Transition Metal Oxides: Geometric and Electronic Structures Introducing Solid State Topics in lnorganic Chemistry Courses Adela Muiioz-Paez Departamento de Quimica Inorganica, Facultad de Quimica, Universidad de Sevilla, Apdo, 553,41012 Sevilla, Spain The Study of lnorganic Solids Although transition metal oxides have been known for a long time, in the last decade there has been a n increasing interest in them due to their important role in heterogeneous catalysis (1)and more recently in material science, where high temperature superconductors have attracted much attention among research groups from all over the world (2).When studying any group of compounds the first questions asked concern the geometric and electronic structures because these determine their physical properties and reactivity. For transition metal oxides much information is available on geometric structure. Diversity in Structure Upon examining this information, a wide variety of geometric arrangements are found, including three-dimensional layered chain molecular This variety produces a wide range in the values of certain physical properties, such as melting points from 2800 'C for HfOz to 6 'C for MnzO7 color, including the entire spectrum electrical pmperties, fmm insulators (CrO) to metallic conductors (TiO) Nevertheless, most of the oxides fall in the first group of structures: the infinite three-dimensional. Thus, they are usually high melting point solids. The last three classes form a negligible fraction of all oxides. Information about the electronic structure of is scarce and can lead to different and often contradictory conclusions, depending on the source. The reason for this lack of information is the special difficulty encoutered in studying the bonds in these compounds. Although most are solids a t room temperature, they do not fit the ionic model well, nor can they be included among the compounds that form infinite covalent networks. Moreover, some properties of these compounds show values close to those of metals, for example, the electrical conductivities of Ti0 and VO. Increased Interest and Use My main purpose for this presentation is the growing value of studying the solid state in inorganic chemistry. The importance of solid state materials in everyday life is reflected in the increasing use of such compounds in diode lasers, semiconductors, or liquid crystals. The preparation and characterization of many of these compounds has been carried out by chemists. Consequently, the ACS Division of Inorganic Chemistry has developed an integrated set ofinstructional materials in solid state chemistry. Additionally, segments of this topic have begun appearing in inorganic chemistry textbooks (351. In the same
trend, solid state chemistry is the subject of several papers that have recently appeared in this Journal (6-8). Even a movement to introduce topics in solid state chemistry in the laboratories has been reported (9). Pedagogical Value of Focusing on These Metal Oxides Transition metal oxides have been selected to achieve this goal for several reasons.
'Agood organizingprinciple is involved: They are binaryeombinations of oxygen and elements fmm the transition series. eonvenientlvillustrates the varietv T h i s emu0 of com~ounds and &m&xity df crystalline struckres possible for inorganic solids. 'These compounds provide a good way of introducing bond theories in solid state chemistry. Unfortunately, chemistry curricula are oRen restricted to the study of bonding in molecular compounds. A serious approach to the study of the electronic structure of inorganic solids thus challenges the student withuew theories. I have divided this paper into two parts. First I deal with geometric structures and offer a summarized description of the crystal structures that appear most otten in these compounds. Because this is intended as only an introduction to the studv of transition metal oxides., onlv " bulk structures of stoichiometric compounds will be examined, thus excluding surface and defective structures. which are by themselvesanother wide topic. The main s o k e s of information for this part are the textbooks from Wells (101, Kung (I), Gutierrez Rios (Il), and Shriver et al. (3). The second half of this uauer is devoted to the studv of the electronic structure ofthi oxides and includes a reAew of bond theories currently amlied in the field. This section is thus more subjective andopen to discussion. The books used to prepare it are by Phillips and Williams (121, Cox (131, Duffy (141, Ziman (151, and West (16). A similar version of this presentation (except for the part dealing with the Hubbard model) has been used with undergraduate students who had previously completed coursework on bonding and crystallography. As a complementary work in the laboratory, the preparation of Y-BaCu superconductors is proposed as well as the study of the Meissner effect. This complex oxide has been chosen because it is one of the most famous compounds involving transition metal oxides. Also, its preparation involves solid state techniques easily available in inorganic chemistry laboratories (171, and the degree of success is easily checked using the Meisnner effect. Determinants of Geometric Structure An examination of the known structures of binary oxides. summarized in Table 1 (1). . .. shows that transition metal oxides exist in many different crystal forms. The classification used in Table 1was chosen to show schematically all the types of structures that appear in these oxides. The oxides are arranged horizontally, as in the periVolume 71 Number 5 May 1994
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Table 1. Crystal Structures of Some Common Transition Metal Oxides
Monoxides
Ti0
Sesquioxides
NaCl Ti203 cor
SC203 c
Dioxides
Ti02
rut a,b
Pentoxides Spinels
VO NaCl v203
Crz03
Vz05
I
SP
COT
Cr02 rut
Cra or
Monoxides
ZrO NaCl
Mn203
c MnOz rut, mt Mn304
wr VOz rut, mono
Trioxides Heptoxides
Sesquioxides
MnO NaCl
FeO NaCl Fez03 cor, sp
COO NaCl
Fe304
Co304 cor, sp
SP
NbO
Yz03
ZrOz
NbOz
f, mono, let rut
Pentoxides
cris CuO RS
W
PdO RS
AgzO cris
CdO NaCl
ZnO
MnnO7 mol
c Dioxides
NiO NaCl
Moon rut
Tc02 rut
Moos I, or
or
Ru02 mt
Rhz03 cor RhOn rut
Nbz05
or Trioxides Heptoxides Tetroxides
TC207 Ru04
mol Monoxides Sesquioxides Dioxides
La203
HgO or
mt
HfOz
Ta02
f, mono, tet rut
Pentoxides Spinels
W02
rut
Taz05 or
Trioxides
Re02 OsO2 mono, rut rut
Ira ~t
ROZ rut %304
cub
Re03 Ephases cub Heptoxides Tetroxides Re& 0504 or mol cris = cristobaiite.NaCl = rock sat, W= wurtzile. C = C-M203.C O =~ corundum. SP =spinel.rui = rutile, a = anatase,b = bmoldte, mono = monoclinic,mt = multiple. I = layer, or = ortharhombic, mol = molecular, f = fluofie, tet = tetragonal, cub = cubic W03
.
odic table, by increasing atomic number of the transition metal. Within a group the oxides are arranged vertically by increasing oxidation state of the metal as you go down the column.
tensive series of oxides, such as Ti,Oz. _ lor Monos,_ with structures related to simple oxides, MOzor M03.
Oxidation States
The organization of Table 1shows the diversity of structures. which a t first does not seem to follow anv svstematic trend. I t also highlights the polymorphism, w&cdis particularly frequent in dioxides. Althoueh - each eeometric structure is stable for a given temperature range, the coexistence of several crystalline ~ h a s e sis still ~ossiblefor a given compound a; mom temperature: The Bctivation energy is hiah for the transformation from the less thermodvnLmically stable structure to the more stable one. For transition metal oxides only one generalization can be made: The ionic radius of a transition metal is usually smaller than that of 0%.Thus, the oxide structures usually show cubic close packing of anions, with the smaller metal cations situated in the octahedral and tetrahedral holes of the oxide network.
In the first transition metal series, oxidation states range from one to seven, in the second from one to eight, d two to eight. The only oxides formed and in the t ~ r from by group 3 metals (Sc, Y, and La) are the sesquioxides Mz03,whereas for gmup 12 metals (Zn, Cd, and Hg) only the monoxides are stable under normal conditions. In the intermediate groups, higher oxidation states are stable. Consequently, the number of oxides from each element increases, with the maximum occurring in group 7 (Mn, Tc, and Re). Simple a n d Complex Formulas This table includes oxides with simple formulas, such as Mz0, MO, MpOs, in which the metal occurs in only one oxidation state. I t also shows oxides of the formula M30a, which contains both M%and M3+.There are numerous examples of more complex stoichiometric oxides that also contain the metal in two oxidation states, for example, MnsOs, Vs013, and Cr5012.Moreover, some metals form ex382
Journal of Chemical Education
Polymorphism and Packing
-
Typical Three-Dimensional Structures A detailed description of structures grouped according to the different atoichiometries follows. Most of the structures included in Table 1are common in other types of sol-
Table 3. Crystalllne Structures for M02 (Rutile Type)
Table 2. Crystalllne Structures for MO (Rock Salt Type)
Transition4 Series
5
first
Ti
V
second
Zr
NbC
6
7
8
9
10
11
12
Mn
Fe
Co
Ni
cub
Zna
pdb
third
Cd HgC
'Assumes wultzie structure. bAssumesPIS structure. 'Assumes other structures. ids (e.g., rock salt, rutile, and flourite), so no detailed description of them will be given here.
&
Cr Mo W
V^ Nb
Ta
Mn' Tc Re'
Ru 0s
Rh
lr
Ft
'Polvmomhic '~doptsfiuo"tetype rather than rutile-typestructure.
Fluorite and Rutile Most transition metal dioxides crystallize with one of the two simple structures-flourite and rutile--shown in Figure 2.The larger MQcations Zr and Hf are octaeoordinated i n the flourite structure, and the smaller are hexacoordinated in the rutile strudwe.
Monoxides MO
Monoxides are known for all elements in the first transition metal series except scandium and chromium. Monoxides are also formed by some elements of the second series (Zr, Nb,Pd, and Cd), as well as Hg in the third. Their crystal structures are summarized in Table 2 and drawn in Figure 1.
FLUORITE
RUTILE
Figure 2. C!ystal structures of fluorite and rutile (a metal cations, 0 oxide anions).
Figure 1. Crystal structures of rock san. FtS, and wurtziie (0 metal cations, 0 oxide anions). Rock Salt, Wurtzite, and PtS Most transition metal monoxides adopt rock salt structure. Oxides can also adopt other structures, including wurtzite (adopted by ZnO) and PtS structure (adopted by CuO and PdO). This structure implies square planar coordination around the metal cation. NhO is unique, having a structure in which both Nb and 0 form four coplanar bonds. occuovine alternate oositions in one of the sirnolest is de3 ~ - c o ~ n e c i enets. h ~ l t e ~ a t i v cthe l ~ structure , scribed as a defect NaCl structure having three h'bO in the unit cell. Distorted Structures and Nonstoichiometrie Forms The classification described in Table 1implies only that the structure of a particular compound is of the general type indicated, and not necessarily the ideal structure with maximum symmetry. Thus, for a given oxide, less symmetric structures in which the atoms slightly deviate from the ideal positions can be found. Additionally, the existence of anionic and cationicvacancies can be observed in some oxides, thus leading to the formation of nonstoichiometric compounds, for example, monoxides of Mn and Fe. Moreover, Ti, V, and Nb monoxides show appreciable ranges of composition, keeping the same type of unit cell but changing the dimensions of it.
In the normal rutile form, each M4+is equidistant from two others in each chain of octahedra. Dioxides of V,Nh, Mo. Tc. W. and Re crvstallize with less svmmetrical variant's of this structwt?, showing successive pairs of metal atoms alwrnativelv closer torrether and further aDart. The close approach of metal atoms has been related td the high electrical conductivity shown by these compounds. Polymorphism As shown in Table 3, polymorphism is very common in this group of oxides. For example, Ti and Zr dioxides are trimorphic a t atmospheric pressure. Titanium dioxide crystallizes in rutile structure above 700 'C in which the coordination group around the metal ion closely appmximates a regular octahedron. It also crystallizes in the less symmetric anatase a t room temperature and in brookite for higher pressures. In ZrOz the normal monoclinic form changes at about 1100 'C to a tetragonal form and a t about 2300 'C to a flourite form. SesquioxidesMZOJ
This stoichiometry is the last one appearing in a wide group of elements. As shown in Table 4, sesquioxides are formed by the metals in the first transition series up to iron, by yttrium and rhodium in the second, and by lanthanum in the third. Corundum and C-Mz03 Crystal structures adopted by these compounds are corundum and type C-Mz03.Corundum can be described as Table 4. Transition Elements That Form Sesquioxides M203 (Corundum Type)
Dioxides MOz ~a~
Many elements from the transition series form dioxides, as indicated in Table 3.
'Assumes C-Mz03 structure. 'Assumes other structures. Volume 71 Number 5 May 1994
383
a hexagonal close-packed array of oxide anions in which M3+cations occupy two-thirds of the octahedral holes. C-M203canbe derived from flourite by removing one-quarter of the anions that generate two types of distorted octahedra. Laz03presents the unusual seven coordination for the metal atom, with the nearest oxygen neighbors around La3+ions forming a monocapped octahedron.
Table 6. Transition Metals That Form Tetroxides M04 and Heptoxides Mz07
&04
Spinels Few elements form oxides with this stoichiometry that dictates that the same element be present in two different oxidation states (Table 5). Mn, Fe, and Co form M304OXides, presenting spinel structure (see below). Stoichiometries That Yield Other Forms
Layers and Chains Mzo5 This stoichiometry is shown by elements of the vanadium group (Table 5). The pentoxide of this element has a layered structure, with a characteristic 5- coordination with one short V-O bond. The pentoxides of Nb and Ta have many crystalline phases, the most remarkable being the chains formed by tantalum pentoxide. M03 This stoichiometry appears in the Cr, Mo, W, and Re oxides (Table 5). Among them, the simplest one is the cubic Table 5. Transition Metals That Form Pentoxides MzOs, Trioxides M03, and Spinels M304
structure of ReO-. directlv related to the oerovskite structure. ~ r ~ ~ f o r m s k i nfhains i t e in which ihe metal cations have tetrahedral coordination with two long and two short Cr-O bonds, whereas Moo3shows a unique layered structure in which each octahedral MOsgroup shares two adjacent edges with similar groups. One of the most remarkable cases of ~olvmornhismis presented by WO.,, which has up to eight phase transitions uo to 900 X!. All the ohases are distorted structures ofthe gghly symmetric cuke Re03.
.. .
Molecular Solids
vapor states. At room temperature they are solids with low meltingr25 and 40 .CI and boilingpoints (100 and 101 'CI. Complex Structures Some mixed oxides adopt uncommon, complex structures, such as perovskite and spinel, that show interesting physical and electronic properties. Perovskovite plays a role in high temperature supercouducting, and spinel is of interst due to the magnetic properties of the oxides that show it. Spinel Spinels form a large class of compounds whose crystal structure is related to that of the mineral spinel itself, MgA1204. The general formula is AB2X4,and the unit ceU contains 32 oxygen atoms in almost perfect cubic closeThe structure can be depacked array (i.e., A&6032) (18). scribed as being built up of alternating cubelets A04 and B404(Fig.3). The four oxygen atoms have the same orientation in the eight cubelets, thus forming the cubic closepacked network. They coordinate cation A tetrahedrally and cation B octahedrally. Depending on the degree of occupancy of the holes in the oxveen network.. soinel . can be normal or inverse. In the n&al spinel structure eight metal atoms (A)occupy onceighth of the tetrahedral sites. and 16 metal atoms (R, occupy one-half of the octahedral sites. In the inverse spinel, A (divalent cation) is placed in the octahedral holes, whereas B is placed in octahedral and tetrahedral holes. Lattice energy calculations based on a simple ionic model indicate that the normal spinel structure should be the more stable. Co and Fe sninel do not conform to this expectation, probably due to t&eeffect of ligand field stabilization energies on the site preferences of the ions. Pernuskite To introduce the pemvskite structure we first examine the related but simpler Re03structure, which consists of corner-shared octahedra (Fig. 4). Alternatively, the unit cell can be described as formed by Re atoms at the corners of a cube and oxide anions on the edges. The presence of corner-shared octahedra can be clarified by extending the structure to more than one unit cell:
Mzo7
.*.--. - -- - - -. -
Oxides of Mn, Tc, and Re show this stoichiometry (Table 6). Mnz07is a deeply colored oil a t room temperature, whereas Tcz07and Re2O7are yellow volatile solids. Mnz07 and Tcz07 are molecular even in the crystalline state. Rez07vaporizes as molecules of the same type, but the crystal has a layer structure in which equal numbers of metal atoms are in tetrahedral and octahedral coordination. SPINEL
There are only two metal tetroxides, Ru04 and 0 ~ 0 ~ .Figure 3. Clystal structuresof spinel structure AB204 (shadedcircles, metal cations; hollow circles, oxide anions). Both exist as regular molecules in the crystalline and 384
Journal of Chemical Education
Figure4. Cubic structure d ReO, (0 ~ e ' +ions, 0 oxide anions).Perovskite structure (0 metal cations. 0oxide anions, @ central atom.)
" .
ReO. structure is verv onen.. and in the center it has a large hole with coordination number 12. Perovskites have the general formula ARO.,, in which the 12-coordinate hole in ReO? is occupied by a large A ion. The 0 inn is generally 0'-or F.The structure is oRen observed to be distorted so that the unit cell is no longer centrosymmetric. Electronic Structure in Transition Metal Oxides ChemicalApproaches The description of the electronic structures of transition metal oxides presents many more problems than that of geometric structures.
The Ionic Model
Interstitial Compounds Moreover, monoxides a t the beginning of the first transition row are in many aspects more like alloys than like ionic compounds due to the existence of metal-metal bonds in the three directions of the lattice. Alternatively, these monoxides have been explained as interstitial compounds in which oxygen would be placed in holes of the metallic network, forming a cubic close-packed structure (12). This group of oxides could then be classified as metallic because they have metal-metal bonds and high electrical conductivities. A direct relationship has been established between these two features assuming the metallic character is the conseouence of the metal-metal bonds. Nevertheless, this sim'ple hypothesis does not explain the high conductivitv of ReO* because metal-metals bonds are not possible there. Band Structure
So far I have discussed the chemical and physical properties of these inorganic solids in terms of covalency and electrovalency. everth he less, these two approaches are not satisfactory when trying to explain electronic properties such as electrical conductivity. For this we need the band theory in which the ionic or covalent character of the bond does not have relevance. Instead. solids are classified a s conductors, semiconductors, or ikulators on the basis of their band gap E, According to this magnitude,
Ti0 is a conductor (Ep= 0). CuzOis a semicondudor ( E p=,2.2eV). MnO, FeO, COO,and NiO are insulators ( E p= 6 eV). In this approach to the study of the bond, the most important features are the energies and the widths of the various bands, the energy gap between bands and the number of electrons involved. To understand the electronic behavior of transition metal oxides, we will look a t band structures of the monoxides of the first transition row.
I n a first aproximation, the ionic model can be used to predict heats of formation and many of the chemical and ohvsical orooerties of transition metal oxides. Thev are dek i e d a s fonii compounds with the formula ~ ~ 0 ~ f o kby n e d Pure Ionic Compounds 0'-anions and M"+catious.where n is 2-8. Usually in a pure ionic oxide (e.g., MgO) each ion has a The simple ionic model with the Born-Lande expression closed-shell electron configuration. Thus, the valence provides goad values of monoxide and dioxide lattice enerband, which forms the highest energy filled level in the gies. solid, should be derived from the highest energy occupied orbitals of the oxide anion. Likewise. the lowest enerw empty level-the conduction band-shbuld be formed frc& the lowest energy empty orbitals of the metal cation. As a consequence all bands are either completely empty or completely full, so ionic compounds are generally insulators. The agreement is particularly good when introducing the correction of the cwstal field solittine (191.a oerturbation Dansition Metal Monoxides that introduces peXodical varikions-in lattik energies as The basic band diagram for transition metal monoxides well as in M-0 distances. (Fig. 5 ) is similar to that of other ionic compounds, showing a valence band, basically an oxygen 2p band, and a conducMolecular and Covalent Crystals tion band of metal character. Because the lowest enerw For higher valent oxides, because electronegativity of unoccupied metal orbitals are the d orbitals, the conduethe metal cation increases with oxidation number, the tion band is usuallv called the d band. As shown below. the metal-oxveen bond becomes nredominantlv covalent. and d band can be split by ligand field effects similar to those the oxidescan no longer be d e k b e d as ion&. ~ccordihgto found in transition metal complexes. Because this band this orediction. oxides with formula M907and MOaare low can be partially occupied with d electrons, transition metal - . melthg moleklar solids. com~oundscan be semiconductor or conductor. &hough the description above is easy to understand, in Although there is good agreement between experimental a rigorous description the valence band is a bonding comand calculated values using the Born-Lande expression, the covalent model more closely describes the bond in secbination of oxygen and metal orbitals, whereas the conduction band is an antibonding combination. Nevertheless. ond and third transition series oxides. In particular, sevthe description of both bands, such as the oxygen 2p and eral low valent oxides a t the end of the series (e.g., ZnO, the metal 3d, is useful because i t denotes the atomic PdO, CuzO, and Ag20)are better described as giant covaorbitals' principal constituents of each level. lent &ol&;les thanas ionic compounds: covalent interactions dominate over electrostatic interactions, and cwrdiAccording to this description T i 0 2 should be semiconductor or insulator because Ti4+has a 3d0 configuration (with nation numbers and geometry are determined by covalent a band gap of 3 em. On the contrary, T i 2 0 3 with a 3d1 coninteractions (12). Volume 71 Number 5 May 1994
385
f
u ( ~ ~ ts ,a p lbands)
that surround it. The overlanoine is not confined to seven atoms hut is spread all over i i e &stal. Thus, this interaction must be considered for all M2+and 0"ions in the crvstal. In this wav an enormous number of molecular orbitals with bonding and antibonding character. will be The number ofmolecular orbitals within bondiue and antibonding sets is extremely high. Thus, the energy separation between them is so small that an enerev continuum is produced, resultingin the band structure p&ted in Figure 6b. As shown in the diagram, the valence band comes from bonding molecular orbitals (e,, t,,, and al,) and the conduction band from nonbonding d levels (tz,). Additionally, there is an empty band of higher energy arising from the antibondmg orbitals e,i, a$, and tyx. The conduction band would be partially filled in conducting oxides, such as TiO, Monoxides with with Ti2+ionsshowing configurations ions having an empty &,band could be either semiconductors or insulators, whereas compounds in which with this band is totally filled (e.g., NiO) with configuration t&, are insulators. ~
Valence band (02,)
Figure 5. Band diagram of a transition metal oxide indicating Valence band, conduction band, and band gap G; (a) d band empty: (b) d partially filled. figuration has one electron per atom in the d band and can show metallic conductivity (at high temperatures). Molecular Orbital and Band Diagrams Another approach to the study of the electronicstructure of monoxides of the first transition row begins with depicting the molecular orbital diagram for the basic unit, that is, the first coordination polyhedra of oxygen anions around metal cations. Then the band structure is obtained by orbital overlapping and delocalization. Although this approach can be considered an oversimplification of the real electronic structure. it reoresents a serious effort to correlate two approaches that generally are olrered w the students as mmpletcly separate: Molecular Orhital theory and bnnd theory. This unifying view proposed by Duffy (141 is an alternative to thc dlchotomv of-bonds or bands" offered by other authors (15,16). As dicussed above transition metal monoxides have the highly symmetric structure of rock salt, the basic polyhedra being MOB,an octahedron of oxide anions around M2+ cations. Using the metal 3d, 49, and 4p atomic orbitals, with symmetry ta and e, (3d orbitals), al, (4s orbital), and tl, (4p orhitals) and taking into account only a overlap with p orbitals from oxide anions, we get the molecular orbital diagram shown in Figure 6a. The crucial differencebetween the structure of three-dimensional monoxides and isolated MO6 units in hexacoordinated complexes (e.g., M(HZO)~~+), is that any particular 02-ion in a M06 unit is also part of five other MO6 units. Therefore, its interactions with one particular M2+ ion is modified by the influence of the other five M" ions
(a) Flg~re6 (a) Mo ecular on, tal scheme showlng a lnteractlon for MO, unrs. (b) ban0 d agram for so o MO wewed in terms of condens ng M 0 6 mo ecdlar orblta s
386
Journal of Chemical Education
~~
~
a.
The Hubbard Model
-
This simole model has manv advantaees because it seems to reconcile microscopic and macroscopic approaches to the bond structure of these compounds, and it satisfactorily explains both physical and chemical properties. Nevertheless. a ~roblemremains. Accordina to this simple band diagram all monoxides with a filled band should show metallic conductivity. Although electronic distributions ib d levels for Mn2+,Fe", and Co2+ions 4 2 are t3 2~tzpe,, ~ and, t&;, all their monoxides are insulator. Electron Repulsions and Localization The reason for this failure of the simple model are the reoulsions between electrons that have been nedected so far. These repulsions tend to keep electrons localized on individual atoms. Metallic properties are then obsewed only when there is a strong tendency to band formation due to an effective overlap. Thus, the high electrical conductivity of Ti0 and VO and the low values of the other monoxides from the first transition row can be attributed to the larger overlap of d orbitals at the beginning of the first transition series, which is much more effective than a t the end. This is a consequence of the contraction of d orhitals along the series due to the increase of nuclear charge. The well known Hartree-Fock approach assumes that electrons move independently in a potential field that includes the average repulsion from other electrons. This ap~roximationworks well in many situations. However, for solids that show small interatomic overlap the effect of electron repulsion becomes more important and cannot be treated as an average potential. The most useful approach to electron repulsion in this case is the Hubbard model, which assumes that the only important repulsion effects occur between two electrons on the same atom. Although reoulsions between electrons on different atoms are not nlgligible, the intraatomic effect seems to be the main reason for the breakdown of the band theorv. For an array of atoms with a single valence electron in which the overlap between orbitals is small, in the ground state each electron will be localized on one atom. This is due to electron repulsion when trying to move the electron to another atom. The energy required to remove one electron from one atomic orbital is the ionization potential IP. Placing this electron in one atom already occupied, we get the electron affmity of the neutral atom EA. The energy required to move the electron is given by
This quantity can be interpreted as the repulsion energy between two electrons on the same atom. Bandwidth and Overlapping The effect of the electron repulsion is to make the partially filled band insulating when the interaction between atoms is small. The bandwidth will increase when the overlapping is more effective. When there is no overlapping, the bandwidth is zero and atomic levels will result. The lower level corresponds to the energy of singly occupied orbitals (IP), and the higher (EA) to the energy obtained aRer putting a second electron in the orbital. The gap, equal to U, is the energy required to excite an electron, displacing it to another atom. This gap, often called Mott-Hubbard splitting, is a wnsequence of electron repulsions. Each level can be broadened by interatomic overlap, and the gap disappears when the bandwidth Wand the repulsion parameter U are equal. Thus, the band theory becomes useful when W > U.
The estimate of the band gap that includes this potential is (see Fig. 7b)
Polarization Effects When dealing with solids we must consider polarization effects, appearingwhenever one electronis moved from one site to another. An approximate estimate of this effect can be made by wnsidering the solid as a continuum with relative dielectric wnstant %. If a charge q spread over a sphere of radius r is moved from the vacuum into the solid, the electrostatic polarization gives an energy change equal to
Global Diagram of Energy Levels, Bands, and Splittings
Mott-Hubbard Splitting
Aglobal approach to the study of the bond in these monoxides that takes into account all the significant wntributions has been included in Figure 7. The f r s t step for calculation of the energy diagram of transition monoxides is to consider the process of forming free ions M2+and 02-. The two levels obtained correspond to the difference in energy between the electron afinity LEA] of oxide anions and ionization potentials of the divalent metal ion (ZIP, (Fig. 7a1:
The polarization of other electrons towards the vacant hole and away from the extra electron lowers the energy required to excite the electron. Consequently, the values of the Mott-Hubbard potential measured in solids are much lower than the values obtained from the difference in energy between the atomic levels. Because the 3d orbitals in a transition metal are partially filled, we must consider the energies due to loosing or gaining one electron.
E,=EA-DP
(3)
Madelung Potential, VM The main correction to this initial estimate is to consider the strong electrostatic potential experienced by the ions in the solid due to the surroundina ions of opposite charpe. We must take into account not oniy the nea;kst.neighb&r ions but also the potential from distant shells of ions in the crystal lattice. This potential involves the Madelung constant of the lattice AM,as in the calculation of lattice energy in the simple ionic model. This potential is thus called Madelung potential and is related to the interionic distance r. by F E E IONS
MAELUNG POTENTIAL
POLARIZATION
OVERLAPPING
m,,
The Mott-Hubbard splitting is the difference between these two energies. In the gas phase it would be the difference between the third (eq 8) and the second (eq 7) ionization energies: about 15 eV. Polarization effects in the solid reduce this magnitude as shown in Figure 7c, the final value being about 3-5 eV. Overlapping of Orbitals Overlapping of orbitals causes the bandwidth shown in Figure I d and 7e. In the first case overlapping is insufficient to overcome the MoteHubbard splitting. Thus, a band gap (E,) appears, making the oxide an insulator (e.g., MnO, NiO, and COO).In this case both the highest energy occupied levels and the lowest energy empty levels have 3d character. Thus, they differ from the other nontramition metal oxides whose highest energy occupied levels have basically 0 2p character, whereas the lowest energy unoccupied levels have metal ns character. Figure 7e shows what happens when, due to a more effective overlap, the d band becomes broader, thus forming a partially filled band and giving a metallic compound. This energy-level diagram is applicable to metallic oxides Ti0 and VO. Conclusion
FlgLre 7 Schemallc oerlvatlon of energy evels for monoxloe MO of an element of the f m trans t on serles. (a)freefons evels. (0) ~onsin tne Madel~ngpotent al. (c) poar zaton ncl-deo showlng the HLObard gap J forlne 3a orblta s, (dl over ap gw ng narrow 30 oand w lh band gap Eg;(e)wide band giving a metallic compound.
The schematic diagram in Figure 7 ends the brief survey of theories currently applied to explain the electronic structure of transition metal oxides. It is our hope that after having shown facts and theories concerning this group of oxides, the idea that transition metal chemistry goes beyond coordination compounds has gained adherents. Note added in Proof: After submission of this article, a book on this topic has been published (20). Volume 71
Number 5
May 1994
387
Acknowledgment The author thanks Spanish DGICYT (project number PB 92.067) for financial and Aria Garcia from the ICMSE for drawing the figures. Literature Cited
9. B ~ I ~ IKWJ: LY, K T J them E ~ U C1.~ ~ 1 . 6 8 . 8 7 5 - 8 7 ~ . 10. Wells,A.FStmduml ImganIcChhmiitry; OxfordUniversity: Oxford. 19% Chap ter 12. ll. G u t i m z Rios, E. Quimimlnorganim; R e v e r b Barcelona, 1978: Chapter 28. 12. Phillips. C. S.G:Williams,R. J. P Inorganic C h i s f r y ; OxfordUniuersity: London, 1965;Chapters 24-26. 13. Cox. P A. The E k f m n i e Structure a d C h i s t r y offhe Sdids; Oxford University: oxf~xf~xfd, 1989;chapter 5.
gapare. 1988. Weat, ;o x fA.~R. ~ BosicSdid ~ ~ ~ ~ i State t y Chemistry; John Wiley: NewYork.1988. 3. S&veqD. F;Athns,P. W ; L m p t o d , c . H . l ~ ~ ~ ~ ~ ~ i ~ c k ~ i ~ t ~16. D.; Mart(ne~-Tamayo,E.; lbaiiez, R.; Beltran-Porter, R.; Folgado, J. 17. Beltm-P&, Oxford, 1990. V.; Escrkd, E. Sdrd Slole I o n h 1988,32133, 116&1166. 4. Jolly. W L. Mdem Inogonlc C h i s f r y ; McCraw-Hill: NearYark, 1984. 18. Greenarwd. N. N.; Earnshaw, A. Chamisfry of f k E k m m f s ; Pergamon: Oxford, 5. Porterfield, w W Inorganic Ckmishy:A Uni~&p~ch;Ad&son-Wesley:Cali. 1984;p 279. fornia. 1984. 19. W s , B. N. Infmducfion toLigondI'lel&; John Wiley: Near York, 1967. 6. Plsanty,A J. Chem. Edue 1891,68,804-808. 7. VerdaLe, J. G. J. Chem E d v c 1991,68,73'%742. 20. C a r , P A . h a s i t i o n MetelOndes:InternatbnaSeriesofMonographsinChemistry. 8. "lnThisleaueSJ. Cham Edue. 1982,69,88. (Soeallauthorsdiacussedonthis psge.1 No. 27,1992.
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