《结构化学 Structural Chemistry》课程教学资源（书籍文献）An article by R.B. King regarding coordination and hybridization

COORDINATIONCHEMISTRYCoordination Chemistry ReviewsREVIEWS197 (2000) 141168ELSEVIERwww.elsevier.com/locate/ccrAtomic orbitals, symmetry, and coordinationpolyhedraR. Bruce King *Department of Chemistry,Uninersity of Georgia,Athens,GA30602,USAReceived 18 February 1999; received in revised form 30 July 1999; accepted 27August 1999This paper is dedicated to Professor Ronald J. Gillespie in recognition of his pioneering work inunderstanding the shape of molecules.Contents142Abstract.142I.Introduction.1432.Properties of atomic orbitals1432.1 Atomic orbitals from spherical harmonics1452.2Valence manifolds of atomic orbitals.1492.3 Hybridization of atomic orbitals1503.Theproperties of coordination polyhedra.1503.1 Topology of coordination polyhedra1523.2 The shapes of coordination polyhedra1543.3Svmmetryforbiddencoordinationpolvhedra1564. Coordination polyhedra for the spherical sp-d5 nine-orbital manifold1564.1 The description of metal coordination by polyhedra1564.2 Coordination numberfour1574.3Coordinationnumberfive1584.4 Coordination number six.1594.5Coordinationnumberseven1604.6 Coordination number eight1614.7Coordinationnumbernine1625. Coordination polyhedra for other spherical manifolds of atomic orbitals1625.1 Coordination polyhedra for the four-orbital sp3 manifold.1645.2 Coordination polyhedra for the six-orbital sd’ manifold.5.3 Coordination polyhedra for the 13-orbital sd'f manifold1661666.Summary167References*Tel.:+1-706-542-1901; fax:+1-706-542-9454.E-mail address: rbking@sunchem.chem.uga.edu (R.B. King)0010-8545/00/S - see front matter 2000 Elsevier Science S.A. All rights reserved.PII:S0010-8545(99)00226-X

Coordination Chemistry Reviews 197 (2000) 141–168 Atomic orbitals, symmetry, and coordination polyhedra R. Bruce King * Department of Chemistry, Uni6ersity of Georgia, Athens, GA 30602, USA Received 18 February 1999; received in revised form 30 July 1999; accepted 27 August 1999 This paper is dedicated to Professor Ronald J. Gillespie in recognition of his pioneering work in understanding the shape of molecules. Contents Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 2. Properties of atomic orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 2.1 Atomic orbitals from spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 2.2 Valence manifolds of atomic orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 2.3 Hybridization of atomic orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3. The properties of coordination polyhedra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.1 Topology of coordination polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.2 The shapes of coordination polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.3 Symmetry forbidden coordination polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4. Coordination polyhedra for the spherical sp3 d5 nine-orbital manifold . . . . . . . . . . . . . . 156 4.1 The description of metal coordination by polyhedra . . . . . . . . . . . . . . . . . . . . . . 156 4.2 Coordination number four . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.3 Coordination number five . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.4 Coordination number six . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.5 Coordination number seven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.6 Coordination number eight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.7 Coordination number nine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5. Coordination polyhedra for other spherical manifolds of atomic orbitals . . . . . . . . . . . . 162 5.1 Coordination polyhedra for the four-orbital sp3 manifold . . . . . . . . . . . . . . . . . . . 162 5.2 Coordination polyhedra for the six-orbital sd5 manifold . . . . . . . . . . . . . . . . . . . . 164 5.3 Coordination polyhedra for the 13-orbital sd5 f 7 manifold . . . . . . . . . . . . . . . . . . . 166 6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 www.elsevier.com/locate/ccr * Tel.: +1-706-542-1901; fax: +1-706-542-9454. E-mail address: rbking@sunchem.chem.uga.edu (R.B. King) 0010-8545/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S0010-8545(99)00226-X

142R.B. King/Coordination Chemistry Reviews 197 (2000) 141-168AbstractThe fundamental ideas on inorganic stereochemistry presented originally by Sidgwick andPowell in 1940 and developed subsequently by Gillespie and Nyholm in 1957 have expandedinto a broad theoretical base for essentially all of coordination chemistry during thesubsequent four decades.A key aspect of this work has been a detailed understanding of thetopology,shape,and symmetry of all of the actual and plausible polyhedra found incoordination chemistry and the relationship of such properties of the relevant polyhedra tothose of the available atomic orbitals of the central metal atom. This paper reviews thepolyhedra for coordination numbers four through nine for the spherical nine-orbital sp'dsmanifold commonlyused intransitionmetal coordinationchemistryaswell aspossibilitiesincoordinationcomplexeshavingotherspherical manifoldsforthecentral atom includingthefour-orbital sp’manifold used by elements without energeticallyaccessible d orbitals, thesix-orbital sd5 manifold used in some early transition metal alkyls and hydrides,and thethirteen-orbital sd5f7manifold used in actinide complexes.2000 Elsevier Science S.A.Allrights reserved.Keywords:Atomic orbitals, Symmetry; Coordination polyhedra1.IntroductionOne of the important objectives of theoretical chemistry is understanding thefactors affecting the shapes of molecules.In the specific area of coordinationchemistry this often corresponds to understanding the coordination polyhedrafavored for particular metals, oxidation states, and ligand sets.In this connection aseminal paper was the 1940 Bakerian Lecture of Sidgwick and Powell [1] onstereochemical types and valencygroups.This paper was the first to develop theidea of therelation between the number of valence electrons,number of ligands,andtheshapeof themoleculeand ledtotheso-calledSidgwick-Powelltheoryofelectron pairrepulsions.By the 1940 publicationdate of this paper,enoughexperimental structural data had been accumulated on key coordination com-pounds and other inorganic molecules using X-ray diffraction as well as absorptionspectra and Raman spectra so that an adequate experimental data base throughouttheperiodictablehadbecomeavailabletotesttheseideas.The next key paper in this area was a review on inorganic stereochemistry byGillespie and Nyholm [2] which introduced the idea that the pairs of electrons in avalency shell, irrespective of whether they are shared (i.e.,bonding) pairs orunshared (i.e., non-bonding)pairs, are always arranged in the same way whichdepends onlyontheirnumber.Thustwopairsarearranged linearly.threepairsinthe form of a plane triangle, four pairs tetrahedrally, five pairs in the form of atrigonal bipyramid, six pairs octahedrally,etc.These ideas were subsequentlydeveloped in more detail in a 1972 book by Gillespie [3] and led to the so-calledvalence-shell electron pair repulsion(VSEPR)theory.This theoryhasprovenparticularly useful over the years in understanding the shapes of hypervalent maingroup element molecules such as SF4, CiF3, etc

142 R.B. King / Coordination Chemistry Re6iews 197 (2000) 141–168 Abstract The fundamental ideas on inorganic stereochemistry presented originally by Sidgwick and Powell in 1940 and developed subsequently by Gillespie and Nyholm in 1957 have expanded into a broad theoretical base for essentially all of coordination chemistry during the subsequent four decades. A key aspect of this work has been a detailed understanding of the topology, shape, and symmetry of all of the actual and plausible polyhedra found in coordination chemistry and the relationship of such properties of the relevant polyhedra to those of the available atomic orbitals of the central metal atom. This paper reviews the polyhedra for coordination numbers four through nine for the spherical nine-orbital sp3 d5 manifold commonly used in transition metal coordination chemistry as well as possibilities in coordination complexes having other spherical manifolds for the central atom including the four-orbital sp3 manifold used by elements without energetically accessible d orbitals, the six-orbital sd5 manifold used in some early transition metal alkyls and hydrides, and the thirteen-orbital sd5 f 7 manifold used in actinide complexes. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Atomic orbitals; Symmetry; Coordination polyhedra 1. Introduction One of the important objectives of theoretical chemistry is understanding the factors affecting the shapes of molecules. In the specific area of coordination chemistry this often corresponds to understanding the coordination polyhedra favored for particular metals, oxidation states, and ligand sets. In this connection a seminal paper was the 1940 Bakerian Lecture of Sidgwick and Powell [1] on stereochemical types and valency groups. This paper was the first to develop the idea of the relation between the number of valence electrons, number of ligands, and the shape of the molecule and led to the so-called Sidgwick–Powell theory of electron pair repulsions. By the 1940 publication date of this paper, enough experimental structural data had been accumulated on key coordination com￾pounds and other inorganic molecules using X-ray diffraction as well as absorption spectra and Raman spectra so that an adequate experimental data base throughout the periodic table had become available to test these ideas. The next key paper in this area was a review on inorganic stereochemistry by Gillespie and Nyholm [2] which introduced the idea that the pairs of electrons in a valency shell, irrespective of whether they are shared (i.e., bonding) pairs or unshared (i.e., non-bonding) pairs, are always arranged in the same way which depends only on their number. Thus two pairs are arranged linearly, three pairs in the form of a plane triangle, four pairs tetrahedrally, five pairs in the form of a trigonal bipyramid, six pairs octahedrally, etc. These ideas were subsequently developed in more detail in a 1972 book by Gillespie [3] and led to the so-called 6alence-shell electron pair repulsion (VSEPR) theory. This theory has proven particularly useful over the years in understanding the shapes of hypervalent main group element molecules such as SF4, ClF3, etc

143R.B.King/Coordination ChemistryReviews197(2000)141-168During the period that these theoretical ideas were developing, additional exper-imental information also accumulated, aided by thegrowing availability of X-raydiffraction methods to elucidate unambiguously the structures of diverse inorganicand organometallic compounds.In thelate1960s,Ibecame interested in exploringthe extent to which elementary concepts from themathematical discipline oftopology could account for the specific coordination polyhedra that were beingdiscovered in inorganic compounds and I summarized my initial observations in a1969 paper [4]. In the three decades since publication of this original paper I haveintroduced a number of additional ideas relating to coordination polyhedra, so thattheapproach of theoriginal1969paper nowappearsvery crude.Ideas whichhaveproven to be useful over the years include the concept of coordination polyhedrawhich are symmetry-forbidden fora given atomic orbital manifold [5] as well as therelationship of the magnetic quantum number ofthe atomic orbitals involved in thehybridization to the shapeof the resulting coordinationpolyhedron [6].This papersummarizes the interplay between these ideas and how theyrelate to the experimen-tallyobserved shapes ofcoordination compounds.2.Properties ofatomicorbitals2.1.Atomicorbitalsfromspherical harmonicsThe shapes of the atomic orbitals of the central atom determine the stereochemistry of the bonding of the central atom to its surrounding ligands, which is basedonthehybridorbitalsformedbyvariouslinear combinationsoftheavailableatomic orbitals.These atomic orbitals arise from the one-particle wave functions ,obtained as spherical harmonics by solution of the following second order differen-tial equation in which the potential energy Vis spherically symmetric:，，28元m8元m"(E-Y=+"(E-V)P=0(1)+++h2h2These spherical harmonics Y are functions of either the three spatial coordinatesx, y, and z or the corresponding spherical polar coordinates r, , and defined bythe equations(2a)x=r sin cos Φ(2b)y=r sin o sin g(2c)z=rcos 0Furthermore, a set of linearly independent wave functions can be found such thatcanbefactoredintothefollowingproduct:(3)Y(r, 0, )=R(r)(0)Φ(Φ)in which the factorsR,,and @ are functions solelyofr,, and ,respectivelySince the value of the radial component R(r)of is completely independent of the

R.B. King / Coordination Chemistry Re6iews 197 (2000) 141–168 143 During the period that these theoretical ideas were developing, additional exper￾imental information also accumulated, aided by the growing availability of X-ray diffraction methods to elucidate unambiguously the structures of diverse inorganic and organometallic compounds. In the late 1960s, I became interested in exploring the extent to which elementary concepts from the mathematical discipline of topology could account for the specific coordination polyhedra that were being discovered in inorganic compounds and I summarized my initial observations in a 1969 paper [4]. In the three decades since publication of this original paper I have introduced a number of additional ideas relating to coordination polyhedra, so that the approach of the original 1969 paper now appears very crude. Ideas which have proven to be useful over the years include the concept of coordination polyhedra which are symmetry-forbidden for a given atomic orbital manifold [5] as well as the relationship of the magnetic quantum number of the atomic orbitals involved in the hybridization to the shape of the resulting coordination polyhedron [6]. This paper summarizes the interplay between these ideas and how they relate to the experimen￾tally observed shapes of coordination compounds. 2. Properties of atomic orbitals 2.1. Atomic orbitals from spherical harmonics The shapes of the atomic orbitals of the central atom determine the stereochem￾istry of the bonding of the central atom to its surrounding ligands, which is based on the hybrid orbitals formed by various linear combinations of the available atomic orbitals. These atomic orbitals arise from the one-particle wave functions C, obtained as spherical harmonics by solution of the following second order differen￾tial equation in which the potential energy V is spherically symmetric: (2 C (x2 + (2 C (y2 + (2 C (z 2 + 8p2 m h2 (E−V)C=92 C+ 8p2 m h2 (E−V)C=0 (1) These spherical harmonics C are functions of either the three spatial coordinates x, y, and z or the corresponding spherical polar coordinates r, u, and f defined by the equations x=r sin u cos f (2a) y=r sin u sin f (2b) z=r cos u (2c) Furthermore, a set of linearly independent wave functions can be found such that C can be factored into the following product: C(r, u, f)=R(r)·U(u)·F(f) (3) in which the factors R, U, and F are functions solely of r, u, and f, respectively. Since the value of the radial component R(r) of C is completely independent of the

144R.B.King/CoordinationChemistryReviews197(2000)141-168angular coordinates and , it is independent of direction (i.e., isotropic) andtherefore remains unaltered by any symmetry operations.For this reason all of thesymmetry properties of a spherical harmonic , and thus of the correspondingwave function or atomic orbital, are contained in its angular component@(0)Φ().Furthermore, each of the threefactors of (Eq.(3))generatesaquantum number. Thus the factors R(r), @(0), and Φ() generate the quantumnumbers n, I, and m, (or simply m), respectively.The principal quantum number n,derivedfromtheradial componentR(r),relatestothedistancefromthecenterofthe sphere (i.e., the nucleus in the case of atomic orbitals). The azimuthal quantumnumber l, derived from thefactor (0)in Eq.(3),relates tothe number of nodesin the angular component (0)-Φ(), where a node is a plane corresponding to azero value of (0)(Φ) or , ie., where the sign of (0)Φ(Φ) changes frompositive to negative.Atomicorbitals for which/=0,1,2,and 3have0,1,2,and 3nodes, respectively, and are conventionally designated as s, p, d, and f orbitals,respectively.For a given value of the azimuthal quantum number I, the magneticquantum number m,or m,derived from the factor Φ()in Eq.(3),maytake on all21+1 different values from +1 to -1. There are therefore necessarily 21+1distinctorthogonal orbitalsforagivenvalueof/ correspondingtol,3,5,and7distinct s, p, d, and f orbitals, respectively.The magnetic quantum number, m, can be related to the distribution of theelectron density of the atomic orbital relative to the z axis.Thus if the nucleus is inthe center of a sphere in which the z axis is the polar axis passing through the northand southpoles,an atomicorbital withm=Ohasits electron densityorientedtowardsthenorthand southpoles ofthe spherewhereas an atomicorbital withthemaximumpossiblevalueofm,ie.,+l,has itsmaximum electrondensityintheequator of the sphere.Inthis waythe angular momentum of theatomic orbitalsinvolved in the hybridization for a given coordination polyhedron can relate to themoment ofinertiaofthatcoordinationpolyhedron.A convenient way of depicting the shapeof an orbital,particularly complicatedorbitals with large numbers of lobes, is by the use of an orbital graph [7]. In suchanorbitalgraphtheverticescorrespondtothelobesoftheatomicorbitalsandtheedges to nodes between adjacent lobes of opposite sign. Such an orbital graph isnecessarily a bipartitegraph inwhich each vertex is labeled withthe sign of thecorresponding lobe and only vertices of opposite sign can be connected by an edge.Table I illustrates some of the important properties of s, p, and d orbitals.SimilarlyTable 2 lists some of the important properties of twodifferent sets ofseven f orbitals.The cubic set of f orbitals is used for structures of sufficiently highsymmetry (e.g.,O,and I)to have sets of triply degenerate f orbitals whereas thegeneral set of f orbitals are used for structures of lower symmetry without sets off orbitals having degeneracies 3 or higher. The g and h orbitals are analogouslydepicted elsewhere [8]; they are not relevant to the discussion of coordinationpolyhedra in this paper.The conventionally used set of five orthogonal d orbitals contains two types oforbitals,namely thexy,xz,yz,and x2-y2orbitals each with four major lobes andthe zorbital with onlytwomajor lobes (Table1).All possible shapes of d orbitals

144 R.B. King / Coordination Chemistry Re6iews 197 (2000) 141–168 angular coordinates u and f, it is independent of direction (i.e., isotropic) and therefore remains unaltered by any symmetry operations. For this reason all of the symmetry properties of a spherical harmonic C, and thus of the corresponding wave function or atomic orbital, are contained in its angular component U(u)·F(f). Furthermore, each of the three factors of C (Eq. (3)) generates a quantum number. Thus the factors R(r), U(u), and F(f) generate the quantum numbers n, l, and ml (or simply m), respectively. The principal quantum number n, derived from the radial component R(r), relates to the distance from the center of the sphere (i.e., the nucleus in the case of atomic orbitals). The azimuthal quantum number l, derived from the factor U(u) in Eq. (3), relates to the number of nodes in the angular component U(u)·F(f), where a node is a plane corresponding to a zero value of U(u)·F(f) or C, i.e., where the sign of U(u)·F(f) changes from positive to negative. Atomic orbitals for which l=0, 1, 2, and 3 have 0, 1, 2, and 3 nodes, respectively, and are conventionally designated as s, p, d, and f orbitals, respectively. For a given value of the azimuthal quantum number l, the magnetic quantum number ml or m, derived from the factor F(f) in Eq. (3), may take on all 2l+1 different values from +l to −l. There are therefore necessarily 2l+1 distinct orthogonal orbitals for a given value of l corresponding to 1, 3, 5, and 7 distinct s, p, d, and f orbitals, respectively. The magnetic quantum number, m, can be related to the distribution of the electron density of the atomic orbital relative to the z axis. Thus if the nucleus is in the center of a sphere in which the z axis is the polar axis passing through the north and south poles, an atomic orbital with m=0 has its electron density oriented towards the north and south poles of the sphere whereas an atomic orbital with the maximum possible value of m, i.e., 9l, has its maximum electron density in the equator of the sphere. In this way the angular momentum of the atomic orbitals involved in the hybridization for a given coordination polyhedron can relate to the moment of inertia of that coordination polyhedron. A convenient way of depicting the shape of an orbital, particularly complicated orbitals with large numbers of lobes, is by the use of an orbital graph [7]. In such an orbital graph the vertices correspond to the lobes of the atomic orbitals and the edges to nodes between adjacent lobes of opposite sign. Such an orbital graph is necessarily a bipartite graph in which each vertex is labeled with the sign of the corresponding lobe and only vertices of opposite sign can be connected by an edge. Table 1 illustrates some of the important properties of s, p, and d orbitals. Similarly Table 2 lists some of the important properties of two different sets of seven f orbitals. The cubic set of f orbitals is used for structures of sufficiently high symmetry (e.g., Oh and Ih) to have sets of triply degenerate f orbitals whereas the general set of f orbitals are used for structures of lower symmetry without sets of f orbitals having degeneracies 3 or higher. The g and h orbitals are analogously depicted elsewhere [8]; they are not relevant to the discussion of coordination polyhedra in this paper. The conventionally used set of five orthogonal d orbitals contains two types of orbitals, namely the xy, xz, yz, and x2 –y2 orbitals each with four major lobes and the z 2 orbital with only two major lobes (Table 1). All possible shapes of d orbitals

145R.B.King/CoordinationChemistryReviews197(2000)141-168can be expressed as linear combinations of these two types of d orbitals by thefollowing equation[9,10]:E(4)d=ap_2+(1-a2)/2Φx2-y20.866025≤a≤12InEq. (4),referstothe wavefunctionofthedatomicorbital andrefers to the function of the d,2-p2 atomic orbital, taken as a representative of oneof the four d orbitals with four major lobes.Two different sets of five orthogonalequivalent d orbitals can be constructed by choosing five orthogonal linear combi-nations of the d,2 and d,2-y2 orbitals using Eq. (4) [9,10] These are called the oblateand prolate sets of five equivalent d orbitals since they are oriented towards thevertices of an oblate and prolate pentagonal antiprism,respectively.Thefive-foldsymmetry of these equivalent sets of five d orbitals makes them inconvenient to usesince relatively few molecules have the matching five-fold symmetry.2.2.Valencemanifoldsof atomic orbitalsValence manifolds of atomic orbitals are the sets of atomic orbitals havingsuitable energies to participate in chemical bonding.The geometry of such valencemanifolds of atomic orbitals relates to contours of the sum over all orbitals inthemanifold.Spherical atomic orbital manifolds arevalencemanifolds of atomicorbitals containing entire sets of atomic orbitals having a given value of theazimuthal quantum number, l, and are isotropic, i.e., they extend equally in allTable 1Properties of s, p, and d atomic orbitalsType[mlNodesAngularPolynomialAppearance and orbitalShapefunctiongraphM00PointIndependent ofSpherically symmetrical0,$110111xPPPsin cos LinearANsin sincOs 02211d２２２2２sin* 0 sin 20Squarexydx2-y2sin* 0 cos 20ax2sincoscosdy2sin cos sind02222p2(3 cos201)Linear8(abbreviated as2)

R.B. King / Coordination Chemistry Re6iews 197 (2000) 141–168 145 can be expressed as linear combinations of these two types of d orbitals by the following equation [9,10]: d=afz 2+(1−a2 ) 1/2 fx 2−y 2 3 2 =0.8660255a51 (4) In Eq. (4), fz 2 refers to the wave function of the dz 2 atomic orbital and fx 2−y 2 refers to the function of the dx 2−y 2 atomic orbital, taken as a representative of one of the four d orbitals with four major lobes. Two different sets of five orthogonal equi6alent d orbitals can be constructed by choosing five orthogonal linear combi￾nations of the dz 2 and dx 2−y 2 orbitals using Eq. (4) [9,10] These are called the oblate and prolate sets of five equivalent d orbitals since they are oriented towards the vertices of an oblate and prolate pentagonal antiprism, respectively. The five-fold symmetry of these equivalent sets of five d orbitals makes them inconvenient to use since relatively few molecules have the matching five-fold symmetry. 2.2. Valence manifolds of atomic orbitals Valence manifolds of atomic orbitals are the sets of atomic orbitals having suitable energies to participate in chemical bonding. The geometry of such valence manifolds of atomic orbitals relates to contours of the sum  c2 over all orbitals in the manifold. Spherical atomic orbital manifolds are valence manifolds of atomic orbitals containing entire sets of atomic orbitals having a given value of the azimuthal quantum number, l, and are isotropic, i.e., they extend equally in all Table 1 Properties of s, p, and d atomic orbitals Type m Nodes Appearance and orbital Polynomial Angular Shape function graph s 0 0 Spherically symmetrical Point Independent of u, f p 1 1 x sin u cos f Linear p y 1 1 sin u sin f p 0 1 z cos u d 2 2 xy sin2 u sin 2f Square 2 2 x2 −y2 d sin2 u cos 2f d 1 2 xz sin u cos u cos f d 1 2 yz sin u cos u sin f 0 (3 cos2 d 2z u−1) 2 −r 2 2 Linear (abbreviated as z 2 )

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-elec￾tron 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 forbid￾den 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 mani￾fold 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 compo￾nent 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-coor￾dination, 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