American Mineralogist, Volume 81, pages 545-549, 1996
Ab initio calculation of electric-field-gradient tensorsof forsterite
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'Mineralogisch-PetrographischesInstitut der Christian-Albrechts Universitiit, Olshausenstrasse40, D 24098 Kiel, Germany 'zInstitutfiir TechnischeElektrochemie. Technische Universitiit Wien. A-1060 Vienna, Austna
Ansrn-lcr Ab initio band-structure calculations basedon the density functional theory have been performed for forsterite to obtain, with a parameter-freemodel, electric-field gradientsfor all nuclei. Calculations basedon the generalizedgradient approximation yield the ratio of the largest components of the calculated electric-field gradients within l0loof the experi- mental value for 25Mgand within 2oloof the experimental value for '7O. The absolute values differ by about 5ol0,depending on which nuclear quadrupole moment is used in the conversion. The asymmetry parameters are also in good agreement with experimental data. Values obtained with a gradient-correctedexchange-correlation potential are better than those based on the standard local-density approximation. The calculated anglesbe- tween the principal axes of the quadrupole coupling tensor and the crystallographicaxes agreevery well with the experimental data. For the Ml site the maximum deviation is 1.8",and for the M2 site the maximum deviation is 0.6". The current calculationsallow an evaluation of three sets of experimental data for therTO electric-field gradient. They also confirm a proposed assignmentof the measuredelectric-field gradient tensors to spe- cific O atoms.
INrnooucrroN V,", which varies between0 (axial symmetry) and 1. Here we follow the sameconventions as in the work of Schwarz In computational mineralogy, predictive microscopic et al. (1990). models are increasingly used to interpret experimental However, QCC and 4 cannot be derived from NMR data for which conflicting phenomenological explana- spectradirectly but must be determined by spectral sim- tions exist. Although reliable quantum mechanicalmodel ulation, which can be problematic for broad, overlapping, calculations require large computational resources,they or weak features. Conventionally, EFGs have been ex- offer the possibility of calculating rather subtle physical plained by point-chargemodels, which contain adjustable properties, including those due to electronic effects. parameters, such as formal chargesor the Sternheimer Therefore, they can be applied to a large variety ofprob- antishielding factors. The predictive power of such par- lems and yield better results than conventional static lat- ameterizedcalculations is limited, and the results are of- tice-energy-minimization calculations. Here we use pa- ten ambiguous. Hence, attempts have been made to de- rameter-free energy-band-structurecalculations to velop parameter-freecalculations. For example, Tossell compute electric-field gradient (EFG) tensors,which are and coworkers [e.g., Tossell (1992) and referencescited experimentally determined by nuclear magnetic reso- therein] usedHartree-Fock calculationson clustersto ob- nance (NMR), nuclear quadrupole resonance,or, M
TABLE1. Atomic coordinates used in the calculations and the proximation (LDA) or the generalizedgradient approxi- corresponding remaining forces mation (GGA). A recent in-depth summary can be found Fujino in the book by Singh (1994).For such calculations,one Hazen Forces Brown Forces et al. Forces Forces of the most accurate schemesis the LAPW method, in (1e76) (GGA) (1e70) (GGA) (1981) (LDA) (GGA) which the unit cell is divided into spherescentered at the Mg2x 0.9915 0.6 0.9896 1.0 0.99169 0.0 0.4 atomic positions and an interstitial region. In the latter -0.1 -0.4 -0.3 -0.4 Mg2y 0.2774 0.2776 0.27739 the basis set consistsof plane waves that are augmented Si x 0.4262 6.1 0 4226 48 6 0.42645 -1.3 2.2 Si y 0.0940 -0.9 0.0945 -4.5 0.09403 -1.9 1.4 by atomic-like solutions (numerical radial functions mul- 01 x 0.7657 18.6 0.7667 -3.4 0.76594 17.0 17.4 tiplied with sphericalharmonics) inside the spheres.The -0.4 01 y 0.0913 2.2 0.0918 1.7 0.09156 1.0 potential within the sphereis not restrictedto have spher- 02 x 0.2215 6.6 0.2202 11 3 0.22164 41 4.7 02 y 0.4474 -21.6 0 4477 -25.9 0.44705 -5.4 -14.8 ical symmetry, as in the older "muffin-tin" calculations 03 x 0.2777 -7.4 0.2781 -13 5 0.27751 -4.5 -5.9 but is allowedto be general,i.e., a full potential without y 03 0.1628 11 .9 0.1633 14.4 0.16310 2.5 8.5 It is well known that 03 z 0.0331 -15.6 0.0337 -23.9 0.03304 -6.0 -12.9 any shape approximation is used. DFT calculations in general give structural parameters given : Nofer Forces in mRyd/au;1 Ryd 13.6 eV, 1 au: 0.529A. within 1-20loof the experimental data. In the presentstudy, the full-potential, linearized, aug- mentedplane-wave package, WIEN95 (Blahaet al. 1995) that in generalFP-LAPW calculations yield EFGs within was used.[WIEN95 is an improved and updatedversion approximately 100/oand asymmetry parameters within ofthe original copyrightedcode by Blaha et al. (1990).1 -0. I of the experimentalvalues. WIEN95 is maintained at the Technical University of Consideringthe widespreaduse of NMR spectroscopy Vienna and includes the capability to employ local or- and the recent progressin quantum mechanical calcula- bitals (Singh 1991) and to calculateresidual forces on tions it is worthwhile to compute electric-field-gradient atoms (Yu et al. l99l: Kohler et al. 1996).The calcula- tensorsfor silicates,which are characterizedby their broad tion of the EFGs is based on the work of Blaha et al. range of bond types and bond strengths. We note that (1985); a more detailed description can be found in NMR spectroscopyis a local technique, and therefore it Schwarzet al. (1990).The precisionand accuracyofthe is often used to investigateshort-range order or local dis- calculationsare controlled by only a few parameters.For tortions. Ab initio calculations, however, exploit the pe- the expansion ofthe chargedensity (potential) in the in- riodicity of a perfectly ordered structure, so that solid terstitialregion a Fourier serieswith lG | = t Z wasused. solutions and the effectsofshort-range order are currently For the wave functions inside the atomic spheresangular approximated rather crudely. Also, the present calcula- momentum componentsup to /: 12 wereincluded, and tions are restricted to ground-state(0 K) properties. up to 18 k points in the irreducible wedgeof the Brillouin For fully ordered structures,theoretical studiesare pos- zonewere used. This correspondsto nearly 150 sampling sible and can be used for the interpretation of measured points in the whole Brillouin zone. The cut-off for the spectra,which is sometimes difficult using experimental plane-wavebasis set was chosenas k-u* : 4.5 ait and results only. Reliable calculations allow evaluations of was checkedby additional calculations,which proved that proposedexperiments, which canbe costlywhen isotopic the results presentedhere are well converged. enrichment is required, becausethey can predict NMR parametersand specify the necessaryresolution. For the Rnsur,rs interpretation of spectraof silicates,correlations between structural and NMR parameters have been established Structure (Sherritretal. l99l; Smith et al. 1983).These studies rely The all-electron ab initio techniques currently in use on available data, which may be scarcefor unusual iso- do not yet allow a constant-pressurerelaxation of low- topes. In theoretical studies, however, structures can be symmetry structures.In fact, even at constant volume a easily deformed, and thus NMR parametersas a function relaxation of an orthorhombic structure containing 28 of structural parameters can be calculated. Also, theory atoms using an all-electron approach would at present can help in evaluating sets of experimental data if the consumea prohibitive amount of computer time. Hence, latter are not in good agreement. in the present study all structural parameterswere fixed Forsterite,MgrSiOo, is a model systemin which the at their experimental values. To investigatethe influence same isotopes occupy different structural environments of small structural distortions on the calculated physical 2sMg and in which the EFGs for both and "O have been propertieswe usedthree setsof internal structural param- studied in detail. The small differencesin the Mg and O eters in our calculations. The atomic coordinates (Table environments seryeas a sensitive test for the accuracvof l) were taken from Hazen (1976), Brown (1970), and the calculations. Fujino et al. (1981),but only one setof latticeparameters was used, namely the very-high-precision data given by CoupuurroNAl APPRoACH Schwaband Ktistner(1977) with a: 4.7540, b : 10.197l, The calculations presentedhere are based on the den- and c -- 5.9806A. ttre WIEN95 code allows the calcu- sity functional theory @FT), using the local-density ap- lation of residual forces acting on the atoms using the WINKLER ET AL.: ELECTRIC-FIELD GRADIENTS OF FORSTERITE 547
TABLE2. Calculated and observed values of V,,l'1021(V lm2l for forsterite
Structure k points M1 M2 M1IM2 01to2 01/o3 Hazen(1976) LDA 6 0.99 0.91 1.09 -1.98 4.00 3.70 3.51 1.08 1.14 Hazen(1976) GGA o 1.08 0.92 1.17 -1.95 4.36 3.92 3.70 1.11 1.18 Hazen(1976) GGA 18 1.08 0.92 1.17 -1.95 4.34 3.92 3.70 1.11 1.17 Brown (1970) LDA o 1.06 0.96 1.10 reo 4.47 3.52 3.24 1.27 1.38 Brown (1970) GGA 6 1.16 0.98 1.18 1.35 4.67 3.71 3.45 1.26 1.35 Fuiinoet al. (1981 ) LDA 6 0.98 0.90 1.09 -2.08 4.12 3.77 3.49 1.09 1.18 Fuiinoet al. (1981 ) GGA 1.08 0.92 1.17 -2.02 4.31 396 3.70 1.09 1.16 Exp' 1.03 089 1.16 Exp'- 4.37 3.79 3.79 1.15 1.15 EXpT 5.41 s.00 4.49 1.08 1.20 Expf 4.48 4.09 3.91 1.09 1.14
A/ofe.Conversion factors O(mS): 0.200; A(O) : -0.02558b. 'Experimentaldata from Derighettiet al. (1978). .* Experimentaldata from Schrammand Oldfield(1984). t Experimentaldata from Muelleret al. (1992).The conversionfrom the stated quadrupoleproduct is based on the asymmetry parametersgiven in Table3. + Experimentaldata from Fritschet al (1986). formalism of Yu et al. (1991).Small residualforces pro- whereasthe M2 site has site symmetry m. The O atoms vide an a posteriori justification for using high-quality Ol and 02 occupy sites with site symmetry m and a structural data in the calculations without any further re- multiplicity of 4, whereas03 occupiesthe general posi- laxation of the internal structural degreesof freedom. tion with a multiplicity of 8. The 25Mg data are those The results are presentedin Table l, which shows that reported by Derighetti et al. (1978). For the '?O NMR the calculatedforces are small. In iterative pseudopoten- data, experimentalvalues were reported by Schramm and tial calculations, where a constant-volume relaxation is Oldfield (1984),Fritsch et al. (1986), and Mueller et al. comparatively simple, a structure is generallyconsidered (1992). Mueller et al. (1992) gave for the o atoms only 1obe relaxedif forcesare lessthan - 15 mRyd/au (-0. I the products(e'Qq,/h)(l + 42/3)'/'.To comparethese with eV/A). Therefore, the atomic parameters from Hazen our calculatedvalues, the asymmetryparameters must be (1976)and Fujino et al. (1981)correspond to very nearly known. Here, we used our calculatedvalues, which agree relaxed structures,whereas this is not quite the casefor well with the experimentaldata of Fritsch et al. (1986). the structureof Brown (1970).The presenttotal-energy It should be noted, though, that the data obtained by calculations confirm that the values for the x coordinate Mueller er al. (1992) are different from those of Schramm of the Si atom of Hazen (1976)and Fujino et al. (1981) and oldfield 0984) and Frirschet al. (1986).The data of arepreferred over the value given by Brown (1970).This Schramm and Oldfield (1984) and Fritsch et al. (1986) diference in the internal parametershas a large effect on are unambiguous with respect to the assignment of the the electric-field gradient, which is discussedbelow. A measuredvalues for the 03 atom only, whereasthe as- comparison of the forces and atomic parameters indi- signmentto O I and 02 could theoreticallybe reversed. cates,however, that generally only changesin the fourth Mueller et al. (1992)did not attempt an assignment. decimal of the fractional coordinateswould be necessary The calculatedresults for the EFGs are given in Tables to relax all structuresfully. 2-4, wherethey are compared to experimental data. The The small differencesin the calculated residual forces betweenLDA and GGA demonstratethat for the internal degreesof freedom both approximations yield very sim- ilar structures.This is consistentwith the findings by oth- TABLE3. Calculatedand observedvalues for the asymmetry er authors (Ozolins and Kdrling 1993) that GGA im- parameter 4 for forsterite proves on the "overbinding" ofLDA and thus leads to a K better agreementof cell parameters,but does not signif- Structure ocints M1 M2 Si 01 02 03 icantly changethe fractional coordinates. Hazen(1976) LDA 6 0.92 o.44 0.74 0.32 0.43 0.19 Hazen(1976) GGA 6 0.93 0.40 0.73 0.30 0.44 0.22 Electric-field gradients Hazen(1976) uuA 18 0.93 0.40 0.73 0.30 0.44 0.22 Brown (1970) LDA 6 0.65 0.38 0.70 0.30 0.52 0.25 The motivation for calculating EFGs of forsterite, Brown (1970) GGA 6 0.70 0.35 0.68 0.29 0.49 0.22 MgrSiOo(space group Pbnm, Z: 4), was mentionedin Fuiinoet al. (1981) LDA 6 0.91 0.42 0.73 0.31 0.45 0.23 0.30 0.43 0.21 the Introduction. Experimentaldata are available for 25Mg Fujinoet al.(1981) GGA 6 0.93 0.39 0.73 17O. Exp. 0.96 0.40 and The calculation of EFG tensors is challenging Exp.- 0.3 1.0 0.2 becauseboth the Mg and the O atoms occupy more than Expt 0.28 0.39 0.18 one site, so relative comparisonscan be made that avoid - Experimentaldata from Derighettiet al. (1978). .. the knowledge of the (rather ill-determined) nuclear- Experimentaldata from Schramm and Oldfield(1984). Experimentaldata from Fritsch et al. (1986). quadrupolemoment. The Ml site has site symmetry l, f 548 WINKLER ET AL.: ELECTRIC.FIELD GRADIENTS OF FORSTERITE
TABLE4. Observedand calculatedangles between the with one noticeable exception, with those of Schramm principalaxes of the quadrupole-couplingtensor and and Oldfield (1984).The one exceptionis the asymmetry the crystallographicaxes at the M1 and M2 sites parameter of the 02 atom, for which all calculationsgive M1 results that are one-half the values reported by Schramm and Oldfield (1984).Hence, we concludethat the data by Fritsch et al. (1986) are preferableto those of Schramm X exp 53.1 125.0 124.0 71.4 18.6 89.4 and Oldfield (1984). X calc 51.4 124.2 122.9 71.8 18.2 90.0 Y exp 137.8 978 131.1 90.0 90.6 0.6 From their single-crystalwork, Derighettiet al. (1978) Ycalc 136.0 98.2 132.8 90.0 90.0 0.0 derived the anglesbetween the principal axesX,Y,Z of Z exp 72.4 36.1 120.4 18.6 108.6 90.2 quadrupole-coupling and the crystallographic Z calc 72.3 35.4 119.6 18.2 108.2 90.0 the tensor axes at the Ml and M2 sites. The calculated orientation Note. Experimentaldata from Derighettiet al. (1978). of the electric-field-gradient tensor with respect to the crystallographic coordinates is in very good ag,reement, with a maximum deviation of <2. For the M2 site the comparison relies on a reliable value for the quadrupole deviation is about as large as the experimental uncertain- moment, Q, which is a constantfor eachisotope. A major ty. The observedvalues are listed in Table 4 togetherwith problem arisesfrom the fact that many quadrupole mo- the calculated values from GGA calculations for the ments are not well known and variations of more than structureofFujino et al. (1981). 100/oare not uncommon. The ratio of V""valuesfor one isotopein differentsites, however, is independentofthe DrscussroN actual value of Q and hence is a good measure of the The present investigation has shown that calculations reliability of the calculations. The calculated EFGs ob- such as those presented here do not necessarilyrely on tainedfor the structureof Brown (1970)cannot be com- constant-pressurerelaxations. Instead, physical proper- pared to experimental values becauseof the poor Si po- ties can be calculated with good accuracyifhigh-quality sition. The presentcalculations with GGA yield V""(Ml)/ structural data are available. The latter can be checked V,,(M2) = | .17, closeto the experimentalvalue of I . 16. by inspection of the residual forces acting on the atoms. The agreement is also excellent for the ralios V""(Ol)/ This allows the prediction of a large variety of physical V""(O2)and V""(Ol)/V""(O3)if the GGA calculationsbased properties, even in the absenceof efficient relaxation al- on the structure of either Hazen (1976) or Fujino et al. gorithms. We expect that for some time this will be the ( 198I ) are compared to the single-crystaldata of Fritsch most common application for all-electron calculations of et al. (1986). The absolute values for the V"" depend complex structures. strongly on the value chosenfor the quadrupole moment. The above calculations have also demonstrated that For O, for which Q is well known, the agreementis sat- modern ab initio calculations are preciseand sufrciently isfactory @etterthan 50/o)for GGA calculations using the accurateto allow the calculation of EFGs and asymmetry data of Schramm and Oldfield (1984) or Fritsch et al. parameters.They are in such good agreement with ex- (1986). The agreementwith the data of Mueller et al. perimental data that they can serve as a reliable tool for (1992) is very unsatisfactory.For Mg, small deviations checking and predicting EFGs. A significant difference occur with Q(Vtg :0.20b, but deviationsare larger(about betweencalculated and observeddata constitutesa strong 100/0)if Qcvte) : 0.22b. The rario of the EFGs differs justification to reexaminethe experimental data and their more for the experimental data for LDA than for GGA analysis.This is shown by a comparison of the three sets calculations, and hence the GGA is preferable in EFG of data for the '7O EFGs; the present calculations are in studies.The LDA results, however, are still significantly very good agreementwith the data by Schramm and Old- better than empirical models. Derighetti et al. (1978) used field (1984) and Fritsch et al. (1986), but they disagree a point-charge model and obtained a ratio of V,"(Ml)/ with the data of Mueller et aI. (1992). The calculations V,"(M2) = L5, which is 300/olarger than the experimen- presentedhere also confirm that the magnitude of a for tally determinedvalue. The presentcalculations show that 02 obtainedby Fritsch et al. (1986) is preferableto the 2,, is positive for both the M I and M2 sites,in agreement value given by Schramm and Oldfield (1984). Further- with the resultsof Derighetti et al. (1978).The effectof more, the current calculationsunambiguously confirm the the poor Si position in the structureof Brown (1970) is assignmentproposed by Fritsch et al. (1986)of the EFGs evident from the calculatedEFGs becausethe sign of the to ol and 02. Si EFG is oppositeto that ofthe other results,and on the From the above calculations of forsterite we conclude basisof the accuracyof the current calculationsthe ratios that for the prediction of the EFG tensors GGA calcula- of the O EFGs differ significantly from the experimental tions provide better results than standard LDA. This is data. consistentwith numerousother examples[Perdew et al. The calculated asymmetry parameters,4, are given in (1992) and referencescited thereinl demonstrating the Table 3. The effectofthe choice ofthe exchangepotential improvement of density functional calculations using on 4 is small. The calculatedvalues are in good agreement GGA. The same formalism can also be used to obtain with the experimentaldata of Fritsch et al. (1986),and, NQR and Mcissbauerparameters. WINKLER ET AL.: ELECTRIC-FIELD GRADIENTS OF FORSTERITE 549
AcxNowr,nocMENTs potential linearized augmented plane wave code WIEN. Computer PhysicsCommunication, 94, 3l-48. B.W. is grateful for financial support from the Deutsche Forschungs- Mueller, K.T., Baltisberger,J.H., Wooten, E.W., and Pines,A. (1992) gemeinschaftunder grant Wil232. P.B. was supported by the Austrian Isotropic chemical shifts and quadrupolar parametersfor oxygen-17 ScienceFoundation Project no. P10847. P.B. and K.S. thank the Hoch- using dynamic-anglespinning NMR. Journal of Chemical Physics,96' schuljubilaeumsstiftungder Stadt Wien 7001-7004 Ozolins, V E., and Kdrling, M. (1993) Full-potential calculations using properties RrnnnnNcns crrED the generalizedgradient approximation: Structural of tran- sition metals.Physical Review B, 48, 18304-18307. Blaha,P., Schwarz,K., and Herzig, P. (1985) 1st principles calculation of Perdew,J.P., Chevary, J.A., Vosko,S.H., Jackson, K.A., Pederson,M.R., the electric field gradient of LiN. Physical Review Letters, 54, I 192- Singh, D.J., and Fiolhais, C. (1992) Atoms, molecules,solids and sur- I 195. faces:Applications of the generalizedgradient approximation for ex- Blaha, P., Schwarz,K., and Dederichs,P.H (1988) lst principles calcu- changeand correlation. Physical Review B, 46, 667 l-6687 . Iation of the electric field gradient in hcp metals. Physical Review B, Schramm, S., and Oldfield, E. (1984) Hieh resolution oxvgen-17NMR of 37,2792-2196. solids. Journal ofthe American Chemical Society, 106, 2502-2506. Blaha, P., Schwarz,K., and Sorantin, P (1990) Full potentral linearized Schwab,R.G., and Kiistner, D. (1977) Priizisionsgitterkonstantenbestim- augmented plane wave programs for crystalline systems. Computer mung zur Festlegungrdntgenographischer Bestimmungskurven fiir syn- PhysrcsCommunication, 59, 399-41 5. thetischeOlivine der Mischkristallreihe Forsterit-Fayalit. Neues Jahr- Blaha,P., Schwarz,K., Dufek, P., and Augustyn,R. (1995) WIEN95. buch fiir Mineralogie Monatshefte, 5,205-215. Technical University of Vienna. Schwaz. K., Ambrosch-Draxl, C., and Blaha, P. (1990) Charge distri- Brown, G.E. (1970) The crystal chemistry of the olivines. Ph.D. thesis, bution and electricfield gradientsin YBarCurO, ,. PhysicalReview B, Virginia Polytechnic Inslitute and State University, Blacksburg,Vir- 42, 205r-2061. ginia (not seen,extracted from Mineralogical Society of America Re- Sherriff,B.L., Grundy, H.D., and Hartman, J.S.(1991) The relationship viewsin Mineralogy,5,275-381). between'?eSi MAS NMR chemical shift and silicate mineral structure. Derighetti, B., Hafner, S., Maner, H , and Rager,H. (1978) NMR of "Si EuropeanJournal ofMineralogy, 3, 751-768' and 25Mgin MgrSiOo with dynamic polarization technique. Physics Singh, D.J. (1991) Ground statesproperties of lanthanum: Treatment of Irtters.66A. I50-I52. extendedcore states.Physical Review B, 43,6388-6392. Fritsch, R., Brinkmann, D., Hafner, S.S.,Hosoya, S., Iorberth, J., and - (1994) Planewaves, pseudopotentials and the LAPW method, I I 5 Roos, J. (1986) Nuclear quadrupole coupling tensorsof'7O in forster- p. Kluwer Academics,Boston. ite, MgrSiOoPhysics Letters A, 118,98-102. Smith, K.A., Kirkpatnck, R.J.,Oldfield, E., and Henderson,D.M (1983) Fujino, K., Sasaki,S., Takeuchi, Y., and Sadanaga,R. (1981) X-ray deter- High-resolution silicon-29 nuclear magnetic resonancespectroscopic mination of electron distributions in forsterite, fayalite and tephroite. study of rock-forming silicates. American Mineralogist, 68, l2}Gl2l5 Acta Crystallographica,B37, 5l3-5 18. Tossell, J.A. (1992) Calculation of the "Si NMR shielding tensor in for- Hazen, R.M. (1976) Effects of temperature and pressureon the crystal sterite. Physicsand Chemistry of Minerals, 19,338-342' structureof forsterite. American Mineralogist, 61, 1280-1293. Yu, R.C, Singh,D, and Krakauer,H. (1991)All-electron and pseudo- Krrkpatrick, R J. (1988)MAS NMR spectroscopyof minerals and glasses. potential force calculationsusing the linearized-augmentedplane wave In MineralogrcalSociety of America Reviews in Mineralogy, 18, 341- method.Physical Review B, 43,64ll-6422' 404. Kohler, B., Wilke, S., Scheffier,M., Kouba, R., and Ambrosch-Draxl, C. MnNuscnrpt REcETvEDSsrrn"rnBr. 14, 1995 (1996) Force calculationand atomic structureoptimization for the full- Ma.NuscnrvrAccEprED JeNu.*v 15, 1996