146R.B. King /Coordination Chemistry Reviews 197 (2000) 141-168Table 2Properties of thef atomic orbitalsJmlLobesShapeOrbital graphGeneral setCubic set36x(x2-3y3)Hexagonnoney(3x2-y)28Cubexyzxyz2(x2-y)x(22-y2),y(22-x2), 2(x2-)?16x(522r)Double squarenoneJ(522_r)3204Linear2(522 r)directions similar to a sphere.The following spherical atomic orbital manifolds(Table 3) are of chemical interest [11]:1.The four-orbital sp’manifold (/=O and 1) involved in the chemistry of maingroup elements including their hypervalent compounds through three-centerfour-electron bonding;2. The six-orbital sd§ manifold (l=0 and 2) involved in the chemistry of earlytransition metal hydrides and alkyls since the p orbitals in such systems mayhaveenergies too high to participate in chemical bonding;3.Thenine-orbital sp'dsmanifold (l=0.1,and 2)involved inmost of thechemistry of the d-block transition metals;4. The 13-orbital sd'f? manifold (l=0, 2, and 3)involved in the chemistry of theactinides.These spherical atomic orbital manifolds are characterized by two numbers(Table 3):1.The total number of atomic orbitals in the manifold designated as x whichcorresponds to themaximumpossible coordination numberusing only two-electron two-center bonding;
146 R.B. King / Coordination Chemistry Re6iews 197 (2000) 141–168 Table 2 Properties of the f atomic orbitals m Lobes Shape Orbital graph General set Cubic set Hexagon x(x2 −3y2 3 6 ) none y(3x2 −y2 ) 2 8 Cube xyz xyz x(z 2 −y2 z(x ), 2 −y2 ) y(z 2 −x2 ), z(x2 −y2 ) 1 6 Double square x(5z none 2 −r 2 ) y(5z 2 −r 2 ) 4 Linear z(5z 2−r 2 ) x3 0 y3 z 3 directions similar to a sphere. The following spherical atomic orbital manifolds (Table 3) are of chemical interest [11]: 1. The four-orbital sp3 manifold (l=0 and 1) involved in the chemistry of main group elements including their hypervalent compounds through three-center four-electron bonding; 2. The six-orbital sd5 manifold (l=0 and 2) involved in the chemistry of early transition metal hydrides and alkyls since the p orbitals in such systems may have energies too high to participate in chemical bonding; 3. The nine-orbital sp3 d5 manifold (l=0, 1, and 2) involved in most of the chemistry of the d-block transition metals; 4. The 13-orbital sd5 f 7 manifold (l=0, 2, and 3) involved in the chemistry of the actinides. These spherical atomic orbital manifolds are characterized by two numbers (Table 3): 1. The total number of atomic orbitals in the manifold designated as x which corresponds to the maximum possible coordination number using only two-electron two-center bonding;
147R.B.King /Coordination Chemistry Reviews 197 (2000) 141-168Table 3Spherical atomic orbital manifoldsElementsManifoldMaximumMaximum coordinationinvolvedcoordinationnumberwith annumber (x)ainversion center(y)spi469206Main groupsdsEarly transition metalssp'dsTransition metals1312nActinidesa Considers only two-center two-electron metal-ligand bonding.2.Themaximum number of atomic orbitals in a submanifold consisting of equalnumbers of gerade and ungerade orbitals designated as y which corresponds tothe maximum possible coordination number for a polyhedron with a center ofsymmetry or a unique reflection plane containing no vertices. For a givenmanifold, such polyhedra with vertices where y<u≤x are symmetry forbid-den coordination polyhedra.Aspecific featureof thechemical bonding in some systems containingthe latetransition and early post-transition metals observed by Nyholm [12] as early as1961 is the shifting of one or two of the outer p orbitals to such high energies thattheynolongerparticipateinthechemicalbondingandtheaccessiblespdvalenceorbital manifold isno longer spherical (isotropic).If onep orbital is soshifted tobecomeantibonding,then theaccessiblespd orbital manifold containsonlyeightorbitals (sp'd)and has thegeometryof a torus or doughnut (Fig.l(a)).The'missing p orbital is responsible for the hole in the doughnut.This toroidal sp'dsmanifold can bond onlyin the two dimensions of theplaneof the ring of thetorusthereby leading only to planar coordination arrangements.Filling this sp'd mani-fold of eight orbitals with electrons leads to the16-electron configuration found in(a)Toroidal sp? manifoldCylindrical spd manifold(Square Planar)(Linear)(b)Toroidal Trigonal PlanarrToroidalPentagonalPlanarFig. 1. (a) The toroidal (sp'd5) and cylindrical (spd’) manifolds; (b) Trigonal planar and pentagonalplanar coordination for the toroidal manifold
R.B. King / Coordination Chemistry Re6iews 197 (2000) 141–168 147 Table 3 Spherical atomic orbital manifolds Manifold Maximum coordination Elements Maximum involved number with an coordination number (x) a inversion center (y) a sp3 Main group 4 2 sd5 Early transition metals 6 0 sp3 d5 Transition metals 9 6 sd Actinides 5 f 7 13 12 a Considers only two-center two-electron metal–ligand bonding. 2. The maximum number of atomic orbitals in a submanifold consisting of equal numbers of gerade and ungerade orbitals designated as y which corresponds to the maximum possible coordination number for a polyhedron with a center of symmetry or a unique reflection plane containing no vertices. For a given manifold, such polyhedra with 6 vertices where yB65x are symmetry forbidden coordination polyhedra. A specific feature of the chemical bonding in some systems containing the late transition and early post-transition metals observed by Nyholm [12] as early as 1961 is the shifting of one or two of the outer p orbitals to such high energies that they no longer participate in the chemical bonding and the accessible spd valence orbital manifold is no longer spherical (isotropic). If one p orbital is so shifted to become antibonding, then the accessible spd orbital manifold contains only eight orbitals (sp2 d5 ) and has the geometry of a torus or doughnut (Fig. 1(a)). The ‘missing’ p orbital is responsible for the hole in the doughnut. This toroidal sp2 d5 manifold can bond only in the two dimensions of the plane of the ring of the torus thereby leading only to planar coordination arrangements. Filling this sp2 d5 manifold of eight orbitals with electrons leads to the 16-electron configuration found in Fig. 1. (a) The toroidal (sp2 d5 ) and cylindrical (spd5 ) manifolds; (b) Trigonal planar and pentagonal planar coordination for the toroidal manifold
148R.B.King/CoordinationChemistryReviews197(2000)141-168square planar complexes of the d transition metals such as Rh(I), Ir(I), Ni(II), Pd(II),Pt(Il),andAu(lll).Thelocationsofthefourligandsinthesesquareplanarcomplexescan be considered tobe points on the surface of thetorus corresponding to the sp'dsmanifold.Thetoroidal sp'd’manifoldcan alsoleadtotrigonal planar and pentagonalplanar coordinationfor three-and five-coordinate complexes,respectively (Fig.l(b)Thex,y, and z axesfor a toroidal sp'd’manifold are conventionally chosen so thatthemissingporbital istheP.orbital.In some structures containing the late transition and post-transition metalsparticularly the 5d metals Pt, Au, Hg,and Tl, two of the outer p orbitals are raisedto antibonding energylevels.This leaves only oneporbital in theaccessible spd orbitalmanifold,which now contains seven orbitals(spd)andhas cylindrical geometryextending in one axial dimension muchfurther than in theremaining twodimensions(Fig. l(a)).Filling this seven-orbital spds manifold with electrons leads to the14-electronconfigurationfound intwo-coordinate linearcomplexesofdiometals suchas Pt(O), Cu(I), Ag(), Au(I), Hg(I1), and Tl(IIl). The raising of one or particularlytwo outerp orbitals to antibonding levels has been attributed to relativistic effects.Thep orbitals which areraised to antibondinglevels as noted above can participateindo→po*ord元-→p*bondingincomplexesofmetalswithtoroidal sp'dandcylindrical spd’manifolds depending on the symmetry of the overlap (Fig.2).Suchbonding was suggested by Dedieu and Hoffmann [13] in 1978for Pt(0)-Pt(0) dimerson the basis of extended Huckel calculations and is discussed in detail in a recentreview by Pyykko [14].This type of surface bonding like, for example, the d-p*backbonding inmetal carbonyls,does not affect the electron bookkeeping in the latetransition and post-transition metal clusters but accounts for the bonding rather thannon-bondingdistancesbetween adjacentmetal vertices in certain compoundsof thecoinage metals, particularly gold, as well as other late and post-transition metals(x2-yA)~x2do-po*bondingXZ→Zd-→p元*bondingFig. 2. Examples of do→pG and dr→pr bonding to the otherwise empty p orbitals in complexes ofmetals with toroidal (sp'd§) and cylindrical (spd’) manifolds
148 R.B. King / Coordination Chemistry Re6iews 197 (2000) 141–168 square planar complexes of the d8 transition metals such as Rh(I), Ir(I), Ni(II), Pd(II), Pt(II), and Au(III). The locations of the four ligands in these square planar complexes can be considered to be points on the surface of the torus corresponding to the sp2 d5 manifold. The toroidal sp2 d5 manifold can also lead to trigonal planar and pentagonal planar coordination for three- and five-coordinate complexes, respectively (Fig. 1(b)). The x, y, and z axes for a toroidal sp2 d5 manifold are conventionally chosen so that the missing p orbital is the pz orbital. In some structures containing the late transition and post-transition metals, particularly the 5d metals Pt, Au, Hg, and Tl, two of the outer p orbitals are raised to antibonding energy levels. This leaves only one p orbital in the accessible spd orbital manifold, which now contains seven orbitals (spd5 ) and has cylindrical geometry extending in one axial dimension much further than in the remaining two dimensions (Fig. 1(a)). Filling this seven-orbital spd5 manifold with electrons leads to the 14-electron configuration found in two-coordinate linear complexes of d10 metals such as Pt(0), Cu(I), Ag(I), Au(I), Hg(II), and Tl(III). The raising of one or particularly two outer p orbitals to antibonding levels has been attributed to relativistic effects. The p orbitals which are raised to antibonding levels as noted above can participate in dsps* or dppp* bonding in complexes of metals with toroidal sp2 d5 and cylindrical spd5 manifolds depending on the symmetry of the overlap (Fig. 2). Such bonding was suggested by Dedieu and Hoffmann [13] in 1978 for Pt(0)–Pt(0) dimers on the basis of extended Hu¨ckel calculations and is discussed in detail in a recent review by Pyykko¨ [14]. This type of surface bonding like, for example, the dppp* backbonding in metal carbonyls, does not affect the electron bookkeeping in the late transition and post-transition metal clusters but accounts for the bonding rather than non-bonding distances between adjacent metal vertices in certain compounds of the coinage metals, particularly gold, as well as other late and post-transition metals. Fig. 2. Examples of dsps and dppp bonding to the otherwise empty p orbitals in complexes of metals with toroidal (sp2 d5 ) and cylindrical (spd5 ) manifolds.
149R.B. King /Coordination Chemistry Reviews 197 (2000) 141-1682.3.Hybridizationof atomic orbitalsConsidera metal complex of thegeneral typeMLin whichMisthe centralmetal atom,Lnreferstonligands surroundingM,and eachligandLisattached toM through a singleatom of L.The combined strengthsof the n chemical bondsformedbyMtothe nligandsLaremaximized if themetal valenceatomicorbitalsoverlap to the maximum extent with the atomic orbitals of the ligands L.Theavailablemetal valenceorbitals maybecombined or hybridized in sucha waytomaximizethis overlap.Consider a light' element of the first row of eight of the periodic table Li-Fsuch as, for example, boron or carbon. The valence orbital manifold of suchelements consists of a single s orbital and the three p orbitals, namely Px, Py, andP-.In the example of methane, CH4, the four hydrogen atoms are located at thevertices of a regular tetrahedron surrounding the central carbon atom. Thestrengths of the four C-H bonds directed towards the vertices of a regulartetrahedron can be maximized if the following linear combinations of the wavefunctions of the atomic orbitals in the spmanifold are used:--(5a)+0:(5b)(5c)P(5d)Y4=-冲+钟2929sIn Eqs. (5a)-(5d) the px, Py, and p orbitals are abbreviated as x, y, and z,respectively,and the hybrid wave functions are represented by and the compo-nent atomic orbitals are represented by .The process of determining the coefficients in equations such as those above isbeyond the scope of this article and can become complicated when the degrees offreedomareincreasedbyloweringthe symmetry ofthecoordinationpolyhedronorbyincreasingthe sizeof thevalence orbital manifoldto included orbitals,as isofinterestforthe transition metalchemistry discussed in this article.Howeverelementary symmetry considerations, as outlined in group-theorytexts [15], can beused todetermine which atomic orbitals have thenecessary symmetryproperties toform a hybrid corresponding to a given coordination polyhedron.For example, thefour atomic orbitals of an spmanifold can form four hybrid orbitals pointingtowards the vertices of a tetrahedron as outlined above. However, the four atomicorbitalsof an sp manifoldare excludedbysymmetry considerations fromformingfour hybrid orbitals pointing towards thevertices of aplanar square or rectangle.Thusif theplane of thesquare orrectangleisthexyplane.thep,orbital is seen tohave no electron density in this plane (i.e., the xy plane is a node for the p: orbital)
R.B. King / Coordination Chemistry Re6iews 197 (2000) 141–168 149 2.3. Hybridization of atomic orbitals Consider a metal complex of the general type MLn in which M is the central metal atom, Ln refers to n ligands surrounding M, and each ligand L is attached to M through a single atom of L. The combined strengths of the n chemical bonds formed by M to the n ligands L are maximized if the metal valence atomic orbitals overlap to the maximum extent with the atomic orbitals of the ligands L. The available metal valence orbitals may be combined or hybridized in such a way to maximize this overlap. Consider a ‘light’ element of the first row of eight of the periodic table LiF such as, for example, boron or carbon. The valence orbital manifold of such elements consists of a single s orbital and the three p orbitals, namely px, py, and pz. In the example of methane, CH4, the four hydrogen atoms are located at the vertices of a regular tetrahedron surrounding the central carbon atom. The strengths of the four C–H bonds directed towards the vertices of a regular tetrahedron can be maximized if the following linear combinations of the wave functions of the atomic orbitals in the sp3 manifold are used: C1=1 2 fs+ 1 2 fx+ 1 2 fy+ 1 2 fz (5a) C2=1 2 fs−1 2 fx−1 2 fy+ 1 2 fz (5b) C3=1 2 fs+ 1 2 fx−1 2 fy−1 2 fz (5c) C4=1 2 fs−1 2 fx+ 1 2 fy−1 2 fz (5d) In Eqs. (5a)–(5d) the px, py, and pz orbitals are abbreviated as x, y, and z, respectively, and the hybrid wave functions are represented by c and the component atomic orbitals are represented by f. The process of determining the coefficients in equations such as those above is beyond the scope of this article and can become complicated when the degrees of freedom are increased by lowering the symmetry of the coordination polyhedron or by increasing the size of the valence orbital manifold to include d orbitals, as is of interest for the transition metal chemistry discussed in this article. However, elementary symmetry considerations, as outlined in group-theory texts [15], can be used to determine which atomic orbitals have the necessary symmetry properties to form a hybrid corresponding to a given coordination polyhedron. For example, the four atomic orbitals of an sp3 manifold can form four hybrid orbitals pointing towards the vertices of a tetrahedron as outlined above. However, the four atomic orbitals of an sp3 manifold are excluded by symmetry considerations from forming four hybrid orbitals pointing towards the vertices of a planar square or rectangle. Thus if the plane of the square or rectangle is the xy plane, the pz orbital is seen to have no electron density in this plane (i.e., the xy plane is a node for the pz orbital)
150R.B.King/Coordination Chemistry Reviews 197 (2000)141-168and thus cannot participate in the bonding to atoms in the plane. In the case ofcoordination polyhedra with larger numbers of vertices, particularly those ofrelatively high symmetry such as the cube and hexagonal bipyramid for eight-coor-dination,theinabilityof certain combinations of atomic orbitalsto form therequired hybrid orbitals is not as obvious and more sophisticated group-theoreticalmethods are required. Such methods are discussed in Section 3.3.3.Theproperties of coordination polyhedra3.1.Topology of coordination polyhedraAkey aspect of the topology of coordination polyhedra is Euler's relationshipbetween the numbers of vertices (v), edges (e), and faces (f), i.e.,(6)u-e+f=2This arises from the properties of ordinary three-dimensional space.In addition the following relationships must be satisfied by any polyhedron:Zifi=2e(7)(l)Relationshipbetween the edges and faces:f=3In Eq. (7), f, is the number of faces with i edges (ie, fs is the number oftriangular faces, f is the number of quadrilateral faces, etc.). This relationshiparises from the fact that each edgeof thepolyhedron is shared by exactly two faces.Since no face can have fewer edges than the three of a triangle, the followinginequality must hold in all cases:(8)3f≤2e(2)Relationship between the edges and vertices:Z iv,=2e(9)=3In Eq. (9), u, is the number of vertices of degree i (i.e., having iedges meeting atthe vertex).This relationship arises from the fact that each edge of the polyhedronconnectsexactlytwo vertices.Since no vertex of apolyhedron can haveadegreeless than three, the following inequality must hold in all cases:(10)3u≤2e(3) Totality of faces:(11)f=-(4) Totality of vertices:(12)2D=U123Eq. (11) relates the fis to f and Eq.(12) relates the v,s to v.In generating actual polyhedra, the operations of capping and dualization areoften important.Capping a polyhedron , consists of adding a new vertex above
150 R.B. King / Coordination Chemistry Re6iews 197 (2000) 141–168 and thus cannot participate in the bonding to atoms in the plane. In the case of coordination polyhedra with larger numbers of vertices, particularly those of relatively high symmetry such as the cube and hexagonal bipyramid for eight-coordination, the inability of certain combinations of atomic orbitals to form the required hybrid orbitals is not as obvious and more sophisticated group-theoretical methods are required. Such methods are discussed in Section 3.3. 3. The properties of coordination polyhedra 3.1. Topology of coordination polyhedra A key aspect of the topology of coordination polyhedra is Euler’s relationship between the numbers of vertices (6), edges (e), and faces ( f ), i.e., 6−e+f=2 (6) This arises from the properties of ordinary three-dimensional space. In addition the following relationships must be satisfied by any polyhedron: (1) Relationship between the edges and faces: % 6−1 i=3 ifi=2e (7) In Eq. (7), fi is the number of faces with i edges (i.e., f3 is the number of triangular faces, f4 is the number of quadrilateral faces, etc.). This relationship arises from the fact that each edge of the polyhedron is shared by exactly two faces. Since no face can have fewer edges than the three of a triangle, the following inequality must hold in all cases: 3f52e (8) (2) Relationship between the edges and vertices: % 6−1 i=3 i6i=2e (9) In Eq. (9), 6i is the number of vertices of degree i (i.e., having i edges meeting at the vertex). This relationship arises from the fact that each edge of the polyhedron connects exactly two vertices. Since no vertex of a polyhedron can have a degree less than three, the following inequality must hold in all cases: 3652e (10) (3) Totality of faces: % 6−1 i=3 fi=f (11) (4) Totality of vertices: % 6−1 i=3 6i=6 (12) Eq. (11) relates the fis to f and Eq. (12) relates the 6is to 6. In generating actual polyhedra, the operations of capping and dualization are often important. Capping a polyhedron P1 consists of adding a new vertex above