IGERT From Bonds to Bands -- How chemistry and physics meet in the solid state.pdf

7/27/2019 IGERT From Bonds to Bands -- How chemistry and physics meet in the solid state.pdf

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How Chemistry and Physics Meet in the Solid StateBy Roald Hoffmann*To make sense of the marve lous electronic proper t ies of the sol id s tate, chemists must learnthe langua ge of sol id-state physics, of band structures. An at tem pt is ma de here to demys-t ify that language, d rawing explici t paral lels to well-known conc epts in theoret ical chemis-try. To the jo in t search o f phys ic i s t s and chemis t s fo r under s tand ing o f the bond ing inextended systems, the chem ist brings a great deal of intui t ion an d som e simple but pow erfulno t ions . M os t impor tan t among these i s the idea o f a bond , a nd the use o f f ron t ier-o rb ita larguments . How to f ind loca l ized bonds among a l l those maximal ly de local ized bands?Interpretat ive constructs , such as the densi ty of s tates , the decomposit ion of these densi t ies ,and crystal orbital over l ap populat ions, al low a recovery of bond s, a f inding of the f rontierorbitals that control s tructure a nd reactivi ty in extended systems as well as discrete mole-cules.

IntroductionThere is no need to p rov ide an apologia pro vita sua f o r

solid-state chemistry. .Macromolecules extended in one,two, or three dim ensions, of biological or natural or igin, orsynthetics , t i l l the world around us. Metals , al loys, andcompos i tes , be they copper or bronze or ceramics, haveplayed a pivotal , shaping role in our culture. Mineralstructures form the base of the paint that colors o u r walls,and the g lass th rough which we look a t the ou ts ide wor ld .Org anic polymers, be they nylon o r wool, clothe us. Newmater ials-ternary inorganic superconductors , conductingorga nic polymers-exhibit unusual electr ic and magneticproper t ies , promise to shape the technology of the future.Solid-state chemistry is im portant , al ive, and growing.

Given the vi tal i ty and at tract iveness of the f ield, I takesom e risk in l is t ing som e mino r problems that I perceive atthe interfaces of sol id-state chemistry with physics andwith the rest of chemistry. This is done not with the intentto f ind fault, but constructively-the remainder of this pa-per tries to resolve sOme of these perceived difficulties.

What is most interest ing about many of the new mater i -als are their electr ical and magnetic proper t ies . Chemistshave to learn to measure the se propert ies , not only to makethe new m ater ia l s an d determ ine the ir s t ruc tu res . The h is -to ry o f the com poun ds tha t a r e a t the cen ter o f today s ex-ci t ing developm ents in high- tem perature superconductivi tymake s this point very well. A nd they must b e abl e to rea-son intel l igently about the electronic s tructure of the com-pound s they make, so tha t they may under s tand how theseproper t ies and structures may be tuned. Heres the f irs tp rob lem then , for such an un derstanding of sol ids perforcemust involve the language of modern solid-state physics,of band theory. That language is general ly not par t of theeducation of chemists . I t should be.

I suspect that physicis ts don t think that chemists havemuch to tel l them ab ou t bon ding in the sol id s tate. I woulddisagree. Chemists have buil t up a great deal of under-

[*] Prof . Roald HoffrnannD ep ar t m en t o f C h em i s t r y an d M a t e r i a ls S c i en ce C en t e rCornel l Univers i ty , Baker LaboratoryI thaca, N Y 14853-1301 ( U S A )

standing, in the intui t ive language of s imple covalent orionic bonding, of the s tructure of sol ids. The chemistsviewpoint is of ten local . Chemists are especial ly good atsee ing bonds or clus ter s , and our l i t e r a tu re and memoryare especial ly well-developed, so tha t we can immedia te lythink of a hundred structures or molecules related to thecom pou nd under s tudy. From much empir ica l exper ience ,a l i t tle s imple theory, chemists have gained much intui tiveknowledge of the what , how, and why molecules hold to-gether. To put i t as provocatively as I can, o u r physicistf r iends know better than we how to calculate the electronicstructure of a molecule or solid , but of ten they do not un-derstand it as well as we do, with all the epistemologicalcomplexity of meaning that understanding somethinginvolves.

Chem ists need no t enter a dialogue with physicis ts withan y infer ior i ty feel ings at al l ; he exper ience of molecularchemistry is t remendously useful in interpret ing complexelectronic s tructure. (Anothe r reason not to feel infer ior :unti l you synthesize that molecule, no one can study i tsproper t ies . Th e synthetic chemist is quite in control .) Thisis not to say that i t wil l not take some effor t to overcomethe skepticism of physicis ts as to the l ikel ihood tha t chem -is ts can teach them someth ing abou t bond ing .

Another inter face is that between solid-state chemistry,of ten inorganic, and molecular chemistry, both organican d inorganic. With one exception, the theoret ical con-cepts that have served solid-state chemists well have notbeen molecular. At the risk of oversimplification, themost important of these concepts have been the idea thaton e has ions (electrostatic forces, Madelung energies) an dthat these ions have a s ize ( ionic radii , packing considera-t ions) . The success of these s imple notions has led solid-state chemists to use these concepts even in cases wherethere is substantial covalency. What can be wrong with anidea that works, that explains s tructure and proper t ies?Wha t is wrong, or can be wrong, i s tha t app lica t ion o f suchconcepts may draw that f ield, that group of scientis ts ,away f rom the hear t of chemistry. At the hear t of chemis-try, let there be no doubt, is the molecule My personalfeeling is that i f there is a choice among explanations insolid-state chemistry, one must choose the explanationwhich permits a connect ion be tween the s t ruc tu re a t hand

846 0 V C H VerlagsgeseNsrhnJr m b H . 0-6940 Wernherm. 1987 0570 -083 3/87/0 909-0 846 3 02 50 /0 Angew Chem Inr. Ed Engl 26 11987) 846-878


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and some d iscre te molecu le , o rgan ic o r inorgan ic . Makingconnections has inherent scientif ic value. I t also makespolitical sense. Again, if I might express myself provo ca-tively, I would sa y that many solid-state chemists have iso-la ted themselves (no w onder tha t the i r o rgan ic o r eveninorganic colleagues aren t interested in what they do) bychoos ing no t to see bonds in the i r materia ls .

Which, or cour se , b rings me to the excep t ion: th e mar -velous and useful Zintl concept . The simple notion, intro-duced by Zintl and popu larized by Klemm, Busmann, Her-bert Sch$er, and o ther s ,] is t h a t i n s o m e co m p o u n d sAXB Y, here A is very elec troposit ive relat ive to a m ain-gro up element B, one cou ld just think, thats al l , think tha tthe A a tom s t r ans fer the i r e lec t rons to the B a toms , whichthen use them to fo rm bonds . Th is very s imp le idea , in myopin ion , i s the s ing le mos t impor tan t theore t ica l concep t(an d how not very theoret ical i t is ) in sol id-state chemistryof this century. And i t is so important , no t just beca use i texplains so much chemistry, but especial ly because i tbu i lds a b r idge be tween so l id - sta te chemist ry and o rga n icor main-gro up chemist ry.

The th ree p rob le ms I have iden t i f ied , and le t me r epea ttha t I th ink they a re r e la tive ly m inor ones fo r a lively field,a r e I ) some lack of knowledge ( therefore fear) of sol id-s ta te physics language on th e par t o f chemis ts , (2) insuffi-cient appreciat ion of the chemists intui t ive feel ing forbonding on the par t o f phys ic i s t s , and (3) n o t en o u g hreach ing ou t fo r connect ion s with molecu lar chemis t ry onthe par t of sol id-state chemists . The character izat ion ofthese as p rob lems r epresen ts a general izat ion on my par t ,with the associated danger that any general izat ion carr ies .Typologies and general izat ions of ten point not so m u ch t orea l i ty as to the weakness o f the mind tha t p roposesthem.

What can a theoret ical chemist contr ibute to the amelio-rat ion of these problems, i f they indeed are real ones? Atheoret ical chemist can, in fact , do very much. With his orher f irm knowledge of sol id-state physics (which heshould, in pr inciple, have, but of ten doesn t) and his feel-ing fo r bond ing and the marvelous boun ty o f s t ruc tu restha t he knows h is chemical co l leagues have made, the the-oret ical chemist should be in a wonderful posi t ion to serveas a br idge between chemistry an d physics. We sh ould cer-ta in ly be ab le to he lp wi th po in t (1) above, showing ourco l leagues tha t ba nd theory i s easy . Po in ts (2) a n d ( 3 ) a r emore d i f f icu l t . We need to push our exper imenta l co l -leagues to see bonds , c lus ters , molecu lar pa t te rns in newspecies. But, they can see these pat terns, without our help,bet ter than we do. An d to convince physicis ts that chemistsare good fo r any th ing excep t making molecu les , tha tchemists in fact understand what the electrons in mole-cu les and so l ids ar e do ing- that w il l t ake some do ing .

In fact , the ef for t has been und er way f rom the theoret i-ca l s ide fo r som e t ime. I would l ike to mention here espe-cial ly the c ontr i butio ns of J e r e m y B ~ r d e t f , ~ . ~ ~ho is re-spons ib le fo r the f i rs t new ide as on what de termines so l id -state s tructures s ince the pioneer ing contr ibution of Paul-i ng , and o f M yung-Hwan W h a n g b ~ , [ ~ . ~ ~hose analys i s o fthe bond ing in low-d imens ional mater ia l s such as the n io -b ium se len ides , t e t r a th iafu lva lene- type o rgan ic con duc-to r s , and molybdenum bronzes has con t r ibu ted much to

ou r knowledge o f the ba lance o f de local iza t ion a nd e lec-t ron r epu ls ion in cond uct ing so lids . On the s ide o f phys ics ,le t me m ent ion the work o f several ind iv iduals who haveshown an unusual sens i t iv i ty to chemis t ry and chemicalways of thinking: Jacques Friedel, Walter A . Harrison,Volker Heine, James C. Phillips, Ole Krogh Andersen, a n dDavid W . Bullef t .

In this paper , I would l ike to work mainly on point (1)ment ion ed abo ve, the teach ing , to chemis t s , o f som e of thelanguag e o f ban d As many connect ions as pos-s ib le to our t r ad i t ional ways o f th ink ing abou t chemicalbon ding will be made- i t is this aspect which shou ld be ofinterest to any physicis ts who might read this ar t icle. Theapproach wil l be s imple, indeed, oversimplif ied in part.Where detai led computat ional results are displayed, theywi l l be o f the ex tended Huckel type , or of its solid-stateanalogue , the t igh t -b ind ing m ethod w i th over lap .

Orbitals and Bands in One DimensionI ts usually easier to work with small , s imple things, an d

one-dimensional inf ini te systems are par t icular ly easy toMuch of the physics of three-dimensionalsolids is there in one dimension. Lets begin with a chainof equal ly spaced H atoms , 1 , or the isomorphic n-systemof a non-bond-al ternating, delocalized polyene 2,stretched out for the moment. And we wil l progress to as tack o f Pt square-p lan ar complexes , 3, [Pt(CN),] or am o d e l [PtH4]20.

H H H H H H

0 0 0 0 0 02

I ,. I ,. I ,,8 I ,. I \\...pt .........PI ....... pt ......... PI . . . . . pt...I I 3

A digression here: every chem ist would have a n intui t ivefee l ing fo r what tha t model chain o f hydrogen a toms , 1 ,wo uld d o if we were to release i t f rom the pr ison of i tstheoret ical construction. At ambient pressure, i t wouldform a chain of hydrogen molecules, 4. This s imple bond-

c 4 - -....H .... ..H....... H ...... H ....................4

H-H H-H H-H4

forming process could be analyzed by the physicis t (wewill d o i t soon) by ca lcu la t ing a band fo r the equal ly

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spaced polymer , then seeing that i t s subject to an instabil-ity, called a Peier ls dis tor t ion. Other words around thatcharacter izat ion would be s trong electron-phonon cou-pling, a pa ir ing distor t ion, or a 2 k F instabili ty . And thephysicis t would come to the conclusion that the ini t ial lyequal ly spaced H polymer would form a chain of hydrogenmolecules. I mention this thought process here to make thepoint , which I wil l d o again and again, that the chemistsintuition is really excellent. But we must bring the lan-guages of our sis ter sciences into correspondence. Inciden-tal ly , whether distor t ion 4 will take place at 2 Mb ar is no tobv ious , an ope n ques t ion .

Lets return to o u r chain of equally spaced H atoms. I ttu rns ou t to be computa t ional ly conven ien t to th ink of tha tchain as a n impercep t ib ly ben t segment o f a large r ing ( thisis cal led applying cyclic boundary condit ions) . The orbi-tals of medium-sized r ings on the way to that very largeone are qu i te wel l known. They are shown in 5 .

pr iate symmetry-adapted l inear combinations cy ( r emem-ber translat ion is jus t as good a symmetry opera t ion as a n yother one we know) are given in 6 . Here a is the latticespac ing ( the unit cell being in one dimension) and k i s anindex which labels which ir reducible representat ion of thet r ans la t ion g roup ty transforms as. We will see in a mo-ment tha t k is much more , bu t fo r now, k i s jus t an indexfor an ir reducible representat ion, just l ike a, e , , an d e2 i nC, re labels .

The process of symmetry adaptat ion is cal led in the sol-id-sta te physics t ra de form ing Bloch f ~ n c t i o n s . ~ ~ ~ ~ ~ . ~ ~oreassure a chemis t tha t one is gett ing what one expectsf r o m 5, lets see what combinations are generated for twospecif ic values of k , k = O a n d k = n / a ( se e 7 . Referring

k = O $o= e0 X = 5 X = X o + X I + z + X J + ..-ex+&cM- ---- ---8 - back to 5 , we see that the wave function corresponding tok = O is t h e m o st b o n d i ng o n e , t h e o n e f or k = n / a t h e t o p

of the band . For o ther va lues o f k we get a neat descr ipt ionof the oth er levels in the ban d. So k counts nodes as well .

- --- -- --- --- - -m v 8v - -- -- --- -- The larger the absolute value of k , the more nodes one hasin the wave funct ion . But on e has t o be carefu l- there is arange o f k a nd i f on e goes ou tside of it, one doesnt get anew wave funct ion , bu t r epeats an o ld one . The un ique va l -u es of k a r e in t he in te rv al - n / a I k < n / a o r I k l 1 d a .This is cal led the f irs t Br i l louin zone, the range of uniquek.

- -- -=- - - -89 - ----v = w= Q 0-5

For a hydrogen m olecu le (o r e thy lene) there is a b o n d -ing o, n) b e l ow a n an t i b o n d i n g o,*(n*).For cyclic H3 orcyclopropenyl we have one orbital below two degenerateones ; fo r cyclobu tad iene the f ami l ia r one be low two belowo n e , an d so on . Excep t fo r the lowest ( and occas ional ly thehighest) level , the orbitals come in degenerate pairs . Then u m b er of node s increases as one rises in energy. Wed ex-pect the sam e for an inf ini te polymer- the lowest levelnodeless , the highest with the maximum number of nodes.In between, the levels sho uld co me in pairs , with a growingnumber of nodes. The chemists representat ion of thatba nd for the polyme r is given at r ight in 5 .Bloch Functions, k, Band Structures

T h er e i s a better way t o wri te out al l these orbitals , mak -ing use of the translat ional symmetry. I f we have a lat t icewhose po in ts a r e labeled by an index n=O, 1, 2, 3, 4, tc.,as shown in 6, nd if on each lat t ice point there is a basisfunction (a H Is orbital) , xo ,x 2 , etc ., then th e ap pro-

How m any values o f k ar e there? As many as the num-ber of t ranslat ions i n the crystal, or , al ternatively, as manyas there are microscopic unit cel ls in the macrosco pic crys-tal. So let us say Avogadros number (N,,), give or t ake afew. There is an energy level for each value of k (actually adegenerate pair of levels for each pair of posi t ive and ne-gative k values) . There is an easi ly proved theorem thatE(k)= E( -k) . Mos t r epresen ta t ions of E(k) do no t g ivet h e r ed u n d an t E( ), but plot E(lk1) an d label it a s E(k)).A lso , the a l lowed values o f k ar e equal ly spaced in thespa ce of k , which is cal led reciprocal or momentum space .The relat ionship between k = 1//2 and momentum der ivesfrom the d e Broglie relat ionship il h/. Remarkab ly , k isnot on ly a symm etry label and a no de counter , but i t is alsoa wave vector , and so measures momentum.

08 k- a / o

So what a chemist draws as a band in 5, repeated at lef tin 8 ( and the chemis t ti r es and d raws =20 lines or just a

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block instead of N A lines), the physicist will alternativelydraw a s an E(k) vs . k d iagram a t r igh t in 8. Recal l tha t k isquantized, and there is a f ini te but large number of levelsin the diagram at r ight . The reason i t looks co n t i n u o u s isthat this is a f ine do t matr ix pr inter-there are N A pointsjammed in there , and so its no w o n d e r w e s ee a line.

Gra phs o f E (k) vs. k ar e ca l led ba nd s t ruc tu res . You c a nbe sure tha t they can be mu ch m ore compl icated tha n th i ss imple one , bu t no mat ter how compl ica ted , they can beunderstood.Band Width

One very impo r tan t f ea tu re o f a band is i ts dispersion, o rband width, the dif ference in energy between the highestand lowest l evels in the band . W hat de termines the wid thof bands? The same th ing tha t de termines the sp l i t t ing o flevels in a dimer , ethylene or H2, namely , the over lapbetween the interact ing orbitals ( in the polymer the over-la p is that between neighbor ing unit cel ls). The greater theover lap be tween neighbors , the g rea ter the b and wid th.Figure 1 i l lustrates this in detai l for a chain of H a t o m s

a . 3 115

I- a - lo...0 ...0 ...0 ... ...0..

a = ~ i

0 k - $ 0 k - $ 0 k - EFig. 1. The band s t r u c tu r e of a chain of H a to ms spaced 3, 2, and I A apart.The energy of an isolated H a to m i s - 13.6 eV.

s p aced 3, 2, a n d 1 A ap a r t . T h a t t h e b a n d s ex t en d u n s ym -metr ical ly around their or igin, the energy of a f ree Ha t o m a t - 3.6 eV, is a conseq uence o f the inc lus ion o fove r lap in the calculat ions. F or two levels , a dimer , the en-erg ies ar e g iven by Equat ion ( a) . The bond ing E , combi-

nat ion i s l ess s tab i l ized than the an t ibond ing one E - isdestabil ized. There are nontr ivial consequences in chemis-

try, for this is the source of four-electron repulsions andsteric effects in one-electron theories.I8 A similar effect ofover lap i s r espons ib le fo r the bands spread ing up in Fig -u r e 1.See How They Run

An othe r interest ing featu re of ban ds is how they run.The lovely mathematical algor i thm 6 ppl ies in genera l ; itdoes no t say any th ing abou t the energy o f the o rb i ta l s a tt h e cen t e r o f t h e zo n e (k=O) r e la t ive to those a t the edge( k = n / a ) . F o r a ch a i n o f H atoms i t is clear thatE ( k =0)


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K2[Pt(CN),] ) indee d show such stacking in the sol id s tate,at th e relat ively uninterest ing R-Pt eparat ion of = 3.3 A.

I+ a 4

12

13

Mo re excit ing are the par t ial ly oxidized mater ials , such asK2[Pt(CN),Clo 4 and K,[R(CN),(FHF), 4. T h es e a r e a l s ostacked, but s taggered, 13,with a much shor ter pt-Pt con-tact of 2.7-3.0 A. T h e R-Pt i s tance had been shown to beinversely related to the degree of oxidation of Pt.l9]

Th e real test of und ersta ndin g is predict ion. So, lets tryto p red ic t the approx imate band s t ruc tu re o f 12 a n d 13without a calculat ion, just using the general pr inciples wehave at hand. Lets not worry about the nature of the l i -gand L-it is usual ly C N e, bu t s ince it i s only t h e s q u a r e -plan ar feature w hich is l ikely t o be essential , lets ima ginea theoret icians gener ic l igand, He . And lets begin with12, because the unit cel l in i t is the chemical PtL, unit,whereas in 13 it is dou bled , [ (RL4)*] .

One a lways beg ins wi th the monom er . W hat ar e i t s fron-t ier levels? The classical crystal f ield or molecular orbital

Y

r y

Fig. 2. Molecular-orbital derivation of the frontier orbitals of a square-planarPtL, complex.

picture of a squa re-pla nar complex (Fig. 2) leads to a four-below-one spli t t ing of the d block.f81 Fo r 16 electrons weha ve dz2, d,,, d,,, an d d,, oc cup ied an d dX2-,2 em pty .Com pet ing with the ligand-field-destabilized d,2-,2 orbitalfor being the lowest unoccupied molecular orbital( LUM O) of the molecule is the metal pz. These two orbi-ta l s can be manipu la ted in understandable ways: n-accept-ors push p z down, n -donor s push i t up . Bet ter o -donor spush d,z-,? UP.

We form the po lymer. Each M O of the m onom er gener -ates a band. There may (wil l) be some fur ther symmetry-

condit ioned mixing between orbitals of the same symmetryin the polymer (e.g., s an d pL an d d,l are of dif ferent sym-metry in the monomer , but cer tain of their polymer MOsare of the same symmetry) . But a good star t is m ad e b yignor ing that secondary mixing, and just developing aband f rom each m onom er level independen t ly .

First , a chemists judg me nt of the ba nd w idths that wil ldevelop (see 14): the b and s that will ar ise f rom dZ2 and pz

14

will be wide, those from d,, an d d,, of medium w idth,those f rom dxZ-, an d d,, narrow. This character izat ionfollows f rom the real ization th at the f irs t set of interact ions(pz, d,z) is o type, thus has a large over lap between unitcells. Th e d,,, d,, set ha s a medium n over lap, and the d, ,and dX?- ,2 orbitals ( the lat ter of course has a l igand admix-ture, but that doesn t change i ts symmetry) are 6.

It is a l s o eas y t o s ee h o w t h e b an d s run . Lets write outthe Bloch funct ions a t the zone cen ter (k=O) and zoneedge k=n/a). O n l y o n e of t h e x a n d 6 functions is repre-sented in 15. The moment one wr ites these dow n, one sees

X

-2

15

that th e dZ2an d d,, ban ds wil l run up f rom the zon e cen-ter ( the k =0 combination is the mos t bond ing) whi le the d ,an d d,, ba nd s will run down ( the k=O combinat ion isthe mos t an t ibonding) .

The predicted band structure, merging considerat ions ofband width and orbital topology, is that of 16. To make areal est imate of band wid th , one would need a n ac tua l ca l -culat ion of the var ious over laps, and these in turn wouldd e p e n d on t h e Pt-Pt separat ion.

The ac tua l band s t ruc tu re , as i t emerges f rom an ex-tended Huckel calculat ion at R-R=3.0 A, is shown in

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A/,=,\16

0 k - a / oFigure 3 . I t matches o u r expectat ions very precisely. The reare, of course, bands below and above the f rontier orbitalsd iscussed- these are R -H o n d o orbitals .

-14-I4= P t - H u

I I0 k - a / n

Fig. 3. Computed band structure of a n eclipsed [RH4]*' stack, spaced at 3 A.Th e orbital marked d,,, d,, is doubly degenerate.

To make a connect ion wi th molecu lar chemis t ry : theconstruction of 16, an ap p r o x i m a t e b an d s t r u c t u re f o r acyanoplat inate s tack, involves no new physics, no newchemistry, no new m athemat ics beyond what every chemista l r eady knows fo r one o f the mos t beau t i fu l ideas o f mod-ern chemistry-Cotton 's construct of the metal-metal q ua-drup le bond . [ ' " ] I f we are asked to exp la in quadrup lebonding, for instance in [Re2C18]2e,what we d o is to d raw17. We f o r m b o n d i n g an d an t i b o n d i n g co m b i n a t i o n s f r o mthe d;(o) , dxZ ,dy r(x) , nd dX2-y2(6) f rontier orbitals ofeach ReCI? f ragment. And we spli t o f rom o b y m o r et h an n f rom n*, which in turn is sp l i t more than 6 a n d 6*.What goes on in the inf ini te sol id is precise ly the samething. Tru e, there a re a few m ore levels , but th e transla-t ional symmetry helps us out with that. I t 's really easy to

7

wri te dow n the sym metry-adap ted l inear combinations , theBloch functions.The Fermi Level

I t 's impor tan t to know how m any e lec t rons one ha s inone's molecule. Fe" has a dif ferent chemistry f rom Fell ' ,and CRF carbocat ions are d i f f eren t f rom C R3 rad ica ls an dCRY anions. In the case of [Re2C18]2e, he archetypicalqu adr up le bond, we have formally Re" ', d4, i .e., a total ofe igh t e lect rons to pu t in to the f ron t ier o rb i ta ls o f th e d ime rlevel schem e, 17. T h ey f i l l the o, w o n, an d t h e 6 level forthe exp l ic i t quadrup le bond . What abou t the [ (PtH4)2e]_po lymer 12? Each m onom er i s dX . I f there a re N A unitcells, there will be N , levels in each band. And each levelhas a p lace fo r two e lec t rons . So the f i r s t four bands aref i l led, the xy, xz, yz, and z2 bands. The Fermi level, thehighest occupied molecular orbital (HOMO), is at the verytop o f the z2 band . (S t ric tly speak ing , there i s an o ther ther -mody nam ic def ini t ion of the Fermi level , appropr iate bothto metals and semiconductors , '" ' but here we wil l use thesimple equivalence of the Fermi level with the HOMO.)

Is there a b ond between th e plat inums in this [{PtH4J2 ']_polymer? W e haven 't intro duc ed, yet , a formal descr ipt ionof the bond ing p roper t ies o f an o rb i ta l o r a band , bu t ag lance a t 15 a n d 16 will show that the bottom of eachband , be i t mad e u p o f z2 , xz, yz , or xy, is bonding, a nd thetop an t ibond in g . F i l ling a band comple te ly , jus t l ike f i l lingbonding a nd an t ibond in g o rb i ta l s in a d ime r ( th ink o f He2 ,th ink o f the sequence N2 , 0 2 , 2, Ne2) provides n o netbonding. In fact, it gives net antibonding. So why does theunoxidized PtL, chain s tack? I t cou ld be van der Waalsat tract ions, not in our quantum chemistry at this pr imit ivelevel. I th ink there is also a contr ibution o f orbital interac-t ion, i .e. , real bonding, involving the mixing of the z 2 an d zbands.'"] We will return to this soon.

The band structure gives a ready explanation for whyt h e R-Pt separa t ion decreases on oxidation. A typical de-gree o f ox idat ion is 0.3 electron per Pt.I9] These electronsmus t com e f rom the top o f the z2 band . T he degree o f oxi-dat ion speci fies tha t 15% of that band is empty. The statesvacated are not innocen t o f bond ing. They are s t rong ly Pt -Pt o antibonding. So it's n o wonder tha t r emoving theseelectrons results in the formation of a par t ial Pt-Pt b o n d .

Th e oxidized m ater ial also has i ts Fermi level in a b a n d ;i .e. , there is a zero ban d ga p between f i l led an d em pty lev-els . The unoxidized cyanoplat inates have a substantial

A n g e w C'hem Inr Ed Engl. 26 (1987) 846-878 85 1


7/33

gap- they are semicondu ctors or insulators . The oxidizedmater ials are good low-dimensional conductors , which is asubstantial par t of what makes them interest ing to physi-cists. 1

I n general , conductivi ty is not a s i m p le p h en o m en o n t oexp la in , and there may be severa l mechan isms impedingthe motion of electrons in a mater ial .[ A prerequisi te forhav ing a good e lec t ron ic conducto r is to have the Fermilevel cu t o ne o r more ban ds ( soon we wi ll use the languageof densi ty of s tates to say this m ore precisely) . O ne ha s tobeware, however , (1) of d i s to rt ions which o pen up ga ps a tthe Fermi level and (2) of very narrow bands cut by theFermi level , for these wil l lead to localized states and nott o g oo d c o n d u ~ t i v i t y . [ ~ - ~ ~

Density of StatesWe have a l r eady r emarked th a t in the so l id , a very large

molecule, one h as to dea l with a very large numb er of lev-els or s tates . I f there a re n a tom ic orbitals (basis functions)in the unit cel l generat ing n molecular orbitals , and if inour mac roscopic crystal there are N unit cel ls (N is a num-b e r t h a t ap p r o ach es N A ) , then we wi ll have N .n crysta llevels . Ma ny of these a re occupied a nd, roughly speaking,they are jam me d in to the sa me energy in terval in which wef ind the m olecular o r unit cel l levels . In a discrete mole-cu le we are ab le to s ing le ou t one o rb i ta l or a small sub-group o f o rb i ta l s (HOMO, LUMO) as be ing the f ron t ier ,or valence, orbitals of the molecule, responsible for i tsgeometry, react ivi ty , etc. There is no way in the world tha ta s ingle level amo ng the myriad N . n orbitals of the crystalwil l have the power to direct a geometry or reactivity.

There is , however , a way to retr ieve a f rontier orbitallanguage in the so l id s ta te . We canno t th ink abou t a s inglelevel , but perhaps we can talk about bunches of levels .There are many ways to g roup levels, bu t on e p re t ty ob-v ious one is to look at all the levels in a given energy inter-val . The densi ty of s tates (DOS) is defined by (b). For aD O S ( E ) d E = n u m b e r of l ev e ls b e tw een E an d E + d E (b)s imple band o f a chain of hydrogen atoms, the DOS curvet ak es o n t h e s h ap e o f 18. Note tha t because the levels a r eequal ly spaced a long the k ax is , and because the E(k)

18

curve, the band structure, has a s imple cosine curve shape,there are mo re states in a given energy interval at the topand bo t tom of th i s band . In genera l , DOS(E) is propor -tional to the inver se o f the s lop e of E(k) vs. k, or to pu t i tin to p la in Engl ish , the f la t te r the band , th e g rea ter the den-sity of states at that energy.

DOS E 1

tl e L

-01

-1 0

-1 2

-1 4

t -0lev1

-1 0

-1 2

-1 4

0 k- 7r/a 0 00s -Fig. 4. a) Band structure and b) density of states (DOS) for an eclipsed[PtH,]O stack. The DOS curves are broadened so that the two-peaked shapeof the x y peak in the DOS is not resolved.

T h e s h ap es o f DOS curves are p red ic tab le f rom the ba ndstructures. Figure 4 shows the DOS curve fo r the [PtH4I2@chain. I t could have been sketched f rom the ba nd structureat lef t . In general , the detai led construction of these is ajob best lef t for computers . The densi ty-of-states curvecou nts levels . The integral of DOS up to th e Ferm i level isthe total num ber of occupied MOs. M ult iplied by two, i tsthe total number of electrons. So, the DOS curves plot thedistr ibution of electrons in energy.

One im por tan t aspect of the DOS curves is that they rep-resent a return f rom reciprocal space, the space of k, toreal space. The DOS is an average over th e B r i l louin zone,over al l k that might give molecular orbitals at the speci-f ied energy. The ad van tage here is largely psychological . I fI may be p ermit ted to general ize, I think chem ists (with theexception of crystal lographers) by and large feel them-selves uncomfortable in reciprocal space. Theyd rather re-turn to , and think in , real space.

The re is anoth er aspect of the return to real spa ce that iss ignif icant: chemists can sketch the D O S of any material ,approximately, intuitively. All thats involved is a knowl-edge o f the a toms , the i r approx im ate ion iza t ion po ten t ia l sand e lect ronegativ it i es , and some judgme nt as to the ex ten tof inter -unit-cell ove r lap (usually appa rent f rom th e s truc-ture).

Lets take the [(PtH,}@], polymer as an example . Themonomer units are clear ly intact in the polymer . At inter -mediate monomer-monomer separat ions (e.g . , 3 A thema jor inter -unit-cel l over lap is between d72 an d p Lorbitals .Next is th e d,,, d,, n-type overlap; all oth er interac tionsare l ikely to be small . 19 is a sketch of what we wouldexpect. In 19, I havent been careful in drawing the inte-g ra ted area s commensura te wi th the ac tua l to ta l num ber o fstates , nor have I put in the two-peaked natu re o f the DOSeach level generates-all I want to do i s to convey therough spread o f each band . Compare 19 to Figure 4.

852 A n g e w . Chem. I n t . Ed. Engl. 26 (1987) 846-878


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Monomer -Pt-t i u

2 -,2-y2 - tEt

7FH-b+Polymer

1

1 u

O S-19

This was easy , because the po lymer was bu i l t up o f m o-lecular mon om er units . Lets try som ethin g inherentlyth ree-d imens ional. The ru t il e s t ruc tu re is a relat ively com-mon type. As 20 shows, the rut i le s tructure has a nice oc-

U20

t ah ed r a l en v i r o n m en t of each meta l cen ter , each l igand(e.g., 0)boun d t o th ree meta ls . There are in f in ite chains ofedge-shar ing M 0 6 octahedra runn ing in one d i r ec t ion inthe crysta l, bu t the meta l -meta l separa t ion is always rela-t ively long.[ There are no monomer un i t s here , jus t an

E [eVII I I I-15

-20

-25IIII

I I II I III

1 I1 II I

r X M r Z

inf ini te assembly. Yet there are quite identif iable octahe-dra l s i tes . At each, th e metal d block mu st spl i t into tZp n dep com bina tions , the classic three-below-two crystal f ieldspli t t ing. The only other thing we need is to real ize that 0has qu i te d i s t inc t 2s and 2p levels, and tha t there is no ef-fective 0 0 or Ti-Ti interact ion in this crystal . We expectsome th ing l ike 21.

mainly TI . pTI-0 antibondingmainly on T it eg Ti -0 antibonding

t Ti -0 nonbonding, perhapss l i gh t l y II antibonding0 p . Ti-0 bondingEZs

DOS-21

Note tha t the wr i t ing down of the approx imate DOScurve i s done bypassing the b and s t ruc tu re ca lcu la t ion perse. Not th at that ba nd structure is very complicated. But itis th ree-d imens ional , and our exercises so far have beeneasy , in one d imens ion . So t h e co m p u t ed b an d s t r u c t u r e(Fig. 5 ) wil l seem complex . The n umbe r o f bands i s doub -led (i.e., twelve 2p, six tzg bands), simply because theunit cell conta ins two formula units, [ (TiO&. The re is notone reciprocal space var iable, but several l ines ( r - + X ,X + M , etc.) which refer to direct ions in the three-dimen-sional Br i l louin zone. These complicat ions of mo ving f romone d imen s ion to th ree we wi ll soon approach. I f weg lance a t the DOS, we see tha t i t doe s r esemble the expec-ta t ions o f 21. There are well-separated 0 2s, 0 2p, Ti tZg,an d eg b an d s . [ i 2 1

-10 k==-20

-30 j-35 DOS -

F i g . 5. a) Band structure andb) density of states for rutile,TiO:. The two Ti-0 distancesare 2.04 .& 2x ), 2.07 A 4 x )in the assumed structure.

A ny r n Chem inr. Ed. Engl . 26 11987) 846-878 853


9/33

Would you l ike to try something a l i t t le (but not much)more challenging? Attempt to construct the DOS of thenew supe rconduc tor s based on t h e L a 2 C u 0 4 a n dYBa2Cu3O7 t ruc tu res . And w hen you have d one so, a n dfoun d that these sh ould be c onductors , ref lect on how thatdoesn t al low you yet , did not al low anyo ne, to predict thatcomp ounds s l igh t ly off these s toichiometr ies w ould b e re-markable superconductors .

The chemist s ab il i ty to wr i te d own approx im ate dens i -ty-of-states curves should not be s l ighted. I t gives us tre-mendous power , and qual i ta t ive under s tand ing , an ob-vious connection to local , chemical viewpoints such as thecrystal or l igand f ield model . I want to ment ion here onesolid-state chem ist , John B . Goodenough, w h o h as s h o w nover the yea rs, an d esp ecially in his prescie nt book,[31 usthow goo d the chemist s approx im ate const ruc tion o f b andstructures can be.

I n 19 a n d 21, the quali tat ive DOS diagrams fo r [P tH4lZQa n d Ti02, here is , however , much more than a guess at aDOS. T h er e is a chemical character izat ion of the localiza-t ion in r ea l spac e of the s tates (are they on Pt, on H ;on Ti,o n 0), a n d a specif icat ion of the i r bond ing p roper t ies( R - H bonding, antibonding, nonbonding, etc.). The chem-ist sees r ight away, or asks-where in space are th e e lec-t rons? Where are the bonds? There mus t be a way tha tthese inheren t ly chemical , loca l ques t ions can be an-swered, even if the crystal molecular orbitals , the Blochfunctions, delocalize the electrons over the entire crystal .

-12 -

Where Are the Electrons?

Pt d band

t eed to add up to I . I t sho uld be real ized th at the M ull ikenprescr ipt ion for par t i t ioning the over lap densi ty , whileuniquely def ined, is quite arbi trary.

would l ike them to be. Lets take the two-center molecularorbital of Equation (c) , where x i is on center I a n d x2oncenter 2, an d lets assum e centers 1 a n d 2 are not identical ,

-1 7-20 --23 --26 --29 ~-32-35

an d t h a t x n d xz r e normal ized , bu t no t o r thogonal .VCIXI +c2x2 c )

[ e V I - I- lo h

-20-23-26--29

--

~

ILLi 0

- 1 T zy2. iy shou ld be normal ized, so that Equ atio n (d) is val id ,

where S I 2s the over lap integral between xi a n d x 2 .Thisi s how one e lec t ron in ty is dis tr ibuted. Now its obvioust h a t c: of it is to be ass igned to cen ter 1 , c: to cen ter 2.2 c i c 2 S i 2 s clear ly a quanti ty that is associated with inter-act ion. I t s cal led the o ver la p populat ion, a nd we wil l soonr e la te i t to the bond o rder . Bu t what a r e we to d o i f wepersis t in wanting to divide up the electron densi ty be-tween centers 1 a n d 2? We want a l l the par t s to add up to 1a n d c: +c: wont do. We must assign, somehow, the over-lap dens i ty 2 c , c 2 S i 2 to the two centers . Mulliken sug-gested (and thats why we cal l this a Mull iken populat ionanalysis1141)a democra t ic so lu t ion , sp l i t t ing 2 c , c,S12equally between centers 1 a n d 2. Thu s cen ter 1 is assignedc : + c , ~ ~ S ~ ~ ,enter 2 c:+c,c,Si2, and the sum is guaran-

/,/__ I Ti-e,__--____----__---

-35- 3 2 L K - - - - - -os-- DOS--Fig. 7. a) Contributions of TI and 0 (dark area) to the total DOS (solid line)of rutile, TiO,. b) The t Z y nd eETi contributions (dark area); their integration(on a scale of 0 to 100%)is given by the dashed line.

854 Angew Chem. Int. Ed. Engl. 26 11987) 846-878


10/33

What a co mpute r does i s jus t a l it tle m ore involved, fori t sums these con t r ibu t ions for each a tom ic o rb i ta l on agiven cen ter ( there are severa l) , over each occ up ied M O(there may be many) . And in the crystal , i t does i t for sev-eral k poin ts in the Br i llou in zone , and then r e tu rns to r ea lspace by averaging over these.[ 51 he net result is a parti-t ioning of the total DOS into contr ibutions to i t by ei thera t o m s or orbitals . In the so l id- s ta te t r ade these are o f tencalled project ions of the DOS or local DOS. What-ever theyre cal led, they divide u p the DOS am o n g t h eatoms . The in tegra l o f these p ro jec t ions u p to the Fermilevel then gives the total electron densi ty on a given a tomor in a speci f ied o rb i ta l. Then , by r eference to som e s tand-ard dens i ty , a ch arge can b e ass igned .

Figures 6 a n d 7 give the par t i t ioning of the electron den -si ty between Pt an d H i n t h e [PtH4I2 s tack , and betweenTi a n d 0 in rut i le. Everything is as 19 a n d 21 pred ic t , asthe chemist knows i t shou ld be-the lower o rb i ta l s a r e lo-calized in the more electronegative l igands (H or o , h eh igher ones on the meta l .

Do we want mo re speci fic in fo rmat ion? In TiO, wemight wan t to see th e crys ta l f i e ld argume nt upheld . So weask fo r the con t r ibu t ions o f the th ree o rb i ta ls tha t mak e upthe t2g (d, ,, dyz rd,, in a local coord inate sys tem) and th etwo orbitals th at ma ke u p the eg (dzz, d ,z-,2) set . This isa l so shown in Figure 7. Note th e very c lear sep ara t ion o fthe t2g and eg orbitals . The eg set has a smal l amount o fdens i ty in the 0 2 s an d 2 p b an d s 0 onding) and the t2gset in the 0 2 p b a n d (n bonding) . Each metal orbital type( t2gor eg) i s sp read ou t in to a ban d , bu t the memory o f then ea r o c t ah ed r a l local crystal field is very clear.

In [PtH,] we cou ld as k the com puter to g ive us the dZ2cont r ibu t ion to the DOS, or t h e pz part (Fig. 8). If we looka t t h e z c o m p o n e n t o f t h e DOS in [RH4],we see a smal lcon t r ibu t ion in th e to p o f the z2 band . Th is i s eas ies tpicked u p by the integral in Figure 8b. The dotte d l ine is as imple in tegra t ion , l ike an N M R in tegra t ion . I t coun ts, o n

E lev1- 8 1

DOS - DOS -Fig. 8.a) d,? and b ) p, contributions (dark area) to the total DOS (dashedline) of an eclipsed [PtH,IZe stack. The dotted line is an integration I of thep, orbital contribution.

a scale of 0 t o 100% at the top, what percent of the speci-fied orbital is filled at a given energy. At the Fermi level inunoxidized [PtH4IZQ, b o u t 4% of the z states a re f il led.

How does th i s come abo u t? There are two ways to ta lkabou t th i s . Local ly , the donor funct ion o f one monomer(dZ2) ca n interact with th e a cce ptor function (pz) of i tsneighbor (22). T h e o v e r l ap is goo d, but the energy match is

22

poor.] So the interaction is small, but its there. Alterna-t ive ly , one cou ld th ink a bou t in terac t ion o f the Bloch func-t ions, or symmetry-adap ted z a n d z2 crystal orbitals. Atk=O and k =n /a , they dont mix . But a t every in terio rpo in t in the Br i l lou in zone , the symmetry g roup o f ly isi s o m o r p h i c t o C4u,1151nd both z a nd z2 -Bloch functionstransform as a , . So they mix. S ome smal l bond ing i s p ro -vided by this mixing. But i t is really small . W hen the s tackis oxidized, the loss of th i s bond ing (which would leng thent h e R-Pt con tac t ) is overcome by the loss o f Pt-Pt an t i -b o n d i n g t h a t is a conseq uence o f the vacated o rb i ta l s be inga t t h e t o p o f t h e z2 b an d .

We have seen tha t we can loca te the e lec t rons in thecrystal . But .Where Are the Bonds?

Local bonding considerat ions (see 19, 21) tr ivially leadus to ass ign bo nd ing charac teri s t ics to cer ta in o rb i ta l s and ,therefo re , bands . There m us t be a way to f ind these bondsin the bands that a ful ly delocalized calculat ion gives.

I t s possib le to ex tend the idea o f an over lap popu la t iont o a crystal . Recall that in the integrat ion of ty2 fo r a two-cen ter o rb i ta l , 2 c , c 2 S I 2 as a character is t ic of bonding. I fthe over lap in tegra l is t aken a s pos it ive ( an d i t can a lwaysb e a r r an g ed so), then this quanti ty scales as we expect of abond o rder : i t is positive (bonding) if c , a n d c2 a r e o f t h esame sign, and negative if cI a n d c2 are of opposite s ign.And th e mag ni tude o f the Mul l iken ove r lap popu la t ion ,fo r tha t is what 2c ,c ,S , , ( summed over a l l o rb i ta ls o n thetwo a tom s , over all o ccu p i ed MOs) is ca l led , d ep en d s o n c, ,c,, a n d SrJ.

Now we move in to the so l id . An obv ious p roce dure i s totake a l l the s ta tes in a cer tain energy interval and interro-gate them as to the i r bond ing p rocliv it i es , measured by theMul l iken over lap popu la t ion , 2c,cJS,,. 141 hat we are de-f ining is an over lap-populat ion-weighted densi ty of s tates .The beg inn ing o f the obv ious acronym (OP WD OS) unfor-tunate ly has been p reemp ted by ano the r common usage insolid-state physics. For that reason we have cal led thisquan t i ty C O O P (pronounc ed co-op) fo r c rys ta l o rb i ta love r lap populat ion.[I6] Th e suggest ion o f orbitals workingtogether to m ake bo nds in the crys tal i s no t acc iden ta l.

To get a feel ing for this quanti ty , lets think what aC O O P curve fo r a hydrogen chain looks l ike . The s impleb an d s t r u c t u r e an d DOS were given ear l ier ; they are re-p ea t ed w i th t h e C O O P cu r v e in 23.

A n g e w . C h e m . I n [ Ed. Engl. 26 (1987) 846-878 855


11/33

anti--bonding banding-

t -8-lev1 -

-10-

-12-

k- DOS- COOP-23

/

To ca lcu la te a C O O P curve , one has to specify a bond .Lets take the nearest-neighbor 1,2 interaction. T he bottomo f t h e b an d is 1,2 bonding , the midd le nonbonding , the topan t ibond ing . The C O O P curve obv ious ly has the shapeshown at r ight in 23. But not all C O O P cu r v es lo o k t h a tway. I f we specify the 1,3 next-nearest-neighbor b ond (sil lyfor a l inear chain, n ot so si l ly if the chain is kinked) , thenthe bo t tom and t h e t o p o f t h e b an d a r e 1,3 bonding , themidd le an t ibond ing . Tha t curve , the do t ted l ine in thedrawing, is dif ferent in shape. And, of course, i ts magni-tude i s much smal ler , because o f the r ap id decrease o f S,jwith d istance.

Note th e genera l charac ter is t ics o f CO OP curves -pos i-t ive regions which are bonding, negative regions which arean t ibond ing . The ampl i tudes o f these curves depend onthe n umbe r o f s ta tes in tha t energy in terval , the magni tudeof the coupling over lap, and the s ize of the coeff icients int h e MOs.

The in tegra l o f the CO O P curve up to the Fermi level i sthe to ta l over lap popu la t ion o f the speci f ied bond . Th ispo in ts us to ano ther way o f th ink ing o f the DOS a n dCO OP curves . These a re the d i f feren t ia l ver sions o f e lec-t ron num ber an d bo nd o rder ind ices in the crys ta l. T he in -tegral of the DOS to the Fermi level gives the total num berof electrons, the integral of the C O O P curve gives the totalover lap popu la t ion , which is not iden t ica l to the b ond or-

C

aDOS (0 b) Pt-H C O O P

-650 00s-

de r but which scales l ike i t. I t is the closest a theoret iciancan get to that i l l -def ined but fantast ical ly useful s impleconcept of a bond order .

To move to someth ing a l it tle more com plicated than thehydrogen o r po lyene chain , l et s examine the CO OP curvesf o r t h e [RH,] chain. Figure 9 shows both the R - H a n dR-Pt CO O P curves. Th e DOS curve fo r the po lymer isalso drawn. The character izat ion of cer ta in bonds a s bond-ing o r a n t ibond ing i s obv ious , and matches fu l ly the expec-ta t ions o f the a pprox im ate sketch 19. (1) T h e b an d s a t4 a n d 15eV are R - H csbonding , the band a t -6 eVR - H antib ond ing ( this is the crystal-f ield-destabil izeddX2-yZ rbital). (2) It is no surpr ise that the mass of d-blocklevels between - 10 a n d - 3 eV doesn t contr ibute any-th ing to Pt-H bonding. But of course i t is these orbitalswhich are involved in R-Pt bonding . The r a ther complexstructure of the - 10 to - 3 eV region is easily under-s tood by th ink ing of i t as a superposi t ion of CT ( d , ~ Z 2),n( (dxz,dyz) dxz,dyz)) , nd 6 ( d x y- xy) bond ing and an t i -bonding, a s shown in 24. Each type o f bond ing genera tes aband , the bo t tom of which is bonding and the top an t i -bond ing ( see 15 and Fig. 3 ) . ( 3 ) T h e 6 contr ibution to theC OO P is smal l , because o f the poor over lap invo lved . Thelarge R-Pt onding r eg ion a t -7 eV is du e to the bo t tomof the Pt z b an d .

We now have a clear representat ion of the R - H a n dPt-Pt bon ding proper t ies as a function of energy. I f we arepresented with an oxidized mater ial , then the conse-quences o f the ox idat ion o n the bo nd ing are crysta l c learf rom Figure 9. Removing e lec trons from the to p o f the zzb an d a t = - 10 eV takes them from orbitals that areR-Pt an t i b o n d i n g an d R - H nonbonding. So we expectt h e Pt-Pt separat ion, the s tacking distance, to decrease, asit does.

The tuning of electron counts is one of the s trategies ofthe solid-state chemists . Elements can be subst i tuted,atoms intercalated, nonstoichiometr ies enhanced. Oxida-t ion and reduction, in sol id-state chemistry as in ordinarymolecular solut ion chemistry, are about as character is t ic(but exper imental ly not always tr ivial) chemical act ivi t ies

C P - P t COOP

Fig. 9. Total density of states a), andR-H (b) and pt-pt c) crystal orbitaloverlap population curves for theeclipsed [PtH4]20 stack.

0COOP-856 Angew. Chem. l n l . Ed. Engl. 26 (1987) 846-878


12/33

I ,o + - o +

-20 -

-25 -

-30 -,

c-2 COOP - ?r-xz,yz COOP - - o + - o +8 - x y COOP - totalCOOP -as one can conceive. The conclusions we reached for thept-Pt chain were simple, easily anticipated. Other cases areguaranteed to be more complicated. The COOP curves al-low one, at a glance, to reach conclusions about the localeffects on bond length (will bonds be weaker, stronger)upon oxidation or reduction.

We showed earlier a band structure for rutile (Fig. 5b,repeated i n Fig. 10a). The corresponding COOP curve forthe Ti-0 bond (Fig. lob) is extremely simple. Note thebonding in the lower, oxygen bands, and antibonding inthe eg crystal-field-destabilized orbital. The tZp s, as ex-pected, T i- 0 n-antibonding.

Lets try our hand at predicting the DOS for somethingquite different from [PtH,IZQor T i02, namely, a bulk tran-sition metal, the face-centered cubic Ni structure. Eachmetal atom has as its valence orbitals 3d, 4s, and 4p, or-dered in energy approximately as at the left in 25. Eachwill spread out into a band. We can make some judgmentas to the width of the bands from the overlap. The s,p or-bitals are diffuse, their overlap will be large, and a wideband will result. They also mix with each other extensively.The d orbitals are contracted, and so will give rise to a re-latively narrow band.

The computed DOS for bulk Ni (bypassing the actualband structure) is shown in Figure 1 1 , along with the Ni sand p contributions to that DOS. What is not s or p is d.

- 5 Bt -loB-

-3500s -

The general features of 25 are reproduced. At the Fermilevel, a substantial part of the s band is occupied, so thatthe ca l c~ la t ed~~i configuration is d9 ss0.62p0.23.

1 DOS-What would one expect of the C OOP curve for bulk Ni?

As a first approximation we could generate the COOPcurve for each band separately (26a, b). Each band in 25has a lower Ni-Ni bonding part, an upper Ni-Ni anti-bonding part. The composite is 26c. The computed COOPcurve is in Figure 12. The expectations of 26c are met rea-sonably well.A metal-metal COOP curve like that of 26c or Figure 12

is expected for any transition metal. The energy levels

ntibonding - 0 + bondingCOOP - Fig. 10. a) DOS and b) Ti-0 COOP for rutile.

857ngew. Chem. l n t . Ed. Engl. 26 11987) 846-878


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a ) I [ I - b ) I [ I -0 100 0 10010 I I

a b C

5

0

-5

-10

-15 . ' - 1 5 ' . . ' .00s- 00s-Fig. I I . Iota1 I)OS dashed line) and 4 (a) and 4p (h) contrihutionr to it inbulk Ni. The dotted line is an integration of the occupation of a specifiedorbital, o n a scale of 0 to 100%given at top.

might be shif ted up, they might be shif ted down, but theirbon ding charac ter is t ics are l ikely to be the sam e. I f we as-s u m e t h a t a s imi lar band s t ruc tu re and COOP curve holdfor al l metals ( in the sol id-state t rade this would be cal ledthe r ig id band model ) , then Figure 12 ga ins t r emendouspower . I t sum marizes, s imply, the cohesive energies of al lmetals . As one mov es across the transi t ion ser ies , the M-Mover lap popu la t ion (which is clear ly related to the bindingor cohesive energy) wil l increase, peaking at a bou t s ix elec-t rons per meta l (Cr, Mo, W). Then i t wil l decrease towardthe en d of the transi t ion ser ies and r ise again for small s ,pe lect ron coun ts . F or more than I4 electrons, a metal is un-l ike ly ; the ne t ove r lap popu la t ion fo r such h igh coord ina-t ion becomes negative. Molecular al lotropes with lowercoord inat ion are f avored . There is much more to cohes iveenergies and the metal-nonmetal t ransi t ion than this , but

a1lo 1

- o + - o + - 0d COOP- s , p COOP- t o t a l COOP-there is much phys ics and chemis t ry tha t f lows f rom thesimple construction of 2 6 .

With a l i t tle ef for t, we have constructed th e tools-den-si ty of s tates , i ts decomposit ions, the crystal orbital over l appopulat ion-which al low us to move f rom a complicated,completely delocalized set of crystal orbitals or Blochfunctions to the localized, chemical descr ipt ion. There isno mystery in this motion. In fact , what I hope I haveshown here is jus t how much power there is in the chem-ists ' concepts . The construction of the approximate DOSand bonding character is t ics of a [(PtH,)"], polymer , orrutile, or bulk Ni, is really easy.

Of course, there is much more to sol id-state physics thanband structures. The mechanism of conductivi ty , the re-markab le phe nom enon of superconduct iv i ty , the m ul t itudeof e lect ric and magnet ic phenom ena tha t a r e specia l to thesolid s tate, for these one needs the tools and ingenuity ofphysics.[61 But as for bonding in the sol id s tate, I think(some will disag ree) there is nothing new, only a dif ferentlanguage.More Than One Electronic Unit in the Unit Cell:Folding Bands

The oxidized cyanoplat inates are not ecl ipsed (27a) , butstaggered (27b). A polyene is not a s imple l inear chain,28a, but, of course, at least s-trans or zigzag, 28b. O r i tco u l d b e s-cis 28c. And obv ious ly tha t does no t exhaus tthe possibi l i ty of ar rangements . Nature always seems to

27

28 b

~ 1 5 I00s- antibonding - 0 +bondingCOOP-Fig. 12. a) The total DOS and h) nearest-neighbor N t - N i COOP in bulkNi.

f ind one we haven ' t thought of . In 27a a n d 28a, the un i tcel l contains on e basic electronic unit , [F'tH,]2e an d a CHgroup, respectively. In 27b a n d 28b, the unit is d o u b l ed ,approximately so in unit cel l dimension, exactly so in

858 Angew. C hem. Int . Ed Engl 26 1987)846-878


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chemical composit ion. In 28c, we have four C H un i t s perunit cell. A pure ly phys ica l approach migh t say each i s acase un to i tself . A chemist is likely to say that probably no tmuch has changed on d o u b l i n g o r q u ad r u p l i n g o r m u l -t iplying by 17 the c onte nts of a unit cel l. I f the geom etr icaldistor t ions of the basic electronic unit that is being re-peated are not large, i t is l ikely that any electronic charac-teristics of that unit are preserved.

T h e n u m b er o f b an d s i n a b an d s t r u c t u re is eq u a l t o t h en u m b er of molecular orbitals in the unit cell. So i f the unitcel l con ta ins 17 t imes as m any a to ms as the bas ic un i t , i tw i l l con ta in 17 t imes as many bands . The band s t ruc tu remay look messy. T he chemists feel ing that th e 17-mer isa smal l per tu rbat ion on the bas ic e lec t ron ic un i t can beused to s implify a complex calculat ion. Lets see how thisgoes, f i rs t for the polyene chain, then for the [ (PtH4JZQ],polymer .

28a, b , a n d c di f fer f rom each o the r no t jus t in the num -ber of C H enti t ies in the unit cell , but also in their geome-try. Lets take these one at a t ime. F i r s t p repare fo r thedistor t ion f rom 28a to 28b by doubl in g the un i t ce ll , andthen, subseque ntly, dis tor t ing. This sequence of act io ns isindicated in 29.

0. . . . . ..-20-: :

29 b

C

but they obv ious ly have the same nodal s t ruc tu re-onenod e every two cen ter s .

I f we now detach our se lves f rom th is v iewpoin t and goback an d cons t ruc t the o rb i tal s o f the one C H per un i t ce lll inear chain 29a, we get 32. The Bri l louin zone in 29b ishalf as long as i t is here, because the unit cell is twice aslong.

32

-aAt this poin t , th e real izat ion hi ts us that , of course, the

orb i ta l s o f these po lymers are the same. Th e po lymers areiden t ica l , i t i s on ly some pecu l iar qu i rk tha t made usch o o s e o n e C H u n i t a s t h e u n i t c el l in o n e ca s e, t w o C Hunits in the other . I have presented the two constructionsindepen den t ly to m ake exp l ic it the iden t ity o f the o rb i -tals.

Wh at we have i s two ways o f p resen t ing the sa me orb i -tals . Band structure 31, with two bands, is identical to 32,with one band . All tha t has happened i s tha t the band ofthe min imal po lymer , one C H per un i t ce ll , has beenfolded back in 32. The process is shown in 33.

Suppo se we cons t ruc t the o rb i ta l s o f 29b, t h e d o u b l e dunit cel l polymer , by the s tandard prescr ipt ion: ( I ) getMOs in unit cell, (2) fo rm Bloch funct ions f rom them.With in the un i t ce l l the MOs o f t h e d i m er a r e n a n d n*. 30.

- 7CZHB -lT

30

Each of these sp reads ou t in to a band , tha t o f the n run-ning up, that of n* runn ing down, 31. The orb i ta l s a r ewr i t ten ou t exp l ic i t ly a t the zone boundar ies in 31. T h i s

31

al lows one to se e tha t the top o f the n b an d a n d t h e b o t t omof the x* b an d , b o t h a t k =n /2a , are precisely degenerate.There i s no bo nd a l te rna t ion in thi s po lyene (ye t) , and th etwo orbitals may have been constructed in a different way,

n = 2033

Th e process ca n be continued. I f the unit cell is tr ipled,the band wi l l fo ld as in Ma. I f i t is quadrupled, we get3 4 b , a n d so on. However , the point of al l this is not just

a = 30 a = 40b34

r edunda ncy , see ing the sam e th ing in d i f f eren t ways . Thereare tw o im por tan t consequences o r u t i l iza t ions of this fold-ing. First, if a unit cell con ta ins more than one e lec t ron icun i t ( and th i s happens o f ten) , then a r ea l iza t ion o f tha tf ac t , and the a t tenda n t mul t ip l ica t ion o f bands ( r emember

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32- 31, 34a, r 34b , l lows a chemist to s implify in his o rher mind the analysis . The mult ipl icity of bands is a conse-que nce of an enlargemen t of the unit cel l . By reversing, inour minds in a model calculat ion, the folding process, byunfolding, we can go back to the most fundam enta l e lec-tronic act- the true monom er .

-1 5- 0 k- R/0' 0 k - - c n/aFig. 13. The band structure o f a staggered [ P t H 4 1 - O stack (a), compared withthe folded-back band s tructure of an eclipsed stack, two [PtH,]'' i n a unitcell (b).

To i l lus t ra te th is po in t , l e t me show the band s t ruc tu re o ft h e staggered [PtH4]*0 chain, 27b. This is don e in Figure13a. The re are twice as man y ban ds in th i s r eg ion as thereare in the case o f the ec l ipsed monomer ( the xy band isdou bly degenerate). This is no surpr i se ; the un i t ce l l in thestaggered polymer is [(PtH4)ZQ]2. ut it 's possible to under-stan d Figure 13 as a small per turbation on the ecl ipsedpolymer. Imagine the thought process 35a + 35b- 3512, .e.,doub l ing the un i t cell in an ec l ipsed po lymer and then ro-tat ing every other unit by 45 a r o u n d t h e z axis. To go

f rom 35a to 35b is tr ivial, a simp le folding back. T he resultis shown in Figure 13b. Figures 13a an d 13b are near lyidentical . There is a small dif ference in the xy band, whichis doubled, nond egenerate, in the folded-back ecl ipsed po-

lymer (Fig. 13b), but dege nerate in the s taggered polymer .What happened here could be s tated in two ways, both theconsequence of the fact that a real rotat ion intervenes be-tween 35b a n d 35c. From a group- theoret ical point ofview, the s taggered polymer has a new, higher symmetryelement, an eightfold rotat ion-ref lect ion axis . Higher sym-metry means more degeneracies. I t is easy to see that thetwo combinat ions , 36, re degenerate.

Except fo r this mino r wrinkle, th e ban d structures of thefo lded-back ec l ipsed po lymer and the s taggered one arevery, very similar. That allows us to reverse the argument,to understand the s taggered one in terms of the ecl ipsedone plus the here minor per turbation of rotat ion of everysecond un i t.

Th e chemist 's intui t ion is that the ecl ipsed an d staggeredpolymers can't be very different. At least until the ligandsstar t bumping into each other , and for such ster ic ef fectsthere is , in turn, much fur the r intui t ion. The b and struc-tures may look dif ferent , for one polymer has one, theother two basic electronic units in the cel l . Chemically ,however , they should be s imilar , and we can see this byreturning f rom reciprocal space to real space. Figure 14,co m p ar i n g t h e DOS of the s taggered (Fig. 14a) andeclipsed (Fig. 14b) polymers, shows just ho w al ike they a rein their distribution of levels in energy.

a) staggered b) eclipsed

LeVl bDOS- DOS-

Fig. 14. A comparison of the 110s of staggered d ) m d cl~psedb) 1PtH,jZQstacks.

There i s ano ther r eason fo r f ee l ing a t home wi th thefolding process. The folding-back construction may be aprerequisi te to understanding a chemically s ignif icant dis-tor t ion of the polymer . To i l lustrate this point , we return tothe po lyene 29.To go f rom 29a to 29b involves no distor-t ion. However , 29b s a way point , a preparat ion for a realdistor t ion to the more real is t ic "kinked" chain, 29c. t be-hooves us to analyze th e process s tepwise if we ar e to un-derstand the levels of 29c.

Of cour se , noth ing much happen s to the n sys tem o f thepo lymer on go ing f rom 29a,b o 29c. If he nearest-neigh-bor distances a re kept con stant , then the f irs t real cha nge is

860 A n g e w . Chem. In . Ed. Engl. 26 11987) 846-878


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in the 1,3 interact ions. These are unlikely to be large in apolyene, s ince the x ove r lap fal ls off very quickly past th ebon ding region. We can est imate what wil l happ en by writ-i ng d o w n s o m e ex p l i ci t p o i n ts i n t h e b an d , an d d ec i d i n gwhether the 1,3 interact ion that is t u r n ed on is s tabil izingor destabilizing. This is do ne in 37. Of course, in a real CH

stab i l ized

37 destab i l ized

stab i l ized

polymer this kinking distor t ion is very much a real thing,b u t t h a t h a s n o t h i n g t o d o w i t h t h e n system, its a re sult ofstrain.

However , there is anoth er distor t ion which the polyenecan and does undergo . Th is i s doub le-bond local iza t ion ,an ex amp le o f the very impo r tan t Peier ls d i s to rt ion , thesolid-state ana logu e of the Jahn-Teller ef fect.Making Bonds in a Crystal

When a chemist sees a molecu lar s t ruc tu re which con-tains several f ree radicals , orbi tals with unpair ed electrons,his incl inat ion is to predict that such a struc ture wil l un-d e r g o a geome try chan ge in which electrons wil l pair up,forming bonds. It is this reasoning, so obvious as to seemalmost subconscious, which is behind the chemists intui-t ion tha t a chain o f hydrogen a toms wi l l co l lapse in to ach a i n of hydrogen molecules.

I f we translate tha t intui t ion into a molecular orbital pic-tu re , we have 38a, a bunch (here s ix) of radicals formingbonds . That p rocess of bond fo rmat ion fo l lows the H zparadigm, 38b, i.e., in the p rocess o f making each bond alevel goes down , a l evel goes up , an d two e lec t rons are s ta-b i li zed by o ccupying the lower , bond ing o rb i ta l .

In so l id -s ta te phys ics , bond fo rmat ion has no t s too d a tcenter s tage, as it has in chem istry. The reasons for this areobvious: the most interest ing developments in sol id-statephys ics have been a round meta l s and a l loys, and in these

of ten close-packed or near ly close-packed substances, byand large localized chemical viewpoints have seemed ir rel-evant . For ano ther large gr oup of mater ials , ionic sol ids, ita l so seemed use less to th ink of bonds. My contention istha t there is a range of bonding, including what ar e usuallycalled metal l ic, covalent , and ionic sol ids, and th at ther e is,in fact , substantial o ver la p between seemingly divergentf r ameworks of descr ib ing the bo nd ing in these th ree typesof crystals . I wil l take the view that the covalent approachis cen t r a l and look for bond s when o th er s wouldn t th inkthey re there . On e r eason for tolerat ing such foolhardinessmigh t be tha t the o ther a pproach es (meta l li c , ion ic) havehad the i r day-why no t give th i s one a chance ? A s eco n dreason, one I ve mentioned ear l ier , is tha t , in th ink ing andta lk ing a bou t bo nds in the crys tal , one makes a psycho log-ical ly valuable connection to molecular chemistry.

To r e tu rn to o ur d i scuss ion o f m olecu lar an d so l id - s ta tebond formation, lets pursue the tr ivial chemical perspec-tive of the beg inn ing of this sect ion. The g uidin g pr inciple,implici t in 38, is: Maximize bonding. There may b e imped i -ments to bonding: electron repulsions, s ter ic ef fects , i .e . ,the impossib il i ty o f two r ad ica l s to r each wi thin bond ingd is tance o f e ach o ther . Obvious ly , the s tab le s ta te i s a com-promise-some bonding may have to be weakened tostrengthen some other bonding. But, in general , a systemwill distort so as t o m ak e b o n d s o u t of radical s i tes . O r totranslate this into the language of densi t ies of s tates: max-imizing bonding in the solid state is connected to loweringthe D O S at the Ferrni level, moving bonding states to lowerenergy, antibonding ones to high energy.

The Peierls Distortion

In considerat ions of the sol id s tate, a natural s tar t ingpoint is high symmetry-a l inear chain, a cub ic or close-pac ked three-dimensional lat t ice. The orbitals of the highlysymmetr ical , ideal ized structures are easy to obtain, buttbey o f ten d o no t co r r espond to s i tua tions of maximumbonding . T hese are less symmetrica l, deform at ions o f thesimplest , archetype structure.

Th e chemists expe r ience is usually the reverse, begin-ning f rom localized structures. However , there is on e pieceof exper ience we have tha t m atches the way of th ink ing o fthe solid-state physicis t . This is the Jahn-Te ller effect,91an d i ts worthwhile t o show i ts working by a s imple exam -ple.

The Huckel R M O s o f a square-p lan ar cyclobu tad ieneare well known. They a re the one-below-two-below -one sets h o w n i n 39. aq 4

We have a typical Jahn-Teller s i tuat ion- two electronsin two degenerate orbitals . (Of course, we need worry

A n g e n . Chem. I n ( . Ed. Engl. 26 (1987) 846-878 86 1


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about the var ious s tates that ar ise f rom this occupation,an d the Jahn-Teller theorem real ly applies to only one.91)The Jahn-Teller theorem says that such a s i tuat ion necessi-tates a large interact ion of vibrat ional and electronic mo-t ion . I t s ta tes tha t there m us t be a t l eas t one normal modeof vibration which will break the deg eneracy and lower theenergy of the system (and , of course, lower i ts symmetry) .I t even specif ies which vibrat ions would accom plish this .

In the case a t hand the mos t e f f ec t ive normal mode isillustrated in 40. I t lowers the symmetry f rom D4,,o DZhrand, to use chemical language, localizes double bonds.

4 0The orbital workings of this Jahn-Teller dis tor t ion are

easy to see. 41 i l lustrates how th e degeneracy of the e orbi-tal is broken in th e two phases o f the v ib rat ion . O n form-ing the rectangle as at r ight in 40 or 41, ryz is s tabil ized:

41

t h e 1-2 a n d 3-4 in terac t ions, which were bo nd ing in thesquare , a r e increased ; the 1-4 an d 2-3 interactions, whichwere an t ibond ing , a r e decreased by the deformat ion . Th ereverse is tru e for ry3-it is destabilized by the distortion a tright. If we follow the op pos i te phase o f the v ib ra tion ( tothe lef t in 40 or 41), ry is stabilized, ry2 destabil ized.

The essence o f the Jahn-Teller theorem is revealed here:a symmetry- lowering deformation breaks a n orbital degen-eracy, s tabil izing one orbital , destabil izing another . Notethe phenomenolog ica l co r r espondence to 38 in the pre-vious sect ion.

On e doesnt need a real degeneracy to benef i t f rom thiseffect. C ons ider a nond egenerate two-level system, 42,

42- A - C- C

with the t wo levels of dif ferent symmetry (here labeled Aan d B) in one geometry. I f a vibrat ion lowers the symmetryso that these two levels t ransform as the sam e ir reduciblerepresentat ion (cal l i t C), then they will interact, mix, repeleach other . For two electrons, the system will b e s tabil ized.The technical name of this ef fect is a second-order Jahn-T e ll er d e f ~ r m a t i o n . ~ ]

The essence of the Jahn-Teller ef fect , f i rs t or second or-der , is : a high-symmetry geometry generates a degeneracyor near degeneracy, which can be broken, with s tabil iza-t ion, by a symmetry- lowering deformation. Note a fur therpoint: the level ,degeneracy is not enough by i tself -oneneeds the r ight electron count . The cyclobutadiene (or anysquare) s i tuat ion of 39 will be stabilized by a DzI , e f o r m a-t ion for three, four , or f ive electrons, but no t for two o r s ix(e.g., s:@).

This f r amework we ca n take over to the so lid . There isdegeneracy and near degeneracy for any par t ial ly f i l ledband . The degeneracy i s tha t a l r eady ment ioned , fo rE(k) =E ( - ) fo r any k i n the zone . The n ear degeneracyis, of course, for ks jus t above o r just be low the speci f iedFermi level . For an y such par t ial ly f i lled ba nd there is , inpr inciple, ava ilable a deformation which will lower the en-ergy o f the sys tem. In the ja rgon o f the t r ade one says tha tthe par t ial f i l ling leads to an electron-phonon couplingwhich ope ns up a ga p jus t a t the Fermi level . Th is i s thePeier ls dis tor t ion,f201 he solid-state c oun terpa r t of t h eJahn-Teller effect.

Lets see how this works on a chain of hydrogen atomsor a polyene) . The or iginal chain has one orbital per unitcell, 43a, and an associa ted s imple band . We prepare i t f o r

/ kEFa W

deformation by doub ling the unit cell , 43b. The band i stypical ly folded. T he Fermi level is hal fway u p the band-the band has room for two e lec trons per o rb i ta l, bu t fo r Ho r CH we have one electron per orbital .

44The phonon or la t t ice v ib ra t ion mode th a t coup les mos t

effect ively with the electronic motions is the symmetr icpair ing vibrat ion, 44. ets exam ine wh at i t doe s to typicalorbitals at the bottom, m iddle (Fermi level), and to p of theb an d , 45.

Y Ik- r112ni862 Angew. Chem. In t . Ed. Engl . 26 1987) 846-878


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At the bo t tom and top o f the band no th ing happens .What is gained (lost) in increased 1-2, 3-4, 5-6, etc. bond-ing (antibonding) is lost (gained) in decreased 2-3, 4-5, 6 -7, etc . bond ing ( an t ibond ing). Bu t in the mid d le o f theband , a t the Fermi level, the ef fec t s a r e d ramat ic . One o fthe degenerate levels there is s tabil ized by the distor t ion,the o the r destab i lized . N ote the phenomenolog ica l s imi lar-i ty to what happ ened fo r cyclobu tad iene .

The ac t ion does no t t ake p lace jus t a t the Fermi level ,b u t i n a second-order way the s tab i l iza t ion penet r a tesinto the zone. I t doe s fal l off with k, a co n s eq u en ce o f t h eway per turbation theory works. A schematic representa-t ion o f what happe ns is shown in 46 ( I a n d I 1 r epresen t the

I = - - - = tt

11

chain before an d af ter the distor t ion) . A net s tabil izat ion ofthe sys tem occu r s fo r any Fermi level, bu t obv ious ly i t i smaximal fo r the ha l f- f il l ed b and , a nd i t i s a t the E~ t h a t t h eb a n d g a p is open ed up . I f we were to summ ar ize what hap-pens in block f orm , wed get 47. Note the r esemblance to38.

i:The polyene case ( today i t would be cal led polyacety-

lene) is especial ly interest ing, for some years ago i t occa-s i o n ed a great dea l o f d i scuss ion . Would an in f in i te po-lyene localize 48)? Eventually , Salem a n d Longuet-Hig-m -8gins dem ons t r a ted tha t i t would . Po lyacety lenes are anexcit ing f ield of mo dern Pure polyacetylene isno t a conducto r . When i t is doped, ei ther par t ial ly f i l l ingt h e u p p e r b an d i n 45 or emptying the lower , i t becomes as u p e r b co n d u c t o r .

There a re m any beau t i fu l in tr icac ies o f the f ir s t- and sec-ond-order and low- or high-spin Peier ls dis tor t ion, and f orthese the re ade r is referred to th e very accessible review by

The Peier ls dis tor t ion plays a crucia l ro le in de term in ingthe s t ruc tu re o f so l ids in genera l ; the one-d imens ionalpair ing distor t ion is only one s imple exam ple o f i t s work-ings. Lets move up in dimensionali ty .

Whangbo.5

On e ubiquitous ternary structure is that of PbFC l (ZrSiS,BiO CI, Co,Sb, FezAs).123.241 ell call i t MAB here, be-cau s e in the phases o f in teres t to us the first element iso f ten a t r ans it ion meta l, the o the r compo nen ts , A , 9, of tenmain-g roup e lements . 49 show s on e view of this s tructure,50 an o t h e r .

949 50

In this s truc ture we see two associated squa re nets of Mand B a toms , separa ted by a square-net l ayer of As. The Alayer is twice as d en s e as the o ther s , hence the MAB s toi -chiometry. Most interest ing, f rom a Zintl viewpoint, is aco n s eq u en ce o f t h a t A layer densi ty , a shor t A-A contact ,typically 2.5 A f o r Si. This is definitely i n the range ofsome bonding . There are no shor t 9 - B contacts .

S o m e c o m p o u n d s in this ser ies in fact retain this s truc-ture. Oth ers distor t . I t is easy to see why. Take GdP S. I f weass ign normal ox idat ion s ta tes o f Gd 3@ nd S O , we cometo a fo rmal charge o f P on the dense-packed P ne t. Froma Zintl viewpoint , Po is like S a n d so shou ld fo rm twobonds per P. This is exactly what i t does. The GdPS struc-

U51

A n g e w . C h e m . In , . Ed. Engl . 26 11987) 846-878 863


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t ~ r e [ * ~ ]s s h o w n i n 51, which is drawn af ter the beautifulrepresentat ion of HuNiger et al.'"] No te the P-P cis chainsin this elegant s tructure.

From the point of view of a band structure calculat ionone m igh t a l so expect bond fo rmat ion , a d i s tor t ion of thesquare ne t . 52 show s a quali tat ive DOS diagram for G d PS .

tE IId dh P Ps 3 P

52

What goes in to the cons t ruc t ion of this diagram is a j u d g -ment as to th e e lec tronegat iv i ti es (G d < P


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60

ing some s ta tes dow n f rom th e Fermi level r egion . But be-cause o f the th ree-d imens ional l inkage i t may no t be possi-ble to remove al l the s tates f rom the Fermi level region.S o m e DOS r emains th ere ; the mater ia l may s t il l be a con-ductor .

The app l ica t ions d iscussed in th i s sec t ion mak e i t c leartha t one mus t know, a t l eas t approx imate ly , the bands t r u c t u re (an d t h e co n s eq u en t DOS) of two- and th ree-d imens ional mater ia l s before one can m ake sense o f the i rmarvelous geometr ical r ichness. The band structures thatwe have discussed in detai l have been one-dimensional .Now le t s look more carefu l ly a t what happens as we in -crease dimensionali ty .

More DimensionsMost m ater ia l s a r e two- o r three-d imens ional , an d whi le

o n e d i m en s i o n i s fun , we must eventually leave i t forh igher d imens ionali ty . Noth in g much new happ ens , excep ttha t we mus t tr ea t as a vec tor , w i th com ponen ts in r ec i-p rocal space , and th e Br i l lou in zone is n o w a two- or three-d imens ional a r ea or volume.[6. ]To introduce some of these ideas, lets begin with asqu are lat t ice, 59, def in ed by the translat ion vectors 5 n d

a,. Su p p o s e t h e re i s an H 1s orbital on eac h lat t ice s ite. I ttu rns ou t tha t the S chrod ing er equat ion in the crys ta l f ac-to r s in to separa te wave equat ions a long the x and y axes ,each o f them iden t ica l to the one-d imens ional eq uat ion fo ra l inear chain. There is a k, a nd a k,, the rang e of eac h is0 kJ, Ik,l5 n/a (a= G,]= / ) . Som e typica l so lu t ions ares h o w n i n 60.

The construction of these is obvious . What th e cons t ruc-t ion a lso shows, very clear ly, is the vector natu re of k . Con-s i d e r t h e (k x, k Y ) = ( n / 2 a , d 2 n ) a n d ( r r / a , n / a ) solutions. Alook a t them reveals tha t they are waves runn ing a long adirect ion which is th e vector sum of k, a nd k,, i.e., on adiagonal . The waveleng th i s inver se ly p ropor t ional to themagnitude of that vector .T h e s p ace o f k h e r e is def ined by two vectors 6 a n d 6;,and th e r ange o f a l lowed k , the Br i llou in zone , is a square.

k , = r / a , k y = O

X

rk,=O, k,=O

k k, = r / ( Z a )

k , , k y = r / a

M

k, = 0, k y = r / ( Z a l

k , = O , k y = r / a

X

Ce rtain special values of k are given nam es: r= O,O)s thezone cen ter , X = d a , )= 0, d a ) , M = d a , a ) . T h es ea r e s h o w n i n 61, and the speci f ic so lu t ions fo r r, X, Mwere so labeled in 60

It is difficult to show the energy levels, E ( l ) or all k: Sowhat one typical ly does is to i l lustrate the evolution of Ealong cer tain l ines in the Br i l louin zone. Some obviouso n es a r e T -+X , r + M , X - M . From 60 it is clear that M isthe h ighes t energy wave funct ion , and tha t X is prettymuch nonbonding , s ince i t has as many bonding in terac-t ions ( a long y ) as i t does a n t ibond ing o nes ( a long x). So wewould expect the band s t ruc tu re to look l ike 62. A co m -p u t ed b an d s t r u c t u re an d DOS f o r a hydrogen lat t ice witha=2.0 A (Fig. 15) conf irms our expectat ions.

62 1

r x M rk -

The chemis t would expect the chessboard o f H a t o m s t od is to rt in to on e o f H 2 molecules (an interest ing problem ishow many d i f f eren t ways there are to accompl i sh th i s ) .The la rge peak in the DOS for the half - f i l led H s q u a r e -lat t ice band would make the physicis t think of a lat t ice vi-b ra t ion tha t would crea te a g a p a t E ~ . ny pairwise defor-mat ion wil l do tha t.

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r 00s-20-~ X MFig. 15. a) The band.structure and b) DOS of a square lattice of H atoms;H-H separation 2.0 A.

Let's now pu t som e p orbitals on the squ are lat t ice, withthe d i r ec t ion perpend ic u lar to the la t ti ce taken as z . The p zorb i ta l s w il l be sep ara ted f rom pr and px by the i r symme-try. Ref lect ion in the plane of the lat t ice remains a g o o dsymmetry opera t ion a t all k. The pz(z) orbitals will give aband s t ruc tu re simi lar to tha t o f the s orbitals , for the top o-logy of the interact ion of these orbitals is s imilar . This iswhy in the one-d imens ional case we cou ld ta lk a t on e andthe sam e t ime abou t chains o f H atoms an d po lyenes .

The p x and p r o rb i ta l s p resen t a somewhat d i f feren tp rob lem. Show n in 63 r e the symmetry-adap ted com bina-t ions of each a t r, X, Y, a n d M. (Y is by symm etry equiva-

ff,7r*

r

X

63

Y

M ff 7r

1

lent to X ; the dif ference is just in the propagation alon g xor y.) Ea ch crystal orbital can be c haracter ized by the p, pCJ or n bonding p resen t . Thus a t the x and y combina-t ions are CJ an t i b o n d i n g an d n bonding ; a t X they are oa n d n bonding (one o f them), and and n an t i b o n d i n g ( t h eother). At M they are bo th CJ bonding, n antibonding. I t isalso clear that the x,y combinations are degenerate at ra n d M ( and i t tu rns ou t a long the l ine r-+M, but fo r tha to n e n eed s a l it tle gro up and nondegene rate at Xa n d Y (an d everywhere else in the Br i l louin zone) .Pu t t ing in the es t imate tha t o onding is more importantt h an n bonding , one can o rder these specia l symmetrypoints of the B r i l louin zone in energy, an d draw a quali ta-t ive ban d structure (Fig. 16). T h e ac t u a l ap p ea r an ce o f an y

p+r x k x

r X M rk-

Fig. 16. Schematic band structure of a planar squar e lattice of atoms bearingns and rip orbitals. The s and p levels have a large enough separation that thes and p bands do not overlap.

real b and structure (e.g . , the P net in Gd PS discussed in thelast sect ion) wil l depend on the lat t ice spacing. Band dis-persions wil l increase with shor t contacts , and complica-t ions due to s ,p mixing wil l ar ise. Roughly, however , anysqu are lat t ice, be i t the P net in GdPS, '241 a squa re over-layer of S a t o m s ad s o r b ed on Ni(100),[28"1he oxygen andlead nets in litharge,'28b1 Si laye r in BaPdS ij,[Zxclwill havethese orbitals .

Three dim ensions real ly in trodu ce li tt le new, except forthe complex i t ies o f d rawing and the wonder s o f g rou p the-ory in the 230 s p ace groups. T h e s, p , and d bands o f acubic lat t ice, or of face-centered or body-centered close-packed structures, are par t icular ly easy to construct .

Let's look a t a three-dimensional case of some complex-i ty, the NiAs- M

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