semiempirical electronic structure calculation on ca and pb apatites

Semiempirical Electronic StructureCalculation on Ca and Pb Apatites

MARIA MATOS,1 JOICE TERRA,2 D. E. ELLIS3

1Departamento de Física, PUC-Rio, Gávea, CEP 22453-970, Caixa Postal 38071, Rio de Janeiro, Brazil2Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil3Department of Chemistry and Institute for Catalysis in Energy Processes, Northwestern University,Evanston, IL 60208

Received 20 May 2008; accepted 17 July 2008Published online 2 October 2008 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.21887

ABSTRACT: A systematic study is made on the electronic structure of stoichiometriccalcium and lead apatites, using the tight binding extended Hückel method (eHT). Theaim is to investigate the applicability of the semiempirical theory to study this family ofcompounds. A10(BO4)6X2 (A = Ca, Pb) apatites, differing by substitutions in the BO4

tetrahedral unit (B = P, As, and V) and X-channel ion (X = OH, Cl), are considered. Thecalculations show that eHT is suitable to describe basic properties especially concerningtrends with atomic substitution and geometry changes. Band structure, Mulliken chargedistribution, and bond orders are in good agreement with results of ab initio densityfunctional theory (DFT) found in the literature. Large variations in the optical gap due tovanadium and lead substitutions are newly found. Changes in the anion X-channel affectthe optical gap, which is in close agreement with DFT results. Analysis involving subnetsare performed to determine the role of halogenic orbitals in the electronic structure ofchloroapatites, showing evidence of covalent Cl bonding. It was also found that Pb–OHbonding in hydroxy-vanadinite Pb10(VO4)6(OH)2, recently synthesized, is weaker thanthat of Ca–OH in vanadate Ca10(VO4)6(OH)2. Arsenium is found to be more weakelybound to the O-tetrahedron than phosphorous, although Ca–O bond is increased with thesubstitution. We investigate, in addition, the electronic structure of a model systemCa10(AsO4)6(OH)2, obtained from direct As substitution in the vanadateCa10(VO4)6(OH)2. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem 109: 849–860, 2009

Key words: apatites; theory; semiempirical; Ca; Pb; electronic structure

1. Introduction

I nterest in the study of hydroxyapatite, Ca10(PO4)6

(OH)2, has increased in the last few years because

Correspondence to: M. Matos; e-mail: [email protected]

of its importance as a biomaterial [1] and the possi-bility of using the material as catalyst and environ-mental agent. Hydroxyapatite (Hap) is notable forits facility of accepting substitutions, either cationic(Ca2+) or anionic (OH−, PO3−

4 ). There is a consid-erable amount of work devoted to understand thephysical and electronic properties of substitutedhydroxyapatite. In recent years, a number of

International Journal of Quantum Chemistry, Vol 109, 849–860 (2009)© 2008 Wiley Periodicals, Inc.

MATOS, TERRA, AND ELLIS

experimental techniques, including EPR, ENDOR,XRD, XANES, EXAFS, infrared, Raman, and Möss-bauer spectroscopies have been used together withatomistic simulation and ab initio density functionaltheory (DFT) calculations to investigate carbonateions [2], Zn2+ [3], Fe2+,3+ [4], and Pb [5] substitutions.In these studies, the preferred site and local geo-metrical environment have been predicted. Divalentcations may substitute at the two calcium sites: com-pact Ca(I) and more open Ca(II), closer to the OH−

channel. It has also been established that the tetra-hedral phosphate group PO4 acts both structurallyand electronically as a compact subunit, remainingpractically unperturbed under changes in the struc-ture. Owing to strong P–O covalent bonds, the unitcan be seen as a structural building block because theoxygen vertex atoms are also bound to the Ca net.

Total cation or anion substitution gives rise todifferent stable compounds that are isostructuralwith hydroxyapatite. An interesting example is thecalcium vanadate apatite Ca10(VO4)6(OH)2. Thismaterial can form solid solutions with the phos-phorus apatite, Ca10(PO4)1−x(VO4)x(OH)2, whichhas shown unexpected catalytic activity, through Vreduction in the presence of hydrogen [6]. Vanadi-nite, Pb10(VO4)6(Cl)2, could as well be seen as a resultof total Ca2+, P5+, and OH− substitution by Pb2+, V5+,and Cl− [7]. The redox properties of V-apatites solidsolutions have stimulated research in another com-pound, the hydroxyvanadinite Pb10(VO4)6(OH)2. Itscrystalline structure was recently refined and thematerial was characterized through X-ray, infrared,and Raman analyses combined with embedded clus-ter DFT electronic structure calculations. Differencesbetween the redox behavior of the Pb and Ca vana-date compounds are not yet well understood [8].Synthesis with arsenic also gives rise to a stablecompound, Ca10(AsO4)6(Cl)2 [9], isomorphic withhydroxyapatite.

Periodic band structure calculations were donefor hydroxy-, halogen-, and oxy-apatites, usingab initio DFT-LDAtheoretical methods [10, 11]. Banddispersion, density of states, and Mulliken chargeand bond order analysis were done [10, 11] com-bined with geometry optimization [10] to study theoptical properties and bonding. Results obtained forthe calculated band gap (valence to conduction bandelectronic transition) in HAp and FHAp, ClHAp,and BrHAp were fairly close, ranging from about5 to 6 eV with HAp in the low and FHAp in the highend. The bond orders in HAp [11] found in the peri-odic model are lower than those obtained by theembedded cluster DFT approach [3], but relations

between P–O, O–H, and Ca–O agree qualitativelywell. Both calculations show that the former twoconstitute strong covalent bonds while the latter ismuch weaker.

Despite the fact that powerful computationalelectronic structure approaches based on first prin-ciple theories are now available and being usedin this family of compounds, we found it worth-while to investigate the applicability of simplerand more efficient theoretical methods in these sys-tems. This has not yet been presented in the lit-erature, at least in a systematic way. Research onrelated and more complex apatite systems suchas e.g. surface processes, nanostructure properties,and contact with surrounding water would ben-efit from a basic qualitative electronic structuredescription. Analysis of trends in chemical bond,charge distribution, and forbidden gap energieswith changes in geometry and substituents couldbe used as starting point for more precise the-oretical calculations. The aim of this work is toperform such an investigation. We use the well-known and widely used extended Hückel methodto study a series of apatite compounds whose mem-bers differ not only through anion c-channel butalso through cation Ca2+ and P5+ substitutions. Themain goal is to investigate the systematic evolutionof electronic properties with composition within theapatite structure under a broader range of chemicalsubstitutions and related structural modifications.We consider for the analysis besides hydroxyap-atite Ca10(PO4)6(OH)2, the substituted compoundsCa10(VO4)6(OH)2, Ca10(AsO4)6Cl2, Pb10(VO4)6(OH)2,and Pb10(VO4)6Cl2. All these materials have knowncrystalline structure. In addition, we examine a modelsystem Ca10(AsO4)6(OH)2, to develop some basicunderstanding of a hypothetical compound withsuch constituents. The model is built by simple Assubstitution in Ca10(VO4)6(OH)2, with no relaxationof the crystal structure. For simplicity, we use thenomenclature ABOH for A10(BO4)6(OH)2, or ABClfor A10(BO4)6Cl2, after the atomic constituents of thecompound’s chemical formula.

2. Theory

Extended Hückel is a widely known semiem-pirical tight binding with overlap method and itsfoundations concerning the electronic structure ofmolecules and solids can be found elsewere [12,13]. Here we point out some aspects of the theo-retical approach which directly concern our study.

850 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 109, NO. 4

EMPIRICAL PARAMETERS FOR THE EXTENDED HÜCKEL CALCULATIONS

TABLE IEmpirical parameters.

O Ca Pb P As Cl H V

n 2 4 6 3 4 3 1 4Hns,ns −32.3 −7.0 −15.7 −18.6 −16.22 −26.3 −13.6 −8.81ζns 2.275 1.2 2.35 1.75 2.23 2.183 1.3 1.30

Hnp,np −14.8 −4.0 −8.0 −14.0 −12.16 −14.2 — −5.52ζnp 2.275 1.2 2.06 1.30 1.89 1.733 — 1.30

Hi ,i in eV; n: principal quantum number. V3d double zeta orbital: H3d,3d = −11.0 eV, ζ1 = 4.75, c1 = 0.456, ζ2 = 1.70, c2 = 0.705.

The hamiltonian matrix is given by diagonal atomicorbital terms Hii and off-diagonal terms Hij, definedas k(Hii + Hjj)Sij, where k is a conveniently cho-sen constant and Sij are overlap integrals betweenvalence basis set Slater type atomic orbitals (STO) iand j. Sij overlaps take account of the geometry ofthe molecule or the unit cell. Empirical parametersnecessary to build the system’s hamiltonian are thediagonal terms Hii and STO exponents. The parame-ters used in this study are chosen from Ref. [14] andgiven in Table I. Hii terms are in general obtainedfrom experimental ionization potentials.

Crystal orbital overlap populations (COOP) arethe crystalline equivalent of the nondiagonal Mul-liken indices pij; these will be used to investigateatom- or orbital-pair interactions [13]; the integrationof appropriately chosen COOP curves, throughoutthe occupied states (total COOP), gives quantitativeestimates of bond orders (BO). To obtain DOS andCOOP curves, a mesh of 120 uniformly spread recip-rocal lattice k points was found to be well suited.A test was made in Ca10(PO4)6(OH)2 by calculatingbond orders with a mesh of 360 hexagonal k pointsand results showed excellent convergence, with adifference smaller than 10−4, when compared withthe 120-mesh set. Mulliken atomic charges are givenby the integration of projected DOS curves.

Extended Hückel calculations were performedwith bind [15] program and the graphics weredrawn with viewkel [16], distributed as part of theYAeHMOP package.

3. Crystal Structure

The apatite family of compounds have the gen-eral formula A10(BO4)6X2, where A = (Ca, Pb), B =(P, As, and V), and X is usually a monovalent halo-gen anion or OH−. These ions have formal chargesA2+, B5+, and X−. The compounds considered in this

study have hexagonal crystal structure with spacegroup P6/3m. The structure contains strongly cova-lent tetrahedral BO4 units which are held togetherthrough interactions between oxygen and surround-ing divalent A cations. The latter are distributedamong two crystal sites, A1 and A2. A1 atoms form acolumn parallel to the c-axis and have sixfold bipyra-midal coordination with two oxygen triangles, theO1 triangle being rotated relative to the O2 triangle.A2 atoms form the walls of another channel, alsoparallel to c, and are disposed in two triangles, form-ing bipyramids with the A-atom at the vertices. TheX− anions are regularly arranged along the interioraxis of the OH-channel. OH− stays very nearly at thecenter of A2 triangles; in CaAsCl, Cl− is located nearA2 triangles but shifts towards the center of the six-fold A2 bipyramidal arrangement in PbVCl. Thereis another oxygen crystalline position O3, so that thephosphate group can be written as BO1O2O32. Thefourth oxygen site, O4, is that of OH. Figure 1 showsthe crystal structure along the c-axis.

FIGURE 1. The crystal structure of the apatites, seenalong the hexagonal c-axis.

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TABLE IIInteratomic distances in several apatites (in Å).

CaPOH CaVOH CaAsCl PbVOH PbVCl

P, V, As–O1 1.54 1.67 1.67 1.70 1.70P, V, As–O2 1.55 1.73 1.69 1.69 1.69P, V, As–O3 1.52/1.50 1.68 1.68 1.70 1.70

Ca1, Pb1–O1 2.41 2.43 2.38 2.50 2.49Ca1, Pb1–O2 2.45 2.53 2.46 2.75 2.75Ca2, Pb2–O2 2.35 2.31 2.32 2.36 2.34Ca2, Pb2–O3 2.50/2.34 2.55/2.32 2.53/2.32 2.73/2.59 2.67/2.57

2.37 — — — —Ca2, Pb2–X 2.38 2.42 2.75 2.70 3.15

O–H 0.95 0.97 — 1.09 —a = b 9.432 9.741 10.076 10.224 10.317

c 6.881 7.004 6.807 7.454 7.338

Table II shows interatomic distances and latticeparameters of the several apatites analyzed in thiswork; the data were taken from X-ray diffractionmeasurements [2, 6–9]. It can be seen that latticeparameters a = b and c are larger in Pb compoundsand show dependence on the several anion–cationbonds for a given A occupancy. Ca–O and Pb–Obond distances in the apatites can be compared withother calcium and lead oxides. In Ca apatites, Ca–Obond distances are very nearly equal to bond dis-tances in calcium oxide [17] (2.40 Å), which hasa sixfold octahedral coordination, and are smallerthan bond-distances in the calcium perovskite [18](2.75 Å), with a 12-fold coordination. Pb–O bond dis-tances in α- and β-PbO [19] are, respectively 2.30 Åand 2.49 Å, similar to bond lengths found in PbVOH(hydroxyvanadinite) and PbVCl (vanadinite) withthe notable exception of Pb1–O2 bond length in bothcases.

It can be seen from Table II that tetrahedral B–Obond distances increase from phosphates to vana-dates, with arsenate (in CaAsCl) close to vanadatesin size. In the calcium compounds there is a con-nection between Ca2–X bond distances and latticeparameter a, with a clear dependence on the X ion.O4–Ca2 bonds increase from CaPOH to CaVOH andare smaller than the Cl–Ca2 bond length in CaAsCl.The same trend is observed in a. In the Pb com-pounds a weaker change in a is noted when X goesfrom OH to Cl, but this might be due to the type ofcoordination of Cl in the Ca2 bipyramid.

The similarity between V–O and As–O bondlengths, when compared with those of the P–O pairs,led us to chose the CaVOH structure to build the

CaAsOH model system. As mentioned earlier, themodel consists in the direct substitution of V by Asin the CaVOH crystal structure.

4. Results

In this section, we investigate the electronicstructure of the materials considered in this arti-cle. To improve understanding on the basic elec-tronic mechanisms, we develop comparative analy-ses between the bulk electronic structure and that ofsubunits and subnets. Results concerning the modelsystem CaAsOH should be regarded as speculative,because we were unable to find any experimentaldata on this compound.

4.1. BAND STRUCTURE

As a first step to understand the effects of sub-stituents, the band structure of calcium and leadapatites have been obtained. Results are shown inFigure 2, in the energy region around the opticalgap. The calculated values of EVB, top of valenceband energy, and of the energy gap �E are given inTable III.

The bands which spread below −14.5 eV are duemainly to O(2p) orbitals. For CaAsCl [Fig. 2(d)] andPbVCl [Fig. 2(f)], a largely dispersive band appearson top of the oxygen band, because of Cl(3p). In Pbapatites, Figures 2(e) and (f), there appear below andright above O(2p), bands arising from Pb(6s)-O(2p)

mixing. O(2s) bands, not shown in Figure 2, areformed far below the valence bands, and have small



FIGURE 2. Band structure of several apatites. (a) CaPOH; (b) model CaAsOH; (c) CaVOH; (d) CaAsCl; (e) PbVOH; (f)PbVCl. Energies are in eV. The energy of the top of the valence band, EVB, is given in Table III. Note that EVB (top ofPb(6s) antibonding bands) is 1.15 eV higher in PbVCl than in PbVOH.

dispersion, around H2s,2s(−32.3 eV). Cl(3s) bands,also not shown, are seen in the energy range from−28 to −26 eV and are mainly due to Cl–Cl inter-action with small amounts of Cl mixing with Ca2or Pb2. The empty bands above −5 eV are mixturesof Ca(4s), Ca(4p), P(3s), P(3p), V(4s), V(4p), H(1s),and Pb(6p). In vanadium apatites, V(3d) forms thecrystal field split bands between −10 and −5 eV. Inthe following, we discuss with more detail the bandsaround the gap (shown in Fig. 2).

The role of Cl could be better understood by exam-ining isolated subnets carved out directly from thecrystal structure. In the pure Cl subnet of CaAsCl oneobtains three groups of Cl(3p) bands. One of them,highly dispersive, was seen to come from 3pz-3pz

interactions along the c-axis. Two narrower ones andpractically degenerate come from Cl(3px,y) whose

dispersion is found in the A2 plane, perpendicular toc. The 3pz bands in the subnet are strikingly similar tothose seen in Figure 2(d), indicating negligible inter-action with the crystal environment of oxygen andcalcium. The larger dispersion along �–A and M−Larise because the reciprocal lattice directions are par-alel to the c axis. These bands are then expected tobe directly related to the lattice constant c, as well asto the size of the halogen ion. In fact, there is a low-ering of ∼1 eV in the E(�) energy of Cl(3p) bandsin vanadinite [Fig. 2(f)], consistent with the latticeconstant c being larger in this compound (Table II).Rulis et al. [11] have found that halogen bands spreadaccordingly with the size of the anion. For example,in bromapatite, dispersion is larger than that of chlo-rapatite and fluorapatite. The transverse Cl(3px) andCl(3py) bands are intermixed with O(2p) but could

TABLE IIIAtomic net charge, optical gap, and EVB of several apatites and for the CaAsOH model system.

CaPOH CaAsOH CaVOH CaAsCl PbVOH PbVCl

q(P, V, As) 2.58 2.89 2.32 2.88 2.43 2.22q(Ca1, Pb1) 1.87 1.74 1.86 1.71 1.75 1.75q(Ca2, Pb2) 1.80 1.66 1.78 1.57 1.68 1.70(1.92)

q(O1) −1.32 −1.34 −1.24 −1.33 −1.21 −1.21q(O2) −1.34 −1.37 −1.30 −1.36 −1.23 −1.24q(O3) −1.34 −1.36 −1.26 −1.37 −1.28 −1.29

q(BO4)(c) −2.77 −2.54 −2.74 −2.55 −2.57 −2.81

q(O4) −1.26 −1.25 −1.26 — −1.31 —q(H) 0.41 0.42 0.42 — 0.45 —

q(OH, Cl) −0.84 −0.83 −0.84 −0.47 −0.86 −0.18�E (eV) 13.2 12.6 6.5 12.1 1.3 0.13EVB(eV) −14.5 −14.6 −14.7 −12.27 −10.6 −9.45



FIGURE 3. The molecular orbital levels of the PO4 subunit and the band structure and projected DOS of CaPOH.Energies are in eV.

be clearly identified in the isolated Cl–Ca2 subnet.As 3px,y lie in the Ca2 plane, interaction with thecation is favored and introduces a covalent characterto Cl–Ca bond. As will be seen, this transverse inter-action affects significantly charge distribution in thechlorine containing apatites.

Oxygen (2p) bands are formed mainly by oxygen-oxygen interaction or by oxygen mixing with thetetrahedral cation. These bands could be betterexamined by considering the isolated molecular unitBO4. In Figure 3, it is shown the molecular orbitallevels distribution of the PO4 unit, carved out fromthe crystal structure of hydroxyapatite (CaPOH), incomparison with the material’s band structure. Itis noted that the band structure of the crystal isvery similar to the isolated PO4 MO levels distribu-tion. Through a closer examination of the molecularorbitals it was seen that the HOMO and molecularlevels nearby are formed by antibonding combina-tions of O(2p) orbitals pointing along the edges ofthe tetrahedron, in a cage-like arrangement aroundphosphorous. In the crystal, the top of the valenceband is seen to be essentially oxygen dependent (seeFig. 3), in accordance with the subunit results. Thesame results were found in the vanadate and arse-nate. The contribution of P, As, or V orbitals to themolecular HOMO was found to be less than 7%. Asshown in Figure 3, contributions of the B cation to thevalence band spread below pure oxygen bands. Theinvariance of the topmost oxygen bands observed inthe whole series of Figure 2 is thus a result of theoxygen-cage effect which makes these band levelsindependent of the B-site constituent.

Anarrow and sparse double band due to O4 can beseen in the vanadate CaVOH, below the main oxygenband [Fig. 2(c)]. By examining the pure OH subnetwe have found that the hydroxyde bands are analo-gous to Cl(3p) but have lower energy, not affectingEVB. Transverse 2px,y bands of O4 do not interactsignificantly with the A2 neighborhood, contrary tothose of chlorine. This suggests that the hydroxy ionis invariant in the crystal structure of the apatites.Nondispersive π -type O4–O4 bands, are seen in thesubnet at −14.8 eV. The hydroxy ion bands are rel-atively low in energy, when compared with thosefound by Rulis et al. [11], which appear above themain oxygen group. This could be due to electron–electron repulsive effects in the negatively chargedO42− ion, which are not explicitly taken into accountin the present one electron approach. The analysis ofthe X ion subnets show a strong similarity betweenCl and OH in what concerns the formation of bandsin the crystal.

In the vanadate CaVOH and in vanadinitesPbVOH and PbVCl [Figs. 2(c),(e) and (f)] the con-duction band onset is shifted downwards by thepresence of 3d vanadium orbitals. In all three com-pounds 3d bands extend from −10 to −5 eV, showinga striking regularity in the VO4 crystal field splitting.Because there are six tetrahedral VO4 groups per unitcell, within this range one finds 2 × 6 = 12 bands inthe lower and 3 × 6 = 18 bands in the higher group,coming from the eg-below-t2g molecular orbital sep-aration, typical of d-orbital splitting in tetrahedralfields. The cubic splitting is, in all cases, ∼0.9 eV.These results show that vanadium substitution at



the B site determines the position of the conductionband, independently of cation and anion substitu-tions at A and X sites. They come predominantlyfrom V(3d) orbitals; in the vanadinites, the higherlying t2g bands are interspersed with bands formedby Pb(6s).

As seen in Figure 2, in vanadinite (PbVCl) andhydroxyvanadinite (PbVOH) there appear threemore groups of bands, because of Pb(6s) and Pb(6p).The latter gives rise to the empty bands extend-ing from −5 eV to 0. Pb(6s), on the other hand,belong to the valence group of levels, as double 6soccupancy in Pb2+ provides complete filling of thecorresponding bands. The group found in the range−18 to −16 eV, below the O(2p) bands, and thatabove them, from −12 to −9 eV [Figs. 2(e) and (f)]come from bonding and antibonding Pb(6s)–O(2p)

mixing, respectively. This is clearly seen throughthe COOP curves of Figure 4. These curves alsoshow how Pb interactions with O change as onegoes from vanadinite (PbVCl) to hydroxy-vanadinite(PbVOH). Note that while Pb1–O bonding remainspractically unchanged by substitution in the X-channel [Figs. 4(a) and (d)], differences are clearlynoted in the Pb2–O COOP [Figs. 4(b), (c), and (e)].This is expected because Pb2 constitutes the walls ofthe X-channel. Panels 4(c), (d), and (e) show strongerPb2–O interaction when OH is substituted by Cl,as noticed by the average upward shift of Pb2–Oin Figure 4(b) respective to Figure 4(e). The shift isinduced by Pb2–Cl coupling [see Fig. 4(c)]. Theseresults are consistent with our analysis above of

the subnets, which showed the hydroxyde ion as apractically independent unit.

Because of double 6s occupancy, EVB is raised byabout 4 eV with Pb substitution at calcium sites.There is also, in the vanadinites, a halogen effectwhich makes EVB increase by 1.15 eV in vanadinite,with respect to hydroxy-vanadinite. The effect has,however, a different origin than that found in thecalcium compounds. As seen in the former para-graph, it comes from Cl(3p) interacting with Pb(6s)rather than through Cl–Cl interaction along the c-axis. Note that direct Cl–Cl interaction exists in thelead compounds as well [Figs. 2(e) and (f)], but,owing to the 6s2+ shift of EVB, it does not affect theoptical gap.

It has been argued that polarization due to sphybridization in Pb2 could explain dislocations ofhalogen ions in the X channel [20]. However, inagreement with DFT calculations [5, 8], we havefound no significant 6s−6p hybridization in the leadapatites. This can be seen in Figure 5, which showsPb(6s)- and Pb(6p)-projected density of states in thevalence band of vanadinite, as an example. It canbe noted that the contribution of 6p is negligible inthe valence band of the material. Spin-orbit effects,not taken into account in this study, could possi-bly influence hybridization. In PbVCl, first principlescalculations are needed to clarify this point.

Large variations in the forbidden gap energy �Eis observed in Figure 2 and Table III, because of sub-stitutions. The main changes arise for vanadium—due to lowering of the conduction band—and lead

FIGURE 4. COOP curves for PbVCl and PbVOH showing bonding (curves pointing right) and antibonding (curvespointing left) Pb–O and Pb–Cl interactions. Energies are in eV.



FIGURE 5. Pb(6s)- and Pb(6p)-orbital projected DOS curves for vanadinite (PbVCl). Energies are in eV.

substitutions—by raising the top of the valenceband. In vanadate (CaVOH), V(3d) bands cause areduction in �E of a factor of 2 with respect to HAp(CaPOH). In the case of PbVOH, reduction is stillmore drastic, of a factor of 5. Cl substitution providesdecreases in the optical gap by ∼1 eV, and is moresignificant in Pb-substituted compounds. Halogenreduction of the optical gap of apatites has also beenobserved by Calderin et al. [10]. Minor changes arefound with As-substitution in the model CaAsOHsystem.

�E in HAp (13.2 eV) is overestimated, in view ofthe experimental value of 5.05 ± 0.30 eV obtained bydeAraujo and Coworkers [21]. Pseudopotential LDA[10] gives between 4 and 5 eV for �E; Rulis et al. [11]using LDA with a Gaussian basis found 4.5 eV for�E. DFT is expected to underestimate experimen-tal optical band gaps. On the other hand, extendedHückel usually tends to overestimate valence to con-duction band excitations. In calcium oxide, wherethe Ca–O distance (2.4 Å) and coordination (six-fold) are the same as for the Ca(1) environmentof HAp, the experimental gap is 7.03 eV [17]. Theeffect of Ca(2), P, and OH on the energy gap canbe determined by examining the lower conductionbands of HAp in Figure 6. It is noted that the bot-tom of the CB is due to O4–H combinations, withthe next contributions coming from Ca(2), less than1 eV above the OH bands. P and Ca(1) contributionsstart 2 eV above the CB bottom. These results wouldsuggest that hydroxyl could be responsible for �Ebeing smaller in HAp than in structurally simplerCaO. We have found, however, through extendedHückel method calculation in calcium oxide, thatEVB = −14.8 eV and �E = 12.8 eV are very similar to

the values obtained in HAp, suggesting that struc-tural differences are not the only factor to explain thedifference in the optical gap of the two compounds.Ca(3d) orbitals have been included in an extendedHückel calculation of elemental calcium to repro-duce the correct Fermi surface of that material [22].By including the cation 3d orbitals in the basis set,one obtains a considerable reduction of the opticalgap to �E = 9.45 eV in hydroxyapatite and 8.51 eV inCaO, the latter being in much better agreement withthe experimental value of 7.03 eV. More detailed firstprinciples calculations on both compounds would bevaluable to understand the nature of the optical gapof HAp and differences relative to calcium oxide.

In vanadinite [Fig. 2(e)] one notices that thereare plenty of possibilities in the band structure toaccount for electronic transitions of ∼2 eV, consis-tently with the crystal’s red-orange color [23].

4.2. MULLIKEN ANALYSIS

Net atomic charges were calculated from Mul-liken population analysis and results are shownin Table III. There is a good general agreementwith results obtained with the embedded clusterDFT approach for CaPOH, PbVOH, and PbVCl[3, 6]. For hydroxyapatite, agreement is excellent,with less than 3% difference for the prediction ofCa, PO4, and OH charges. Individual net chargesshow more polarization within DFT calculations,with q(cation) + q(anion) slightly greater than whatis given in Table III.

It can be seen that atomic charges at the B sitedecrease from As to P to V. The difference betweenq(P) in CaPOH and q(V) in CaVOH is 6%, whereas



FIGURE 6. The conduction band of hydroxyapatite, showing integration curves of several orbital-projected DOS.Energies in eV.

it is twice as big when one goes from P to As inCaAsCl and the model CaAsOH. Arsenic electronloss is compensated by and effective electron trans-fer to Ca1 and Ca2. V substitution, on the other hand,leads to effective electron flow from oxygen to vana-dium, with q(Ca) practically unchanged when onegoes from CaPOH to CaVOH. Atomic charges ofPb1 and Pb2 are smaller than those of Ca1 and Ca2in hydroxyapatite and CaVOH, but are very simi-lar to those of calcium in the arsenates. However,in the lead compounds, electron gain at the A siteis associated with electron loss in oxygen, with nosignificant decrease at the B cation, differently fromwhat is observed with As substitution. The effectiveelectron transfer from oxygen to lead in the vana-dinites denote ionic-covalent bond mainly at Pb2–O

pairs, as can be seen in Table IV (see comment below).Charge of Ca2 and Pb2 is consistently more sen-sitive to changes from OH to Cl than at site A1,more distant to the X-channel. OH net charge andthat of its constituents are fairly uniform among thewhole series of compounds, confirming the invariantidentity of the hydroxyde ion.

Calculated chlorine charges in the arsenateCaAsCl and in vanadinite are high, when comparedwith oxidation state −1. Electron loss from Cl shouldbe expected as a result of covalency. DFT calcula-tion gives q(Cl) = −0.68 in vanadinite [6], indicatingionic-covalent bonding; such a sign of covalency isconsistent with the transverse chlorine bands seenabove to arise from Ca–Cl mixing. Therefore, in theapatite structure, there seem to be conditions for

TABLE IVBond orders of next-neighbor pairs in several apatites and in the model system CaAsOH.

CaPOH CaAsOH CaVOH CaAsCl PbVOH PbVCl

P,V,As–O1 0.73 0.63 0.79 0.63(0.63)a 0.65 0.64(0.66)a

P,V,As–O2 0.72 0.56 0.71 0.59(0.59) 0.62 0.61(0.62)

P,V,As–O3 0.74 0.61 0.78 0.61(0.61) 0.64 0.61(0.63)

Ca1,Pb1–O1 0.076 0.094 0.059 0.105(0.105) 0.080 0.081(0.080)

Ca1,Pb1–O2 0.032 0.053 0.023 0.060(0.060) −0.013 −0.014(−0.014)

Ca2,Pb2–O2 0.074 0.123 0.080 0.106(0.106) 0.157 0.145(0.160)

Ca2,Pb2–O3 0.018(L) 0.041(L) 0.011(L) 0.042(0.044)(L) 0.032(L) 0.044(0.040)(L)

0.109(S) 0.148(S) 0.101(S) 0.138(0.140)(S) 0.053(S) 0.049(0.046)(S)

Ca2,Pb2-X 0.093 0.095 0.102 0.281(0.196) 0.018 0.107(0.031)

O–H 0.65 0.64 0.64 — 0.55 —

a With Cl3p contracted orbital; S and L: Short and Large bond lengths.



stronger bonds with Cl. The present halogenic elec-tron loss is probably overestimated, however, andcould be due to parameter choice. More experimen-tal or theoretical data would be required on the Clapatites to define new values for chlorine parametersor possibly neighboring atoms. Despite this limita-tion, we have made a preliminary analysis on thesubject by arbitrarily changing the exponent of a ClSlater orbital and investigating for comparison theeffects of the same changes in another system, theionic NaCl salt.

Because covalent bonds are sensitive to spreadingof atomic orbitals, we have considered a contrac-tion of the Cl(3p) STO by making ζ3p = ζ3s (seeTable I); with contraction one gets smaller over-lap density and bonding. Consistently, we obtainfor q(Cl) the values −0.63 in CaAsCl and −0.68in PbVCl after contraction, in surprising agreementwith results found in Ref. [6]. To check the con-sistency of the procedure, we have examined thebehavior of sodium chloride under the same contrac-tion; as ionic bonding does not expectedly dependon the extent of atomic orbitals, no big differencesshould be obtained in cation–anion charge transferin the salt. Indeed, for NaCl, we have found thatq(Cl) is very nearly −1 regardless of the choice ofζ3p, with only a 6% variation under contraction. Notethat, for the apatites, changes as high as 32% and280% for CaAsCl and vanadinite, respectively, werefound with contraction. These results clearly pointtowards the existence of covalent chlorine bonds inthe apatite structure.

Aside from being a useful analytical tool, thenew parameterization for Cl could be consideredconvenient for the apatite crystalline environment.For example, when using ζ3p = 2.183, calculatedcharges of Pb1, V, and oxygen, remain practicallyunchanged. More significant differences occur, asexpected, in q(Ca2) = 1.61 and q(Pb2) = 1.63,because of the proximity with chlorine, but thesechanges are only ∼4%. Vanadium charge in PbVClpresents a small increase to 2.42, becoming moresimilar to that of vanadium in PbVOH, more consis-tent with the independent nature of the VO4 group.Cl(3pz) bands are less dispersive, in better agreementwith results found by Calderin et al. [10] and by Ruliset al. [11]. As for the optical gap in vanadinite, con-traction leads to �E = 1.22 eV, as a result of loweringantibonding bands near the top of the valence band(see Fig. 4). Despite being comparatively larger thanthe former value, it is still too small to representsignificant changes in the material’s characteriza-tion. Therefore, contraction of the chlorine orbital is

physically consistent, besides improving calculatedresults in the apatite system.

Bond orders were calculated for the various com-pounds and results are shown in Table IV. Forhydroxyapatite (CaPOH) extended Hückel calcu-lated values are in good agreement with DFT cal-culations [3]. P–O and O–H bond orders denotestrong covalent character in both theoretical descrip-tions and with the same trend, that is, P − O3 >

P − O1 > P − O2 > O − H. There is also good agree-ment between the absolute bond order values in bothcalculations. Bond orders for Ca1- and Ca2-oxygenindicate that calcium atoms are more loosely boundthan the tetrahedral B–O pairs. This result is also con-sistent with DFT-calculated bond orders which werefound to be negative for Ca–O pairs. We also foundthe same trend with interatomic bond lengths in cal-cium oxygen pairs. This corresponds to Ca2 − O3 >

Ca2 − O2 and Ca1 − O1 > Ca1 − O2, inverselyrelated to bond lengths (Table II). Results obtainedwith contracted Cl(3p) are also shown in Table IV.

More detailed behavior can be seen by compar-ing the several compounds. It can be noted that Ca1is more losely bound in the vanadate CaVOH thanin other Ca apatites, with a decrease of 30% relativeto hydroxyapatite (CaPOH). In opposite behavior,arsenic substitution in HAp causes an increase ofabout 40% in Ca1–O bond orders of CaAsCl and ofmodel CaAsOH. A similar behavior is noted in Ca2-oxygen pairs under As substitution, with no impor-tant changes due to vanadium at the B site. Ca2–Xbonding is fairly constant in hydroxyde compoundsbut increases considerably with Cl substitution inCaAsCl. However, with 3p contraction this effect islowered.

The VO4 subunits in vanadinites are more weaklycovalent than in vanadate CaVOH. From Table IV, itcan be noted that V–O bond orders in the calciumcompound are about 20% higher. A1–O1 bonding iscomparable with those of HAp and much strongerthan those of A1–O2 pairs. This is in part connectedto bond strenghtening between O2 and Pb2, whichis twice as big in the vanadinites than in hydrox-yapatite, as can be noted from quantities shown inTable IV. This variation is directly related with bondlengths as, in both materials, d(Pb1 − O2) > d(Pb2 −O2). It can be observed that bond orders of the Cl-apatites, calculated by using the contracted Cl(3p),are barely modified, being thus robust relatively tothe change in the atomic orbital exponent.

Comparison between bonding in vanadateCaVOH and hydroxy-vanadinite PbVOH is of inter-est, because of the distinct behavior showed by the



two compounds in the presence of hydrogen [6]. Ithas been argued that this might be due to differencesin the nature of the cation or in charge transfer mech-anisms. The present results show that bonding atA–O pairs does not present significant changes whengoing from calcium to lead. However, a strickingdifference is observed between the two compoundsrelated to the X-channel, as Ca2–OH bond order isfound to be one order of magnitude higher than thatof Pb2–OH. It is also interesting to note that theoptical gap in PbVOH is smaller than in CaVOH,so that empty vanadium bands are more likely tobe occupied in the former compound. This is aninteresting result because it is related to charge trans-fer and could explain differences between the twocompounds.

In the model system CaAsOH, the tetrahedralarsenate group forms a molecular unit practicallyidentical to that found in CaAsCl, as seen fromMulliken As, O, and AsO4 charge and bond orders(Tables III and IV). q(Ca) in CaAsOH show negligi-ble differences from those in CaAsCl; bond orders areslightly different but have the same trend as CaAsCl.The model system could then be thought as a OH-substituted CaAsCl. The band structure of CaAsOHis very similar to that of hydroxyapatite as phos-phorous and arsenate do not contribute to the lowerconduction band levels.

The results obtained above show that the theoret-ical extended Hückel approach provides a reliabledescription of basic electronic properties in apatitematerials, especially concerning bonding and trendsrelative to chemical substitutions and site geometry.Although the calculated optical gap in hydroxiap-atite is found to be high, as compared with availableexperimental data, a considerable improvement isfound with the inclusion of Ca(3d) orbitals in thebasis set. It should also be taken into account thatpowder measurements by using reflectance tech-niques [21] could be influenced by surface states.To check for the presence of such states, we havedone calculations on Hap 001 slabs. In this prelimi-nary account, oxygen-lacking exposed planes, withor without relaxation, created surface states withenergies inside the bulk optical gap which couldsignificantly reduce �E.

5. Conclusion

In this work we have presented a systematic elec-tronic structure study of a broad series of apatitematerials, classified by chemical substitution in all

crystalline sites, A, B, and X of hydroxyapatiteCa10(PO4)6(OH)2. The results have shown that thetight binding extended Hückel method could beregarded as a convenient and reliable theoreticaltool to understand basic properties of these sys-tems, especially concerning trends. It is expected thatthe method could also describe trends concerninglocal geometry changes due to distortions, latticerelaxation around impurities, surfaces and inter-faces. This would, however, require further exper-imental inputs and/or elaboration of a total energyscheme.

Calculations have shown that charge distributionand bond orders are in excellent agreement with DFTab initio results for hydroxyapatite, hydroxyvana-dinite, and vanadinite. It is expected that the trendin the optical gap among the series is well repro-duced. Large variations in this quantity were found,mainly due to vanadium and lead substitutions. X-channel ions produce smaller changes in the opticalgap, which decreases from OH to Cl in both calciumand lead apatites. Halogen reduction of the opticalgap in the calcium apatites were found to be veryclose to those obtained in the literature by using theDFT method. We have observed that arsenic is morelosely bound in the BO4 tetrahedra than P, but Assubstitution leads to strengthening of Ca–O bonds.A regularity was found in the energy of the top-most oxygen 2p bands in the whole series os apatites.Empty V(3d) bands are also invariant with substitu-tions in A, B, and X sites and stay ∼5 eV above O(2p)

higher levels. It is suggested that more theoreticaland experimental studies are necessary to character-ize the optical gap in Hap by also considering thepossible effects due to surface states.

Analysis of Cl interactions in the chloroapatiteshave shown evidence of covalence in chlorine bondsmainly with Ca2, which surrounds the X-channel. Aconvenient set of empirical parameters for chlorinewas found for these systems and could be suitedto apatite crystalline environments in general. Amore systematic study including other halogen ionswould however be needed to bring more certaintyon this result.

We have also investigated the electronic structureof a hypothetical system, formed by direct As subti-tution in Ca10(VO4)6(OH)2. The model system CaA-sOH was seen to behave as a OH-substituted CaAsClwith band structure very similar to that of hydrox-yapatite. The optical gap of the model CaAsOH is∼0.6 eV lower than that of the OH-apatite.

The results presented above enables the extendedHückel method as a valuable tool to investigate



basic properties of this family of compounds, espe-cially concerning trends with geometry changes andatomic substitutions. The flexibility of the theoreticalmethod could, in addition, be useful in a qualitativeanalysis of electronic mechanisms when the calcula-tion requires large super-cells, as in nonstechiometriccompounds and surface studies.

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semiempirical electronic structure calculation on ca and pb apatites

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