American Mineralogist, Volume 76, pages 7 33-742, 1991

Bonding and elasticity of stishovite SiO, at high pressure:Linearized augmented plane wave calculations

RoNar,o E. ConnNGeophysical laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road, NW, Washington, DC 20015, U.S.A.

ABSTRACT

The crystal structure, equation-of-state, elasticity, A,, Raman mode, and electronicstructure of stishovite (rutile-structured SiOr) are studied at high pressure from first prin-ciples using the linearized augmented plane wave (LAPW) method. Excellent agreementis found with the experimentally known properties. The pressure dependence of threeelastic constants is predicted. The charge density and band structure indicate a mixture ofionic and covalent bonding in stishovite, and the ionicity is quantitatively estimated. Nomajor changes in bonding are found between zero pressure and 150 GPa.

INrnonucrroN are complementary to experimental studies. The purposeof these calculations is not merely to compare results with

Stishovite is the simplest silicate with octahedrally co- experimental data but to make predictions and to in-ordinated Si and thus is of great interest, both theoreti- crease understanding ofthe origin ofthe behavior ofsol-cally and experimentally, as a prototype for high-pressure ids. Only recently has it become possible to study thesilicate phases that are important in Earth's lower mantle. elastic, optical, and electronic properties of complex crys-Furthermore, it is of crystal chemical interest to under- tals such as stishovite, making no approximations exceptstand the bonding and electronic structure and, in partic- for a local approximation for the exchange and correla-ular, the ionicity and covalency of the Si-O bond, in a tion interactions between electrons. Although this localdense octahedrally coordinated phase. A theoretical study density approximation (LDA) is a rather severe approx-of stishovite is particularly suited for this honorary vol- imation, in general it gives excellent results for groundume, because the existence of a rutile-structured form of state properties, with errors of up to a few percent inSiO, was predicted by James B. Thompson, Jr. in the latticeparametersandusuallybetterthan l5o/ointhebulkearly 1950s (Birch, p. 234 and 274, 1952). Such a phase modulus. The occupied electronic bands are usually inwas not synthesized until 196l (Stishov and Popova, very good agreement with experiment. The main prob-196l), shortlyafterwhichthemineralwasdiscoveredat lems with the LDA occur in systems such as FeO andMeteor Crater, Arizona (Chao et al., 1962). La"CvOo, where the LDA incorrectly predicts metallic

Numerous experimental data are available for stisho- rather than insulating behavior because antiferromagnet-vite at various pressures, including Raman frequencies ic susceptibility is underestimated.(Hemley et al., 1986; Hemley, 1987), X-ray difraction Incontrasttomodelcalculations,noassumptionsaboutdata (Liu et al., 1974; Sato, 1977; Endo et al., 1986), such characteristics as ionicity and covalency are made,precise electron density maps derived from X-ray dif- and the only inputs are the nuclear charges and masses.fraction (Spackman et al., 1987), and single-crystal elastic With only these inputs, the total static cryslal energy canconstants determined from Brillouin scattering (Weidner be calculated as a function of the nuclear positions. If oneet al., 1982); all ofthese can be compared with results of is interested in the properties of a given (either experi-first-principle calculations. Theoretical studies of stisho- mentally known or hypothetical) crystal, one uses thevite include empirical models (Erikson and Hostetler, symmetry constraints for the given space group to sim-1987), ab initio models such as the potential induced plifythe calculation. The choice of symmetry, however,breathing (PIB) model (Cohen 1987a), models based on is not an essential feature, and symmetry constraints canmolecular orbital theory (Burdett, 1985; Lasaga and be relaxed completely, except for translation symmetry,Gibbs, 1987; Tsuneyuki et al., 1988, 1989), linearized when studying a bulk crystalline solid. Vibrational andaugmented plane wave (LAPW) calculations investigat- elastic properties can be studied by calculating the energying possible high-pressure phase transitions (Park et al., as a function of strain or internal structural parameters.1988), and Hartree-Fock crystal calculations at zero pres- Here the elastic and vibrational properties that can besure (Nada et al., 1990). Recently a reversible high-pres- obtained by distortions of stishovite without lowering thesure phase transition from stishovite to a CaCl, structure space group symmetry, Pnnm, are considered. By induc-was claimed by Tsuchida and Yagi (1989) at pressures of ing such strains and distortions, the fully symmetric, Alg,80-100 GPa in a laser-heated diamond anvil cell. Raman frequency and its Gruneisen parameter, 7, the

First-principlecalculationsofthepropertiesofcrystals elastic constants C,, * Crr, Crr, Cr, (which also deter-

0003404x/91/0506-o733$02.00 733

734 COHEN: STISHOVITE AT HIGH PRESSURE

mine the bulk modulus Kand the linear compressabilitiesk"and k,), the structural parameter c/a, the O positionalparameter :r(O) [the O equipoint is at (xxO)], and thevolume can be determined as functions of pressure. Thesequantities can be compared with experimental data wheresuch data are available, and they form first-principle pre-dictions in the absence of data. For example, the high-pressure elastic properties of stishovite are unknown. Inaddition to the elastic, vibrational, and structural quan-tities that can be calculated, both the charge density andthe band structure can be investigated as functions ofpressure. Both the total energy and the charge densityresults can be used as benchmarks for the formulation ofbetter models that could then be used to calculate ther-moelastic properties of stishovite.

LDA lNo rnn LAPW METHoD

The calculations discussed here are based on the den-sity functional theory ofHohenberg and Kohn (1964) andKohn and Sham (1965). The basic idea is that the exactground state properties of a substance, including theground state energy, are functionals ofthe charge densityalone. Kohn and Sham showed how to calculate theground-state charge density and total energy by solving aset of single-particle SchrOdinger-like equation s, H rtt, (k): e,(k)r!;(k) self-consistently, with a Hamiltonian givenby

h 2H: -::-v2 + v"*, + v*" (1)

zm

where 2.., is the nuclear electrostatic potential and 4" isthe exchange-correlation potential. The total energy isgiven by

E : T + U + E * . ( 2 )

where ?"is the kinetic energy ofa system ofnoninteract-ing electrons with the same density as the interacting sys-lem, U is the electrostatic (Hartree) energy including theelectrostatic energy between the nuclei, and E-. is the ex-change-correlation energy. The beauty of the method isthat the exact, many-body ground state energy can befound by solving a set of single-particle equations. Un-fortunately, the exact solution requires the exact ex-change-correlation functional, which is unknown and un-doubtedly extremely complicated and nonlocal.Fortunately, a simple functional based on the homoge-neous electron gas works very well, at least for nonmag-netic systems. In this approximation, one simply uses alocal exchange-correlation potential that corresponds ateach point in space to the exchange-correlation functionalappropriate to a homogeneous electron gas with a densitycorresponding to the charge density at that point in space.This is known as the local density approximation (LDA).The Hedin-Lundqvist (1971) parameteization of the ex-change-correlation functional is used here. Reviews ofthe

LDA are given by Pickett (1985) and Schluter and Sham(1982).

The LAPW method (Wei and Krakauer, 1985) is oneof the most accurate for solution of the Kohn-Sham equa-tions within the local density approximation. All types ofcrystals, whether bonding is ionic, metallic, or covalentand no matter which elements in the periodic table theycontain, can be treated on the same footing. This is anall-electron method that makes no shape approximationsfor the charge density or the potential and that can beessentially firlly converged. Space is divided into two typesof regions: the volumes inside spherical "muffin tins" sur-rounding each nucleus and the interstitial volume be-tween the muffin tins. Within the muffin tins, the radialfunctions are represented as linear combinations of so-lutions ofthe radial Schrddinger's equation for an energy-8, for each orbital and its energy derivative in a sphericalpotential. Different energy parameters -8, can be used inseparate energy windows. Nonspherical terms are repre-sented in terms of sums of spherical harmonics. In theinterstitial region, a plane wave expansion is used. Theplane waves are joined smoothly onto the expansion in-side the muffin tins. The one approximation inherent inthe method is that the basis functions inside the sphereare not a complete set. The basis functions in each sphereare the eigensolutions ofthe sphericalized potential, rath-er than the actual nonspherical crystal potential insidethe spheres. (Note that this is only to form the basis func-tions; the full nonspherical potential is used to diagonal-ize the problem within the LAPW basis set.) Also, onlya single set of orbitals with a given angular momentumis used inside a given energy window. This can causeproblems when highJying semi-core states with a givenangular mom€ntum are present below the valence bandwhere the same angular momenta are important; the low-er states can be picked up below the energy window, ef-fectively reducing the variational freedom to describe thevalence states. This is a technical issue that leads to great-er complexity in the calculations but fortunately does notarise at all in SiOr. The former approximation is notknown to cause problems even in the most nonsphericalsystems.

The time-consuming steps are the setup of the Ham-iltonian (I1, : \ilHlj>) and overlap (Or: (i l7)) matricesand the diagonalization ofthe secular equation

4 uur,k):

) e{r) ouf,G).

The charge density is given by

re: I dk ) f,*(t)9,(t) (3)

where the integral ,r"o.r., tie srittorrin zone and is re-placed by a discrete sum over a set ofspecial k-points inthe irreducible wedge of the Brillouin zone, i is the bandindex, and the sum is for occupied states. Typically thereare 70-150 basis functions per atom in the unit cell. A

TaBLE 1. Equation-of-state, structural parameters, and A,e Ra-man frequency for stishovite

v v E + 1 7 5 5 P(Bohr3) (A1 (Ryd) (Gpa) cta

COHEN: STISHOVITE AT HIGH PRESSURE

v

x ( c m ' )

x o z o

P

735

240260280300314320

35.56 -0.0915 143 0.654838.53 -0.2440 84 0.653541 .49 -0.3290 43 0.651644.45 -0.3660 13 0.646046.53 -0.3708 -2.5 0.640447.42 -0.3686 -8.2 0.6361

I

rtg>g

0.3002 11020.3013 9830.3025 8730.3042 8080.3057 7510.3066 727

setup and diagonalization has to be performed for eachk-point in the sum over the Brillouin zone. For maxi-mum accuracy, at least two windows are used, with aseparate basis set for each window. The valence states aretreated in one window and the semicore states are treatedin another window. (This still allows the states to hybrid-ize but in general increases the accuracy by linearizingthe basis functions around energy parameters in eachwindow.) The deep core states are treated by solution ofthe Dirac equation in the potential of all of the electrons.The core states are thus treated fully relativistically, andthe semicore and valence stat€s are treated semi-(scalar)relativistically. Approximately ten iterations are requiredto reach self-consistency.

The LAPW method has been used to study the ener-getics of several oxides. Mehl et al. (1988) studied theequations-of-state of MgO and CaO and predicted a phasetransition from the NaCl structure (Bl) to the CsCl struc-ture (B2) in MgO at 510 GPa. Cohen et al. (1989a, 1990a)calculated vibrational frequencies in LarCuOo, Cohen etal. (1990b) calculated all 15 Raman frequencies and ei-genvectors in YBarCurO,, and Cohen et al. (1989b) cal-culated the equation of state of MgSiO, perovskite usingthe LAPW method. Park et al. (1988) studied possiblephase transitions in stishovite using the LAPW. Mehl etal. (1990) demonstrated that elastic constants can be re-liably calculated in intermetallic alloys using the LAPWmethod.

Torn, ENERGv cALcuLATroNS AND RESuLTS

For stishovite, the O 2s states were treated in a separatesemicore energy window and the hybrid O 2p-Si valencestates in a valence window. (The use of two windows isonly to decrease any errors caused by the linearization ofthe basis functions with respect to energy. It does notassume that the semicore and valence states do not in-teract. Also, the presence ofO 2s and hybrid O 2p andSi band states is never assumed. That is a product ofthecomputations and agrees with chemical intuition.) Thedeeper states were treated as core states. A convergenceparameter RK--. : 7 was found to be sufficient, whichgives about 540 (9O/atom) LAPW basis functions at zeropressure. The nonspherical potential and charge densitywere expanded to an angular momentum I : 8. To per-form the Brillouin zone sums such as Equation 3, a (444)special k-point mesh was used which gives six specialk-points for each energy window. The total energy was

rnrt0 50 100 150

P (cPo)Fig. 1. Equation-of-state of stishovite. The solid curve is the

LAPW equation-of-state, and the symbols represent_the follow-ing experimental data'. +, Liu et al. (1974'11' A Sato (1977); x ,Tsuchida and Yagi (1989).

calculated as a function of c/a and x(O) for six volumesfrom 35.56 to 47.42 Ar. Calculations for at least threevalues of x wer€ perforrned at each value of c/ a and forat least three values of c/a at each volume.

Sfiucture and equation-of-state

The calculated energies at each volume and c/a werefitted to a polynomial as a function of the parameter r,and then the minimum energies with respect to r werefitted as functions of c/a at each volume. In this way, theminimum energy as a function of volume as well as thestructural parameters c/a and x were determined (Tablel). The total energies as a function of volume were thenfitted to a third-order Birch equation (Birch, 1978) toobtain the equation-of-state and the parameters Vo, Ko,and K"' . The resulting equation-of-state gives a zero pres-sure volume V, of 46.16 A', bulk modulus K, of 324 GPa,and bulk modulus pressure derivative K,' of 4.04. A fitto the Murnaghan equation-of-state gives V,: 46.16 L3,K,: 329 GPa, and K"' : 3.66. Figure I shows the cal-culated equation-of-stato compared with experimentaldata. The higher pressure data of Tsuchida and Yagi(1989) are systematically displaced from both the pre-dicted equation-of-state and extrapolated equations-of-state of the lower pressure studies. This may reflect sig-nificant nonhydrostaticity, as suggested in their paper, orerror in the lattice parameters derived from four to sixdiffraction lines, or both. A Murnaghan equation-of-statefit to their data gives K.: 407 GPa and K"' : 3.2 (K,:376 GPa if K'is fixed at 4). This bulk modulus is signif-icantly higher than that obtained by other studies (Table2\.

The present LAPW results differ somewhat from theLAPW results of Park et al. (1988), who found V.: 47.51L', x":288 GPa, and K"' : 3.14 using a Murnaghanequation-of-state. This difference may simply be becausePark et al. (1988) fitted their results over a different range

736 COHEN: STISHOVITE AT HIGH PRESSURE

-10 0 10 20 30 40 50 60 70 80 90 100 110 120130140150P (GPo)

Fig.2. The axial ratio c/a vs. pressure. Symbols are the sameas for Figure l, plus o, which represents data from Ross et al.0990).

of volume, used one energy window, or used a differentexchange-correlation functional.

The present LAPW equation-of-state is in good agree-ment with experiment (Table 2). It is evident from thelarge error bars that there is essentially no informationon the bulk modulus derivative K,'. Even the very highpressure equation-of-state to 75 GPa determined in thediamond anvil cell by Tsuchida and Yagi (1989) cannotconstrain K,' well because of the nonhydrostatic condi-tions and the relatively large uncertainty in the volumeand pressure at the highest pressures. The LAPW staticzero-pressure bulk modulus, 324 GPa, is slightly higherthan the experimental room temperature values, but atthe experimental zero-pressure volume, the present LAPWvalue of 314 GPa is quite close to the experimental valuederived from the single crystal elastic constants of 306 +4 (Weidner er al., 1982).

TABLE 2. Comparison of equation-of-state and zero-pressurestructure parameters with experimental values

LAPW LAPWPresent Park et al. Experiment

study static (1988) static (300 K)

oO)

@(I'

S*G)ct

r)roo

ooN

v. A"K" GPa

46.16324

4.O4

0.6400.306

47.51288

3.'14

46.60A344 ! 27^292 + gec294 + 14oc306 + 4E3.1 + 3.7s 'c3.2 + 3.3oc2.8 + 0.2F

0.638c0.306c

-r0 0 10 20 30 40 50P (GPo)

Fig. 3. Pressure dependence ofthe A,, Raman frequency forstishovite. The experimentddata (n) are from Hemley (1987).

The structural parameters c/a and x (O) agree well withexperiment at zero pressure. Figure 2 compares the cal-culated with experimental c/a ratio as functions of pres-sure. The lattice parameters of Tsuchida and Yagi seemto be less reliable, perhaps because oflarge nonhydrostat-ic stresses compared with either the trends in lower pres-sure data or with the LAPW results. The observed andpredicted c/a values could be combined with the elasticconstants (derived below) to estimate the nonhydrostaticstresses. The calculated decrease in c/a with increasingvolume is consistent with the high-temperature study ofIto et al. (1974), in which c/a deqeased by 0.290lo from291Kto 873 K.

Ar*Raman frequency

Table I and Figure 3 show the calculated A,. Ramanfrequency as a function ofpressure. The zero-pressure Arsfrequency is calculated to be 751 cm-r, in perfect agree-ment with the experimental value of 753 cm-' (Hemley,1987). (Here and below, when the LAPW results are com-pared with zero-pressrue experimental quantities, the de-rived LAPW values are presented for the cell volume46.53 43, which corresponds to -2.5 GPa, to account forzero point and thermal pressure, which is probably roughly2-3 GPa. This is a minor correction in any case.) TheGruneisen parameter T : -(d ln v/dln V) was obtainedby fitting ln v to a second order polynomial in ln Z(with

K"',

clax(o)

A Liu et al. (1974).B Sato (1977).c Reanalyzed by Bass et al. (1981).D Olinger (1976).E Weidner et al. (1982).F Ross et al. (1990).G Spackman et al. (1987).

COHEN: STISHOVITE AT HIGH PRESSURE 737

TABLE 3. Derived values of elastic properties for stishovite

Y (Bohr3) & x 1 o o k " x 1 0 1 K 1l(2k. + k) C"" Cr + Cr2

240 5.15260 4.95280 7.47300 10.6314 12.7320 13.6

840638489376314290

3.085.22c . / o5.655.425.31

551470327255221219

246188173157158145

747661483372323307

1 4051024889812807763

161 215241 087807693650

an rrns error of 0.lolo). The derived value T : 1.40 is inexcellent agreement with the experimental value 1.38 +0.04 obtained by Hemley (1987). The second derivative(d'zlnv/dln V2): -(d ln 7/d ln Z) is ofinterest since thethermodynamic Gruneisen parameter z, which is impor-tant in the thermal equation of state, is related to theindividual mode parameters, /,, and few or no experi-mental data are available to estimate its magnitude. Thesecond derivative obtained is 0.14, which gives ̂ y: 1.47at 143 GPa.

Elasticity

In a crystal with Raman-active modes, the Ramanmodes couple with the elastic constants. Thus elastic con-stants cannot be obtained by simple homogeneous shearsof the crystal lattice. It is necessary either to use the long-wavelength limit of lattice dynamics, as was done usingthe PIB model for stishovite by Cohen (1987a), or tominimize the energy as a function of internal atomic co-ordinates for each value of strain. Here the latter ap-proach is used. The c/a strain at constant volume can berepresented as the following strain tensor:

The new lattice parameters are given by a' : (e * I)awhere l is the identity matrix. The parameter 6 is relatedto c/aby (c/a): (c/a),(L + 6) where (c/a)"is the equilib-rium value of c/ a at the given volume. Strains at constantvolume (Eq. 4) were used here to find elastic constants,so no pressure terms arise in the analysis. Equation 5 wasused to derive the values of the elastic constants, whichis correct for the second-order elastic constants. The re-lationship between the second derivative of the energywith respect to strains is found from A'E/V: t/22 C,,e;ei,using the Voigt notation (Nye, 1985). The result for a

tetragonal crystal and the above strain is AE/V : (l/

9XC,, + 2Cr, + Co - 4CB)62 : C,62. The resulting val-

ues of C" are given in Table 3 as functions of pressure.The elastic constants Ctt + Cn, Crr, artd C,, can be de-coupled by using two of the following which have alsobeen determined: the a and c linear compressibilities, k,

and h defined as for example k.: -dln a/dP, and thebulk modulus K. These quantities are related by K :

l/(2k" + k ). Here the linear compressibilities were ob-tained by fitting a third-order polynomial in the pressure

to the natural logarithm of the lattice parameters, withthe pressures obtained from the third-order Birch fit tothe energies. This procedure is less reliable for the highestpressures because of the limited number of volumes stud-ied. The bulk modulus K and l/(zk" + k ) are comparedin Table 3, and one finds reasonable correspondence be-tween these two estimates. The linear compressibilitiesare related to the following elastic compliances (Nye,

l 98 5):

k , : s , , + s r 2 + s r lk": 2sr., * srr. (6)

The elastic compliances are related to the quantities C,k", and k,by

srr: (4k" + 2k, + C,lA/d

srr: (-2k. - k" + C"k,k,)/d

.s,, + sr2 : Qk" + k" + zc"le)/d (7)

where d : 2C,(k" + k"). The elastic constants can befound by inversion of the compliances. The results areshown in Table 3 and Figure 4.

The derived elastic constants can be compared with thesingle crystal Brillouin scattering results of Weidner et al.(1982) (Table 4). The elastic constants derived from theLAPW calculations are within a few percent of the ex-perimental values, which is probably the limit of numer-ical accuracy of the calculations. Probably the LDA worksso well for stishovite because it is rather stiffin terms ofstrains in volume and c/a, so that relatively large changesin energy are involved. There is no reason for the accu-racy ofLDA to decrease as pressure is increased, and infact its accuracy may increase because the charge densitybecomes more uniform in the limit of very high pressuresand energy differences become much larger.

The very small pressure derivative for C,, which is ap-proximately 0.2, is interesting because it may imply thatthe pressure dependence ofthe average shear modulus is

/ 3 - l 0

( 6 + 1 ; - " ' - t

0 ( 0 +

(4)

which is to linear order in D:

738 COHEN: STISHOVITE AT HIGH PRESSURE

Fig. 4. Derived elastic constants for stishovite as functionsof pressure.

small. The upper bound (Voigt approximation) for theaverage shear modulus for stishovite is G, : 0.3C" +0.1(C,, - C,r) + 0.4C44 + 0.2C66 (see Schreiber et al.,1973). The value of C"' :0.2 is smaller than the valueof 1.3 predicted by the fully ionic PIB model calculations(Cohen, 1987a). The pressure derivative (Cr, - C,r), iscalculated using the PIB model also to have a tiny pres-sure derivative (0.2), and (C,, - C,r) is actually expectedto soften at high pressures because of the softening of theB,, Raman mode (Hemley, 1987; Cohen, 1987a). Usingthe present value of C"' and the PIB values of the otherpressure derivatives gsves Gri : 1.3. The lower bound(Reuss bound) would be significantly less than I (the pIBlower bound for the shear modulus pressure derivative is

TABLE 4. Comparison of calculated zero-pressure elastic prop-erties with experimental values (GPa)

LAPWWeidner et al. Ross et al

(1982) (1990)

!

gu o

. f A X Y M D f A Z U R T A S Z

Fig. 5. Band structure for stishovite at (top) zero pressureand (bottom) 143 GPa. The labeled k-points are in units of (z/as/bs/c),I (0,0,0), X (1,0,0), M (1,1,0), z(0,0,r), R (1,0,1), andA (1 , r , 1 ) .

0.7), and the Reuss bound will go to zero as the B,u modesoftens before C,, - C,, goes to zero. For comparison,the estimated G' is | .7 for MgSiO. perovskite and 2.2 forMgO using the PIB model (Cohen 1987b). The smallpressure derivatives ofthe shear modulus caution one notto rule out the possibility that stishovite exists in thelower mantle because of its relative stiftress at zero pres-sure; in other words, stishovite may stiffen more slowlywith increasing pressure than other possible lower mantlephases.

BoNurNc AND ELECTRoN srRUcruRE

Band structure and hybridization

Figure 5 shows the calculated band structure for sti-shovite at low and high pressures. The only qualitativechanges in the band structure with pressure are generalwidening of the bands. The valence band consists pri-marily of O 2p states and is quite wide (l1.2 eY at P :0), but the dispersion of individual bands in the manifoldis quite small. The small dispersion implies a localizedionic or atomic character, but the very large band widthimplies strong covalency, so that the mixed ionic-cova-

€-gLd

40 60 80 100 120 140P (GPo)

c. 158cr + c12 693c"" 907c," 221*" x 10 12.7k" x 'l0a 5.42K. 314

15606477620313.26.0

308

11 .96.8

313

\ ?S

-?

>

--='

z4t o 2 9

=4

:\

O 2 e

\ <-

COHEN: STISHOVITE AT HIGH PRESSURE 739

lent character of SiO, is evident in the stishovite bandstructure. The Si hybridization in the valence wavefunc-tions has s, p, and d character, and the decomposition ofthe resulting valence charge density atzero pressure con-sists of 0.19 s, 0.26 p, and 0.11 d electrons in each Siatomic sphere.

The O 2s band is also quite wide (4.4 eV), so one mightbe concerned about treating the O 2s states as core statesin, for example, a pseudopotential calculation. The O 2sand valence states are well separated by a gap of 7.0 eV.The calculated band gap (which should be an underesti-mare wirh LDA (Pickett, 1985) is a direct gap atl of 5.7eV. The band widths are in good agreement with the X-rayphotoemission study of Wiech (1984) on stishovite; Wiechobtained an O 2s band width of 5.5 + 0.6 eV and an Ovalence band width of 12 + 0.6 eV, with a gap betweenof 5.5 t 0.6 eV.

The valence band width increases from ll.2 at zeropressure to l3.l eY at 143 GPa, the O 2s width increasesfrom 4.4 to 6.8 eV, the gap between O 2s and O 2p de-creases from 7.0 to 6.2 eY, and the band gap increasesfrom 5.7 to 6.1 eV.

The latter observation is important because it indicatesthat the band gap ofstishovite will not close at pressuresexisting in the earth. The band gap of MgO is also cal-culated to increase with pressure, but the gap of CaOdecreases (Mehl et al., 1988). The band gap also increaseswith pressure in diamond (Fahey et al., 1987), but inmost semiconductors the band gap shrinks with increas-ing pressure. Fahey et al. (1987) explained the positivepressure derivative of the diamond band gap by the ab-sence of d-states with the same principal quantum num-ber as the upper valence states. The same explanationmight hold for SiO, and MgO. One cannot predict wheth-er any silicates or oxides metallize in the earth at highpressures because ofthe presence ofCa and Fe, but met-alization of magnesium silicates is unlikely.

The Hartree-Fock density of states (Nada et al., 1990)is qualitatively similar to the LDA and experiment butdiffers quantitatively, as expected. The Hartree-Fock va-lence band width is 14, compared with 11.2 eV, and thegap between the valence and O 2s bands is 10, ratherthan 7 eV. Nevertheless, the Hartree-Fock total energiesgive a good zero-pressure crystal structure for stishoviteand give qualitatively similar charge density deformationmaps.

Charge density

A more intuitive picture of bonding in stishovite canbe obtained by examining the self-consistent charge den-sity. It is difficult to obtain crystal charge densities ex-perimentally because the interesting part of the chargedensity, that involved in the bonding, is superposed overspherical core densities that usually dominate the diffrac-tion. Spackman et al. (1987) obtained a roliable experi-mental charge density for stishovite by using a combi-nation of powder and single-crystal diffraction data andtook great care to eliminate or correct for systematic er-

rors. The atomic motions were removed from the exper-imental map by an anisotropic Debye-Waller treatment.They refined the nonspherical parts ofthe charge densityalone, using standard sphericalized atomic densities for

the spherical parts. Thus the experiment can contributelittle regarding the relative ionicity of Si and O but can

contribute much about the covalent bonding features. The

experimental deformation densities are compared withthe present calculations in Figure 6. Figures 6a and 6bshow the axial and apical Si-O bonds, and Figures 6c and6d show the O-O shared edge.

Very good qualitative agreement is found between theexperimental deformation maps and the LAPW results.The main difference is that the calculation shows excesscharge on the O and less on the Si than neutral atomicdensities, as was assumed in the experimental study. Boththe theoretical and experimental deformation densitiesshow nonbonding O density perpendicular to the Si-Obond, and both give rise to about the same value of thepeak bond charge density. It should be pointed out thatthe LAPW difference map was calculated using LDAspherical atoms as a reference density so that systematicdifferences in LDA and Hartree-Fock charge densities donot dominate the differences. Since the spherical part ofthe charge density was not determined in the experimentin any case, the choice of neutral spherical reference is

somewhat arbitrary.To explore the bonding in stishovite further, it is useful

to have a reference density, unlike the reference neutralions, that takes the ionicity into account. Furthermore, it

is of interest to know what the effective ionicity is in

stishovite in order to improve model calculations for high-pressure silicates. Cohen (1987a) used the nonempiricalPIB model, with fully ionized Sio* and O2-, and obtainedelastic constants significantly higher than experimentalvalues. One possible source ofthe large discrepancies wasthe neglect of covalency and the use of fully ionized spe-cies. Therefore, the self-consistent charge density is com-pared with PIB charge densities generated using O charg-

es from -2 to -1.2 and Si charges to compensate. TheSi ion charge densities were calculated with partial oc-cupancy of the Si 3s states and calculating a sphericallyaveraged charge density self-consistently. Each O ioncharge density was calculated self-consistently as in thePIB model (Cohen et al., 1987) using a Watson spherecharged opposite that ofthe O ion charge and was spher-ically averaged with partially occupied 2p states. The ra-dius of the Watson sphere was chosen to give the Ma-delung potential at the O nucleus in the chosen chargeconfiguration.

Figure 7 shows the diference in integrated charge ineach 1.5 Bohr sphere centered around each ion betweenthe PIB charge density and the self-consistent LAPWcharge density. There is not a single value of ionicity thatagrees perfectly with the self-consistent inte$ated charg-es, but an O charge between -1.4 and -1.5 is the bestfit. (By separately varying the Watson sphere potential

rather than using the Madelung potential, a slightly better

740 COHEN: STISHOVITE AT HIGH PRESSURE

Fig. 6. Deformation densities for stishovite. (a) LAPW results for axial-equatorial bond plane (5 A by 5 A). O) Experimentalmodel (Spackman et al., 1987, Fig. 7a), same section as a. (c) LAPW result for shared edge between two SiOu octahedra g Aby 4A1. 1a; nxperimental model (Spackman et al., 1987, Fig. 7b), same section as c. Contour interval is 0.05 e/A,'.

fit could be obtained.) Figure 8 shows the differences inthe valence band eigenvalues between the potential gen-erated by the PIB charge density and the self-consistentpotential at the six k-points that were used for self-con-sistency. The differences converge at an O charge between-1.2 and -1.4, shifted somewhat to lower ionicity butgenerally consistent with the ionicity derived from Figure7. It is impossible to draw more precise conclusions onthe ionicity of stishovite by comparing charge densities,but it is clear that the ionicity is reduced from full O,and Si4* to something like O'a and Sirs+. This is very

close to the Mulliken charges of -1.2 and +2.4 derivedfrom Hartree-Fock calculations (Nada et al., 1990). Notethat these static charges should not be confused with thedynamic efective charges appropriate for lattice dynam-ics (Martin and Kunc, l98l).

Also of interest is how the ionicity changes with pres-sure. It appears that no major change in ionicity occursbetween zero pressure and 143 GPa. At the latter pres-sure, the best fit obtained for the integrated muffin tincharges is bracketed by O charges of -1.4 and O -1.5,very similar values for zero pressure.

COHEN: STISHOVITE AT HIGH PRESSURE 7 4 1

oof

o(tI

n-2 -1.5

Oxygen lon ChorgeFig. 7. Difference in muffin tin charges between PIB and

LAPW for different assumed O and Si ion charges at zero pres-

Diference maps show that the O atoms are very non-spherical. O charge moves out ofthe shared edge regioninto the direction perpendicular to the [1 l0] plane oftheshared edge and the Si, i.e., out of the bonding regions.This feature is similar to so-called lone pairs but consistsof much less than two electrons. To model this featureaccurately would require more than a simple shell model,even with multiple shells.

CoNcr-usroNs

For the first time, elastic constants for a complex crys-tal with Raman-active modes are calculated using a self-consistent, full potential method, with no uncontrolledapproximations other than the LDA. Excellent agreementwilh experiment is found for the structural properties, theequation-of-state, the linear compressibilities, three elas-tic constants, and the Raman active A,, mode. Predic-tions are made for the pressure dependence of some elas-tic constants. The feasibility ofaccurately studying elasticand vibrational properties at high pressures from firstprinciples is demonstrated. The discrepancies between thecalculated and observed structures at very high pressures(Tsuchida and Yagi, 1989) suggest a reinvestigation ofthe possible 100-GPa phase transition in stishovite, es-pecially since the interpretation of a phase transition inthe diffraction data was based on a small shift in onediffraction line.

olr)

E

4 ot - | r )\ > lo

-2 -1.5

Oxygen lon Chorge

Fig. 8. Differences in valence band eigenvalues ofthe poten-tial generated by the PIB charge density and the LAPW self-consistent potential at zero pressure.

Both the charge density and the band structure indicatemixed ionic and covalent character in stishovite, withbest fit static charges of O'o- and Si28+. The challengenow is to take the results ofthe total energy calculationsand the observations about the charge density to con-struct a model that can be used to accurately calculatethermal properties and the rest of the elastic constantsand vibrational frequencies at high pressures.

AcxNowr.encMENTs

I owe many thanks to R.J. Hemley for many suggestions and helpfuldiscussions and to M.J. MehI for discussions regarding elasticity theory.Helpful discussions with W.E. Pickett, D. Singh, H. Krakauer, L.L. Boyer,G.V. Gibbs, G.D. Price, N. Nada, N. Ross, and C.R.A. Catlow are alsoacknowledged. Special thanks are due to J.B. Thompson, Jr. for suggestingthat I "may have to consider the electrons." The computations were per-

formed on the Cray 2 Computer at the National Center for Supercom-puting Applications under the auspices of the National Science Founda-tion.

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Mnr.ruscnrp,r REcHVED Frnnueny 14, 1990Memrcnrrr AccEPTED Mencu 7, 1991