An orbital interpretation of Pauling's rules

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American Mineralgist, Vlume 69, pages , 1984 An rbital interpretatin f Pauling's rules Jenr,uv K. Bunerrl en Trursy J. MclennnN Department f Chemistry The University f Chicag, Chicag, Illinis 60637
American Mineralgist, Vlume 69, pages , 1984 An rbital interpretatin f Pauling's rules Jenr,uv K. Bunerrl en Trursy J. MclennnN Department f Chemistry The University f Chicag, Chicag, Illinis Abstract Pauling's rules, first cllected tgether ver fifty years ag, have traditinally been assciated with electrstatic, inic ideas f bnding in slids. In this paper we use a purely rbital mdel t tackle sme f the structural predictins made by the rules. As a result f band structure cmputatins n several mdel structures we present a scheme which fcuses attentin n the lcal anin crdinatin. We als use simple rbital based perturbatin theretical ideas t present a unifying explanatin fr Pauling's electrstatic bnd strength sum rule and fr the first time ne fr Baur's extensin f it. In almst all areas the predictins f the rules and thse f the cvalent apprach are similar, but their explanatins are very different. The ideas presented lead t a new level f understanding cncerning the structures f slids. These ideas have clse ties with current mdels f mlecular sterechemistrv. Intrductin ln 1929, Pauling prpsed a cllectin f five rules gverning the gemetries f inic crystals (Pauling, 1929). These rules and the bdy f additins and emendatins surrunding them passed rapidly int the accepted flklre f crystal chemistry, and fr mre than 50 years they have remained imprtant and largely successful guides fr mineralgists and slid state structural chemists. In their riginal frmulatin, Pauling's rules were presented primarily as ad hc generalizatins useful in develping the hypthetical gemetric mdels which were necessary t the slutin f cmplex crystal structures using the experimental methds available in the 1920's. While several f the rules are given interpretatins based n the inic mdel, their prf is clearly in their applicability t real structures rather than in the details f smewhat vague inic arguments. Further, nninic ways f lking at crystals enter implicitly even in 1929 (see Pauling's remark that 6-membered rings ccur in beryl because the bnd angle at the xygen atm in such rings apprximates the tetrahedral angle). In his later writings Pauling (1960) is quite explicit in claiming that even quite cvalent materials may bey rules ismrphus t sme f thse applying in inic crystals. Despite effrts t distinguish between gemetric precepts and a particular methd f cmputing crystal energies, Pauling's rules have frequently been identified with the inic mdel, and the success fthse rules has ften been used I Fellw f the Alfred P. Slan Fundatin and Camille and Henry Dreyfus Teacher-Schlar KE4/070E-060I$ r as an argument fr the inic nature f the bnding in such cmpunds. Since abut 1970, Gibbs and Tssell and their cwrkers (Gibbs et al., 1981; Tssell and Gibbs, 1978) have made substantial prgress in clarifying the rigins f Pauling's rules by shwing that many f his gemetric predictins, mst ntably the shrtening f shared plyhedral edges, are cnsequences nt nly f inic arguments but als f mre r less cvalent effects which can be studied by perfrming mlecular rbital calculatins. At present the results f such cmputatins have led t predictins f bnd angles, the variatins f bnd strengths with bnd angles, and s n, which in general seem in very gd agreement with bservatin. Unfrtunately, the methds f Tssell and Gibbs have required them t perfrm calculatins nt n crystals as a whle but n mlecular fragments chsen t mimic lcal regins within a crystal. This is usually nt a limitatin in investigating lcal gemetric preferences, but it des mean that the energies f crystal structures which differ in basic atmic tplgy and cnnectivity cannt be cmpared. The mst bvius way t discuss theretically the electrstatic valence rule r the rules invlving the destabilizing character f shared plyhedral elements is t cmpare the energies f real and hypthetical structures sme f which bey and sme f which disbey these rules. Calculatins n small mlecular clusters will nt lead t such infrmatin. In this paper, we present an interpretatin f Pauling's rules based n the results f band structure calculatins using the tight binding apprximatin. These are just cvalent mlecular rbital calculatins applied nt t such clusters but t the crystal as a whle. Using these @2 BT]RDETT AND McLARNAN: ORBITAL INTERPRETATION OF PAULING,S RULES results we will btain an electrnic understanding f these rules cmpatible with the arguments presently used t ratinalize a large number f mlecular structures. This understanding has a simple gemetric cntent based n the hybridizatin and crdinatin gemetry f the anins in the crystal. The structures we will cnsider can be rdered energetically by cnsidering anin crdinatin spheres. While ur calculatins and interpretatin generally supprt Pauling's rules, we mentin several interesting pints at which ur predictins differ frm his. The structure types we shall discuss are largely either wurtzite derivatives r diplar tetrahedral structures (Mclarnan and Baur, 1982). This chice is mtivated by the desire t cnsider a large family f clsely related types which are typical f mineral structures with 4- crdinated anins. We expect ur majr cnclusins t apply t a far larger class f structures than thse actually studied. Similarly, mst f ur calculatins use atmic parameters fr Be and O, nt because f the verwhelming mineralgical imprtance f brmellite (wurtzite-type BeO), but because Be and O lie near the tp f the peridic table and BeO seems a typical inic wurtzitetype cmpund. Nne fur cnclusins depend n this precise chice f parameters. In the final sectin we mentin sme structures with tw- and three-crdinate anlns. In discussing ur calculatins, we shall ften use the wrds anin and catin as a familiar shrthand fr electrnegative and electrpsitive atms, respectively. This terminlgy is used slely fr linguistic cnvenience, and is nt intended t have any implicatin regarding the physical nature f the bnding in any cmpund. Pauling's First Rule: inic radii A crdinated plyhedrn f anins is frmed abut each catin, the catin-anin distance being determined by the radius sum and the crdinatin number f the catin by the radius rati. (Pauling, 1929). This is perhaps the best knwn f Pauling's rules and intrduces the cncept f atmic (r rathei inic) size. Because apprximate mlecular rbital methds predict bnd distances prly, we cannt directly use ur band calculatins t cast light n this rule. Hwever we d wish t pint ut the prgress made in recent years using Phillips' Mendeleyevian philsphy in the area f structural mapping (St. Jhn and Blch, 1974;Zunger and Chen, 1978; Blch and Schatteman, l98l; Burdett et al., 1982). An imprtant result btained by the use f pseudptential radii (r1) has been the much clser definitin f atmic size . Mre specifically the identificatin f 11 rwith an electrnegativity has allwed a clse crrespndence t be made between billiard-ball theries based n sphere packings and mdern electrnic ideas based upn rbital cncepts. In this paper we will fcus n the remaining Pauling rules. Rule 2: electrstatic valence sums In a stable crdinatin structure the electric charge f each in tends t cmpensate the strength f the electrstatic valence bnds reaching t it frm the catins at the centers f the plyhedra f which it frms a crner; that is, fr each anin (: 1=7,, (l) (Pauling, 1929). Here -( is the charge f an anin, i runs ver all catins crdinated t that anin, zi and 4 are the charge and crdinatin numbers, respectively, f catin i, and s1 : z;lta. This electrstatic valence rule, while nly apprximately satisfied by sme cmpunds, has prven an enrmus aid t mineralgists and crystallgraphers. It is, fr example, the primary methd fr distingishing 02- frm (OH)- and H2O in cmplex mineral structures. Further, it was regarded by Bragg (1930) as the mst imprtant and innvative f Pauling's rules. Pauling himselfjustified this rule by saying that cnfiguratins satisfying it placed highly charged anins at sites f large psitive ptential, and hence f high bnd strength sums; thugh this argument des little t prve the equality f charge and bnd sum. On the ther hand, the validity f the rule fr simple crystals fllws frm purely tplgical cnsideratins: any binary cmpund in which all atms f each type have the same crdinatin number must satisfy the rule. Since frmal charge is nt a quantity appearing in the mlecular rbital frmalism, (althugh, given the mlecular wavefunctin we can calculate an atmic charge frm a ppulatin analysis) it des nt seem pssible t use these calculatins t test the valence sum rule numerically (i.e., t predict the allwable departure f Is1 frm (). It is pssible, hwever, t ask whether such calculatins predict qualitatively that mre highly charged anins shuld ccupy sites with larger bnd sums. T d this we must cnsider a structure with mre than ne type f anin and with tw r mre catin species difering either in charge r crdinatin number r bth. The nly knwn such structure which is a derivative f the wurtzite type is that adpted by a-lision (Laurent et al., 1980; Laurent et al., l98l;o'keeffe et al., l98l), and prbably by MgAION and ZnAlON. This structure, shwn in Figure l, has a (2, l) unit cell; that is, it has c = 2as, b : bs and c = c0, where as, bs, cs are the axes f the smallest rthhexagnal cell f a hexagnal clse packing. Als shwn in Figure I are three ther hypthetical structures representing the nly ways t rder equal numbers f tw catins ver the metal sites in the (l,l) wurtzite supercell. The ntatin emplyed fr these types is that f Mclarnan and Baur (Mclarnan and Baur, 1982; Baur and Mclarnan, 1982) wh refer t a-lision as the W-Pbc2; (2,1) type. Nne fthese fur structure types exactly satisfies the BURDETT AND McLARNAN: ORBITAL INTERPRETATION OF PAULING,S RULES 603 t determine the charge n each atm in this mdel crystal via a ppulatin analysis (Burdett, 1980; Chen and Hffmann, 1976).In the real cmpund, the mre electrnegative O atms can be expected t ccupy the sites with the larger electrn density (i.e., larger negative charge). In rder t avid cnfusing effects due t metal charge with thse due t metal size, we have carried ut such calculatins fr MgAION rather than fr LiSiON, and have further chsen identical Slater expnents fr the tw metals. Crrespndingly, all calculatins have been () (b) perfrmed with equal catin-anin distances. The resulting charges are given in Table l, which shws that indeed sites with small bnd strength sums, which accrding t the secnd rule shuld be ccupied by O rather than N, are significantly mre electrn rich. A further test f this is given in Table 2, which shws the site charges and energies f all fur f these structures calculated nt with averaged anins but with real O and N atms. Values are shwn fr bth the rdering expected frm Pauling's rule and the ppsite ne. The crrectly rdered structures lie lwest in energy, fllwed by thse in which O and N ccupy sites with equal electrstatic bnd strength sum, fllwed by thse in which N ccupies sites with lwer (d) bnd sum than O. Further, in the structures with the Fig.. ldealized crystal structures f a-lision : W-Pbc2r Pauling's rule-predicted rdering, there has been a migratin f charge away frm bth N and Mg, the mre (2,1)(a) and three hypthetical arrangements f the same stichimetry: W-Pmn21(l,l) (b), tn-p3m (l,l) (c) and \N-Pmc21electrpsitive anin and catin, tward bth Al and O, (l,l) (d). The anins are hexagnally clse packed and the the mre electrnegative pair. catins fill tetrahedral hles. Filled squares : Si(0),N(i); empty The valence sum rule, hwever, says mre than just squares : Li(0),O(il; filled circles = Si (),N(0; empty circles : that very electrnegative catins, i.e., thse with small L(t,O(il.Only the catin tetrabedrat z: 0 are drawn. Structures negative frmal charges, shuld be lcated in sites f b{ and their antistructures are the nly nes gemetrically pssible fr LiSiON with a (l,l) rthhexagnalsmall electrstatic valence sums. It als says that the cell. bnd strength sum shuld equal the negative fthe frmal charge. In ur cvalent language, this means that anins f the same electrnegativity shuld be fund in sites f electrstatic valence rule (Fig. 2). In three f them, a- equal bnd sum. Thus, in calculatins in which all anins LiSiON, W-Pmn21 (l,l) and W-P3m (l,l), half f the are taken t be identical, the W-Pmczt (l,l) structure in anins are crdinated t 3 atms f metal A and ne f which every anin receives the same bnd strength sum metal B, and the ther half are crdinated t 3B + la. shuld be favred ver the W-P3m (l,l), W-Pmnzt (l,l) Fr (A,B) : (Li,Si) this prduces bnd strength sums f 3.25 and 1.75, while fr (Mg,Al) the bnd strength sums are 2.75 and While the bnd strength sums are nt Li Li Si Si equal t 3 and2, it is easy t see that they cme as clse t these values as is pssible fr wurtzite derivatives it where all metal atms f each kind are crystallgraphical- LiI Si Sii Li tt ly equivalent. In the W-PmcZ1(l,l) type, every anin is crdinated by 2A + 28, and hence has Is; : 2.5. We wuld like t shw that in an ABON crystal with the w-pbc21 Q,l), w-pmn2' (1,1) r w-p3m (l,l) srructure, the nitrgen atms will ccupy the sites f higher bnd sum, and that these structures all lie lwer in energy than the W-Pmc21(1,1) type in which O and N atms see the same bnd sum. The standard way f predicting site preferences f this srt using mdern electrnic ideas is t calculate the rbital wavefunctin f a material in which O and N atms are bth replaced by hypthetical atms intermediate in size and electrnegativity, and then Mg tt ebs=1.75 Mg Msi Ar tt ebs=2.25 ebs= 3.25 AI AI,l Ms ebs= 2.75 Fig. 2. Electrstatic bnd sums fr LiSiON and MgAION in the structure types W-Pmn2r (l,l) r tn-p3m (l,l). 6(X BURDETT AND McLARNAN: ORBITAL INTERPRETATION OF PAULING,S RULES Table l. Electrnic charges in MgAl(O,N)z structuresa ts Structure fl-pbc2l(2,1) U-Pnn2t(l,l) W-P3m(1,1) W-Pnc2t{1,1) l,l-pmc2r ( 1,1 ) t{-pbc21 (2,1 ) H-hrn2t(1,1) H-P3n(1,1) t.)j a Ttre number, q, f valence eleclrns n slles rllh dtfferent electrstatic bnd strensgh sums ln l Al(0,N)r' Fur dlfferent 6Lrucbures vtlh this cpsitin are cn-- sldered. fn these calculatlns all anins {ere glven identical paraneters lneermediate between thse l 0 and N, s bhat varlatlns ln q reflect dlfferences assclated wlth the sites, nt uith ghe nature f the anins. In lhls and ther tables in the texl se have deliberately aeprled Bre sisnificant fisures than perhaps abslutely necesaary, andw-pbc2 (2,1) type s in which half the anins have Is; = 2.25, and half have Is1 = 2.7t. That this is s is shwn in Table 3, which gives the energies f these fur structures with all the anins given equal parameters midway between thse f O and N. We shall return belw bth t this calculated preference f identical anins fr similar envirnments and t the significant fact that the energy difierences in Table 3 are smaller than thse in Table 2. One ther imprtant variatin n Pauling's secnd rule is the Zachariasen-Baur extensin f it (Baur, 1970) t state that versaturated anins shuld frm unusually lng bnds t their catin neighbrs, and that undersaturated anins shuld frm unusually shrt bnds. As shwn by Baur (1981a) nearly all f the variatin f individual Si-O bnd lengths which can be attributed t any lcal effect is due t this relatin between pe, the electrstatic bnd strength sum at an xygen atm in the structure, and d(si-o). Figure 3 plts the bnd verlap ppulatins f all crystallgraphically distinct Mg-O, Al- O, Mg-N and Al-N bnds in the eight structure types f Table 2 as a functin f the electrstatic bnd strength sum at the anin. These bnd verlap ppulatins are a measure f the calculated bnd strength between tw atms, and hence nrmally crrelate inversely with bnd lengths. Thus, the calculated decreases in the bnd verlap ppulatins f all fur types f bnds with increasing anin saturatin represents a predictin by mlecular rbital methds f exactly the qualitative bnd length variatins bserved generally by Baur. We emphasize that these numbers cme frm cmputatins which generate the band structure f the infinite crystal and nt frm mlecular fragments. It is als imprtant t nte that all f the calculatins are perfrmed n structures with identical anin-catin distances s that there is n gemetrical bias initially built int the prblem. The effects we are seeing are purely electrnic nes. All f these results can be understd using the methds f perturbatin thery (Burdett, 1980; Hffmann, l97l) using well-knwn techniques. The ccupied rbitals in MgAION are largely lcated n the anins, s t evaluate the bnding energy f a particular structure fr MgAION we must ask hw the anin levels are perturbed in the crystal frm their energies in a free atm. These perturbatins will arise n ur mdel primarily frm the strngest rbital interactins experienced by the anins, namely thse with the neighbring metal atms. Cnsider therefre the interactin f a single anin rbital with energy H11 with a single catin rbital with energy H22 H11 as in Figure 4. The magnitude f the interactin between these tw rbitals is measured by their verlap integral S12. It is therefre reasnable t write (Burdett, 1980) the energy fthe perturbed anin rbital as a pwer Table 2. Bnd sums, charges and energies f MgAION variants' St ructu re w-pbc2t(2,1) W-Pmn2r(1,1) W-P3m(1,1) ) s.(vu) q (e,l '1.76\'t 1.365\ : A1 tls Energy (ev ) 0.0ll rr \ w-pmc2l(1,1) l,l-pnc2., ( 1,1)* \3\9 O ? tl-pbc2., (2,1)* W-Pnn2t(1,'!)r w-plm(1,1)r 2, ? ? ?3q ? r q \3 BURDETT AND MCLARNAN: ORBITAL INTERPRETATIAN OF PAULING,S RUTES 605 Table 3. Energies f sme averaged MgAION structures Bnd verlp ppultins t IN Struclure Energy (ev/fnla -untt ) W-Rrc21 ( 1,1 ) w-p3d( l, I ) W-Pmn21(1,1) W-Pbczt (2,1) r 'Tlre energies f 4 psslble structureg fr llaaln, cnputed flith bth 0 and N replaced by an averaged anin wlth atf,ic parameters midvay betveen thelrs. The fist stable structure type tn thlr calculatin, W-Rrc2t(1,1) ia the ne in which all anlns aee equal bnd strdngth suns f 2.5. c ) AI series in Srz. As lng as Hrr,Hzz and H22-H11 are f the same rder f magnitude, the leading terms in this series ar E1 = Hn + H12S?2/(Hrr - H2). Thus : HTzS?zl (Hrr - Hz) 0 is the first rder energetic stabilizatin f the filled anin rbital. Often the numeratr f this expressin is sufficiently cnstant s that ne can write 5166 a(h11 - Hzz)-'. Thus, the stabilizatin resulting frm tw interacting rbitals is inversely prprtinal t their energy difference. Further, these secnd rder energy terms are additive s that if the anin is crdinated by several catins f energies Hzz, HLz, Hiz,... ne has G31a6 a(h11 - Hz)-t + (Hrr - HL)-t + (Hrr - Hiz)-l The metal rbital is pushed up in energy, i.e., it is destabilized. Since it is unccupied it will nt cntribute energetically t the prblem and we will mentin it n further. The atmic rbitals in MgAION increase in energy in the rder O N Al Mg. In structures rdered accrding t Pauling's secnd rule, the mst numerus nearest neighbr interactins are between O and Mg and between N and Al. The cmmn pairs f neighbrs in structures with the ppsite rdering are O-Al and N- Mg. Let H1 be the energy f an atm f element i. Then the efect f interchanging the anins (Fig. 2) in a structure satisfying Pauling's pstulate is t replace cntributins t the crystal binding energy prprtinal t (H - Hrr)-t + (HN - Ha)-r with terms prprtinal t (He - HeD-r i (HN - Hr*)-t. T determine whether r nt this change is stabilizing, we evaluat Ae,66 = (H - H,)-l + (HN - H r)-' - (H - HMJ-I - (HN - HeJ-' = [(H - HeD-' - (H - HrJ-t] + [(HN - HMJ-r - (HN - -^J*tl. Expanding this yields (Her - Hve) {[(H - HeJ (H - Hr,aJ]-t - [(HN - HaJ (Hr., - Hue)l-t). Since H6 - Hnr ( HN - Hel and Hq - Hr*le ( HN - Hr re and all fur f these terms are negative; the expressin in braces is negative and Ae,66 0. Thus the arrangement with the rdering predicted by Pauling's rule shuld be the mre stable ne, as bserved and as calculated. Mre generally, the cnsequence f this reasning is that stable crystal structures shuld allw as many interactins as pssible between electrnegative ca
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