S bond distances. Howevehigher applied pressures, the rate of compression of Fand S–S bonds becomes increasingly similar~see Fig. 8!since, in general, as bond lengths are decreased, the eneics of compression are increasingly dominated by the repsive component of their interaction curves. Changes inpyrite structure with pressure have several significant effeon the band structure~see Fig. 9!. In general, there is anincreased dispersion of bands with pressure, as expectedto the enhanced coupling between adjacent unit cells wharises from a decrease in cell volume. It was also foundoccupied bands are lowered in energy with respect tovalence band maximum~VBM !, while the energies of unoccupied bands are raised. These effects are attributed tohanced covalency between Fe and S as the Fe–S bondtance is decreased. The formation of a steep band~shown inbold in Fig. 9! at theG point was found at around 0 GPawhich becomes the conduction band minimum~CBM! atpressures greater than 0 GPa. At theG point, this band low-ers in energy with respect to the VBM with increasing presure, causing reduction of the band gap. Thus, in terms oflattice parameters, a decrease in xs causes reduction in theband gap due to lowering of the CBM at theG point. Thisbehavior may be explained in terms of interactions betwS2 dimers in the pyrite ‘‘sub-lattice’’ model, given by Eyeret al.48 The band at CBM is predominantly S 3p in charactand is directly related to the S–S bond distance. AccordinEyert, at constant cell volume, decrease in xs ~increase in

s

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FIG. 6. The total densities ostates for FeS2 computed usingB3LYP at a range of pressures.
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S–S bond length! causes increased coupling between2dimers, resulting in increased band dispersion at the CBOur results show that there is a decrease in S–S bond lewith increasing pressure. However, as explained above, sthe Fe–S bonds compress more readily than S–S bondscell volume decreases more rapidly than S–S bond lenwith pressure. Thus there is an overall increased coupbetween S2 dimers, leading to increased dispersion of S

FIG. 7. The cell parameters of FeS2 at various pressures acomputed using B3LYP.

FIG. 8. The change of Fe–S and S–S bond lengths with apppressure as computed using B3LYP.

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band at theG point and a lowering of the CBM. Note thathis result is contrary to the expectations pointed out by Eert et al.,48 but is consistent with the calculations of Opahet al.20 The rate of energy lowering of the CBM slowshigher pressures, since the Fe–S bonds~and hence cell vol-ume! and S–S bonds compress at more similar rateshigher pressures, leading to a lower rate of increase in inactions between S2 dimers. Although our results show thathe band gap decreases with applied pressure, the blueof the optical band gap that is observed experimentally mbe explained by considering the DOS. The band at theGpoint that causes the band gap reduction only contributesvery shallow tail in the DOS~see Figs. 6 and 9! in agreementto earlier studies,20 while the antibonding Fe 3d and S 3states immediately above this band are raised in energyrespect to increasing pressure VBM. Neglecting the stband at the CBM, our calculations predict an optical blshift of around 0.016 eV per GPa of pressure applied,excellent agreement with the value of around 0.02 eVGPa obtained by Batlogget al.61,62 from pressure experi-ments. Our results therefore suggest that this band maneglected in interpretation of the optical phenomena ofrite.

We have investigated possible causes for the poor destion of the S–S bond length by theCRYSTAL98 calculations.Comparing the S–S bond distance computed using basiswith and without d-symmetry polarization functions on thsulphur ions, we find that the lack of polarization functioresults in an increase in the S–S distance of up to 9% relato the S–S distance computed with polarization functioClearly, polarization functions are very important in descring the S–S bond. To examine these effects further, we hperformed additional tests using theGAUSSIAN98 code63 todetermine how important d-symmetry and higher angumomentum polarization functions are in describing S2 mol-ecules and S2

22 dimers. These calculations were performusing the 6-3111G basis set which does not include anpolarization functions, the 6-3111G~d! basis which includesone d-symmetry function on each sulphur atom and, finathe 6-3111G~3df! basis set which includes three d-symmefunctions and an additional f-symmetry function. Althougthe systems modeled by the Gaussian calculations werthe gas phase, the results, shown in Table VII, illustrate td-symmetry functions are essential for a reasonable desd

FIG. 9. The influence of pressure on the band structure of F2

computed usingCRYSTAL98 with the B3LYP functional.

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TABLE VII. Bond distances computed for S2 dimers in Å. TheCRYSTAL98 results relate to the S–S bondistance computed in pyrite whereas the Gaussian data are for S2 and S2

22 dimers in the gas phase.

Method Basis set HF LDA GGA B3LYP

S222 bond distance in FeS2

CRYSTAL98 no d 2.196 2.422 2.461 2.386d 2.143 2.274 2.295 2.197

Experiment 2.162~Ref. 67!

S222 dimer

GAUSSIAN98 6-3111G 2.329 2.352 2.384 2.3886-3111G~d! 2.180 2.191 2.225 2.231

6-3111G~3df! 2.153 2.133 2.170 2.181

S2 dimerGAUSSIAN98 6-3111G 2.017 2.069 2.083 2.073

6-3111G~d! 1.879 1.922 1.937 1.9276-3111G~3df! 1.863 1.897 1.911 1.903

Experiment 1.887~Ref. 68!

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tion of the S2 and S222 dimers and that f-symmetry polariza

tion functions may be necessary to fully converge the bodistances with respect to the basis set. However, as discuin Sec. II, basis functions with f- or g-symmetry are navailable in theCRYSTAL98 code. This may also explain paof the discrepancy between theCRYSTAL98 and CASTEP re-sults.

V. CONCLUSIONS

In the current study, we have performed calculationsFeS2 using a variety of theoretical treatments, in particulHF, DFT and hybrid-functional treatments of exchange acorrelation, and computational approaches using theelectron LCAO technique and the plane wave pseudopotial methodology.

The computed results are influenced by the selectionthe treatment of exchange and correlation. Correlation effmust be taken into account in order to yield the correct etronic ground state and structrual parameters. The HFproximation provides a poor description of pyrite FeS2 giv-ing an incorrect ground state. In contrast, calculatioperformed with DFT and hybrid-functional~B3LYP! meth-ods provide a reasonable description of FeS2. Calculationsperformed within the LDA generally underestimate the cvolume whereas GGA corrects for this effect. The tests incurrent study reveal that sulphur undergoes significant poization effects. Calculations performed using atomic-orbbasis sets must take account for these effects and inc

,

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higher angular momentum polarization functions.The current DFT and B3LYP calculations give a go

description of the occupied electronic states. The positand width of the valence bands are in good agreement wexperiment. However, LDA and GGA underestimate tband gap and predict pyrite to be a conductor. B3LYP callations yield a band gap of about 2 eV, significantly largthan the accepted experimental value of 1 eV. We belithat a possible reason for the discrepancy is due to thethat experimentally, pyrite is usually sulphur deficient acan be better described by the formula FeS22x where x canvary from 0.05 to 0.25. This large deviation from ideal stichiometry leads to the formation of new defect states witthe band gap and may give rise to the low experimentadetermined value.

The current study provides an understanding of how dferent computational methodologies affect the descriptionthe structural and electronic properties of FeS2 and serves tounderpin further studies of the surface structure and chemtry of pyrite FeS2.

ACKNOWLEDGMENTS

We thank Accelrys Inc. for provision of aCASTEPlicenseduring this study and to Professor Stephen Fletcher~Univer-sity of Loughborough! for initiating the collaboration be-tween CSIRO Minerals and RMIT. The calculations weperformed on Silicon Graphics workstations and a paraPC cluster.64

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Documents

First-principles studies of the structural and electronic properties of