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 compounds 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 experimental 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 experimentally 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 stereochemistry 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 differential 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 combinations 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 )