structure and bonding of calcite: a theoretical study · structure and bonding of calcite: a...

American Mineralogist, Volume 79, pages 205-214, 1994

Structure and bonding of calcite: A theoretical study

ANonnw J. SxrNNnnr* JonN P. L.c,FrNrrNaMolecular Science Research Center, Pacific Northwest laboratory, P.O. Box 999, Richland, Washington 99352, U.S.A.

Hnnnr J. F. J,cxsnNPhysics Department, Oregon State University, Corvallis, Oregon 97331, U.S.A.

Ansrru.cr

The ground-state structural and electronic properties of bulk calcite (CaCOr) are cal-culated from first principles by working within the full-potential linearized augmentedplane wave implementation of density functional theory. Calculated charge density plotsfor bulk calcite show clearly the mixed ionic and covalent bonding in this mineral. Fromthese plots the relative size of the constituent atoms are determined by calculating bondedradii. In calcite, the O atoms are shown to be smaller than expected from ionic measuresof the atomic size and of a similar size to the Ca atoms. Examination of the one-electroneigenfunctions and the computed density ofstates reveals the character oflocal bonding.Comparison of the computed density of states with recent X-ray photoemission dataallows the orbital character of the experimental valence band peaks to be assigned. Thehighest occupied states are of O 2p character, with Ca 3p states mixing slightly withcarbonate group states in the valence region. The calculated value of the band gap incalcite is consistent with the experimental value, as measured by reflection electron energy-loss spectroscopy. Lattice parameters are reproduced to within 50/o of their experimentalequilibrium values. Under pressure the carbonate group is shown to act as a rigid structuralunit, with the C-O bond length decreasing by approximately 3 mA, for a 5olo compressionin volume.

INrnonucrronr

Calcite (rhombohedral calcium carbonate) is an envi-ronmentally important mineral. It acts as a buffer in seawater (Winchesuer, 1972), forms the major inorganic partof sea shells, and is secreted by some organisms, such ascorals, to form solid skeletal structures (Schmalz, 1972).As a common component of soil, waste sludge, and groundwater, it mediates the subsurface transport ofheavy metalcontaminants (Davis et al., 1987). Transparent calcite (e.9.,Iceland spar), with its large birefringence, is importanttechnologically, having applications in polarized opticaldevices (Hirano et al., l99l).

Our interest in calcite is in understanding its role inthe transport of heavy metal contaminants. A microscop-ic knowledge of the bulk and surface chemistry of calciteand other carbonate minerals is critical to many ground-water contamination issues (Davis et al., 1987) and isimportant to the long-term cleanup of nuclear and chem-ical waste sites (Hochella and White, 1990). Althoughdivalent metal cations are known to adsorb selectively oncalcite in solution (Z,achara et al., l99l), the process bywhich this happens on an atomic scale is not well under-stood (Hochella and White, 1990; Baeret al., l99l). Does

t Present address: Complex Systems Theory Branch Code 6690,Naval Research Laboratory, Washington, DC 20375-5000,U.S.A.

0003-o04x/94l03044205$02.00

this adsorption occur through direct cation substitutionor through complex formation at the interface betweenground water and calcite? What role does the electroniccharacter of the constituent atoms play in this process?To interpret experimental results, a detailed knowledgeof the electronic structure, local bonding, and relative sizeof the atoms in calcite is needed. This information is alsocritical in developing semiempirical models based onquantum mechanics to simulate these complex geochem-ical processes on the atomic scale.

Experimental measurements of the electronic proper-ties of calcite are made more complex by surface chargingand decomposition problems. Those spectral measure-ments that do exist have mostly been determined withX-ray photoemission spectroscopy (XPS) and have con-centrated on the core-level photopeaks (Hochella, 1988;Stipp and Hochella, l99l; Baer et a1., l99l; Baer et al.,1992). Only recently have detailed lower binding-energyX-ray-generated valence-region photospectra been col-Iected (Baer and Moulder, 1993). He(I), He(II) (Connoret al., 1978), and X-ray photoemission spectra (Calabreseand Hayes, 1975) of weakly bound carbonate salts (KrCO,and LirCOr) have given additional valence informationon the carbonate anion. Anisotropic X-ray emission hasallowed the o and zr valence electron spectral contribu-tions of the carbonate group to be separated in calcite(Tegeler et al., 1980). X-ray diffraction electron charge-density studies ofcalcite (Peterson et al.,1979)have shown

205

206 SKINNER ET AL.: CALCITE STRUCTURE AND BONDING

N a = 6c t= @

v d = L ,

c = @O = o

(a) b)

Fig. l. Comparison of (a) the NaCl rock salt unit cell with(b) the calcite rhombohedral unit cell.

that charge builds up between the C and O, indicatingthe covalent nature ofthe bond. However, because ofthedifficulty of deconvoluting the thermal motion of the car-bonate group, no detailed charge density plots for calciteexist in the literature (Gibbs, personal communication).

Theoretical quantum-mechanical calculations of theelectronic and structural properties of calcite are scarce.All have been within a Hartree-Fock formalism and havebeen used only to calculate effective point charges as in-put to electrostatic models (Ladd, 19721- Yuen et al., 1978)or to estimate energy differences between various CaCO,polymorphs (Yuen et al., 1978). These calculations havebeen limited by the use of finite clusters and small Gauss-ian basis sets. More recently, good values of the refractiveindices of calcite were calculated working within a ran-dom phase approximation method and using largeGaussian basis sets (Tossell andLazzeretti, 1988). How-ever, these calculations must be considered preliminary,as calcite was only modeled by a collection of indepen-dent Ca2+ and CO3- ions. The only other related quan-tum-mechanical calculations have been on the CO3 an-ion, either isolated (Connor et al., 1972; Random, 1976;Tossell, 1976; Konovalov and Solomonik, 1983; Tossell,1985; Hotokka and Pyykko, 1989) or embedded in a lat-tice of point charges to mimic the calcite crystal environ-ment (Julg and L6toquart,1977).

In this paper, we calculate the ground-state electronicand structural properties of bulk calcite using densityfunctional theory (DFT) (Hohenberg and Kohn, 1964;Kohn and Sham, 1965) as implemented within the full-potential, linearized augmented plane wave (FLAPW)method (see Jansen and Freeman, 1984). The only inputsto the calculation are the atomic number and position ofthe constituent atoms in the periodic rhombohedral unitcell. Our results are checked against experimental resultswhere available. The calculated density of states (DOS),in particular, is directly compared with detailed valenceband XPS data for calcite (Baer and Moulder, 1993) and

reflection electron energy-loss spectroscopy (REELS)measurement of the band gap (Baer and Blanchard, 1993).These latter two measurements were taken especially forcomparison with calculations in this paper and are part

of a combined experimental and theoretical effort to studythe role of calcite in the transport of contaminants in theenvironment. Study of the calculated charge density re-veals the mixed nature of bonding within calcite and therelative sizes of the constituent atoms. Experimentallythe carbonate group can be considered isolated from itscrystal environment. For example, C-O bond lengths arealmost identical in many carbonates (Znrnann, l98l) andonly change by a small amount under compression (Rossand Reeder, 1992). Current empirical atomic scale mod-els of carbonates (e.g., Dove et al., 1992) exploit this bytreating the carbonate group as a rigid, incompressibleunit. In this paper, we further test the environmental in-dependence of the carbonate group by calculating thechange in the C-O bond length under compression forcalcite.

Bur-x srnuc:tunB

The structure of calcite has the space group R3c andcan be thought of as a distorted rock salt structure. Todemonstrate this, we start with the undistorted NaCl rocksalt structure given in Figure la. For this structure theprimitive basis translations are the vectors n,, ar, and at,where the vectors are ofequal length and are at angles of33.6" to each other. The primitive rhombohedral unit cellof calcite (Fig. lb) can be obtained by replacing Na atomswith Ca atoms, each Cl atom with the C atom at thecenter of each carbonate group, and uniformly compress-ing this structure along the threefold symmetry axis Il I l]so that the translational vectors make angles ofapproxi-mately 46" with one another.

The primitive rhombohedral unit cell of calcite con-tains ten atoms (two CaCO. formula units) and consistsof alternating (lll) planes of Ca atoms and carbonategroups. Ca atoms are situated at 0 and t/z along Il I l], andcarbonate groups are located at Vq and 3/ along this vector.Each carbonate group forms an equilateral triangle lyingin the (l I l) plane, and carbonate groups in adjacent car-bonate layers are rotated by 60" with respect to one an-other. Three structural parameters are needed to describethe calcite structure: a, the length of each translationalvector; a, the angle between them; and u, a parameterproportional to the C-O bond length. In Table I we listthe experimental equilibrium rhombohedral structuralparameters for calcite (Reeder, 1983), together with theequilibrium C-O bond leneth d(C-O) and the primitiveunit-cell volume Zo. These values were determined fromthe alternative hexagonal lattice description ofcalcite (p.8, Reeder, 1983). To denote the calcite cleavage plane,instead of using the rhombohedral indices (2ll) we usethe hexagonal indices (10T4), as that has become the stan-dard notation quoted in the literature.

DrNsrrv FUNCTToNAL THEoRy wITHIN THE LocALDENSITY APPROXIMATION

General rnethod

To calculate the zero-temperature ground-state prop-erties of bulk calcite, we use density functional theory(DFT) (Hohenberg and Kohn, 1964; Kohn and Sham,1965), modeling exchange-correlation effects within thelocal-density approximation (LDA) (Hedin and Lund-qvist, l97l). As a first principle, or ab initio, method, theonly inputs to a static calculation at a fixed geometry arethe position and atomic number of each atom in the unitcell. Within the DFT-LDA method there are no restric-tions on the types of atom, bonding, or material that canbe considered. For mixed bonding systems, such as cal-cite, that is essential, as the ionic and covalent bondingmust be treated on equal terms. Working within the LDA,ground-state properties and valence band spectra are wellproduced. Structural parameters and elastic moduli typ-ically can be computed to within 5 and 20o/o of the cor-responding experiment values. For an excellent review ofdensity functional theory, consult the paper by Jones andGunnarsson (1989).

Within the DFT formalism, the total energy of a manyelectron system can be written exactly as a functional ofthe electron charge density, p (Hohenberg and Kohn,1964). To calculate the ground-state electron density, andhence the ground-state total energy, of a particular sys-tem, a series of coupled eigenvalue equations (Kohn andSham, 1965),

(-v'+ V*lpDr": Eil/, (l)

need to be solved. For these one electron equations, eachelectron moves in an effective potential, V.nlpl, that de-scribes its interaction with the fixed atomic nuclei and allthe other remaining electrons. The charge density,

can then be constructed from the eigenfunctions, ry',, bysumming over all occupied states. Because the effectivepotential, %n[p], depends explicitly on the charge density,p, Equations I and 2 must be solved self-consistently.

To solve these one electron equations, we use theFLAPW method (see Jansen and Freeman, 1984). Thisis the most precise implementation of DFT. No shapeapproximations are made for the potential V.nlpl, and allcore and valence electrons are treated explicitly. Each oneelectron equation is solved using a linearized, augmentedplane wave (LAPW) basis (Andersen, 1973; Koelling andArbman, 1975). These LAPW basis functions are definedas follows. In spherical regions centered about each atom,atomic basis functions are used, whereas in the remaininginterstitial region, a plane wave is used. The solutions arethen matched at the boundary, and the energy depen-dence, which is inherent in the traditional augmentedplane wave (APW) method (Loucks, 1967), is removedby linearizing about a typical energy for the band ofinterest.

207

TrsLe 1. Comparison of calculated equilibrium lattice parame-ters, C-O bond length, unit-cell volume, and bulk mod-ulus with exPerimental values

Theory(this study) ExPeriment

SKINNER ET AL.: CALCITE STRUCTURE AND BONDING

a (A)o (')

ud(c-o) (A)% (4.)Ko (GPa)

6.20148.30.26511.346

121 .9657

6.375-46.1-

0.2568-1.281'

122.62-71 .07-., 71 .56f, 73.1 5*

- Derived from hexagonal description of calcite (see Reeder' 1983)'t' Measurements were taken at 298 K for a single calcite crystal (Singh

and Kennedy,1974).t Kaga (1968).+ Bridgman (1925).

Within the FLAPW method, the numerical error in thetotal energy scales with the number of basis functionsused (Jansen and Freeman, 1984). The total energy foran infinite basis can then be extrapolated from finite sizebasis results. The FLAPW method thus allows energydifferences between distorted geometries or differentpolytopes to be calculated precisely. Realistic estimatesof absolute and relative errors in the total energies canalso be obtained. This implementation of density-func-tional theory has been used to study metals (Jansen andFreeman, 1984), semiconductors (Massidda et al., 1990),ionic solids (Jansen and Freeman, 1986), and molecularsolids (Min et a1., 1986). Recent applications of thismethod include calculating the Fermi surface for new cu-prate high-temperature superconductors (Pickett et al.,1992) and the elastic moduli of Al-rich Al-Li compounds(Mehl, 1993). Equations of state and electronic propertiesof minerals such as MgSiO, perovskite (Cohen, 1989),stishovite SiO, (Cohen, 1991), and CaO and MgO (Mehlet al., 1988) have also been studied. This applicability tosuch a wide range of materials, together with the abilityto obtain realistic estimates of errors incurred, gives usconfidence that this method is appropriate for the studyof calcite.

Although the FLAPW method is a numerically precisemethod for solving the DFT-LDA equations and calcu-lating total energies, it is also one of the more computa-tionally intensive. The exact central processing unit (cpu)time depends on the type and number of atoms in theunit cell, the number of basis functions, and the numberof integration k points used. For calcite, with ten atomsin the unit cell, a large basis of 126 functions per atom,and 20 integration k points, each bulk total energy cal-culation takes approximately 13 cpu h per geometry ona Cray Y-MP.

Details of calcite calculation

For the FLAPW calculations described in this work forbulk calcite, the C and O 2s and 2p orbitals along withthe Ca 3p and 4s orbitals are treated as valence states.All the remaining electrons are defined to be core elec-trons. For the valence electrons, one electron eigenvaluesare calculated within a semirelativistic approximation


TABLE 2. Comparison of the calculated bonded radii with ionicand covalent radii for Ca. C. and O atoms

lonic radius. Covalent radius'(A) (A)

Bonded radiusAtom (A)

1 .18"0.4011 .1 8'- 0.88t

1 . 3 5 2

1 . 3 5 0

1 . 3 4 8

1 . 3 4 6

1 . 3 4 4

1 . 3 4 2

oI

oo

0 9 90 1 51 .40o

0.770.66

. Kitrer (1986).'- Calculated from the Ca-O bond.t Calculated from the C-O bond.

(MacDonald et al., 1980) and using a single energy win-dow, whereas for core electrons the one electron eigen-values are treated fully relativistically. Janak's (l 978) pa-rameterized form of the local density approximation isused throughout. A periodic primitive rhombohedral unitcell (Fig. lb) for calcite is used for all our calculations (nofinite cluster approximations were made to simulate thebulk system). Atomic units of energy, the Rydberg (Ry),and ofdistance, the Bohr, are used (l Ry : 13.6058 eVand I Bohr : 0.529 A;. However, in Tables I and 2distances are given in dngstrdms.

The principal numerical convergence parameters with-in the FLAPW calculation are the number of k pointsused to perform Brillouin zone integrations and the num-ber of augmented plane waves used to expand the oneelectron eigenfunctions in Equation l. After extensiveconvergence tests, 20 k points and over 126 basis func-tions per atom (a cutoff of k^ - : 4.5) were used for theelectronic structure and charge-density calculations re-ported here. This yields relative errors in the total energydifferences of <2 mRy per atom. To calculate the equi-librium geometry and bulk modulus, we used a smallerbasis set on the order of 60 basis functions per atom (k-"-: 3.5), together with l0 k points. In this case, relativeerrors were larger, on the order of 5 mRy per atom. Thedetails of the convergence test and values of various se-ries cutoffs used in this calculation are given in theappendix.

Rnsur-rsShuctural parameters

To determine the equilibrium structure and bulk mod-ulus for calcite as predicted within the DFT-LDA meth-od, we calculated the total energy of bulk calcite at 0.9,1.0, and l.l of the observed experimental equilibriumvolume Vo. At each volume, the energy was calculatedfor three values of the angle a and for three C-O bondlengths. In total, the energies of 27 distorted calcite struc-tures were determined. A quadratic fit was then used todetermine the optimal angle and C-O distance thal min-imized the energy at each volume. To calculate the energyat each geometry, a basis set of about 60 functions peratom (h.. : 3.5) was used, and integration was per-formed with l0 k points. A basis set of this size must beconsidered small for these types of calculation, and, al-though structural properties are well produced, properties

1 . 3 4 00 . 8 5 0 . 9 0 0 . 9 5 1 . 0 0 1 . 0 5 1 . 1 0 1 . 1 5

v/vo

Fig. 2. Calculated change in the C-O bond lenglh with volume.

that depend on the accurate determination ofenergy dif-ference are less accurate. Our calculation of the bulkmodulus to follow must therefore must be considered asa first estimate.

In Figure 2 the calculated change in the C-O bond lenglhas a function of volume is plotted. Under compressionthe C-O bond length decreases. For a 5olo decrease in vol-ume, we calculated this change to be small, approximate-ly 3 mA @r 0.2o/o). This result compares well with recentstructural refinements determined from high-pressureX-ray intensity data for the rhombohedral carbonates do-lomite and ankerite (Ross and Reeder, 1992). In thatstudy, a 40lo decrease in volume reduced the C-O bondlength by S + 3 mA. Over this volume range the carbon-ate group in calcite acts as an incompressible structuralunit. Treating the carbonate group as a rigid object withinempirical models of calcite (e.g., Dove et al., 1992; Skin-ner and LaFemina, in preparation) therefore is a goodapproxrmatron.

The calculated equilibrium structural parameters, C-Obond length, unit-cell volume, and isothermal bulk mod-ulus (Ko) are given in Table I as determined from a qua-dratic fit. Structural parameters are reproduced to within5olo of their experimental values, as given by the recentstructural refinement in Reeder (1983). The error in thebulk modulus, as calculated at the predicted LDA vol-ume, is approximately 20o/o larger. Thermal correctionsto our zero temperature calculation would only increasethe bulk modulus by approximately 2 GPa, givinga l7o/oerror. The major numerical causes of the error in thecalculated bulk modulus are due to the small basis setbeing used, for which the relative error in the total energyis 5 mRy per atom, and the small number of volume datapoints that were used. More extensive calculations at alarger number of volume points, using a larger basis set,on the order of 120 functions per atom (k-". : 4.5), and20 integration k points, would reduce this error. Furtherimprovements may also be obtained if the calculationswere restricted to a smaller volume regime. That wouldallow direct comparison with the experimental measure-


o)

o,o

UJ

15.0

5 .0

-5.0

-15 .0

-25.0

--=

EF

-

Fig. 3, Electronic band structure for calcite along selectedhigh-symmetry directions as defined by the points A (0.5, 0.0,0 .0 ) , I ( 0 .0 ,0 .0 ,0 .0 ) , Z (0 . s ,0 .5 ,0 .5 ) , and D (0 .5 ,0 .5 ,0 .0 ) i nthe first Brillouin zone (see Fig. l6b of Koster, 1957), wherefractional coordinates in parentheses refer to reciprocal latticevectors. The Fermi energy (.E") is defined to be at the valenceband maximum.

ment of the bulk modulus that uses data only over a l.60lovolume compression regime (Singh and Kennedy, 1974)before calcite transforms to the monoclinic CaCO.-IIphase (Merrill and Bassett, 1975). However, to do this,energy differences on the order of 2 mRy must be calcu-lated very accurately. That would require very large basissets, which are currently computationally prohibitive.

Electronic structure

To calculate the electronic band structure (and thecharge density), we worked at the experimental equilib-rium geometry for calcite (Reeder, 1983), solving the oneelectron equation (Eq. l) with a basis set ofover I 20 basisfunctions per atom and 20 integration k points. Figure 3shows the resultant band structure of calcite for selectedhigh-symmetry lines in the first Brillouin zone. The bandsare flat, with the highest occupied band completely filledand the lowest unoccupied band completely empty (con-sistent with calcite being an insulator). The smallest bandgap is calculated to be indirect from D + Z, with a valueof 4.4 + 0.2 eY as extrapolated from careful finite sizebasis set studies. The experimental value ofthe band gapis 6.0 + 0.35 eV (Baer and Blanchard, 1993), as measuredby reflection electron energyJoss spectroscopy (REELS).Our calculated value of the band gap is consistent withother DFT calculations, which show that the local densityapproximation typically underestimates the experimentalband gap in solids. The relatively large 270lo error in thecalculated band gap compared with experiment is alsoexpected, as errors can range from 25 to 800/o or more(Pickett, 1986). An improved description of the ex-change-correlation effects, beyond the local density ap-proximation, can partially correct for this (Jones andGunnarsson, 1 989).

o@@

thin the calcite

Ca=

A

209

aII

oo-

Fig. 4. The relative arrangement of atoms wi(10T4) cleavage plane (dashed line).

In the range ofenergy shown in Figure 3, four occupiedbands of well-defined symmetry exist below the Fermienergy (.Er), defined as the valence band maximum. Low-est in energy, at approximately -20 eV, is an atomiclikeband of predominantly O 2s character. The next highestband, at -17 to -16 eV, comprised the Ca 3p states,which are raised in energy relative to their atomic levelsand mix appreciably with O 2s states in the carbonategroup. The broad band at -4 eY is composed of hybrid-ized O 2s, O 2p, C 2s, and C 2p states and represents thebonding states within the carbonate group. Last, the bandimmediately below the Fermi energy is essentially pureO 2p in character, except for a slight mixing of Ca 4p and3d states. A detailed discussion of the symmetry char-acter of these bands is given in the next section, alongwith a discussion of the charge density and the compo-nent eigenfunction contour plots.

Charge density and the density of states

X-ray diffraction studies for calcite (Peterson et al.,1979) showed a buildup ofcharge between the C and Oatoms in the carbonate group, indicating its covalent na-ture. Studies on the rhombohedral carbonate dolomite(Effenberger et al., 1983) showed a similar buildup ofcharge in the C-O bond and also observed a buildup ofcharge in the O lone pair regions. However, attempts atproducing detailed X-ray diffraction charge-densitydistributions for calcite have been unsuccessful (Petersonet al., 1979). This is primarily due to the problem ofdeconvoluting the translational, librational, and (large)screw motion of the carbonate group (Gibbs, personalcommunication).

In Figure 4 we show the arrangement of atoms in theunrelaxed calcite (10T4) cleavage plane (dashed line). Foreach carbonate group, one C-O bond lies within the plane,whereas the remaining two O atoms lie equidistant aboveand below the plane, respectively. All Ca atoms lie in theplane. In Figure 5, the calculated bulk charge densitywithin the calcite cleavage plane is given. From this plotthe mixed nature ofbonding in calcite can be clearly seen.To obtain a quantitative measure of this bonding, thevalue of the charge density is determined at the points P,

MklgJlJ=a-/


TleLe 3. Net no. of valence electrons within spheres centeredon the atoms and in the remaining interstitial (l) region

No. of electrons Sohere radius

Atom densities. Bulk calcite-'

z . J

o.71 . 5

Ca

oI

Total

12.17010.512

26.34620.97260.000

1 2.6531o.523

30.04416.78060.000

1.3230.370o.794

Fig. 5. Charge density for bulk calcite within the (1014)cleavage plane. Two unit cells are shown. Contours are spacedat 0.05 Bohr 3 and are in the range 0.05-0.5 Bohr 3.

Q, and R, the midpoints between C-O, Ca-O, and Ca-Caatoms, respectively. For these points the charge densityhas the value 0.3, 0.04, and 0.02 Bohr-3, respectively.For an ionic solid such as NaCl, the charge density at themidpoint of the Na-Cl bond is on the order of 0.01 Bohr-3.Comparing our results for calcite with this ionic refer-ence, the C and O atoms are covalently bonded withinthe carbonate anion, whereas the carbonate group is ion-ically bonded to the Ca cation. This buildup of charge inthe C-O bond is also borne out by calculations on theisolated COI anion (Tossell, 1985).

Our calculations thus confirm the mixed ionic and co-valent bonding within calcite that has been commonlyassumed within the mineralogy community. To modelaccurately the bonding in calcite, it is crucial that thecovalent bonding within the carbonate group be included.Point charge models have tried to mimic this covalentbonding by representing the carbonate group charge dis-tribution by a collection of point charges (Ladd, 1972;Yuen et al., 1978). More recently, empirical models forcalcite (Dove et al., 1992) have included this covalentcharacter with bond-bending terms. We believe, howev-er, that semiempirical models based on quantum me-chanics need to be developed for calcite that explicitlyinclude the covalent nature of bonding within the car-bonate group. That would be especially important whensimulating processes where electronic effects are expected

'Distribution obtained from an overlapping of atomic densities placedat the atomic sites.

" Distribution for bulk calcite.t Ca 3D electrons are treated as valence electrons and included in this

sum.

to be crucial, for example, the substitutional adsorptionof Mn cations on the calcite surface (Baer et al., l99l).

An estimate of the sizes of the atoms in calcite can alsobe obtained from the charge-density plots in Figure 5, bycalculating appropriate bonded radii (Gibbs et al., 1992,and references therein). These bonded radii are definedas the distance along an atomic bond from the point ofminimum electron density to the atom centers. For cal-cite, the bonded radii along the C-O bond within thecarbonate group are r(C) : 0.40 and r(O) : 0.88 A,whereas along the bond formed between each Ca atomand the nearest O atom the bonded radii are r(Ca) : l. I 8and r(O) : 1.18 A. In Table 2 these values are comparedwith the standard ionic and covalent radii for these at-oms. From these values, it is seen that the radii of the Cand O atoms in calcite fall between these two limits. TheO anions are 16-370/o smaller than those expected basedpurely upon an ionic picture of atomic size, whereas theCa cations are 20o/o bigger. At the calcite cleavage surface,the sizes of these ions control the approach of solvatedcontaminant ions and the loss of their waters of solvation.If the bonded radius of O determined from the Ca-Obond of l.l8 A is taken as the measure of the size of anO ion at a calcite surface, then the O anion is the samesize as the Ca cation. The comparable size of these ionscould therefore have significant consequences for thechemical reactivity of the calcite surface.

Table 3 gives the net number of valence electrons with-in spheres centered on the atoms and in the interstitial(I) region for bulk calcite. These are compared with thenumber of valence electrons obtained within these sameregions from overlapping Ca, C, and O atomic densitiesplaced at the atomic sites. As with all population analy-ses, there is no quantitative significance associated withthe values of these numbers, since they depend on detailsof the procedure used to obtain them (in our case thechoice of sphere radii). However, a qualitative under-standing ofhow charge is rearranged in the unit cell canbe obtained. From Table 3, it is clear that the majorredistribution ofcharge is from the interstitial region toinside the O spheres. This results in a net increase ofcharge within each O sphere of approximately 0.62 elec-trons, compared with the neutral atom case. For Ca, there

XPS

V

IVil l


15 .0

2tl

5.0

IO)!-

ocul

- 1 5 . 0

a

1 5 . 0

-5.0

EF

5.0

1 5 . 0

-25.0 25.O

Fig. 6. Comparison of the calculated density of states (left)with experimental X-ray photoemission spectrum (XPS) (right).To compare bands, the XPS zero ofenergy has been shifted by5.5 eV to align the theoretical and experimental valence bandmaxima. ktters a-d denote the position of the energy eigen-values corresponding to the eigenfunction components plottedin Fig. 7.

is also a net increase in charge ofabout 0.24 electrons peratom. That conflicts with the classical ionic picture, usedin most electrostatic models, of Ca being a divalent calion(Yuen et al., 1978; Singh et al.,1987; Dove et al.,1992).In fact, the atomic Ca 4s orbital is long-range, extendingto about 4.0 Bohr (2.116 A), so that 960/o of the Ca 4selectronic charge is within the interstitial region. For theC atom, there is a net increase in charge within the Csphere of 0.006 electron per atom. This change is twoorders of magnitude smaller than both the Ca and O casesand is not significant within this simple analysis. In con-trast, most single-point ion representations ofthe carbon-ate group, obtained through either fitting charge param-eters in electrostatic models (Yuen et al., 1978; Singh etal., 1987; Dove et al., 1992) or by Mulliken populationanalysis in ab initio SCF-MO calculations (Yuen et al.,1978; Connor et al., 1912), flnd a positive charge on theC atom of approximately + l. The difference between thesetwo analyses is, in part, simply a matter of perspective.In our calculation, the small C sphere radius of 0.7 Bohr(0.370 A) focuses the analysis near the C atom, where anegligible change of charge occurs. On the other hand,within the single-point ion descriptions of the carbonategroup, negative charge is constrained to cling to the atom-ic sites, as opposed to between the C and O atoms.

From Equation2 it can be seen that the charge density

Fig. 7 . Real-space contour plots within the (l I 1) carbonateplane of four (a-d) eigenfunction components 19"(r)1'? of thecharge density, as calculated at I(k : 0). In each panel a singlecarbonate group is shown consisting ofthree O atoms positionedat the corners of an equilateral triangle and one C atom at thecenter of the triangle. Contours are spaced at 0.02 Bohr 3 andare in the range 0.02-0.2 Bohr 3.

p can be decomposed into the occupied eigenfunctioncomponents l{,(r)l'. Although the eigenfunctions 9,(r)have no formal physical significance within DFT, they doreveal the symmetry of various states as a function ofen€rgy and yield a qualitative insight into the orbitalcharacter of the electronic band structure and density ofstates shown in Figures 3 and 6. To obtain a more de-tailed understanding of the covalent bonding within thecarbonate group, in Figure 7 contour plots of lrl"(r)1'z, ascalculated at I (k : 0), are given as a function ofpositionr. Each panel (a-d) shows a single carbonate group withinthe (l I l) carbonate plane. The positions ofthe associatedone-electron energy eigenvalues -8, are marked (a-d) onthe density of states (DOS) in Figure 6.

For the lowest energy eigenvalue at a, a symmetricbonding combination of mainly O 2s (480/o) and the C 2sand 2p states in the interstitial region is formed. A smallcontribution (l0o/o) of O 2p is also present. This resultsin charge piling up between the C and O atoms in a co-valent manner. In contrast, for the eigenvalue aI b, a sym-metric nonbonding combination of mainly O 2p (360/o)and O 2s (200/o) with C 2s and 2p states is obtained. Atthat value, charge builds up outside the C-O bond, re-sulting in orbital lobes on the O atoms pointing awayfrom the central C atom. These lobes are consistent withthe observed X-ray diffraction electron-density maxima

a

2 1 2 SKINNER ET AL.: CALCITE STRUCTURE AND BONDING

in the O lone pair region for the rhombohedral carbonatedolomite (Effenberger et al., 1983). At c, a degeneratestate, hybrid orbitals are formed between O 2p (49o/o) andC states, with some mixing of O 2s states (120lo). Last, forthe highest occupied eigenvalue at 4 which is positionedat the Fermi energy, the states are nonbonding and havenearly pure O 2p (72o/o) character, except for a slight mix-ing of Ca 4p states (2o/o). ln our analysis, the percentagebreakdown of the wave functions given in brackets shouldbe taken as only a qualitative measure of orbital charac-ter, as these values ultimately depend on the choice ofsphere radii used to partition space.

In Figure 6, the valence band X-ray photoemissionspectrum (XPS) for bulk calcite (Baer and Moulder, 1993)is compared with the calculated density of states. Withallowances for finite temperature broadening of the XPSspectrum, the relative band widths and peak positionsare well reproduced within our zero-temperature ground-state calculation. An assignment of the orbital characterof the experimental XPS peaks can thus be made by di-rect comparison with the calculated band structure anddensity of states. The lowest XPS band at a binding en-ergy of approximately 20 eV is mainly bf O 2s character(peak I), with some mixing of Ca 3p states (peak II). Atapproximately 5 eV, peaks III and IV are due to the mix-ing of C and O 2s and 2p states within the carbonategroup, whereas the highest occupied band at about 2 eV(peak V) is almost pure O 2p states. Peak intensities com-pare poorly with experimental results, as they ultimatelydepend on scattering effects and the different cross-sec-tional areas ofs and p electrons; these factors are exclud-ed from the ground-state density-functional formalismused in this paper. However, the excellent agreement inpeak positions and band widths demonstrates the impor-tant role that ab initio calculations can play in interpret-ing complex experimental data.

These results, together with previously discussed struc-tural calculations, are currently being used to parameter-ize computationally economical tight-binding models forthe study of calcite surfaces and steps (Skinner andLaFemina, in preparation).

AcxNowr,nocMENTS

The authors would like to thank D.R. Baer, R.E. Cohen, D.L. Blan-chard, T.J. Godin, M.J. Mehl, D.M. Sherman, and M.F. Stave for enlight-ening discussions and practical comments during the course of our re-search and L. Rivenbark for careful proofreading of this manuscript. Inaddition, we would like to thank D.R. Baer and D.L. Blanchard for al-lowing us to use their XPS and REEIS data for calcite prior to publica-tion. Correspondenc€ with G.V. Gibbs, S.C. Nyberg, and P.S. Yuen con-cerning their work is also appreciated. J.P.L acknowledges the support ofthe Geosciences Research Program, Division of Engineering and Geosci-ences, U.S. Department of Energy, under contract number DE-AC06-76RLO 1830 with Battelle Memorial Institute, which operates the PacificNorthwest Laboratory. For computing facilities we would like to acknowl-edge the support of the National Energy Research Supercomputing Center.

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M*.n-rscnrrr RECEIvED Mll 7, 1993MnNuscmrr AccEprED Nover"rsER 19, 1993

ApprNmx 1. Nulynmc,ll, coNvERGENCE TEsrs

In this appendix, we present the numerical details ofthe cal-culations and discuss the convergence of the total energy withthe number of k points and basis functions.

At each calcite geometry, nonoverlapping spheres of radii 2.5Bohr (1.323 A), 0.2 sohr (0.370 A), and 1.5 Bohr (0.794 A)centered on the Ca, C, and O atoms, respectively, were chosento divide up space. The radius of the C sphere was smaller thanwhat we would have liked to use. Ho\ilever, our choice was re-stricted by the small C-O bond length of 1.281 A. If the sphereswere not to overlap for the range of volumes of interest, thenthe sum of the C and O sphere radii had to be less than thisvalue. Ultimately, the radii of these spheres do not affect theconverged value of the energy, bur they do control the rate ofconvergence. The charge density p and the one electron potentialV"rftl are then expanded in terms of spherical harmonics I,-within the spheres and plane waves e's'in the interstitial regionbetween the spheres. For calcite, we truncated these expansionsat l:6 and for all reciprocal lattice vectors g such that lgl <

k^, : 9 .O au. Higher expansions in / of the potential and densitytypically change the total energy by <0.5 mRy (Jansen and Free-man, 1984). (If we had used only the /: 0 component and asingle plane wave at k : 0, that would correspond to a mufrn-tin approximation for the potential.)

Each angular momentum component Z of the LAPW basisfunction inside the sph€res is characterized by an energy param-

eter EL. For calcite, all functions from Z : 0 through to I : l0were included, giving 33 energy parameters to be determined.For the well-defined Ca 3p, O 2s, and O 2p bands, the corre-sponding energy parameters, E"[Ca], E"[O], and Eo[O], were cho-sen to be at the center ofeach band. All other energy parameterswere set to the energy O p level, ,8"[O], as that is the largestsource of electrons. The number of basis functions used in ourcalculation was controlled by a cutoffparameter k-.,. For a par-

ticular integration point k, all basis functions labeled by the re-ciprocal lattice vector g were generated such that lk - g | < k-...

To determine the appropriate numbers of k points and basisfunctions to use, extensive convergence tests were performed.

For k-.. : 3.5, the number of k points was varied from l0through 80. For 20 k points, the total energy was well convergedto within an asymptotic regime, in which the relative errors were<2.0 mRy per atom. To investigate the convergence of the totalenergy with the number ofbasis functions, k* was varied from



3.5 (approximately 60 basis functions p€r atom) to 4.5 (over 120basis functions per atom). The associated total energies, E.,(h-)were then linearly frtted to an equation of the form t,",(k-",) :.8.,(oo) + A / (k^"")4, 5 . Comparison of the finite basis results withthe extrapolated total energy E.,(oo) allowed estimates of theabsolute and relative errors in the energy to be obtained. Fork^..: 4.5, although the absolute error in the total energy waslarge, on the order of 50 mRy per atom, relative errors in thetotal energy between the experimental geometry and a distorted

calcite structure were <2 mRy per atom. [The parameters forthe distorted calcite structure were a : 35", volume : 0.9Vo,and d(C-O): 1.281 A.l

For the calculation ofthe electronic structure and charge den-sity, 20 k points and a cutoff of k-., : 4.5 were chosen. Thischoice of parameters resulted in a basis set size of over 120functions per atom and principal energy parameters ofEo[Ca] :-r.1936, E"lOl : -1.2224, and E,[O] : 0.032 Ry at the ex-perimental geometry.

structure and bonding of calcite: a theoretical study · structure and bonding of calcite: a...

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