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1.3 Summary of Operations, Symmetry Elements, and Groups. axis. A rotation by 360˚/n that brings a three-dimensional body into an ^ equivalent configuration comprises a Cn symmetry operation. If this operation is performed a ^ ^ ^ 2 second time, the product CnCn equals a rotation by 2(360˚/n), which may be written as Cn . If n ^ ^ m is even, n/2 is integral and the rotation reduces to Cn/2. In general, a Cn operation is reduced ^ 6 ^ 2 by dividing m and n by their least common divisor (e.g., C9 = C3 ). Continued rotation by

360˚/n generates the set of operations: ^ ^ 2 ^ 3 ^ 4 ... ^ n Cn, Cn , Cn , Cn , Cn ^ n ^ ^ n+m ^ m where Cn = rotation by a full 360˚ = E the identity. Therefore Cn = Cn . Operations resulting from a Cn symmetry axis comprise a that is isomorphic to the of order n. If a contains no other symmetry elements than Cn, this set constitutes the symmetry for that molecule and the group specified is denoted Cn. When additional symmetry elements are present, Cn forms a proper subgroup of the complete symmetry point group. that possess only a Cn symmetry element are rare, an example being + Co(NH2CH2CH2NH2)2Cl2 , which possesses a sole C2 symmetry element.

N Cl N Co

Cl N N

^ One special type of Cn operation exists only for linear molecules (e.g., HCl). Rotation by any around the internuclear axis defines a symmetry operation. This element is called ^ f C• axis and an infinite number of operations C• are associated with the element where f denotes rotation in decimal degrees.

We already saw that molecules may contain more than one rotation axis. 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F F

B

F currentpoint 192837465

triangular projection. Three C2 axes containing each B-F bond lie in the plane of the molecule

to the three-fold axis. The rotation axis of highest order (i.e., C3) is called the

principal axis of rotation. When the principal Cn axis has n even, then it contains a C2 operation

associated with this axis. Perpendicular C2 axes and their associated operations must be denoted

with prime and double prime superscripts.

Reflection planes. Mirror planes or planes of are symmetry elements whose associated operation, reflection in the plane, inverts the projection of an object to the mirror plane. That is, reflection in the xy plane carries out the transformation (x,y,z) ∅ (x,y,-z). Mirror planes are denoted by the symbol s and given the subscripts v, d, and h according to the

following prescription. Planes of reflection that are perpendicular to a principal rotation axis of

even or odd order are named sh (e.g., the plane containing the B and 3F atoms in BF3). Mirror

planes that contain a principal rotation axis are called vertical planes and designated sv. For

example, in BF3 there are three sv planes, each of which contains the boron atom, fluorine atom,

and is perpendicular to the molecular plane. Notice that for an odd n-fold axis there will be n sv

mirror planes which are similar. That is if we have one sv plane, then by operation of Cn a total

of n times, we generate n equivalent mirror planes. The n sv operations comprise a conjugate

class. For even n the sv planes fall into two classes of n/2 sv and n/2 sd planes, as illustrated

earlier in the chapter.

Rotoreflection axes or axes. This new operation is best thought of as the stepwise product of a rotation, and reflection in a plane perpendicular to the rotation axis. ^ ^ We need to introduce it since a product such as C6sh is not equal to either a normal rotation or

reflection. To satisfy group closure a new symmetry operation must be introduced. An

21 22

^ ^ inversion operation is the simplest rotoreflection operation and is given the name S2 or i . That ^ ^ ^. ^ ^ . S2 = i is immediately apparent because i (x,y,z) = (-x,-y,-z) and C2(z)s(xy) (x,y,z) = ^ . ^ ^ ^ ^ ^ C2(z) (x,y,-z) = (-x,-y,-z). Because Cn and sh commute we have Sn = Cnsh for the general ^ n ^ ^ m ^ rotoreflection operation. Since sh = E for even m and for m odd sh = sh, an Sn axis requires simultaneous independent existence of a Cn/2 rotation axis and an inversion center when n is even. For example, an S6 axis generates the operations for the point group S6. They are shown below. ^ ^ 2 ^ 2^ 2 ^ 2 ^ 3 ^ ^ ^ ^ 4 ^ 2 ^ 5 ^ 5^ ^ 6 ^ S6, S6 = C3 sh = C3 , S6 = S2 = C2sh = î, S6 = C3 , S6 = C6 sh, S6 = E.

An odd order rotoreflection axis requires independent existence of a Cn axis and a sh symmetry element. An S7 axis generates the following operations. ^ ^ 2 ^ 2 ^ 3 ^ 4 ^ 4 ^ 5 ^ 6 ^ 6 ^ 7 ^ 7^ 7 ^ ^ 8 ^ 8 S7, S7 = C7 , S7 , S7 = C7 , S7 , S7 = C7 , S7 = C7 sh = sh, S7 = C7 = ^ ^ 9 ^ 2^ ^ 10 ^ 3 ^ 11 ^ 4^ ^ 12 ^ 5 ^ 13 ^ 6^ ^ 14 ^ C7, S7 = C7 sh, S7 = C7 , S7 = C7 sh, S7 = C7 , S7 = C7 sh, S7 = E

The S7 axis generates a cyclic group of order fourteen and requires the independent existence of

C7 and sh. This group is denoted C7h and not S7.

Conjugate Symmetry Elements and Operations. In the of BF3 the ^ -1^ ^ ^ sequence of operations C3´ svC3 means to rotate by 120 ˚ perform sv, then rotate back 120˚.

This sequence of operations is equivalent to reflection in a plane rotated 120˚ from the initial ^ 2 -1^ ^ 2 one, as shown below. The transformation (C3 ) svC3 generates an operation, which is the third vertical mirror plane. The three equivalent sv symmetry operations belong to the same conjugate class. Symmetry equivalent symmetry operations (i.e., operations belonging to symmetry elements that are interchanged by symmetry operations of the group) belong to the same conjugate class. In BF3 all the operations associated with the equivalent C2 elements ^ -1 ^ ^ belong to the same class. This may be verified by the similarity transformations C3 C2C3 and ^ 2 -1 ^ ^ 2 (C3 ) C2C3 operating on any of the three C2 axis.

22 23

s sn sn n

1 2 ^ ^ 2 ^ -1 C3 sv C3

3 2 1 3 3 1

sv 1 3 generated conjugate sv from a conjugate sv 1 2 3 2

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l0lp grnprxdv/m2 s/b1-1 px OB1lt{pp exec}{alr}b/x{epy slxlt{Ar1}{gsaL r0.5 dv5ac -10cmsmCA 2L/sg/setgray-0.40 sscnp lp0g}ie x L/cw/currentlinewidthnppx 1 dv12rcurvetocma3 32x}if 0 sgsm dx180OA}{120cwslW x/lWaRradpy27 1x 0py pwF py-0.4OB cY d/rO{4 smfill d}b/cmrly lxn/dx2 bWlWppstsmvlW -8 ew rOCA pm m}if/bds/ly mey bW5exx/bW gr st}{pys 2 2p l dp270 dppx 1oac.135 2o DA}{1cms dv dvcmmvne{lyx ZLBCAne{bW}{bd 0roxlW dvom2 scL/sl/se 2- are possible. Consider the planar ion PtCl4 , which contains one four-fold axis normal to the molecular plane and four perpendicular C2 axis as shown. Clearly the pair of axis labeled C2´

C2" C2' C2' Cl Cl

Pt C2" Cl Cl y

x currentpoint 192837465 are equivalent, as are the pair of axis C2" since the four-fold rotation interchanges elements among the pairs; however, the C2´ and C2" axis are not equivalent. There is no operation that will interchange these elements. Primes designate C2 axes perpendicular to an even principal ^ 2 ^ axis. The C4 operation associated with the four-fold axis is called C2. You might be surprised to find that a C4 axis and one perpendicular C2´ axis generate the remaining three "primed" two- ^ ^ fold axes. Action of C4 on one C2´ axis generates the other C2´ axis. With the application of C4 ^ ^ ^ or C2 no new axes are generated; however, consider the product of operators C4C2´ acting on the point (x,y,z). Take one C2´ axis to lie along the x axis and choose C4 coincident with the z

23 24

^ ^ axis. The operation C2´ on the general point (x,y,z) produces a new point at (x,-y,-z). The C4 ^ operation does not change z and rotates the molecule in the x,y plane. So C4 (x,y,z) ∅ y (-y,x,z) • ^ • C4 x x

^ ^ ^ (-y,x,z) and the product C4C2´(x,y,z) = C4(x,-y,-z) = (y,x,-z) is equivalent to rotation around the ^ C2" axis bisecting1361940[1]dgr [01end the 983900 I 6120 1700xy 77 quadrant 6020 294 17607200 40 3900 of 6140DSt 1296 the 7180][1coordinate 12 20I 3900Ar 20 3720 chemdict system7180 6080 becauseDSt begin 3900Db C2 6080"SP (x,y,z) 6 ≠Ar (y,x,-z). 3220 6660 2380 6660 2 Ar /bs[[1 1 1700 6040 1700 7200] %userdict/chemdictL/gr/grestoreL/tr/transformxl0pplSARA}{6-1st}b/OrA{py-8dppycpnpgsfilleo5awyInx w 02 dp/cYcwpy1-1gbWChemDrawCopyRight dpp cmfillpA sc8DA}{cwgr-1pp}{sqrt0m2p2oix dv1p l 2 -9.6r lt{1-2mv m p2px scsmv180 cmHA}{dL0gr}{pp}{gs a}b/PT{8-1dp exec}{aldvsms{dp dvm gmvdv dprO dv-.6 x WIpy -1 exec0sc smnegb2 lp 12stCAnp wF5 0pn mvpy0 at bdp LB 1.2scpxneg grm a}{ex1986, pmvest 2.251-9.6Laser plOA}{1emarcn 2L/gs/gsavep clip}b/Ct{bspp naLneg}if/pypexlcpgssl0L/xl/translatenpOBatmvgr 5 pmst}b/HA{lW lwy12 0op mv 1 3.375ypx-1 0 fillmvn/ex81ro gsnSA16.8 90lppy npl1451st}{0 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C2" (x,y,z) • x

^ ^ ^ This proves that the combination of a C2´ operation with C4 generates the C2" operation, and ^ that the two C2´ belong to the same conjugate class. You may prove that the two C2" comprise a second class. There are likewise two classes of vertical reflection planes. The two planes containing the C4 axis and one C2´ axis are called sv. Those mirror planes containing the C2" axis are denoted sd. This illustrates an interesting consequence of a vertical reflection plane. Consider ^ ^ ^ the effect of sv or sd on rotation around the principal axis. The effect of sv is to change the sense of

sv

m -m Cn Cn currentpoint 192837465

24 25 rotation from counterclockwise to clockwise. This means that rotation by 360˚/n and -360˚/n are ^ m ^ -m equivalent. In other words a rotation Cn and its inverse Cn belong to the same conjugate class when a mirror plane contains the rotation axis. For example, the operations of a C7 axis 6 2 5 3 4 5 fall into the classes (C7, C7 ); (C7 , C7 ); (C7 , C7 ) and for a C6 axis the classes are (C6, C6 ); 2 (C3, C3 ); (C2) when a sv or sd mirror plane also exists.

Molecular symmetry groups. When you search for elements, look for , reflections and rotoreflection elements, which interchange equivalent atoms in a molecule and leave others untouched. The short hand notation for the various point groups uses abbreviations, which often specify the key symmetry elements present. The complete set of symmetry operations of a molecule defines a group, since they satisfy all the necessary mathematical conditions. When confronted with a molecule whose symmetry group must be determined, the following approach can be used.

Special symmetry groups usually are self-evident. The group D•h is the point group of homonuclear diatomic molecules and other linear molecules that possess a sh plane (e.g., CO2).

For linear molecules an infinite number of sv planes always exist. The symbol D in a group name always means that there are n-two-fold axes perpendicular to the n-fold principal axis. The presence of sh and sv planes in a molecule always requires the presence of n^C´2 axes, since ^ ^ ^ one can show that shsv = C´2; however, the converse is not true. Therefore, there are an infinite number of C2 axes perpendicular to the C• axis of the molecule. The C• axis and sh generate f ^ an infinite number of S• operations; one is S2 = î.

If a linear molecule does not contain a sh plane, e.g., HCl or OCS, then it belongs to the

C•v point group. This point group contains a C• axis and an infinite number of sv planes.

Clearly C•v is a proper subgroup of D•h.

A free atom belongs to the symmetry group Kh, which is the group of symmetry operations of a . We shall not be concerned with this point group; however, a

25 26 consequence of spherical symmetry is the requirement that the angular wavefunctions of atoms behave like spherical harmonics.

There are seven special symmetry groups, which contain multiple high order (Cn, n > 3) axes. These are the only possibilities for 3-dimensional objects. The groups may be derived from

Oh Td Oh 3 C 4 4 C 3 3 C 4

Ih Ih 12 C 5 axes 12 C 5 axes

Figure 1.16. The Five Platonic Solids and Their Point Groups, with The High Order C n Axes Listed.

26 27 the symmetry operations of the five Platonic solids (octahedron, tetrahedron, cube, icosahedron, dodecahedron). Platonic solids are solids, shown in figure 1.16, whose edges and verticies are all equivalent and whose faces are all some regular . The symmetry operations of the tetrahedron comprise the group Td. The cube and octahedron possess equivalent elements and their operations define the Oh group.

The pentagonal dodecahedron and icosahedron likewise define Ih. We will restrict our discussion to the Oh group and its subgroups, O, Td, Th, and T, since Ih and its subgroup I are encountered less frequently. The collection of groups {O, Td, Th, and T} are often referred to as cubic groups. The symmetry elements belonging to an octahedron are: three C4 axes that pass through opposite verticies of the octahedron; four C3 axes that pass through opposite triangular faces of the octahedron; six C2 axes that bisect opposite edges of the octaedron; four S6 axes coincident with the C3 axes; three S4 axes coincident with the C4 axes; an inversion center or S2 axis coincident with C4; three sh which are perpendicular to one C4 axis and contain two others.

Therefore, a single plane defined as sh, also can be thought of as sv with respect to the other C4 axis. Similarly, there are six planes called sd, which are also sv planes containing a C3 axis.

Subgroups of Oh are generated by removing the following symmetry elements:

O = lacks i, S4, S6, sh and sd and is called the pure rotation subgroup of Oh.

Td = this group lacks C4, i and sh and is the group of tetrahedral molecules, e.g., CH4.

Th = this uncommon group is derived from Td by removing S4 and sd elements.

T = the pure rotation subgroup of Td contains only C3 and C2 axes.

If a molecule belongs to none of these special groups, then on the highest order rotation axis. Those rare molecules that possess only one n-fold rotation axis and no other symmetry elements belong to the Cn symmetry group, which is cyclic. If only a rotation axis exists and no mirror planes or other elements are obvious, a last check should be made for an Sn axis of order higher than the obvious rotation axis. When a higher order Sn axis can be found,

27 28

^ n ^ then the Sn groups (n = even) are obtained. Naturally n must be even since Sn = sh when n is odd and both Cn and s h exist independently. These groups (n = odd) are conventionally designated Cnh. Also, when n = 2, the S2 group (remember S2 = i) is conventionally called Ci.

If we add to a Cn group nsv (n odd) or n/2sv and n/2sd (n even) mirror planes, but no other rotation axes, then the Cnv groups are generated. If to the elements of Cn we add a horizontal ^ ^ mirror plane, then we generate the Cnh groups. The product of Cn and sh also generate an Sn symmetry element and its associated operations.

When additional n-two-fold axes perpendicular to Cn are present, but no mirror planes, then the Dn groups are generated. Addition of a sh plane to the rotation symmetry elements of a

Dn group generates the Dnh groups. Similar to the Cnh groups an Sn symmetry axis is also ^ ^ produced (if n is even an inversion center will be present). The product C2sh (where C2 is associated with n ^ C2 axes in the Dnh group) generates vertical mirror planes so that the Dnh group also contains nsv planes (n odd) or n/2sv and n/2sd planes (n even). If in addition to the elements of a Dn group an S2n axis is present, then the Dnd groups are obtained. The product ^ ^ ^ SnC2 (C2 belongs to a C2 axis perpendicular to the n-fold axis) yields the operation sd so that an additional n dihedral mirror planes are required. If a molecule contains as the only symmetry element a mirror plane, then the group is called Cs. Finally, if no symmetry elements, other than E, exist, then the molecular symmetry group is the trivial one called C1. Table I.1 summarizes the groups we just discussed and their associated elements. It provides a useful reminder of all the necessary elements when assigning symmetry point groups.

28 29

Table 1.1 Common Point Groups and Their Symmetry Elements

Point Group Symmetry Elements Present

C1 E

Cs E, sh

Ci E, i

Cn E, Cn

Dn n = odd E, Cn, n^C2 n n Dn n = even E, Cn, /2^C2´, /2^C2´´

Cnv n = odd E, Cn, nsv n n Cnv n = even E, Cn, /2sv, /2sd

Cnh n = odd E, Cn, sh, Sn

Cnh n = even E, Cn, sh, Sn, i

Dnh n = odd E, Cn, sh, n^C2, Sn, nsv n n n n Dnh n = even E, Cn, sh, /2^C2´, /2^C2´´, Sn, /2sv, /2sd, i

Dnd n = odd E, Cn, n^C2, i, S2n, nsd

Dnd n = even E, Cn, nC2´, S2n, nsd S n = even only E, S , C and i if n/2 odd n n n/2

T E, 4C3, 3C2

Th E, 4C3, 3C2, 4S2n, i, 3sh

Td E, 4C3, 3C2, 3S4, 6sd

O E, 3C4, 4C3, 6C2

Oh E, 3C4, 4C3, 6C2, 4S6, 3S4, i, 3sh, 6sd

I E, 6C5, 10C3, 15C2

Ih E, 6C5, 10C3, 15C2, i, 6S10, 10S6, 15s Kh E, infinite numbers of all symmetry elements

29 30

1.4 Site Symmetry of Molecules and Ions in Solids Most of the elements, as well as most inorganic compounds, exist as solids. The ideas about symmetry point groups can be applied to molecules or ions in crystal lattices in the context of site symmetry and pseudosymmetry. A crystal results from the periodic repetition of a specific arrangement of atoms through space. For a regular infinite array of objects the act of translation in space can be a valid symmetry operation, in addition to those described previously. Consider the infinite one-dimensional ---,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,--- Translation to the right or left by an integral multiple of the lattice spacing yields an indistinguishable physical configuration. This operation is important in considering atomic arrays in crystals. Seven three dimensional boxes (or parallelpipeds) used to summarize three- dimensional packing are given in Table I.2. Each box can be classified by the of its three edges a, b, and c, which are assigned using the convention for a right-handed . ______Table I.2: The Seven Crystal Systems System Conditions on Cell Parameters Lattice Symmetry _ triclinic a ≠ b ≠ c; a ≠ b ≠ g Ci or 1 monoclinic a ≠ b ≠ c; a = g = 90˚, b > 90˚ C2h or 2/m orthorhombic a ≠ b ≠ c; a = b = g =90˚ D2h or mmm tetragonal a = b ≠ c; a = b = g = 90˚ D4h or 4/mmm _ * trigonal (rhombohedral setting) a = b = c; a = b = g ≠90˚ D3d or 3 m hexagonal a = b = c; a = b = 90˚; g = 120˚ D6h or 6/mmm cubic a = b = c; a = b = g = 90˚ Oh or m3m ______*A hexagonal setting is also possible. The between the vectors are defined as a = between b and c, b = between a and c and g = between a and b. These unit cells summarize the fundamental packing patterns possible in

30 31 solids. The rightmost column in this table lists the symmetry point group of the in conventional (Schoenflies) notation, along with the Hermann-Maugin equivalent used by crystallographers and solid state chemists. In the latter notation Cn rotation axes are denoted by n, mirror planes by m and sh mirror planes are denoted n/m. An inversion center is specified by a bar above the principal rotation axis. In a formal approach to space groups the addition of centered lattices and the possible subsymmetries within each would be developed to yield the 230 space groups. For an excellent discussion of these considerations we recommend the text by Stout and Jensen listed in the references for this chapter. Extensive tabulations for the symmetry elements of each can be found in the reference work, International Tables for X-ray , available in most libraries. The discussion here will be limited to the relation between the lattice symmetry for the solid and the point group symmetry of an individual ion or molecule in the unit cell. It is important to recognize that for molecules in crystals there is not necessarily a correlation between molecular symmetry and the symmetry of the unit cell. Highly symmetrical molecules may crystallize in low symmetry unit cells and molecules of low symmetry may crystallize in unit cells of high symmetry. Atoms, molecules, or ions in a crystal may occupy two types of positions in a unit cell. The first is called a general position, which exists for all unit cells. There is no crystal symmetry associated with a general position. When a high symmetry molecule, e.g., an octahedral metal complex, occupies a general position then it rigorously possesses no symmetry because the environment of the crystal about it is asymmetric.

The asymmetry of the local environment distorts the site symmetry to C1 or 1; however, an octahedral pseudosymmetry remains. The low symmetry crystal environment can often be detected spectroscopically from a splitting of degenerate E or T energy levels. Perturbation of a molecule's symmetry in the crystal from its isolated symmetry, as in the gas phase, may be great or small. The magnitude of the perturbation depends on the strength of interactions between the molecule and its environment. In a molecular crystal, where the forces will primarily be Van der Waals in nature, the deviation from the idealized gas phase molecular symmetry will be small.

31 32

In ionic crystal, where strong electrostatic forces are present, the distortion of the idealized symmetry is expected to be greater. The other kind of position in a unit cell is called a special position. These special positions generally have unit cell (x, y, z) coordinates of 0, 1/2, 1/4, or 1/3 and are listed for each space group in the International Tables for X-ray Crystallography. When a molecule or ion occupies a special position then it possesses a rigorous site symmetry, which is determined by the crystal environment. Several different site are possible in a unit cell, the highest being the maximum symmetry of the cell. Thus cubic unit cells may have special positions of

Oh symmetry, which an octahedral molecule could occupy. If an octahedral molecule occupies such a position, then the symmetry is rigorously Oh and splitting of degenerate energy levels by a low symmetry crystal environment does not occur. Coupling between molecules in the unit cell may, however, give exciton splittings, as noted later. A simple example illustrates the essential points. Consider the case of triclinic crystals. _ Two space groups are possible. One, denoted P1, has the maximum symmetry indicated in _ Table I.2. The other P1, a subgroup of P1, has no symmetry. In P1 every position in the unit cell has a crystal environment with complete asymmetry. There is generally only one molecule (more rigorously 1 asymmetric unit since a pair or e.g., a hydrogen bonded cluster of molecules _ can sometimes form a packing unit) per unit cell in this space group. In P1, which possesses an inversion center at the origin of the unit cell, an arbitrary point or general position (x, y, z) is transformed to (-x, -y, -z) by the inversion operation. Thus, there must be two molecules (asymmetric units) per unit cell if they occupy a general position. It is possible for molecules that contain an inversion center to occupy the special position (0. 0, 0). Then there need be only one asymmetric unit per cell, with the added condition that it possess rigorous inversion symmetry. Similar conditions hold for higher symmetry space groups, but the basic concept resembles that illustrated above. Another concern, beyond that of the site symmetry in a crystal, is the possibility of coupling between asymmetric units in a crystal. In ionic solids the forces that interconnect all

32 33 the asymmetric units in a cell makes it imperative that these couplings be considered if one wants to understand vibrations in the solid state. Even molecular crystals can exhibit unusual splittings from such phenomena. One example occurs for coupling between excited states in crystals and this is often referred to as exciton coupling. For example, the excited states q1 and q2 of two _ asymmetric units in P1 could interact to yield composite sum and difference net states (q1 + q2 and q1 - q2). In this way a single excited state of the isolated molecule gives rise to two crystal states of different energy. The student with a serious interest in solid-state will need to delve into this in more detail. For most readers this section serves as a caveat should they attempt to interpret spectra of solids.

33 34

Additional Readings: Atkins, P. W.; Child, M. S.; Phillips, C. S. G. Tables for ; Oxford University Press: London, 1970. Bunker, P. R. Molecular Symmetry and Spectroscopy; Academic Press, New York, 1979. Burdett, J. K. Molecular Shapes: Theoretical Models of Inorganic Stereochemistry; Wiley: New York, 1980. Burns, G. Introduction to Group Theory with Applications; Academic Press: New York, 1977. Cotton, F. A. Chemical Applications of Group Theory; 2nd Ed., Wiley-Interscience: New York, 1971. A good introduction for chemists, with many worked examples. Donaldson, J. D.; Ross, S. D. Symmetry and Stereochemistry; Wiley: New York, 1972. Describes space groups and solid state symmetries. Dorain, P. B. Symmetry in Inorganic ; Addison-Wesley: Reading, Mass., 1965. Fackler, J. P., Jr. Symmetry in Chemical Theory; the Application of Group Theoretical Techniques to the Solution of Chemical Problems; Dowden, Hutchinson and Rossi: Stroudsburg, Pennsylvania, 1973. Fackler, J. P., Jr. Symmetry in Coordination Chemistry; Academic Press, New York, 1971. Ferraro, J. R.; Ziomek, J. S. Introductory Group Theory and its Application to Molecular Structure; 2nd Ed., Plenum Press: New York, 1975. Griffith, J. S. The Irreducible Tenson Method for Molecular Symmetry Groups; Prentice-Hall: Englewood Cliffs, New Jersey, 1962. Hall, L. H. Group Theory and Symmetry in Chemistry; McGraw-Hill: New York, 1969. Hamermesh, M. Group Theory and its Application to Physical Problems; Addision-Wesley: Reading, Massachusetts, 1962. A rigorous mathematical development with advanced applications to atomic and nuclear energy levels. Develops symmetric and continuous groups as well as irreducible tensor methods. Hochstrasser, R. M. Molecular Aspects of Symmetry; W. A. Benjamin: New York, 1966.

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Jaffe, H. H.; Orchin, M. Symmetry in Chemistry; R. E. Krieger Pub. Co.: Huntington, New York, 1977. Kettle, S. F. A. Symmetry and Structure; Wiley: New York, 1986. Lax, M. J. Symmetry Principles in Solid State and Molecular Physics; Wiley: New York, 1974. McWeeny, R. Symmetry; An Introduction to Group Thoery and its Applications; Pergamon Press: Oxford, 1963. Piepho, S. B.; Schatz, P. N. Group Theory in Spectroscopy: With Applications to Magnetic Circular Dichroism; Wiley: New York, 1983. Schonland, D. S. Molecular Symmetry: An Introduction to Group Theory and Its Uses in Chemistry; Van Nostrand: Princeton, New Jersey, 1965. Slater, J. C. Symmetry and Energy Bands in Crystals; Dover: New York, 1972. Applications of Group Theory to Wavefunctions of Crystalline Solids. Stout, G. H.; Jensen, J. H. X-ray Structure Determination: A Practical Guide; 2nd Ed., Wiley: New York, 1989. Contains a discussion of crystal symmetries with applications to X-ray crystallography. Tinkham, M. Group Theory and Quantum Mechanics; McGraw-Hill: New York, 1964. A concise development of theory with applications to atomic, molecular, and solid state electronic wavefunctions. Vincent, A. Molecular Symmetry and Group Theory: A Programmed Introduction to Chemical Applications; Wiley: New York, 1977. Wells, A. F. Structural , 5th ed., Clarendon Press: Oxford, 1984. Wherrett, B. S. Group Theory for Atoms, Molecules, and Solids; Prentice-Hall International: Englewood Cliffs, New Jersey, 1986.

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36

1.6 Problems 1) Determine which of the following are groups. For those, that are not groups, specify the missing conditions. a) The set of all positive integers with group multiplication defined as normal addition. b) The set of all integers (± and 0) with group multiplication defined as normal addition. c) The set of all integers (± and 0) with group multiplication defined as normal multiplication. 2) Prove that all groups of prime order are isomorphic to the cyclic group of that order. 3) Give the point group to which the following molecules belong. 2- a) MnO4 c) WF6

b) Rh6(CO)16 d) Mo6Cl12 (8 face capping, 4 terminal Cl)

e) f)

g) h) Mn2(CO)10 2- - i) Re2Cl8 j) OsCl4N

k) H2S l)

H B H N N

B B N

H currentpoint (planar conformation 192837465 ignore H on carbons) 3+ m) Co(en)3 n) cis-MoCl2(CO)4

o) trans-MoCl2(CO)4 p) fac-Mo(CO)3(PPh3)3 (coordination symmetry only,

i.e., ignore Ph groups)

q) mer-Mo(CO)3(PPh3)3 r) Al2Cl6

s) BrCl2F t) CO2

36 37

u) S8 (cyclic crown structure) v) PPh3 (with propeller conformation of phenyl rings)

w) HOCl x) cyclo-(BCl)4(NCH3)4 2- y) Ce(NO3)6 (the coordination environment of Ce is icosahedral-each nitrate is bound in

a - planar bidentate fashion such that trans NO3 groups are eclipsed).

z) PF5

4) Consider the 2-dimensional packing of regular of three-fold through eight-fold symmetry. Which ones can form a close-packed planar network with no overlapping or gaps in the plane? 5) Based on work in 4 what are the allowed axes in unit cells of 3- dimensional crystalline solids? ^ 6) Consider the S2 operation around the z axis. Show that its effect on the vector (x, y, z) is

the same as the inversion operation.

37 38

Key to Problems 1) a) Not group - identity element missing and inverse missing. b) Yes c) Not group - inverse elements missing. 2) Proof: Consider how one generates a group of prime order. Lagrange's theorem tells us the elements of a group must be an integral divisor of the order of the group. If n = prime then the elements are of order n. Since the # of elements = n, then one element a, and its powers generate the group a, a2, ..., an This is just the cyclic group of order n. Q.E.D.

3) a) Td; b) Td; c) Oh; d) Oh; e) Td; f) Oh; g) D6h; h) D4d; i) D4h; j) C4v;k)

C2v; l) C3h; m) D3; n) C2v; o) D4h; p) C3v; q) C2v; r) D2h; s) C1; t) D•h; u) D4d;

v) C3; w) Cs; x) S4; y) Th; z) D3h

4) It is possible to pack , , and in a close packed 2-dimensional array but not : See figure on following page. ^ 2 ^ 5) C2, C3, C4, and C6; note: the C4 requires the possibility of C4 = C2.

38 39

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