simetría

symm_etry_incoord_|nat|onchemistry

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Contents

Preface . . _ . . . . . . . . . . . . . _ _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . isAcirnowiedgrnicnts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . xi

1. Molecular Symmetry and Point Groups

Molecular Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Coordination Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 4

Symmetry Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . _ .. . . . . . . . . . . . . 12

Symmetry Point Groups. . . . . . . . . . . _ . . . . . . . . . . . . . . . . . .. . . . . _ . . . . . . 13

Multiplication of Symmetr;-,r Operations. . , . . . . . . . . . . . . . . . . . . . . . . . . . 14

Point Group Symmetry . . . . . . . . . . . . I . . . . . . . . . . . . . . . . .. . . . . I . . . . . . 1?

Supplementary Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2. Atomic and Electronic Properties of cl-Block Transition Elements

Transition Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 23

Periodic Build-Up . . . . I . . . . , , , . . . . . . . . . . . . . . . . . . . . . . . . , . , . . . . . . .. 25

Eiectrenie Properties of the atoms and Ions . . . . . . . . . . . . . . . . . . . . . . . . 26

Energy Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 31

Atomic Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 33

Russel1Saunders Classication . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 34

Magnetic Properties _ . . . _ . . . . . . . . . . . . . . _ . . . . . . . . . . . . . . . . . . . . . . . . . 39

Supplementary Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3. Orbital Symmetries and Their Representation

Orbital Symmetry . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . .. . . _ . . . , . . . _ . . . 43

Representations of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . .. -15

Reducihle Representations . . . . . . . . . . . . . . . . . . . I . . . . . . . . . . . . . . , . , . . . 5|]

Reduction of Reducible Representations . . . . . . . . . . . . . . . . . . . . .. . . . . . . 53

Supplementary Reading . . . . . . . . . . . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . . . 61

vii

viii Contents

4. Symmetry Applied to Molecular Vibrations

Representing Vibrational Symmetry . . . . . . _ . . . . . . . . . . . . . . . . . . . . . . . . 63

Types of Molecular Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66

Selection Ruies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T2

Infrared-Active Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 73

Structural Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . T5

Supplementary Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T?

5. Chemical Bonding in Transition Metal Compounds

Crystal Field Theory _ . . . . . . . . . . . . . . . . . . . . . . . . . _ . . .. . . . . . . . . . . . . . TSLigand Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 95Molecular Orbital Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 9?Supplementary Heading , . . . . . . . . . . _ . _ . . . . . . . . . . . . . . . . . . _ . . . . . . . . . 109

6. The rl-Block Transition Elements and Their Chemistries

First Transition Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 111

JahnTeller Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . _ . . 113

Second and Third Transition Series . . . . . . _ _ . . . . . . . . . . . . . _ _. . . _ . . . . . 131

Miscellaneous Coordination Compounds . . . . . . . . . . . . . . . . . . . . . .. . . . . . 124

Appendix Some Important Character Tables . . . . . . . . . . . . . . . . . . . .. . . . . 129

Subject hides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . _ 13?

Preface

With the popularization of the symmerty rules designedby R. B. Woodward and Roald Hoffmann to describe certain organicreaction mechanisms it has become increasingly common to present thesymmetry properties of molecules in undergraduate organic chemistrycourses. In inorganic coordination chemistry, symmetry properties longhave been used to provide a simplied description of bonding and struc-tural properties. Since these concepts are so valuable I feel that an elemen-tary knowledge of molecular symmetry should be taught to all chemistrystudents. With this thought in mind I began to write this book, whichwill, I hope, bridge the gap between the elementary ideas of bonding andstructure learned by freshmen and those more sophisticated conceptsused by the practicing chemist. In practice, I have found the materialmost helpful to supplement the traditional course work of juniorseniorinorganic students. It is for them that the problems and examples have beenchosen.

I make no apology for excluding some importantsymmetry-related material in this book. However, I have attempted topresent topics which clearly emphasise the use of symmetry in describingthe bonding and structure of transition metal coordination compounds.I feel that within this framework an early exposure to the beautifullyintricate and symmetric micromolecular world dealing with molecular

structure and bonding can be achieved. This exposure should stimulate thestudent to think about how steric and electronic properties of molecules cancontrol their chemical reactivity. Once a student begins to think aboutchemistry in terms of reactions of discrete molecular species with deniteshapes, he is well on the way to discovering a whole new, exciting, andesthetieally pleasing chemical world.

ix

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Acknowledgments

Stimulation for writing this book can be attributeddirectly to attempts to teach this material to undergraduates. I have taughtseminars on symmetry in chemistry to college freshmen and long havefelt that the gifted beginning college student can grasp the concepts ofsymmetry and apply them to chemical problems. in fact, high schoolscience students also could prot from an early exposure to the concepts ofsymmetry as applied to the molecular world. However, abstract grouptheory, the underlying mathematical basis for the application of symmetryto chemical problems, usually is presented as such an esoteric subject thateven the practicing scientist rarely wishes to devote the time necessary tofully grasp the material. As a result, consequences of symmetry are oftengrossly neglected.

Certainly this book would not have been written if Ihad not learned about symmetry and its application to chemical problemsas a graduate student working with F. it. Cotton. His Chemical Applica-tions of Group Theory [Wiley (Interseience), New York, 1963] remainsthe best book available for the practicing chemist seeking knowledge of thesubject.

Thanks are given to all the various students of mine whohave read and criticized the manuscript but especially Charles Cowmanand James Smith. Helen Bircher really made the whole thing possible bytyping the original manuscript from a poor handwritten copy.

Finally, I would like to thank the International Unionof Crystallography for permission to reproduce the gures appearing inthe Appendix.

xi

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Molecular symmetry

and point groups

1

A complete theory of chemical bonding needs to makeno assumptions regarding chemical structure. Unfortunately, except forthe simplest compounds such as H2, HD, etc., a detailed theory is notcurrently available and is unlikely to be fully developed for many yearsto come. However, by using empirical and semiempirical structural rela-tionships that have evolved over a number of years, we often can correctlypredict molecular structures which usually can be conrmed or rejectedby spectroscopic techniques. These predicted or experimentally observedstructures can provide a basis for meaningful comments about bonding.

One approach to chemical structure starts with theassumption that similar types of ligands produce similarly structured

molecules. If, for example, the anion Coli is tetrahedral, it is reasonableto assume that CoBr,," also is tetrahedral. This approach has its obviouslimitations. However, additional relationships do occur to the extent thata student acquainted with modern concepts of bonding and structure islikely to guess that PtCl' is planar and Nl(0H2}:+ is octahedral. Hemight even correctly suggest that the uoride atoms surrounding manga-nese in KgNaMnF,, are at the corners of a distorted octahedron. Yet inspite of many recent successes in predicting structures, no one supposed

1

2 1 Molecular Symmetry and Point Groups

Figure 1-1 T rtpeuet prismatic structure as observed for Re(SgCgR2)a, where R = Celia.

that 1?.e[SgC2(C.;.H5.)2)_, would have a structure [Figure 1-1) in which thesix sulfur atoms surrounding the rhenium are arranged at the corners of atrigonal prism. Nor did anyone predict that the anionic dimer RegClwould have the cul:-ie" structure pictured in Figure 1-2, and contain aquadruple rnetal-metal bond. A description of the bonding in such sys-tems has had to await structural determinations.

Figure 1-2 Ecttpsed str ueture of ReCl3_.

Molecular Symmetry 3

MOLECULAR SYMMETR Y

The most commonly found structural arrangement forcoordination compounds is one in which the groups attached to the centralmetal atom appear at the corners of an octahedron. Associated with thisstructure are certain very important symmetry features. There are, forexample, three mutually perpendicular axes, 3:, y, and .2, about which arotation by u.(21r/4) rad, where n = 1, 2, 3, . . ., a (90, 180, 2?0', etc.],makes the molecule indistinguishable from the original species. If theligands along the 1: axis are identical to each other, but different from thefour identical ligands along y and s, only one such four-fold rotation axiswould exist. The octahedral molecule also contains a center of symmetry.This means that each atom has its counterpart at a position with coordi-nates of identical magnitude and opposite sign {Figure 1-3).

Since symmetry operations such as rotation, inversion,etc., produce no change in the molecule, any model of bonding we choosealso must be invariant with respect to these same symmetry operations.

I

Figure 1-3 An octahedral complex.


If, for example, we describe the bond between the metal atom and oneof the ligands in an octahedral complex in a particular way, the bondingof the metal to the other five ligands must be described similarly. Thebonding model we use must be consistent with the overall structure ofthe compound. Thus it is apparent that a knowledge of molecular struc-tures can be very important to an understanding of bonding in coordina-tion chemistrya chemistry noted for its numerous structural types [seeTables 6-3 and 6-4).

Ligand atoms which are appreciably distorted fromtheir free-atom or molecule symmetry by bonding to the metal atom

(they are said to be polnrteebte) often produce complexes with nonocta-hedral structures. For example, with the polarisable chloride ion, tetra-hedral complexes of the type MCIE" form for M = Mn, Fe, Co, Ni,Cu, and Zn. A trigonal bipyramidal CuCl also is known. When themetal is bonded to carbon as in cyanides or carbonyls, or to sulfur, phos-phorus, arsenic, and several other easily polarised atoms such as metalatoms themselves, unusual structures appear to be the rule rather thanthe exception. Chemists are just beginning to nd out why.'

COORDINA TION NUEFIBERS

In a systematic way we will now explore some of thedifferent symmetries that are observed when varying numbers of ligandsare attached or coordinated to one metal ion.

COORDINATION NUMBER TWO

Compounds of the type ML; may be linear or bent.In the former case all bonding properties must be the same on rotationby any angle about the molecular axis and on inversion through the centerof the molecule (Figure 1-4). With a bent molecule, rotation producingan indistinguishable structure is restricted to 21:7? rad {180'') about anaxis through M and bisecting the LML angle. No center of symmetryexists. Molecularly isolated, condensed-phase {solid or liquid) transition*metal compounds having linear or bent shapes are quite rare. However,some vapor-phase species such as gaseous Z1112 are known by structural

* Transition elements are dened in Chapter 2.

Cuurdinatiun Numbers 5

,7-.

Center 01' _E.-syrmrruetry. r

I. K

Figure 1-4 Linear Don. and beat Ch. lriatonric ML, compounds.

studies to be linear. Gaseous MCI; species with M = Mn, Fe, Co,Ni, and Cu are also thought to be linear from spectroscopic studies.Nontransition metal compounds of both structural types are known; H33and F20 are examples of bent compounds, while C02 and CS2 are repre-sentative of the linear M14 shape.

COORDINATION NUMBER THREE

Either a trig-anal planar or a pyramidal shape is ex-pected for ML; (Figure 1-5). Both structures have an axis about which a

L I3/I

/U\'L-)/ LHas

Figure 1-5 Three-coordinate planar and pyramidal structures.


rotation by either 2-.rr/3 or 4-.-r/3 (l20 or 240) leads to an equivalentstructure. The trigonal planar compound also has a syinrnetry plane which

contains all four atoms. Three-coordinate transition metal compounds alsoare rather uncommon in the condensed phase. Some complexes of Cu,Ag, and Hg are thought to contain essentially a trigonal planar coordina-tion. The species BF; and PF; are examples of ncntransitien metal com-pounds with established planar and pyramidal geometries, respectively.

COORDINA TI ON N UMBER F O UR

Two idealised structures are observed for the coordina-tion number four. These are the tetrahedral and square planar arrange-ments cf ligands (Figures 1-6 and 1-7). Since the tetrahedron can beinscribed in a cube, the structure is called cubic. This always implies thepresence of considerable syrmnetry. In particular, there are four three-foldrotation axes in every cubic structure (Figure 1-6}. These are the principalaxes of a tetrahedron. Rotation about these axes by 2-.-r/3 rad or multiplesthereof makes the rotated species indistinguishable from the original.There is no center of symmetry in the tetrahedron. Maily tetrahedraltransition metal compounds are known, especially with anionic halide or

Figure 1-15 Telrahedral and octahedral coordination showing the presence of fourthree-fold axes.

Coordination Numbers 7

ii" W.--:3 I I' .--"""' .T L) -J /L .._) ".r'| /. ' ,4" Ie , c .- F)Symmelr-ll sf l Symmetry Icenter * center '-c, c,

Figure 1-? Square planar ML; and trans planar IvIL1L._ coordination.

pseudohalide ligands, as indicated earlier. Nickel tetracarbonyl, Ni(CO}.:,also is tetrahedral.

A planar ML; stereochemistry is very commonly ob-served with bivalent platinum, palladium, copper, and nickel, and nearlyalways with trivalent gold. Certain ehelate* ligands such as

l '-12...

R/

tend to promote the formation of planar compounds with many metalions.

The square planarf ML. structure is characterised by aprincipal four-fold rotation axis (2.u-/4) with four two-fold rotation axesperpendicular to it (Figure 1-7). A center of symmetry also exists. Oftenplanar compounds are found in which three two-fold axes (two in theplane, one from C4), the symmetry plane, and the center are preserved,but the four-fold rotation axis and two two-fold axes of the square aremissing. an example is trans-Pt (NH-,.)gClg, neglecting hydrogen atoms.

' The word chelate, xelta, from the Greek, means claw of a crab.1 While square" requires planar," the use of "square planar by chemists is socommon that no effort will be made to change this practice here.


Exercise 1-1 A square planar structure can be achieved from atetrahedral one (and vice verse) by a twist or rotationabout the two-fold axis of two ligands relative to theother two ligands. A clockwise twist may lead to adifferent isomer from that produced by a counterclock-wise twist. Consider a tetrahedral Mabcd complex withCa. bisecting angle a-Mb, Ca bisecting cMb,and C2,; bisecting bMd. How many different planarisomers are produced from the two (optical) tetra-hedral isomers? Draw them.

COORDINATION NUMBER FIVE

A ve-fold rotation axis (2-.r/5) has not been found ina compound in which the metal has a coordination number of live. How-ever, the square pyramidal structure (Figure 1-8) is present in sometransition metal compounds. While few transition metal compounds withve identical ligand atoms have this shape, several compounds of thetype MAB; are known which approximate it. An X-ray structural deter-mination has shown, for example, that VO(AcAc)g. (Figure 1-9] hasnearly a square pyramidal arrangement of the oxygens about the metal.The interaction of planar Cu compounds with bases such as pyridine,C5H,=,N, may give similarly structured compounds.

A well-known example of a trigonal bipyramidal five-coordinate transition metal compound (Figure 1-10) is iron pentacarbonyl,Fe(CO),=,. In this compound, the principal rotation axis is three-fold. Inaddition to three twofold axes perpendicular to the threefold axis, thereare four mirror planes, each containing three carbonyls and the iron. Thethree ligands at the corners of the equilateral triangle perpendicular to thethree-fold axis are called equatorial ligands; the other two are called axial.

COORDINATION NUMBER SIX

In addition to the octahedral structure which is verycommon, the trigonal prismatic structure (Figure 1-1) is also known. The

"" One type of Ni(CN]' ion in {Cr(l*lH1CH:CH:NH,}3][Ni(CN):.]- 1.5 Hi0 has 11.Square P1 1'3-Tidl Nil-as structure, while another is a distorted trigonal bipyramicl.

Coordination Numbers

4'r' -lily;as

Figure 1-9 Vanadytacetylacetonate, VO(Ac.ll.c);.

Cs..l;;.uI

/guy

Figure 1-10 Trigonal liipyrarniital structure.


best example of this arrangement is found in crystalline MoS2. A few sulfurligand complexes besides REESgC2{C5H5lg]3 also are thought to have asimilar arrangement of atoms around the metal. In this structure, thereare three two-fold rotation axes perpendicular to the three-fold axis; foursymmetry planes are also present.

COORDINATION NUMBER EIGHT

Three import.ant structural arrangements have beenfound for compounds in which the coordination number of the metalatom is eight, with identical ligands. These are the square antiprismaticconguration typied by Tall: , the dodecahedral structure found inMo(CN) (Figure 1-11) and the cube. A cubic structure has been ob-served recently for the anions in Na3MF5, M = U, Pa, Np. In the dodeca-hedral structure there exist three two-fold rotation axes and two mirrorplanes. In the square antiprismatic conguration the principal rotationaxis is four-fold. No inversion center is present in either structure.

The symmetry operations of a cube are identical tothose of the octahedron. A tetrahedron displays only one-half of theseoperations.

Figure I-ll Square antiprisrnatic and dodecahedral ML; structures.

Exercise I-2 By means of a molecular model, prove to yourself thatan inversion center combined with all the symmetry

Coordination Numbers ] 1

Figure 1-12 The symmetric, facecentered trigonal prism ML; structure of the ReHEanion in K2RBHg.

present in a tetrahedron produces the symmetry opera-tions of an octahedron.

Hint Remember that a tetrahedron can be inscribedin a cube.

COORDINATION NUMBEIES SEVEN, NINE, AND HIGHER

Since the number of transition metal compounds havingthese coordination numbers denitely established is limited [some ex-amples are NbF, TaF$, and Nb(H2O)3+:|, the various idealisedstructures observed will not be described here. However, it is apparentthat a trigonal prismatic compound containing atoms in the centers of therectangular faces represents a reasonably high symmetry (Figure 1-12).The anion ReH has such a structure.

12 1 Nlolecular Symmetry and Point Groups

SYMMETR Y CLASSIFICATION

As we have observed, the presence of rotation axes,inversion centers, etc. in a molecular structure implies that certain move-ments of the molecule, symmetry operations, produce structurally identicalorientations. These symmetry operations are a direct result of the presenceof various symmetry elements, the rotation axes, inversion center, etc.With the square planar PtCl:", anion, the presence of the four-fold rotation

axis perpendicular to the plane of the molecule (see Figure 1-?) meansthat the molecule can be rotated around this axis by 90 (2-r/4), 180(2 X 2s/4), 2T0 (3 )< 2i-/4), and 360 (4 X 2r/4) to produce an equiva-lent or, in the latter case, an identical conguration. This axis is labeledC4 or simply 4. In general, a 0.. or n operation* implies rotation by 2r/nrad.

In addition to the symmetry elements we have alreadyconsidered, there is one additional element called an improper rotation axis.We speak of n-fold improper rotation axes, S... (The proper rotation axisis C...) The operation implied by the improper rotation axis is dened asa rotation by 2:-r/n rad about the axis followed by reection through aplane perpendicular to this axis. If an equivalent structure results, an Saxis is present.

If we examine one of the three-f old rotation axes presentin an octahedron (Figure 1-6), we see that it is also a six-fold improperrotation axis, S.-,. Rotation about this axis by 60 (21r/6) followed byreflection in a (hypothetical) plane perpendicular to this axis produces anequivalent structure. Repeating this process six times brings the gureback to its original conguration. This would not be true, however, ifn were odd. In such cases, we must repeat the operation 2n times beforethe structure returns to the original conguration.

Table 1-1 contains a summary of the symmetry ele-ments used to describe molecular structures, together with their requiredsymmetry operations. Two additional symmetry elements are necessaryto completely describe three-dimensional crystalline solids, since transla-tional symmetry may be present. To discuss molecular bonding, however,we need consider only those operations which do not change the positionof the molecule in space. This means that some point in the molecule,real or imagined, is not moved by any of the symmetry operations associ-

"' Spectroscopists use On. the Schoenies notation, while crystallographers use theArabic numeral n, the HermannMauguin notation, to describe an n-fold axis.

Sym metry Point Groups 13

Table 1-1 Symbols (Hermann-Mauyuin and Schoenies) for symmetry elements andtheir implied operations

Element Operation

n, 05 Rotation by 211-in rad

I, i Inversion through a symmetry center

.8... Rotation by 211',/ft rad followed by reflection through a plane perpendicular

to S... (Schoenies notation only)

11 Rotation by 2:-x'n rad followed by inversion (Hermann-l'vIauguin notationonly}m, c Reflection through the symmetry plane

ated with the moleculethus, we speak of a point symmetry. In the caseof the planar PtCll,, this point and the Pt atom coincide. In fact, withmany transition metal compounds, the transition metal itself is at thispoint. When translations are also considered, we speak of space symmetry.

SYMMETRY POINT GROUPS

A collection of elements related by certain specific rulesconstitutes an abstract mathematical group. We will state explicitly whatthese rules are and leave a detailed discussion of their origin to more ad-vanced texts.*

I. The product of any two elements in the group and the square ofeach element must be an element in the group. (ctosons)

2. One element in the group must commute with all the others andleave them unchanged. (IDENTITY)

3. The associative law of multiplication must hold. (assoelarlve)

4. Every element must have a reciprocal which is also an element ofthe group. (HECIPRDCAIJ

In classifying molecules into point groups it is neces-

sary to realise that an element as referred to in the above rules means

" Since this is a chemistry book, the statements of F. .41. Cotton as found in ChemicalApplications of Group Theory," Wiley (Interscience), New York, 1963, have been used.

I-2|. 1 loleeular Symmetry and Point Groups

a sgrnrnetrg operation--in other words, action. It should not be confusedwith a static element of symmetry discussed previously which indicatessome particular symmetry operation or operations such as a Cu axis. Theelements of rule 1 for a group containing only a 0 axis are the rotationsC,,, Ci, Ci, Ci, . . . , Cf, C3. The mathematical terms such as product,commute, associative, and reciprocal will be familiar to mostreaders. However, to avoid confusion they will be used only in the followingmanner in this book.

The product of the multiplication of etenients is the reeuttobtained by carrying out the iniptied operations in the order specied. Hencethe product of rotation by 2-ir/4 with itself is consecutive rotation twiceby 2r/4 or rotation by 1l'.TTll1S,'Cd >< C4 = C3 = C2.

The order in which we carry out the operations may heiinportant since not att operations witt cornrnute. With commuting opera-tions, AB equals BA. However, a reflection followed by rotation will notalways give the same product as a rotation followed by a reflection (unlessthe plane of reflection is perpendicular to the rotation axis). Hence theseoperations do not always commute, AB ;-5 BA.

An associative taw means that we can arrange the etenientstogether in whatever way we choose providing the order is preserved. HenceA >-(B)-

To see how the rules of group theory can be appliedto molecules and also to introduce some of the symbolism which is associ-ated with point groups, we consider the symmetry found in the interestingsandwich structure of ferrocene {Figure 1-13}. This structure, a twistedpentagonal prism, contains a ve-fold principal rotation axis coincidentalwith the axis labeled e. The presence of this C5 axis requires that rotationby in X (21:-/5), where in = 2, 3, 4, and 5, gives congurations indistin-guishable from the first one. The last rotation, C2, is symbolized by E,the identity operation, since it leaves the molecule unchanged. Along with

Prlultiplieation of Symmetry Operations 15

F V:

I-

he

52 air

Top view

Side lH'E"I'|l

Figure 1-13 The structure offerrocene, (I-C5H5)2FE.

the ve-fold rotation axis, 2 contains a ten-fold improper rotation axis.Rotation by 27:-/10 (36) followed by reflection in a plane perpendicularto this axis carries the molecule into an equivalent, indistinguishablestructure. This ten-fold rotationreection axis* is labeled Sm. As with C5,various products of 81.1 with itself must lead to indistinguishable congura-tions. However, some of the (n 1), nine, new operations we might ex-pect to nd are not labeled as muitiples of Sm. For example, S ten takenve times, 3,5,, gives the same result that would be obtained by inversion,i. This operation is labeled T, bar one in the crystallographic notation.The presence of the symmetry element requires the operation and vice

* Or it is a five-fold rotationinversion axis, 5 (bar ve").


versa. We will use the rotation1-eection terminology throughout most ofthis text, but the reader should be able to work with either system.

Parallel to the C5 (or Sm) axis in ferrocene are vemirror planes. The presence of any one such plane containing a (7.. axisrequires that (n 1) additional symmetry planes also contain this axis.These n planes are labeled or... or ad, depending on whether the planes con-tain two-fold axes perpendicular to C3,. (or...) or bisect the angle betweensuch axes (as). Rotation of a a... plane by C will produce the other asplanes. If as planes are present, rotation by 0.; will generate each of thesefrom an initial one. Only one set of vertical planes is present in ferrocene(Figure 1-13). These are org planes, since they bisect the angle betweenthe C2 axes. In planar PtCl' (Figure 1-7} both as and as planes are pres-ent. A plane perpendicular to the principal rotation axis is given the specialsymbol ah (horizontal). No such plane is present in ferrocene, but thereis one in PtClE.

The presence of one two-fold axis perpendicular to theC5 axis in ferrocene requires that four more C: axes be present. Rotationby 2-.r/ 5 about e carries one such C1 axis into another.

In summary, the following twenty symmetry operationsare present in the structure of the molecule ferrocene: E, C.=,( = 3,), CE,

0:, Cl, Sm, S3,, i(.S,"',;.}, Sf, Sf',,, ve cafe, and ve Cifs. Taken together,these constitute the symmetry operations of the point group labeled D5,1.*By appropriate combination of these operations, it is easy to construct amultiplication table and prove to yourself that the operations listed forD5,; constitute a mathematical group. Table 1-2 is the multiplication tablefor the point group Cit. Here each of the four operations leads to Ewhen combined with itself. Thus E appears on the diagonal. In con-structing multiplication tables, it is assumed that the product operationappearing at the intersection of any column with any row transforms an

Table 1-2 Multiplication Table for Ca

Group C E Ci ar..+(:..-'s) crt(gs)E E C: We II:C: C: E at -in.owlirzll on; at E 01at (ya) at as C: E

"" In point group tables or character tables, the elements are grouped accordingto theclasses to which they belong.

Point Group Symmetry 1'1

object in exactly the same way that the object would be transformed bysequentially performing the operations given at the top of the columnand the left-hand side of the row. In the C3,. point group all operationscommute with each other, for example, ::r..(:t:s) C2 = (72-c*..(.1.*s), and suchagroup is called Abelian. However, most groups of interest to us do nothave this property.

POINT GROUP SYMMETRY

The point group to which a particular molecular struc-tu re belongs now can be determined in a systematic manner. The line chartFigure 1-14 indicates the procedure we will follow. First look for specialtypes of symmetry such as are characterised by linear, cubic, and icosa-hedral structures. The linear molecule which contains a center of symmetrybelongs to the point group D,,,.,,; otherwise it is in C,,.,.. The innity symbolimplies that an innite number of vertical planes may contain the mole-cule. Since the cubic groups T, 0, Ta (tetrahedral), Oh (octahedral), andT1, require the presence of four tetrahedrally oriented three-fold axes, thissymmetry is readily recognised, and so is the regular icosahedron, I;,,which has six ve-fold axes.

Molecules which do not display any type of rotationaxes belong to (3,, Ci, or C1. In the latter case there is no symmetry at all;for the others only one element of symmetry is present in addition to theidentity element. These point groups are groups of order two; that is,they have only two elements.

If the only symmetry operations present in additionto an n-fold rotation axis are 2n-fold improper rotations, the point grouplS .8211.

The presence of is two-fold rotation axes perpendicularto the pririciipot o;ri's* places the molecule in a D point g'oupeitherD,,1,, Dad, or D,,. The subscripts indicate the symmetry planes, if any,that are present. Without the two-fold axes, the molecule belongs to Cub;C,,,., or C, and again the subscripts indicate the presence or absence ofsymmetry planes.

The Schoenies notation for a point group specifies theminimum symmetry required to dene a particular point group. This

" Sometimes the principal axis is not unique, as for example in D: or T. In suchcasesthe choice of axis to be called principal is arbitrary.

Sp-EC-n:.n groups

L inc-or; Cw D, Ierrnhedrol. T.Tm T,,. I:!||:1I:Il"rE!I:IH.'.I|', D,{.!I,, H:+:I-so-

heurcl. !|,_.-:-FoI::I nznnhon mus. No -|:|HJ|:|El' I:|x-isEels,-qr principal axis'5-ymrrl-Elry ponl53,, :" C: -'Crllglr 5?" lI'l Eld-lllD-2'1r: rvro-loll: nice No '-A-IJ-EDIE one-3 1m.ngr5.p.n cg:-113;I IPeroeno-color !o C... llNo srmrrrchrHonroruol clone, or" Herlzenlul alone. :5, C,Elan C'11- -.-e-r1:-ml planes, era :1 -.-ern-:oI planes, or.H 4 r 4No prunes of D, No plane of C,.,..,symmetry Q, 5.yn1rneIry E,

Figure 1-14 Linc c.'uu'l ide:-Hicaon of point group symmetry.

Point Group Symmetry 19

fact is recognised quickly by the use of a mnemonic called a stereographicprojection. (Stereographic projections originally were developed to describecrystal faces, but these details need not concern us here.) We will usestereographic projections simply to represent the symmetry operationspresent in point groups.

As an example of stereographic projection constructionwe consider the point group D, (Figure 1-15). The symmetry present

-er

Figure 1-15 Construction of the stereographic projection for the symmetry operationsfound in Dih. (Reproduced by permission of the International Union ofCrystallography.)


will be indicated by symbols on a circle (Figure 1-15a). The z axis projectsfrom the center of the circle. It contains the four-fold rotation axis and isgiven the symbol of a solid square, the appropriate regular polygon sym-metric to rotation by 2r/4 (Figure 1-15b). it two-fold axis is specied bya solid ellipse, a three-fold axis by a solid triangle, etc.

The point group De, requires the presence of a two-foldaxis (Figure 1-15c) and a mirror plane perpendicular to the four-fold axis.The mirror plane is indicated [Figure 1-15d) by the heavy circle. Now,by placement of a general point C) (e:,y,s) as in Figure 1-15d, the threesymmetry elements already listed cause the generation of fteen otherpoints (Figure 1-15c). Eight of the sixteen points lie above the plane of

the paper and are labeled -:3, while those below are given the symbol X.It is now easy to add the other symmetry operations suggested by thearrangement of Us and )

compressed relative to the equilibrium bond length?

Carbon monoxide is linear, as is CO2. What symmetryfeature distinguishes these two molecules? Would thisbe true if CO: were CO-0?

A bunch of marbles on a tray can be made to pack to-gether with all marbles in maximum contact. Maximum

22 I llrloleculur Symmetry and Point Groups

density is achieved with a very symmetric structure.What is the highest-order rotational axis found per-pendicular to the tray? What others are present?Can a ve-iold axis appear in any packing arrangementwhich has a well-defined number of marbles in a groupwhich repeats as a unit on the whole tray?

Exercise I-3 By means of models, deduce the symmetry operationspresent and the point group for the following molecularspecies:

(a) Tctrahedral NiBrCl"

lb) Octahedral cis-CoCl3F:

(c) Clctahedral trans-CoCl;;F'

(Cl) Linear HgClg

(e) Bent HES

(f) Tctrahedral CoCli,

lg) Planar cis-PtCl2Br{

(h) Planar iTii-3-PtClgBT:_

(i) Trigonal bipyramidal PF5

(j) Trigonal bipyramidal I.'3iS-PF3Cl2(k) Trigonal bipyrimical transPF 3C1:(1) Square pyramidal CuCl"

Exercise 1-9 Sketch a stereographic projection for the point groupD3. Add a center of symmetry to the gure. What newpoints are developed? To what point group does thenew gure belong?

SUPPLEMENTARY READING.

F. A. Cotton, "Chemical Applications of Group Theory. Wiley nterscicncc), NewYork, l93.

F. A. Cotton and G. Wilkinson, Advanced Inorganic Chemistry," 2nd ed., pp. 11~122.Wiley llntcrsciencel, New York, lil.

F. C. Phillips, "An Introduction to Crystallography," 3rd ed. Wiley, New York, 1964.

G. II. Stout and L. H. Jensen, X-Ray Structure Determination," Chap. 3. Macmillan,Nitw York, 1953.

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